<figure><img src="image_24.png" /><figcaption>Pic. 2.2.6. Coincidences between the heralding detector and each of the detectors A (red) and B (blue) with timelike separation, measured in a window of 1ns during a time period of 10 minutes. $R_{HA} = 9.90 \times 10^4/10$ min, $R_{HB} = 6.22 \times 10^4/10$ min. Adapted from [53].</figcaption></figure>

The raw TDC data is shown in Figures 5-6 and the results are summarized in Table I.

<table><caption>Spacelike separation</caption><tbody><tr><th scope="row">R<sub>HA</sub><br>(94.8 ± 0.3) · 10<sup>3</sup></th><th scope="col">R<sub>H(A)</sub><br>(5570 ± 2) · 10<sup>3</sup></th><th scope="col">P<sup>SL</sup><sub>A</sub> = R<sub>HA</sub>/R<sub>H(A)</sub><br>(1.703 ± 0.006) · 10<sup>-2</sup></th></tr><tr><th scope="row">R<sub>HB</sub><br>(63.8 ± 0.2) · 10<sup>3</sup></th><th scope="col">R<sub>H(B)</sub><br>(5860 ± 2) · 10<sup>3</sup></th><th scope="col">P<sup>SL</sup><sub>B</sub> = R<sub>HB</sub>/R<sub>H(B)</sub><br>(1.090 ± 0.004) · 10<sup>-2</sup></th></tr><tr><th scope="row">R<sub>HAB</sub><br>4 ± 2</th><th scope="col">R<sub>H(AB)</sub><br>(17145 ± 4) · 10<sup>3</sup></th><th scope="col">P<sup>SL</sup>(1, 1) = R<sub>HAB</sub>/R<sub>H(AB)</sub><br>(2.3 ± 1.2) · 10<sup>-7</sup></th></tr><tr><th scope="row">R<sub>HN</sub><br>50 ± 7</th><th scope="col">R<sub>H(N)</sub><br>(5500 ± 2) · 10<sup>3</sup></th><th scope="col">P<sup>SL</sup><sub>N</sub> = R<sub>HN</sub>/R<sub>H(N)</sub><br>(9.0 ± 1.3) · 10<sup>-6</sup></th></tr></tbody></table>

<table><caption>Table I. Summary of results. Values obtained for the different counting rates and corresponding probabilities defined in the text, measured with spacelike and timelike separation. Adapted from [53].</caption><caption>Timelike separation</caption><tbody><tr><th scope="row">R<sub>HA</sub><br>(99.0 ± 0.3) · 10<sup>3</sup></th><th scope="col">R<sub>H(A)</sub><br>(6130 ± 2) · 10<sup>3</sup></th><th scope="col">P<sup>TL</sup><sub>A</sub> = R<sub>HA</sub>/R<sub>H(A)</sub><br>(1.616 ± 0.005) · 10<sup>-2</sup></th></tr><tr><th scope="row">R<sub>HB</sub><br>(62.2 ± 0.2) · 10<sup>3</sup></th><th scope="col">R<sub>H(B)</sub><br>(6100 ± 2) · 10<sup>3</sup></th><th scope="col">P<sup>TL</sup><sub>B</sub> = R<sub>HB</sub>/R<sub>H(B)</sub><br>(1.019 ± 0.004) · 10<sup>-2</sup></th></tr><tr><th scope="row">R<sub>HAB</sub><br>4 ± 2</th><th scope="col">R<sub>H(AB)</sub><br>(18345 ± 4) · 10<sup>3</sup></th><th scope="col">P<sup>TL</sup>(1, 1) = R<sub>HAB</sub>/R<sub>H(AB)</sub><br>(2.2 ± 1.1) · 10<sup>-7</sup></th></tr></tbody></table>

The number of counts given by detector noise and twophoton events can be estimated by looking at the counts away from the peak. As an example, for the space-like configuration (Figure 2.2.5.) in a window of 1ns the noise rate is on average $R_{HN} = 50$ (7) for a 10 minutes integration time. This corresponds to a noise probability $P_N = 9 \cdot 10^{-6}(1.3 \cdot 10^{-6})$. From the values in Table 1 one derives the following probability values for spacelike separation:
$$ 
\begin{aligned}
\mathbf{P}_A^{SL} \cdot \mathbf{P}_B^{SL} &= 1.86 \pm 0.01 \cdot 10^{-4}, \\
\mathbf{P}_{A \wedge B}^{SL} &= 0.002 \pm 0.001 \cdot 10^{-4}.
\end{aligned} 
\qquad (2.2.1) $$

For timelike separation one derives the values:

$$ 
\begin{aligned}
\mathbf{P}_A^{TL} \cdot \mathbf{P}_B^{TL} &= 1.65 \pm 0.01 \cdot 10^{-4}, \\
\mathbf{P}_{A \wedge B}^{TL} &= 0.002 \pm 0.001 \cdot 10^{-4}.
\end{aligned} 
\qquad (2.2.2) $$

For the probability $\mathbf{P}_N^{SL}$ that A and B detect photons coming from different pairs (noise) one derives the value:

$$ 
\begin{aligned}
\mathbf{P}_N^{SL}(1,1) &= \mathbf{P}_N^{SL} \cdot \mathbf{P}_A^{SL} + \mathbf{P}_N^{SL} \cdot \mathbf{P}_B^{SL} \approx \\
&0.0025 \pm 0.0026 \cdot 10^{-4}
\end{aligned} 
\qquad (2.2.3) $$

**Definition 2.2.1.** [54]. A measure algebra $\mathcal{F} = (\mathbf{B}, \mathbf{P})$ with a probability measure $\mathbf{P}$, is a

Boolean algebra $\mathbf{B}$ with a countably additive probability measure.

**Definition 2.2.2.** (i) A measure algebra of physical events $\mathcal{F}^{ph} = (\mathbf{B}, \mathbf{P})$ with a probability measure $\mathbf{P}$, is an Boolean algebra of physical events $\mathbf{B}$ with an countably

additive probability measure.

(ii) A Boolean algebra of physical events can be formally defined as a set $\mathbf{B}$ of elements

$a, b, \dots$ with the following properties:

1. $\mathbf{B}$ has two binary operations, $\wedge$ (logical AND, or "wedge") and $\vee$ (logical OR, or "vee"),

which satisfy:

the idempotent laws:

(1) $a \wedge a = a \vee a = a,$

the commutative laws:

(2) $a \wedge b = b \wedge a,$

(3) $a \vee b = b \vee a,$

and the associative laws:

(4) $a \wedge (b \wedge c) = (a \wedge b) \wedge c,$

(5) $a \vee (b \vee c) = (a \vee b) \vee c.$

2. The operations satisfy the absorption law:

(6) $a \wedge (a \vee b) = a \vee (a \wedge b) = a.$

3. The operations are mutually distributive

(7) $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c),$

(8) $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c).$

4. $\mathbf{B}$ contains universal bounds **0** and **1** which satisfy

(9) $\mathbf{0} \wedge a = \mathbf{0}$

(10) $\mathbf{0} \vee a = a$

(11) $\mathbf{1} \wedge a = a$

(12) $\mathbf{1} \vee a = \mathbf{1}.$
5. B has a unary operation $\neg a$ (or $a'$) of complementation (logical negation), which obeys the laws:

$$ (13) \quad a \wedge \neg a = \mathbf{0} $$

$$ (14) \quad a \vee \neg a = \mathbf{1} $$

All properties of negation including the laws below follow from the above two laws alone.

6. Double negation law: $\neg(\neg a) = a$

7. De Morgan's laws: (i) $\neg a \wedge \neg b = \neg(a \vee b)$, (ii) $\neg a \vee \neg b = \neg(a \wedge b)$.

8. Operations composed from the basic operations include the following important examples:

The first operation, $a \to b$ (logical material implication):

$$ (i) \quad a \to b \triangleq \neg a \vee b. $$

The second operation, $a \oplus b$, is called exclusive. It excludes the possibility of both a and b

$$ (ii) \quad a \oplus b \triangleq (a \vee b) \wedge \neg(a \wedge b). $$

The third operation, the complement of exclusive or, is equivalence or Boolean equality:

$$ (iii) \quad a \equiv b \triangleq \neg(a \oplus b) $$

9. B has a unary predicate $\mathbf{Occ}(a)$, which means that event $a$ has occurred, and which obeys the laws:

$$ (i) \quad \mathbf{Occ}(a \wedge b) \Leftrightarrow \mathbf{Occ}(a) \wedge \mathbf{Occ}(b), $$

$$ (ii) \quad \mathbf{Occ}(a \vee b) \Leftrightarrow \mathbf{Occ}(a) \vee \mathbf{Occ}(b), $$

$$ (iii) \quad \mathbf{Occ}(\neg a) \Leftrightarrow \neg \mathbf{Occ}(a). $$

**Remark 2.2.4.** A probability measure $P$ on a measure space $(\Omega, \Sigma)$ gives a probability

measure algebra $\mathcal{F} = (\Omega, \Sigma, \mathbf{P})$ on the Boolean algebra of measurable sets modulo null sets.

**Definition 2.2.3.** (i) Let $\mathbf{B}$ be a Boolean algebra of physical events. A Boolean algebra $\mathbf{B}_{M_4}$

of physical events in Minkowski spacetime $M_4 = \mathbb{R}^{1,3}$ that is cartesian product

$$ \mathbf{B}_{M_4} = \mathbf{B} \times M_4. $$

(ii) Let $\mathbf{B}_{M_4}$ be a Boolean algebra of physical events in Minkowski spacetime. A measure

algebra of physical events $\mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ in Minkowski spacetime that is a Boolean

algebra $\mathbf{B}_{M_4}$ with a probability measure $\mathbf{P}$.

(iii) Let $\mathbf{B}_{M_4}$ be a Boolean algebra of all physical events in Minkowski spacetime and let

$\mathcal{F}_{M_4}^{ph}$ be a measure algebra $\mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ with a probability measure $\mathbf{P}$. We denote such
physical events by $A(\mathbf{x}), B(\mathbf{x}),$ etc., where $\mathbf{x} = (t, x_1, x_2, x_3) \in M_4$ or $A, B, \dots$ etc.
(iv) We will be write for a short $A^{Oc}(\mathbf{x}), B^{Oc}(\mathbf{x}),$ etc., instead
**Occ($A(\mathbf{x})$), Occ($B(\mathbf{x})$), \dots etc.

**Definition 2.2.4.** Let Aut<sub>P</sub>(B<sub>M<sub>4</sub></sub>) be a set of the all measure-preserving automorphism of

$B_{M_4}$. This is a group, being a subgroup of the group Aut($B_{M_4}$) of all Boolean automorphism

of $B_{M_4}$. Let $P_+$ be Poincaré group.

**Remark 2.2.5.** We assume now that: any element $\Theta = (\Lambda, a) \in P_+$ induced an element

$\tilde{\Theta} \in \mathrm{Aut}_P(\mathbf{B}_{M_4})$ by formula $\tilde{\Theta} = \Theta[A(\mathbf{x})] = A(\Lambda\mathbf{x} + \mathbf{a}) \in \mathbf{B}_{M_4}$.

**Definition 2.2.5.** Given two events $A$ and $B$ from the algebra $\mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ the conditional probability of $A$ given $B$ is defined as the quotient of the probability of the joint of events $A$ and $B$, and the probability of $B$:

$$ \mathbf{P}(A|B) = \frac{\mathbf{P}(A \wedge B)}{\mathbf{P}(B)} = \frac{\mathbf{P}_{A \wedge B}}{\mathbf{P}_{B}} = \mathbf{P}_{A|B}, \qquad (2.2.4) $$

where $\mathbf{P}(B) \neq 0$.

**Definition 2.2.6.** (i) Events $A$ and $B$ from the algebra $\mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ are defined to be statistically independent or uncorrelated iff

$$ \mathbf{P}_{A \wedge B} = \mathbf{P}_A \cdot \mathbf{P}_B, \qquad (2.2.5) $$

where $\mathbf{P}_B \neq 0$, then this is equivalent to the statement that $\mathbf{P}_{A|B} = \mathbf{P}_A$. Similarly, if $\mathbf{P}_A$ is not zero, then $\mathbf{P}_{B|A} = \mathbf{P}_B$ is also equivalent.

(ii) Events $A$ and $B$ from the algebra $\mathcal{F} = (\mathbf{B}_{M_4}, \mathbf{P})$ are defined to be statistically almost independent or almost uncorrelated iff

$$ \mathbf{P}_{A \wedge B} \approx \mathbf{P}_A \cdot \mathbf{P}_B, \qquad (2.2.6) $$

**Remark 2.2.6.** Note that

$$ \mathbf{P}_{A \vee B} = \mathbf{P}_A + \mathbf{P}_B - \mathbf{P}_{A \wedge B}. \qquad (2.2.7) $$

Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations such as the subjective theory, conditional probability is considered a primitive entity. Further, this "multiplication axiom" introduces a symmetry with the summation axiom for mutually exclusive events, i.e.

$$ \mathbf{P}_{A \wedge B} = \mathbf{P}_A + \mathbf{P}_B - \mathbf{P}_{A \vee B}. \qquad (2.2.8) $$

**Definition 2.2.7.** (i) Events $A_1, A_2, \dots, A_n \in \mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ are said to be exactly mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining $n-1$ events. Therefore, two mutually exclusive events cannot both
occur. Formally said, the conjunction of each two of them is 0 (the null event):
$A \land B = 0$. In consequence, exactly mutually exclusive events A and B have the property:

$$ P(A \land B) = 0. \qquad (2.2.9) $$

(ii) Events $A_1, A_2, \dots, A_n \in \mathcal{F}_{M_4}^{ph} = (\mathbf{B}_{M_4}, \mathbf{P})$ are said to be almost mutually exclusive if

$A_1, A_2, \dots, A_n$ have the property:

$$ \begin{gathered} \mathbf{P}(A_1 \land A_2 \land \dots \land A_n) \approx 0, \qquad (2.2.10) \\ \mathbf{P}(A_1 \land A_2 \land \dots \land A_n) \ll \mathbf{P}(A_1) \cdot \mathbf{P}(A_2) \cdot \dots \cdot \mathbf{P}(A_n). \end{gathered} $$

In consequence, almost mutually exclusive events A and B have the property:

$$ \begin{gathered} \mathbf{P}(A \land B) \approx 0, \qquad (2.2.11) \\ \mathbf{P}(A \land B) \ll \mathbf{P}(A) \cdot \mathbf{P}(B). \end{gathered} $$

**Remark 2.2.7.** Let $A^{ph}, B^{ph}$ be events such that detectors A,B detect photon at an instants $t_1$ and $t_2$ correspondingly. Note that (2.2.1) and (2.2.2) show that whether the separation between the detectors is timelike or spacelike, the number of coincidences is three orders of magnitude smaller than what would be expected had the events been statistically almost uncorrelated, i.e., $\mathbf{P}_{A\land B} \approx \mathbf{P}_A \cdot \mathbf{P}_B$, see Def.2.2.5(ii).

**Remark 2.2.8.** Let $A^{ph}, B^{ph}$ be events such that detectors A,B detect photon at an instants $t_1$ and $t_2$ correspondingly. Note that:

(i) from Eq.(2.2.1) follows probability value for spacelike separation:

$$ \mathbf{P}_{A^{ph} \land B^{ph}}^{SL} = 0.002 \pm 0.001 \cdot 10^{-4} \neq 0, \qquad (2.2.12) $$

(ii) from Eq.(2.2.2) follows probability value for timelike separation:

$$ \mathbf{P}_{A^{ph} \land B^{ph}}^{TL} = 0.002 \pm 0.001 \cdot 10^{-4} \neq 0. \qquad (2.2.13) $$

Therefore in both cases the property (2.2.9) are violated, i.e. $\mathbf{P}_{A^{ph} \land B^{ph}} \neq 0$ but however in both cases the property (2.2.11) is satisfied

$$ \begin{align} 0.002 \pm 0.001 \cdot 10^{-4} &= \mathbf{P}_{A^{ph} \land B^{ph}}^{SL} \ll \mathbf{P}_{A^{ph}}^{SL} \cdot \mathbf{P}_{B^{ph}}^{SL} = 1.86 \pm 0.01 \cdot 10^{-4}, \tag{2.2.12} \\ 0.002 \pm 0.001 \cdot 10^{-4} &= \mathbf{P}_{A^{ph} \land B^{ph}}^{TL} \ll \mathbf{P}_{A^{ph}}^{TL} \cdot \mathbf{P}_{B^{ph}}^{TL} = 1.65 \pm 0.01 \cdot 10^{-4} \notag \end{align} $$

and therefore in both cases the events $A^{ph}, B^{ph}$ are almost mutually exclusive events.

**Beamsplitter transformation.**

A beamsplitter is the most simple way to mix two modes, see Figure 2.2.7. From classical electrodynamics, one gets the following amplitudes for the outgoing modes:

$$ \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}_{\text{in}} \mapsto \begin{bmatrix} A_1 \\ A_2 \end{bmatrix}_{\text{out}} = \begin{pmatrix} t & r \\ r' & t' \end{pmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}_{\text{in}}. \qquad (2.2.13) $$
<figure><img src="image_25.png" /><figcaption>Pic.2.2.7.Mixing of two modes by a
beam splitter.</figcaption></figure>

The recipe for quantization is now: 'replace the classical amplitudes by annihilation operators'. If the outgoing modes are still to be useful for the quantum theory, they have to satisfy the commutation relations:

$$
[A_i(out), A_j(out)] = \delta_{ij} \tag{2.2.14}
$$

These conditions give constraints on the reflection and transmission amplitudes,
for example $|t'|^2 + |r'|^2 = 1$. We are now looking for an unitary operator $\mathcal{S}$ [the
$\mathcal{S}$-matrix] that implements this beamsplitter transformation in the following sense:

$$
A_i = S^\dagger a_i S, i = 1, 2. \tag{2.2.15}
$$

Let us start from the general transformation (summation over double indices)

$$
a_i \mapsto A_i = B_{ij} a_i, \vec{a} \mapsto \vec{A} = \mathbf{B} \vec{a} \qquad (2.2.16)
$$

where we have introduced matrix and vector notation. Using this S-matrix one can
also compute the transformation of the states: $|out\rangle = S|in\rangle$. For the unitary
transformation, we make the ansatz

$$
S(\theta) = \exp(i\theta J_{kl} a_k^\dagger a_l) \quad (2.2.17)
$$

with $J_{kl}$ a hermitean matrix (ensuring unitarity). The action of this unitary on the
photon mode operators is now required to reduce to

$$
a_i \mapsto A_i(\theta) = S^\dagger(\theta) a_i S(\theta) = B_{ij} a_j. \tag{2.2.18}
$$

We compute this ‘operator conjugation’ by using a differential equation:

$$
\frac{dA_i(\theta)}{d\theta} = iJ_{ki}A_i(\theta). \tag{2.2.19}
$$

This is a system of linear differential equations with constant coefficients, so that
one obtains a solution

$$
\vec{A}(\theta) = \exp(i\theta\mathbf{J}). \tag{2.2.20}
$$

We thus conclude that the so-called generator **J** of the beam splitter matrix is fixed
by equatuon

$$
\mathbf{B} = \exp(i\theta\mathbf{J}). \tag{2.2.21}
$$

If the transformation **B** is part of a continuous group and depends on θ as a
parameter, we can expand it around unity. Doing the same for the matrix exponential, we get

$$ \mathbf{B} = 1 + i\theta\mathbf{J} + \dots \qquad (2.2.22) $$

Equation (2.2.22) explains the name generator for the matrix J: it actually generates a subgroup of matrices $\mathbf{B} = \mathbf{B}(\theta)$ parametrized by the angle $\theta$. The unitary transformation we are looking for is thus determined via the same generator J. For the two-mode beam splitter, an admissible transformation is given by

$$ \mathbf{B}(\theta) = \begin{pmatrix} t & r \\ r' & t' \end{pmatrix} = \begin{pmatrix} \cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta \end{pmatrix} \qquad (2.2.23) $$

The factor $i$ is just put for convenience so that the reflection amplitudes are the same for both sides, $r = r'$, as expected by symmetry. Expanding for small $\theta$, the generator is

$$ \mathbf{J} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \sigma_1 \qquad (2.2.24) $$

so that the unitary operator for this beamsplitter is

$$ S(\theta) = \exp[i\theta(a_1^\dagger a_2 + a_2^\dagger a_1)]. \qquad (2.2.25) $$

Therefore, the effective Hamiltonian of the beam splitter is given by

$$ \hat{H}_{\text{eff}} = a_1^\dagger a_2 + a_2^\dagger a_1. \qquad (2.2.26) $$

## Splitting a two-photon state

Let us consider two single photon states $|in\rangle = |1, 1\rangle$ incident on the beam splitter such that mentioned above. Then

$$ \begin{aligned} |\psi(\theta)\rangle &= |out\rangle = S|in\rangle = Sa_1^\dagger S^\dagger Sa_2^\dagger S^\dagger S|0,0\rangle = \\ & (a_1^\dagger \cos\theta + ia_2^\dagger \sin\theta)(a_2^\dagger \cos\theta + ia_1^\dagger \sin\theta)|0,0\rangle = \\ & (|2,0\rangle - |0,2\rangle) \frac{\sin\theta}{\sqrt{2}} + |1,1\rangle \cos\theta. \end{aligned} \qquad (2.2.27) $$

Let $\mathbf{H}$ be a complex Hilbert space such that

$$ \begin{aligned} & \forall \theta[|\psi(\theta)\rangle]_{cl} \in \mathbf{H}], \\ & \forall \theta \forall \delta(\delta \in (0,1]) \forall \varepsilon(\varepsilon \in (0,1]) [|\psi_{\delta,\varepsilon,x}^i(\theta)\rangle]_{cl} \in \mathbf{H}], \\ & |\psi_{\delta,\varepsilon,x}^i(\theta)\rangle]_{cl} = L_x^i(\delta, \varepsilon)|\psi(\theta)\rangle]_{cl}. \end{aligned} \qquad (2.2.28) $$

By postulate C.I (see Appendix C) quantum system with Hamiltonian given by Eq.(2.2.26) is identified by a set $\Xi \triangleq \langle \mathbf{H}, \hat{H}_{\text{eff}}, \mathfrak{I}, \mathfrak{R}, \mathcal{L}_{2,1}(\Omega), \mathbf{G}, |\psi_t\rangle \rangle$, where

(i) $\mathbf{H}$ that is a complex Hilbert space defined above,

(ii) $\mathfrak{I} = (\Omega, \mathcal{F}, \mathbf{P})$ that is complete probability space,

(iii) $\mathfrak{R} = (\mathbb{R}^n, \Sigma)$ that is measurable space,
(iv) $\mathcal{L}_{2,1}(\Omega)$ that is complete space of random variables $X: \Omega \to \mathbb{R}^n$ such that

$$ \int_{\Omega} \|X(\omega)\| d\mathbf{P} < \infty, \quad \int_{\Omega} \|X(\omega)\|^2 d\mathbf{P} < \infty, \qquad (2.2.29) $$

(v) $\mathbf{G}: \mathbf{H} \to \mathcal{L}_{2,1}(\Omega)$ that is one to one correspondence such that

$$ |\langle\psi|\hat{Q}|\psi\rangle| = \int_{\Omega} (\mathbf{G}[\hat{Q}|\psi\rangle](\omega)) d\mathbf{P} = \mathbf{E}_{\Omega}[\mathbf{G}[\hat{Q}|\psi\rangle](\omega)] \qquad (2.2.30) $$

for any $|\psi\rangle \in \mathbf{H}$ and for any Hermitian operator $\hat{Q}: \mathbf{H} \to \mathbf{H}$,

(vi) $|\psi_t\rangle$ is a continuous vector function $|\psi_t\rangle: \mathbb{R}_+ \to \mathbf{H}$ which represented the canonical evolution of the quantum system $\Xi$.

**Remark 2.2.9.** Note that $\mathfrak{I}_{M_4}^{ph} = \mathcal{F} \times M_4 = (\Omega, \Sigma, \mathbf{P}) \times M_4$, where $\mathcal{F}$ is a probability measure

algebra $\mathcal{F} = (\Omega, \Sigma, \mathbf{P})$ on the Boolean algebra of measurable sets modulo null sets, see

Remark 2.2.4.

Let $\mathbf{B}_{M_4}$ be Boolean algebra of the all physical events in Minkowski spacetime $M_4$ and let

$\tilde{\mathfrak{F}}_{M_4}$ be an measure algebra $\tilde{\mathfrak{F}}_{M_4} = (\mathbf{B}_{M_4}, \tilde{\mathbf{P}})$ with a probability measure $\tilde{\mathbf{P}}$, see

**Definition 2.2.2 (vii).**

We assume now that there exist subalgebra $\mathfrak{F}_{M_4}^{\#} \subsetneq \tilde{\mathfrak{F}}_{M_4}$ and isomorphism

$\lambda[\cdot]: \mathfrak{F}_{M_4}^{\#} \to \mathfrak{I}_{M_4}^{ph}$ such that for any event $A(\mathbf{x}) \in \mathfrak{F}_{M_4}^{\#}, \mathbf{x} = (t, x_1, x_2, x_3) \in M_4$ (see
**Definition 2.2.2):**

$$ \lambda[A(\mathbf{x})] = \lambda[A](\mathbf{x}), \qquad (2.2.31) \\ \tilde{\mathbf{P}}(A(\mathbf{x})) = \mathbf{P}(\lambda[A](\mathbf{x})) \triangleq \mathbf{P}(A_{\lambda}(\mathbf{x})). $$

**Proposition 2.2.1.** Suppose that $A$ and $B$ are events in measure algebra $\tilde{\mathfrak{F}}_{M_4} = (\mathbf{B}_{M_4}, \tilde{\mathbf{P}})$. Then following properties is satisfied:

$$ \begin{aligned} 1. & \mathbf{P}(A|B) > \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) > \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \wedge B) > \mathbf{P}(A)\mathbf{P}(B) \\ 2. & \mathbf{P}(A|B) < \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) < \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \wedge B) < \mathbf{P}(A)\mathbf{P}(B) \end{aligned} \qquad (2.2.32) $$

$$ 3. \mathbf{P}(A|B) = \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) = \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \wedge B) = \mathbf{P}(A)\mathbf{P}(B) $$

**Proposition 2.2.2.** Suppose that $A$ and $B$ are events in measure algebra $\mathfrak{I}_{M_4} = (\Omega, \mathcal{F}, \mathbf{P})$.

Then following properties is satisfied:

$$ \begin{aligned} 1. & \mathbf{P}(A|B) > \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) > \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \cap B) > \mathbf{P}(A)\mathbf{P}(B) \\ 2. & \mathbf{P}(A|B) < \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) < \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \cap B) < \mathbf{P}(A)\mathbf{P}(B) \end{aligned} \qquad (2.2.33) $$

$$ 3. \mathbf{P}(A|B) = \mathbf{P}(A) \Leftrightarrow \mathbf{P}(B|A) = \mathbf{P}(B) \Leftrightarrow \mathbf{P}(A \cap B) = \mathbf{P}(A)\mathbf{P}(B) $$

**Definition 2.2.8.** In case (1), $A$ and $B$ are said to be positively correlated.

Intuitively, the occurrence of either event means that the other event is more likely.
In case (2), A and B are said to be negatively correlated.

Intuitively, the occurrence of either event means that the other event is less
likely.

In case (3), A and B are said to be uncorrelated or independent.

Intuitively, the occurrence of either event does not change the probability of the
other event.

**Remark 2.2.10.** Suppose that A and B are events in measure algebra
$\mathcal{F}_{M_4} = (\Omega, \mathcal{F}, \mathbf{P})$.

Note from the result above that if $A \subseteq B$ or $B \subseteq A$ then $A$ and $B$ are positively correlated. If

A and B are disjoint then A and B are negatively correlated.

**Proposition 2.2.3.** Suppose that A and B are events in measure algebra
$\tilde{\mathcal{F}}_{M_4} = (\mathbf{B}_{M_4}, \tilde{\mathbf{P}})$. Then:

(i) A and B have the same correlation (positive, negative, or zero) as $\neg A$ and $\neg B$.

(ii) A and B have the opposite correlation as A and $\neg B$ (that is, positive-negative, negative-positive, or zero-zero).

**Proposition 2.2.4.** Suppose that A and B are events in measure algebra
$\mathcal{F}_{M_4} = (\Omega, \mathcal{F}, \mathbf{P})$.

Then:

(i) A and B have the same correlation (positive, negative, or zero) as $A^c$ and $B^c$.

(ii) A and B have the opposite correlation as A and $B^c$ (that is, positive-negative, negative-positive, or zero-zero).

**Definition 2.2.9.** Let $A(\mathbf{x}_1) = A(t_1, \mathbf{r}_1)$ and $B(\mathbf{x}_2) = B(t_2, \mathbf{r}_2)$ be an events
$A(\mathbf{x}_1) \in \mathcal{F}_{M_4}^\# $ which occurs at instant $t_1$ and $B(\mathbf{x}_2) \in \mathcal{F}_{M_4}^\#$ at instant $t_2$
correspondingly.

Let $\mathbf{x}_{1,2}$ be a vector: $\mathbf{x}_{1,2} = \{c(t_1 - t_2), \mathbf{r}_1 - \mathbf{r}_2\} = (ct_{1,2}, \mathbf{r}_{1,2}), t_{1,2} = t_1 - t_2, \mathbf{r}_{1,2} = \mathbf{r}_1 - \mathbf{r}_2$.

Vectors $\mathbf{x}_{1,2} = (ct_{1,2}, \mathbf{r}_{1,2})$ are classified according to the sign of $c^2t_{1,2}^2 - \mathbf{r}_{1,2}^2$. A
vector is

(i) timelike if $c^2t_{1,2}^2 > \mathbf{r}_{1,2}^2$, (ii) spacelike if $c^2t_{1,2}^2 < \mathbf{r}_{1,2}^2$, and null or lightlike if (iii)
$c^2t_{1,2}^2 = \mathbf{r}_{1,2}^2$.

Pairs of events {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$} $\in \mathcal{F}_{M_4}^\# \times \mathcal{F}_{M_4}^\#$ are classified according to the
sign of

$c^2t_{1,2}^2 - \mathbf{r}_{1,2}^2:$

(i) a pair {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$} is timelike separated if $c^2t_{1,2}^2 > \mathbf{r}_{1,2}^2$,
and we denoted such pairs by {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$}$_{t.l.s.}$

(ii) a pair {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$} is spacelike separated if $c^2t_{1,2}^2 < \mathbf{r}_{1,2}^2$,
and we denoted such pairs by {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$}$_{s.l.s.}$

(iii) a pair {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$} is null or lightlike separated if $c^2t_{1,2}^2 = \mathbf{r}_{1,2}^2$.
and we denoted such pairs by {$A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)$}$_{l.l.s.}$

**Definition 2.2.10.** (i) Let $[\mathcal{F}_{M_4}^\#, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{t.l.s.}$ be a set of the all timelike separated
pairs ${A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)}_{\text{t.l.s.}}$ which are corresponding to a given vector
$\{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\} \in M_4 \times M_4, \text{i.e.,}$

$$
\begin{equation} \tag{2.2.34.a}
\begin{aligned}
& [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}} = \\
& \quad \{\{(A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2))\} \in \mathcal{F}_{M_4}^{\#} \times \mathcal{F}_{M_4}^{\#} | c^2 t_{1,2}^2 > \mathbf{r}_{1,2}^2\}.
\end{aligned}
\end{equation}
$$

(ii) Let $[\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}}$ be a set of the all spacelike separated
pairs $\{A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)\}_{\text{s.l.s.}}$, which is corresponding to a given vector
$\{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\} \in M_4 \times M_4, \text{i.e.,}$

$$
\begin{equation} \tag{2.2.34.b}
\begin{aligned}
& [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}} = \\
& \quad \{\{(A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2))\} \in \mathcal{F}_{M_4}^{\#} \times \mathcal{F}_{M_4}^{\#} | c^2 t_{1,2}^2 < \mathbf{r}_{1,2}^2\}.
\end{aligned}
\end{equation}
$$

**Remark 2.2.11.** Let $[\mathcal{F}_{M_4}^{\#}, \{(t, \mathbf{r}_1), (t, \mathbf{r}_2)\}]_{\text{s.l.s.}}$ be a set of the all pairs
$\{A(t, \mathbf{r}_1), B(t, \mathbf{r}_2)\}$

which is corresponding to a given vector $\{(t, \mathbf{r}_1), (t, \mathbf{r}_2)\} \in M_4 \times M_4, \mathbf{r}_1 \neq \mathbf{r}_2, \text{i.e.}$,

$$
\begin{align}
& [\mathcal{F}_{M_4}^{\#}, \{(t, \mathbf{r}_1), (t, \mathbf{r}_2)\}]_{\text{s.l.s.}} = \nonumber \\
& \quad \{\{(A(t, \mathbf{r}_1), B(t, \mathbf{r}_2))\} \in \mathcal{F}_{M_4}^{\#} \times \mathcal{F}_{M_4}^{\#} | 0 < \mathbf{r}_{1,2}^2\}, \tag{2.2.35} \\
& \mathbf{r}_{1,2} = \mathbf{r}_1 - \mathbf{r}_2. \nonumber
\end{align}
$$

Such pairs obviously is spacelike separated. Note that

$$
\forall t \forall \mathbf{r}_1 \forall \mathbf{r}_2 (\mathbf{r}_1 \neq \mathbf{r}_2) \{[\mathcal{F}_{M_4}^{\#}, \{(t, \mathbf{r}_1), (t, \mathbf{r}_2)\}]_{\text{s.l.s.}} \neq \emptyset\}. \quad (2.2.36)
$$

**Definition 2.2.11.** Let $A^{t_1} \triangleq A(\mathbf{x}_1) = A(t_1, x_A)$ and $B^{t_2} \triangleq B(\mathbf{x}_2) = B(t_2, x_B)$ be a
symbols such that $A^{t_1}$ and $B^{t_2}$ represent there is detection events $A(\mathbf{x}_1) \in \mathcal{F}_{M_4}^{\#}$ at
instant $t_1$ and $B(\mathbf{x}_2) \in \mathcal{F}_{M_4}^{\#}$ at instant $t_2$ correspondingly, where symbols $x_A$ and $x_B$
represent the locations of the detectors $A$ and $B$ correspondingly (see Pic.2.2.4).
We assume that

$$
\{A^{t_1}, B^{t_2}\} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, x_A), (t_2, x_B)\}]_{\text{s.l.s.}}. \qquad (2.2.37)
$$

**Remark 2.2.12.** We assume now without loss of generality that $t_1 = t_2 = t$, note that such assumption valid by properties: $A(\Lambda\mathbf{x}_1) \in \mathcal{F}_{M_4}^{\#}$ and $B(\Lambda\mathbf{x}_2) \in \mathcal{F}_{M_4}^{\#}$, required above, see Remark 2.2.5.

**Einstein's 1927 gedanken experiment resolution**

In classical case considered by A. Einstein in his 1927 gedanken experiment,
by postulates of canonical QM, both events $A^t \in \mathcal{F}_{M_4}^\# $ and $B^t \in \mathcal{F}_{M_4}^\# $ cannot occur
simultaneously, i.e. that is mutually exclusive events with a probability = 1, and
therefore $A^t \wedge B^t = 0$. Such exactly mutually exclusive events have the property:

$$
\tilde{\mathbf{P}}(A^t \wedge B^t) = 0, \tag{2.2.38}
$$

see Definition 2.2.6.

We remind that the probability density $p^{\text{ph}}(x, \delta, \epsilon)$ for the occurrence of a photon
localization at point $x$ is assumed to be
$$ p^{\text{ph}}(x, \delta, \varepsilon) = \left\| \left| \psi_{\delta, \varepsilon, x}^{\text{ph}}(\theta) \right\rangle \right\|^2, \qquad (2.2.39) \\ \delta \in (0, 1], \varepsilon \in (0, 1], $$

where

$$ \left| \psi_{\delta, \varepsilon, x}^{\text{ph}} \right\rangle_{\text{cl}} = L_x(\delta, \varepsilon) \left| \psi^{\text{ph}}(\theta) \right\rangle_{\text{cl}}. \qquad (2.2.40) $$

and where the localization operators $L_x(\delta, \varepsilon)$ have been chosen to have the form:

$$ \hat{L}_x(\hat{q}, \delta, \varepsilon) = \begin{cases} \left(\frac{1}{\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{2\delta}(\hat{q}-x)^2\right] & \text{iff } |\hat{q}-x| \le \varepsilon \ll 1, \\ 0 & \text{iff } |\hat{q}-x| > \varepsilon. \end{cases} \qquad (2.2.41) $$

see subsection I.2.4.

**Remark 2.2.13.** Note that: (i) from (2.2.28) follows that $|\psi_{\delta, \varepsilon, x}^{\text{ph}}(\theta)\rangle_{\text{cl}} \in \mathbf{H}$, (ii) from (2.2.39) and (2.2.41) where $\delta \ll 1$ follows that

$$ \begin{aligned} p^{\text{ph}}(x, \delta, \varepsilon) = & \left\| |\psi_{\delta, \varepsilon, x}^{\text{ph}}(\theta)\rangle \right\|_{\text{cl}}^2 = \int dq \langle \psi_{\delta, \varepsilon, x}^{\text{ph}}(\theta) |q\rangle \langle q| |\psi_{\delta, \varepsilon, x}^{\text{ph}}(\theta)\rangle = \\ & \int dq \langle \hat{L}_x(\hat{q}, \delta, \varepsilon) \psi^{\text{ph}}(\theta) |q\rangle \langle q| \hat{L}_x(\hat{q}, \delta, \varepsilon) \psi^{\text{ph}}(\theta) \rangle = \\ & \int dq L_x^2(q, \delta, \varepsilon) \langle \psi^{\text{ph}}(\theta) |q\rangle \langle q| \psi^{\text{ph}}(\theta) \rangle = \|\langle x|\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}\|^2 + O(\delta) \asymp \\ & \asymp \|\langle x|\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}\|^2, \\ & \delta \ll 1, \varepsilon \in (0, 1], \end{aligned} \qquad (2.2.42) $$

From postulate C.I.3 follows that there exist unique random variable $X(\omega; ||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}})$ given on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and a measurable space $(\mathbb{R}^n, \Sigma)$ by formula

$$ X(\omega; ||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}) \triangleq X_{||\psi^{\text{ph}}(\theta)\rangle}(\omega) = \mathbf{G}[||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}] \qquad (1.2.43) $$

The probability density of random variable $X_{||\psi^{\text{ph}}(\theta)\rangle}(\omega)$ we denote by $p_{||\psi^{\text{ph}}(\theta)\rangle}(q), q \in \mathbb{R}$.

**Remark 2.2.14.** From postulate Q.II.2 (see subsection I.7.1) follows that for the system in state $||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}$ the probability $P(q, q+dq; ||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}})$ of obtaining the result $q$ lying in the range $(q, q+dq)$ on measuring observable $\hat{q}$ given by

$$ P(q, q+dq; ||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}) = p_{||\psi^{\text{ph}}(\theta)\rangle}(q)dq = |c_{||\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}}(q)|^2 = |\langle q|\psi^{\text{ph}}(\theta)\rangle_{\text{cl}}|^2 \qquad (2.2.44) $$

Now we go to explain Einstein's 1927 gedanken experiment. Let $A^{\text{ph}}(t, x_A)$ and $B^{\text{ph}}(t, x_A)$ be events such that detectors A,B detect photon at an instant $t$ correspondingly. By properties (2.2.31) we obtain
$$ 
\begin{aligned}
\mathbf{P}(A_{\lambda}^{ph}(t, x_A)) &\triangleq \mathbf{P}(\lambda[A^{ph}](t, x_A)) = \tilde{\mathbf{P}}(A^{ph}(t, x_A)), \\
\mathbf{P}(B_{\lambda}^{ph}(t, x_B)) &\triangleq \mathbf{P}(\lambda[B^{ph}](t, x_B)) = \tilde{\mathbf{P}}(B^{ph}(t, x_B)).
\end{aligned}
\qquad (2.2.45) $$

Note that

$$ 
\begin{aligned}
A^t &\triangleq A_{\lambda}^{ph}(t, x_A) = \{\omega | x_A - \epsilon \le X_{|\psi^{ph}(\theta)|}(\omega) \le x_A - \epsilon\}, \\
B^t &\triangleq B_{\lambda}^{ph}(t, x_B) = \{\omega | x_B - \epsilon \le X_{|\psi^{ph}(\theta)|}(\omega) \le x_B - \epsilon\}, \\
\epsilon &\in (0, \gamma], \gamma \ll 1,
\end{aligned}
\qquad (2.2.46) $$

where a small parameter $\epsilon \ll |x_A - x_B|$ dependent on measuring device. Thus by general definition of random variable one obtains directly

$$ A_{\lambda}^{ph}(t, x_A) \cap B_{\lambda}^{ph}(t, x_B) = \emptyset \qquad (2.2.47) $$

and therefore

$$ \mathbf{P}(A_{\lambda}^{ph}(t, x_A) \cap B_{\lambda}^{ph}(t, x_B)) = 0 \qquad (2.2.48) $$

The property (2.2.48) follows directly from (2.2.45).

<figure><img src="image_26.png" /><figcaption>Pic. 2.2.8. The plot of the random variable $X_{|\psi^{ph}(\theta)|}(\omega)$.</figcaption></figure>

$$ A^t = A_{\lambda}^{ph}(t, x_A), B^t = B_{\lambda}^{ph}(t, x_B), A^t \cap B^t = \emptyset. $$

**Remark 2.2.15.** Let $[\mathcal{F}_{M_4}^{\#}, \{(t, x_A), (t, x_B)\}]_{\text{s.l.s.}}$ be a set of the all pairs $\{A(t, x_A), B(t, x_B)\}$

which is corresponding to a given vector $\{(t, x_A, 0, 0), (t, x_B, 0, 0)\} \in M_4 \times M_4$, $x_A \ne x_B$, i.e.,

$$ [\mathcal{F}_{M_4}^{\#}, \{(t, x_A), (t, x_B)\}]_{\text{s.l.s.}} = \{\{A(t, x_A), B(t, x_B)\} \in \mathcal{F}_{M_4}^{\#} \times \mathcal{F}_{M_4}^{\#} | 0 < (x_A - x_B)^2\}. \qquad (2.2.49) $$

Such pairs obviously is spacelike separated. Note that

$$ \forall t \forall x_A \forall x_B (x_A \ne x_B) \{[\mathcal{F}_{M_4}^{\#}, \{(t, x_A), (t, x_B)\}]_{\text{s.l.s.}} \ne \emptyset\}. \qquad (2.2.50) $$
Now we go to explain non zero result $\tilde{\mathbf{P}}(A^t \wedge B^t) \neq 0$ given above by (2.2.1) and (2.2.2):

$$ 
\begin{aligned}
\tilde{\mathbf{P}}_{A^t \wedge B^t}^{TL} &= 0.002 \pm 0.001 \cdot 10^{-4}, \quad \mathbf{P}_{A^t}^{TL} \cdot \mathbf{P}_{B^t}^{TL} = 1.65 \pm 0.01 \cdot 10^{-4}, \\
\tilde{\mathbf{P}}_{A^t \wedge B^t}^{SL} &= 0.002 \pm 0.001 \cdot 10^{-4}. \quad \mathbf{P}_{A^t \wedge B^t}^{SL} = 0.002 \pm 0.001 \cdot 10^{-4}.
\end{aligned}
\qquad (2.2.51) $$

We consider this problem in general case.

**Remark 2.2.16.** Note that: (i) a probability density $p(x, \delta_A, \epsilon, \cdot)$ for the occurrence of a localization inside interval $[x - \epsilon, x + \epsilon]$ in arm with detector **A** (see Pic. 2.2.2) is given by formula (see Remark 1.2.4)

$$ p(x, \delta_A, \epsilon) = \frac{\| |\psi_{\delta_A, \epsilon, x} \rangle_{cl} \|^2}{\Delta(\delta_A, \epsilon)}, \qquad (2.2.52) $$

where

$$ 
\begin{aligned}
\| |\psi_{\delta_A, \epsilon, x} \rangle_{cl} \|^2 &= \left( \frac{1}{\delta_A \pi \delta_A} \right)^{1/2} \int_{|q-x| \le \epsilon} dq |\psi(q)|^2 \exp \left[ -\frac{1}{\delta_A} (q-x)^2 \right], \\
\psi(q) &= \langle q \| \psi \rangle, \\
\Delta(\delta_A, \epsilon) &= \int_{-\infty}^{\infty} \| |\psi_{\delta_A, \epsilon, x} \rangle_{cl} \|^2 dx,
\end{aligned}
\qquad (2.2.53) $$

and where parametr $\delta_A$ depend on arm with detector **A**.

(ii) a probability density $p(x, \delta_B, \epsilon, \cdot)$ for the occurrence of a localization inside interval $[x - \epsilon, x + \epsilon]$ in arm with detector **B** (see Pic. 2.2.2) is given by formula (see Remark 1.2.4)

$$ p(x, \delta_B, \epsilon) = \frac{\| |\psi_{\delta_B, \epsilon, x} \rangle_{cl} \|^2}{\Delta(\delta_B, \epsilon)}, \qquad (2.2.54) $$

where

$$ 
\begin{aligned}
\| |\psi_{\delta_B, \epsilon, x} \rangle_{cl} \|^2 &= \left( \frac{1}{\delta_B \pi \delta_B} \right)^{1/2} \int_{|q-x| \le \epsilon} dq |\psi(q)|^2 \exp \left[ -\frac{1}{\delta_B} (q-x)^2 \right], \\
\psi(q) &= \langle q \| \psi \rangle, \\
\Delta(\delta_B, \epsilon) &= \int_{-\infty}^{\infty} \| |\psi_{\delta_B, \epsilon, x} \rangle_{cl} \|^2 dx,
\end{aligned}
\qquad (2.2.55) $$

and where parametr $\delta_B$ depend on arm with detector **B**.

**Remark 2.2.17.** Note that parametr $\delta$ in formula (1.2.27) (see Remark 1.2.4) of course depend on measurement device and there no exist two equivalet devices such that $\delta_A = \delta_B$.

We assume now that
$$
\begin{align}
\delta_A &\approx \delta_B \ll 1, \nonumber \\
0 &< |\delta_A - \delta_B|, \tag{2.2.56} \\
\int_{-\infty}^{\infty} [|\psi(x)|^2]'' dx &< \infty, \nonumber
\end{align}
$$

From Eq.(2.2.53) and (2.2.56) by using laplace method [24], we obtain:

$$
\begin{align}
\|\psi_{\delta_A, \epsilon, x}\|_{cl}^2 &= \left(\frac{1}{\delta_A \pi \delta_A}\right)^{1/2} \int_{|q-x|\le\epsilon} dq |\psi(q)|^2 \exp\left[-\frac{1}{\delta_A}(q-x)^2\right] \approx \nonumber \\
&\approx |\psi(x)|^2 + \delta_A O\left([|\psi(x)|^2]''\right) = |\psi(x)|^2 + \delta_A c_1^A [|\psi(x)|^2]'', \tag{2.2.57} \\
\Delta(\delta_A, \epsilon) &= \int_{-\infty}^{\infty} \|\psi_{\delta_A, \epsilon, x}\|_{cl}^2 dx = 1 + c_2^A \delta_A, c_2^A = O\left(\int_{-\infty}^{\infty} [|\psi(x)|^2]'' dx\right). \nonumber
\end{align}
$$

From Eq.(2.2.55) and (2.2.56) by using laplace method [24], we obtain:

$$
\begin{align}
\|\psi_{\delta_B, \epsilon, x}\|_{cl}^2 &= \left(\frac{1}{\delta_B \pi \delta_B}\right)^{1/2} \int_{|q-x|\le\epsilon} dq |\psi(q)|^2 \exp\left[-\frac{1}{\delta_B}(q-x)^2\right] \approx \nonumber \\
&\quad |\psi(x)|^2 + \delta_B O\left([|\psi(x)|^2]''\right) = |\psi(x)|^2 + \delta_B c_1^B [|\psi(x)|^2]'', \tag{2.2.58} \\
\Delta(\delta_B, \epsilon) &= \int_{-\infty}^{\infty} \|\psi_{\delta_B, \epsilon, x}\|_{cl}^2 dx = 1 + c_2^B \delta_B, c_2^B = O\left(\int_{-\infty}^{\infty} [|\psi(x)|^2]'' dx\right). \nonumber
\end{align}
$$

From Eq.(2.2.52) and Eq.(2.2.56) we obtain

$$
p(x, \delta_A, \epsilon) = \frac{\|\psi_{\delta_A, \epsilon, x}\|_{cl}^2}{\Delta(\delta_A, \epsilon)} = \frac{|\psi(x)|^2 + \delta_A c_1^A [|\psi(x)|^2]''}{1 + c_2^A \delta_A}. \quad (2.2.59)
$$

From Eq.(2.2.54) and Eq.(2.2.57) we obtain

$$
p(x, \delta_B, \epsilon) = \frac{\|\psi_{\delta_B, \epsilon, x}\|_{cl}^2}{\Delta(\delta_B, \epsilon)} = \frac{|\psi(x)|^2 + \delta_B c_1^B [|\psi(x)|^2]''}{1 + c_2^B \delta_B}. \quad (2.2.60)
$$

**Definition 2.2.12.** We define now the probability measures $\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle(A^t)}$ and $\mathbf{P}_{|\psi_{\delta_B, \epsilon, x}\rangle(A^t)}$ by formulae

$$
\begin{align}
\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle(A^t)} &= \int_{A^t} p(x, \delta_A, \epsilon) d\mu(x), \nonumber \\
\mathbf{P}_{|\psi_{\delta_B, \epsilon, x}\rangle(A^t)} &= \int_{A^t} p(x, \delta_B, \epsilon) d\mu(x), \tag{2.2.61}
\end{align}
$$

where $A^t \in \Sigma_{a,b}$ and $d\mu(x)$ is the Lebesgue measure and $\Sigma_{a,b} = B([a,b])$ is the Borel algebra on a set $[a,b]$, (see subsection I.6 Definition 1.6.3).

**Definition 2.2.13.** We assume now that $\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle} \ll \mathbf{P}$ and $\mathbf{P}_{|\psi_{\delta_B, \epsilon, x}\rangle} \ll \mathbf{P}$, i.e. $\mathbf{P}_{|\psi\rangle}$ is absolutely continuous with respect to $\mathbf{P}$ (see subsection I.6, Eq.(1.6.5)).By
Radon-Nicodym theorem we obtain for any $A^t \in \Sigma_{a,b}$:

$$
\begin{align}
\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle}(A^t) &= \int_{A^t} X_{|\psi_{\delta_A, \epsilon, x}\rangle}(\omega) d\mathbf{P}, \nonumber \\
X_{|\psi_{\delta_A, \epsilon, x}\rangle}(\omega) &= \frac{d\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle}}{d\mathbf{P}}, \tag{2.2.62} \\
\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle}(A^t) &= \int_{A^t} X_{|\psi_{\delta_A, \epsilon, x}\rangle}(\omega) d\mathbf{P}, \nonumber \\
X_{|\psi_{\delta_A, \epsilon, x}\rangle}(\omega) &= \frac{d\mathbf{P}_{|\psi_{\delta_A, \epsilon, x}\rangle}}{d\mathbf{P}}. \nonumber
\end{align}
$$

We write below for a short

$$
X_1(\omega) \triangleq X_{|\psi_{\delta_A, \epsilon, x}\rangle}(\omega), X_2(\omega) \triangleq X_{|\psi_{\delta_B, \epsilon, x}\rangle}(\omega). \quad (2.2.63)
$$

**Remark 2.2.18.** We assume now without loss of generality that

$$
X_2(\omega) - X_1(\omega) \geq 0 \text{ a.s.} \tag{2.2.64}
$$

see Pic.2.2.9.

Let us consider now the quantity

$$
\eta_{1,2} = \int_{\Omega} |X_1(\omega) - X_2(\omega)| d\mathbf{P} = \int_{\Omega} [X_2(\omega) - X_1(\omega)] d\mathbf{P}. \quad (2.2.65)
$$

We assume now that

$$
\int_{-\infty}^{\infty} x |\psi(x)|^2 dx < \infty, \int_{-\infty}^{\infty} x [ |\psi(x)|^2 ]'' dx < \infty, \quad (2.2.66)
$$

From Eq.(2.2.65) by using Eq.(2.2.59) and Eq.(2.2.60) we obtain
$$
\begin{align*}
\eta_{1,2} &= \int_{\mathbb{R}} xp(x, \delta_B, \epsilon)dx - \int_{\mathbb{R}} xp(x, \delta_A, \epsilon)dx \\
&= \frac{1}{1+c_2^B \delta_B} \int_{\mathbb{R}} x[|\psi(x)|^2 + \delta_B c_1^B [|\psi(x)|^2]'']dx \\
&\quad - \frac{1}{1+c_2^A \delta_A} \int_{\mathbb{R}} x[|\psi(x)|^2 + \delta_A c_1^A [|\psi(x)|^2]'']dx \\
&\approx (1-c_2^B \delta_B) \int_{\mathbb{R}} x[|\psi(x)|^2 + \delta_B c_1^B [|\psi(x)|^2]'']dx \\
&\quad - (1-c_2^A \delta_A) \int_{\mathbb{R}} x[|\psi(x)|^2 + \delta_A c_1^A [|\psi(x)|^2]'']dx \\
&= (2.2.67) \\
&\qquad \delta_B c_1^B \int_{\mathbb{R}} x[|\psi(x)|^2]''dx - c_2^B \delta_B \int_{\mathbb{R}} x|\psi(x)|^2dx - \delta_B^2 c_1^B c_2^B \int_{\mathbb{R}} x[|\psi(x)|^2]''dx \\
&\quad - \delta_A c_1^A \int_{\mathbb{R}} x[|\psi(x)|^2]''dx + c_2^A \delta_A \int_{\mathbb{R}} x|\psi(x)|^2dx + \delta_A^2 c_1^A c_2^A \int_{\mathbb{R}} x[|\psi(x)|^2]''dx \\
&= (\delta_B c_1^B - \delta_A c_1^A - \delta_B^2 c_1^B c_2^B + \delta_A^2 c_1^A c_2^A) \int_{\mathbb{R}} x[|\psi(x)|^2]''dx + (c_2^A \delta_A - c_2^B \delta_B) \int_{\mathbb{R}} x|\psi(x)|^2dx \\
&\approx \alpha_1(c_2^A \delta_A - c_2^B \delta_B) + \alpha_2(\delta_B c_1^B - \delta_A c_1^A),
\end{align*}
$$

where

$$
\alpha_1 = \int_{\mathbb{R}} x|\psi(x)|^2 dx, \alpha_2 = \int_{\mathbb{R}} x[|\psi(x)|^2]'' dx. \tag{2.2.68}
$$

**Lemma 2.2.1.** Let $(\Omega, \Sigma, \mathbf{P})$ be a measure space, and let $f$ be an real-valued measurable function defined on $\Omega$. Then for any real number $t > 0$:

$$
\mathbf{P}\{\omega \in \Omega | |f(\omega)| \geq t\} \leq \frac{1}{t} \int_{|f(\omega)| \geq t} |f(\omega)| d\mathbf{P}(\omega). \quad (2.2.69)
$$

From inequality (2.2.69) and Eq.(2.2.67) we obtain

$$
\begin{align}
\mathbf{P}\{\omega \in \Omega : |X_1(\omega) - X_2(\omega)| \ge t\} &\le \frac{1}{t} \int_{|X_1(\omega) - X_2(\omega)| \ge t} [|X_1(\omega) - X_2(\omega)|] d\mathbf{P}(\omega) \tag{2.2.70} \\
&< \frac{1}{t^2} \int_{\Omega} [X_1(\omega) - X_2(\omega)] = \frac{\eta_{1,2}}{t} \approx \frac{\alpha_1(c_2^A \delta_A - c_2^B \delta_B) + \alpha_2(\delta_B c_1^B - \delta_A c_1^A)}{t}. \nonumber
\end{align}
$$

We define now

$$
\begin{align}
A^t &= A_{\lambda}^{ph}(t, x_A) = \{\omega | x_A - \epsilon \le x_A(\omega) \le x_A - \epsilon\}, \tag{2.2.71} \\
B^t &= B_{\lambda}^{ph}(t, x_B) = \{\omega | x_B - \epsilon \le x_B(\omega) \le x_B - \epsilon\},
\end{align}
$$

and chose in (2.2.68) number $t = x_B - x_A \gg 1$.
<figure><img src="image_27.png" /><figcaption>Pic. 2.2.9. The plot of the random variables $X_1(\omega)$ and $X_2(\omega)$.</figcaption></figure>

$$A^t = A_{\lambda}^{ph}(t, x_A), B^t = B_{\lambda}^{ph}(t, x_B), A^t \cap B^t = B^t.$$

Note that $\mathbf{P}(A^t \cap B^t) \le \mathbf{P}(B^t)$, see Pic. 2.2.9. From (2.2.68) follows that

$$\mathbf{P}(A^t \cap B^t) < \frac{\alpha_1(c_2^A \delta_A - c_2^B \delta_B) + \alpha_2(\delta_B c_1^B - \delta_A c_1^A)}{(x_B - x_A)^2} \ll 1. \qquad (2.2.70)$$

### III. Schrödinger's Cat paradox resolution

In this section we shall consider the problem of the collapse of the cat state vector on the basis of two different hypotheses:

(A) The canonical postulate of QM is correct in all cases.

(B) The canonical interpretation of the wave function $\psi = c_1\psi_1 + c_2\psi_2$ is correct only when the supports the wave functions $\psi_1$ and $\psi_2$ essentially overlap. When the wave functions $\psi_1$ and $\psi_2$ have separated supports (as in the case of the experiment that we are considering in section II) we claim that canonical interpretation of the wave function $\psi = c_1\psi_1 + c_2\psi_2$ is no longer valid for a such cat state, for details see Appendix C.

#### III.1. Consideration of the Schrödinger's cat paradox using canonical von Neumann postulate

Let $|s_1(t)\rangle$ and $|s_2(t)\rangle$ be
$$|s_1(t)\rangle = |\text{undecayed nucleus at instant } t\rangle, \qquad (3.1.1)$$

$$|s_2(t)\rangle = |\text{decayed nucleus at instant } t\rangle.$$

In a good approximation we assume now that

$$|s_1(0)\rangle = \int_{-\infty}^{+\infty} \Psi_{II}^{\#}(x)|x\rangle dx \qquad (3.1.2)$$

and

$$|s_2(0)\rangle = \int_{-\infty}^{+\infty} \Psi_{I}^{\#}(x)|x\rangle dx. \qquad (3.1.3)$$

**Remark 3.1.1.** Note that: (i) $|s_2(0)\rangle = |\text{decayed nucleus at instant } 0\rangle = |\text{free } \alpha\text{-particle at instant } 0\rangle$. (ii) Feynman propagator of a free $\alpha$-particle are [52]:

$$K_2(x, t, x_0) = \left(\frac{m}{2\pi i\hbar t}\right)^{1/2} \exp\left\{\frac{i}{\hbar}\left[\frac{m(x-x_0)^2}{2t}\right]\right\}. \qquad (3.1.4)$$

Therefore from Eq.(3.1.3),Eq.(2.1.9) and Eq.(3.1.4) we obtain
$$
\begin{align*}
|s_2(t)\rangle &= \int_{-\infty}^{+\infty} \Psi_I^{\#}(x,t)|x\rangle dx, \\
\Psi_I^{\#}(x,t) &= \int_{-\infty}^{0} \Psi_I^{\#}(x_0) K_2(x,t,x_0) dx_0 = \\
&\quad (\pi r_c^2)^{-1/4} \times \left(\frac{m}{2\pi i \hbar t}\right)^{1/2} \times \int_{-\infty}^{0} \theta_1(x_0,l) \exp\left(-\frac{x_0^2}{2r_c^2}\right) \exp\left(-i \frac{2\pi}{\hbar} \sqrt{2mE} x_0\right) \times \\
&\qquad \times \exp\left\{\frac{i}{\hbar} \left[\frac{m(x-x_0)^2}{2t}\right]\right\} dx_0 = \tag{3.1.5} \\
&\quad (\pi r_c^2)^{-1/4} \times \left(\frac{m}{2\pi i \hbar t}\right)^{1/2} \times \int_{-l}^{0} \theta_1(x_0,l) \exp\left(-\frac{x_0^2}{2r_c^2}\right) \times \\
&\qquad \times \exp\left\{\frac{i}{\hbar} \left[\frac{m(x-x_0)^2}{2t} - \pi \sqrt{4mE} x_0\right]\right\} dx_0 = \\
&\quad (\pi r_c^2)^{-1/4} \times \left(\frac{m}{2\pi i \hbar t}\right)^{1/2} \times \int_{-l}^{0} \theta_1(x_0,l) \exp\left(-\frac{x_0^2}{2r_c^2}\right) \times \exp\left\{\frac{i}{\hbar}[S(t,x,x_0)]\right\} dx_0,
\end{align*}
$$

where

$$
S(t,x,x_0) = \frac{m(x-x_0)^2}{2t} - \pi \sqrt{8mE} x_0. \tag{3.1.6}
$$

We assume now that

$$
\hbar \ll 2r_c^2 \ll l^2 < 1. \tag{3.1.7}
$$

Oscillatory integral in RHS of Eq.(3.1.5) is calculated now directly using stationary
phase approximation. The phase term $S(x,x_0)$ given by Eq.(3.1.6) is stationary
when

$$
\frac{\partial S(t, x, x_0)}{\partial x_0} = - \frac{m(x - x_0)}{t} - \pi \sqrt{8mE} = 0. \quad (3.1.8)
$$

Therefore

$$
\begin{align}
& -\frac{m(x-x_0)}{t} - \pi\sqrt{8mE} = 0, \tag{3.1.9} \\
& -(x-x_0) = \pi t \sqrt{8E/m}, \nonumber
\end{align}
$$

and thus stationary point $x_0(t,x)$ are
$$x_0(t,x) = \pi t \sqrt{8E/m} + x. \tag{3.1.10}$$

Thus from Eq.(3.1.5) and Eq.(3.1.10) using stationary phase approximation we obtain

$$\begin{gathered} |s_2(t)\rangle = \int_{-\infty}^{+\infty} \Psi_I^{\#}(x,t)|x\rangle dx, \\ \Psi_I^{\#}(x,t) = (\pi r_c^2)^{-1/4} \times \theta_1(x_0(t,x),l) \exp\left[-\frac{x_0^2(t,x)}{2r_c^2}\right] \times \exp\left\{\frac{i}{\hbar}[S(t,x,x_0(t,x))]\right\} + O(\hbar), \end{gathered} \tag{3.1.11}$$

where

$$S(x, x_0(t,x)) = \frac{m(x - x_0(t,x))^2}{2t} - \pi \sqrt{8mE} x_0(t,x). \tag{3.1.12}$$

From Eq.(3.10) we obtain

$$\overline{\Psi_I^{\#}(x,t)} \Psi_I^{\#}(x,t) \approx (\pi r_c^2)^{-1/2} \times \theta_1(x + \pi t \sqrt{8E/m}, l) \exp\left[-\frac{(x + \pi t \sqrt{8E/m})^2}{r_c^2}\right]. \tag{3.1.13}$$

**Remark 3.1.2.** From the inequality (3.1.7) and Eq.(3.1.13) follows that $\alpha$-particle at each instant $t \ge 0$ moves quasiclassically from right to left by the law

$$x(t) = -\pi t \sqrt{8E/m}, \tag{3.1.14}$$

i.e. i.e., estimating the position $x(t, x_0, t_0; \hbar)$ at each instant $t \ge 0$ with final error $r_c$ gives $|\langle x \rangle(t) - x(t)| \le r_c, i = 1, \dots, d$ with a probability

$$P\{|\langle x \rangle(t, 0, 0; \hbar) - x(t)| \le r_c\} = 1.$$

**Remark 3.1.3.** We assume now that a distance between radioactive source and internal monitor which detects a single atom decaying (see Pic.1.1.1) is equal to $L$.

**Proposition 3.1.1.** After $\alpha$-decay at instant $t = 0$ the collapses:

$|\text{live cat}\rangle \rightarrow |\text{death cat}\rangle$ arises at instant

$$T = \frac{L}{\pi \sqrt{8E/m}} \tag{3.1.15}$$

with a probability $P_T(|\text{death cat}\rangle)$ to observe a state $|\text{death cat}\rangle$ at instant $T$ is $P_T(|\text{death cat}\rangle) = 1$.

**Proof.** Note that. In this case Schrödinger's cat in fact permorm the single measurement of $\alpha$-particle position with accuracy of $\delta x = l$ at instant $t = T$ (given by
Eq.(3.1.15)) by internal monitor (see Pic.1.1.1.). The probability of getting a result L with accuracy of $\delta x = l$ given by

$$ \int_{|L-x| \le l/2} |\langle x || s_2(T) \rangle|^2 dx = 1. \qquad (3.1.16) $$

Therefore at instant T the $\alpha$-particle kills Schrödinger's cat with a probability $P_T(|\text{death cat}\rangle) = 1$.

**Remark 3.1.4.** Note that. When Schrödinger's cat has permormed this measurement the immediate post measurement state of $\alpha$-particle (by von Neumann postulate Q.III.5, see subsection I.7.1) will end up in the state

$$ |\Psi_T\rangle = \frac{\int_{|L-x| \le l/2} |x\rangle\langle x||s_2(T)\rangle dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x||s_2(T)\rangle|^2 dx}} = \int_{|L-x| \le l/2} |x\rangle\langle x||s_2(T)\rangle dx \in S_\Theta, \Theta = \{x||L-x| \le l/2\} \quad (3.1.17) $$

From Eq.(3.1.17) one obtains

$$ \langle x' || \Psi_T \rangle = \int_{|L-x| \le l/2} \langle x' || x \rangle \langle x || s_2(T) \rangle dx = \int_{|L-x| \le l/2} \delta(x' - x) \langle x || s_2(T) \rangle dx = \Psi_I^{\#}(x', t). \quad (3.1.18) $$

Therefore the state $|\Psi_T\rangle$ again kills Schrödinger's cat with a probability $P_T(|\text{death cat}\rangle) = 1$.

Suppose now that a nucleus n, whose Hilbert space is spanned by orthonormal states $|s_i(t)\rangle$, $i = 1,2$, where $|s_1(t)\rangle = |\text{undecayed nucleus at instant } t\rangle$ and $|s_2(t)\rangle = |\text{decayed nucleus at instant } t\rangle$ is in the superposition state

$$ |\Psi_t\rangle_n = c_1|s_1(t)\rangle + c_2|s_2(t)\rangle, |c_1|^2 + |c_2|^2 = 1. \qquad (3.1.19) $$

**Remark 3.1.5.** Note that: (i) $|s_1(0)\rangle = |\text{undecayed nucleus at instant } t = 0\rangle = |\alpha$-particle iside region $(0, l]$ at instant $t = 0\rangle$. (ii) Feynman propagator of $\alpha$-particle inside region $(0, l]$ are [52]:

$$ K_2(x, t, x_0) = \left(\frac{m}{2\pi iht}\right)^{1/2} \exp\left\{\frac{i}{\hbar}[S(t, x, x_0)]\right\}, \qquad (3.1.20) $$

where

$$ S(t, x, x_0) = \frac{m(x - x_0)^2}{2t} + mt(U_0 - E). \qquad (3.1.21) $$

Therefore from Eq.(2.1.11)-Eq.(2.1.12) and Eq.(3.1.20)-Eq.(3.1.21) we obtain
$$
\begin{gather*}
|s_1(t)\rangle = \int_{-\infty}^{+\infty} \Psi_{\Pi}^{\#}(x,t)|x\rangle dx, \\
\Psi_{\Pi}^{\#}(x,t) = \int_{0}^{l} \Psi_{\Pi}^{\#}(x_0) K_2(x,t,x_0) dx_0 = \tag{3.1.22}
\end{gather*}
$$

$$
\left( \frac{m}{2\pi i \hbar t} \right)^{1/2} \int_0^l E(x_0, l) \Psi_{\Pi}(x_0) \theta_l(x_0) \exp \left\{ \frac{i}{\hbar} [S(t, x, x_0)] \right\} dx_0,
$$

where

$$
\theta_l(x) = \begin{cases} 1 & \text{for } x \in [0, l] \\ 0 & \text{for } x \notin [0, l] \end{cases}
$$

**Remark 3.1.6.** We assume for simplification now that

$$
k' \le 1. \tag{3.1.23}
$$

Therefore oscillatory integral in RHS of Eq.(3.1.22) is calculated now directly using
stationary phase approximation. The phase term $S(x,x_0)$ given by Eq.(3.21) is
stationary when

$$
\frac{\partial S(t, x, x_0)}{\partial x_0} = - \frac{m(x - x_0)}{t} = 0. \qquad (3.1.24)
$$

and thus stationary point $x_0(t,x)$ are

$$
\begin{align}
-x + x_0 &= 0 \tag{3.1.25} \\
x_0(t,x) &= x. \nonumber
\end{align}
$$

Thus from Eq.(3.1.22) and Eq.(3.1.25) using stationary phase approximation we
obtain

$$
\begin{align}
\Psi_{\Pi}^{\#}(x,t) ={}& E(x_0(t,x),l)\Psi_{\Pi}(x_0(t,x))\theta_l(x_0(t,x))\exp\left\{\frac{i}{\hbar}[S(t,x,x_0(t,x))]\right\} + O(\hbar) = \nonumber \\
&= E(x,l)\Psi_{\Pi}(x)\theta_l(x)\exp\left\{\frac{i}{\hbar}[mt(U_0-E)]\right\} + O(\hbar) = \tag{3.1.26} \\
&= E(x,l)\theta_l(x)O(1)\exp\left\{\frac{i}{\hbar}[mt(U_0-E)]\right\} + O(\hbar). \nonumber
\end{align}
$$

Therefore from Eq.(3.22) and Eq.(3.26) we obtain

$$
|\Psi_{\Pi}^{\#}(x,t)|^2 = E^2(x,l)\theta_l(x)O(1) + O(\hbar). \tag{3.1.27}
$$

**Remark 3.1.7.** Note that for each instant *t* > 0 :
$\mathbf{supp}(\Psi_{\mathbf{II}}^{\#}(x,t)) \cap \mathbf{supp}(\Psi_{\mathbf{I}}^{\#}(x,t)) = \emptyset$.

**Remark 3.1.8.** Note that. From Eq.(3.1.11),Eq.(3.1.13), Eq.(3.1.19), Eq.(3.1.22)-Eq.(3.1.27) and Eq.(A.13) by Remark 3.1.7 we obtain

$$ 
\begin{aligned}
\mathbf{n}\langle\Psi_t|\hat{x}|\Psi_t\rangle_{\mathbf{n}} &= |c_1|^2\langle s_1(t)|\hat{x}|s_1(t)\rangle + |c_2|^2\langle s_2(t)|\hat{x}|s_2(t)\rangle + \\
&\qquad c_1c_2^*\langle s_2(t)|\hat{x}|s_1(t)\rangle + c_1^*c_2\langle s_2(t)|\hat{x}|s_1(t)\rangle^* =
\end{aligned} 
\quad (3.1.28) 
$$

$$ |c_1|^2\langle s_1(t)|\hat{x}|s_1(t)\rangle + |c_2|^2\langle s_2(t)|\hat{x}|s_2(t)\rangle = |c_1|^2l + |c_2|^2T\pi\sqrt{8E/m} $$

**Proposition 3.1.2.** (i) Suppose that a nucleus **n** is in the superposition state $|\Psi_t\rangle_{\mathbf{n}}$ ($|\Psi_t\rangle_{\mathbf{n}}$-particle) given by Eq.(3.19). Then the collaps: $|\text{live cat}\rangle \rightarrow |\text{death cat}\rangle$ arises at instant

$$ T_{\text{col}} \approx \frac{L \pm l}{|c_2|^2 \sqrt{8\pi^2 E/m}}. \quad (3.1.29) $$

with a probability $\mathbf{P}_{T_{\text{col}}}(|\text{death cat}\rangle)$ to observe a state $|\text{death cat}\rangle$ at instant $T_{\text{col}}$ is $\mathbf{P}_{T_{\text{col}}}(|\text{death cat}\rangle) = |c_2|^2$.

(ii) Assume now a Schrödinger's cat has performed the single measurement of $|\Psi_t\rangle_{\mathbf{n}}$-particle position with accuracy of $\delta x = l$ at instant $T = T_{\text{col}}$ (given by Eq.(3.1.29)) by internal monitor (see Pic.1.1.1) and the result $x \approx L \pm l$ is not observed by Schrödinger's cat. Then the collaps: $|\text{live cat}\rangle \rightarrow |\text{death cat}\rangle$ never arises at any instant $T > T_{\text{col}}$ and a probability $\mathbf{P}_{T>T_{\text{col}}}(|\text{death cat}\rangle)$ to observe a state $|\text{death cat}\rangle$ at instant $T > T_{\text{col}}$ is $\mathbf{P}_{T>T_{\text{col}}}(|\text{death cat}\rangle) = 0$.

**Proof.** (i) Note that for $t > 0$ the marginal density matrix $\rho(t)$ is diagonal

$$ \rho(t) = \begin{pmatrix} |c_1|^2 \int |\Psi_{\mathbf{II}}^{\#}(x,t)|^2 dx & 0 \\ 0 & |c_2|^2 \int |\Psi_{\mathbf{I}}^{\#}(x,t)|^2 dx \end{pmatrix} \qquad \text{In this} $$

case a Schrödinger's cat in fact perform the single measurement of $|\Psi_t\rangle_{\mathbf{n}}$-particle position with accuracy of $\delta x = l$ at instant $t = T_{\text{col}}$ (given by Eq.(3.1.29)) by internal monitor (see Pic.1.1.1). The probability of getting a result $L$ at instant $T \approx T_{\text{col}}$ with accuracy of $\delta x = l$ given by

$$ 
\begin{aligned}
\int_{|L-x| \le l/2} |\langle x || \Psi_T \rangle_{\mathbf{n}}|^2 dx &= \int_{|L-x| \le l/2} |\langle x | c_1 | s_1(T) \rangle + c_2 |s_2(T) \rangle|^2 dx = \\
\int_{|L-x| \le l/2} |c_1 \langle x || s_1(T) \rangle + c_2 \langle x || s_2(T) \rangle|^2 dx &= \\
\int_{|L-x| \le l/2} |c_1^2 \Psi_{\mathbf{II}}^{\#2}(x, T) + c_2^2 \Psi_{\mathbf{I}}^{\#2}(x, T) + 2c_1 c_2 \Psi_{\mathbf{I}}^{\#}(x, T) \Psi_{\mathbf{II}}^{\#}(x, T)| dx &=
\end{aligned} 
\quad (3.1.30) 
$$
From Eq.(3.1.30) by Remark 3.1.7 and Eq.(3.1.13) one obtains

$$ \int_{|L-x| \le l/2} |\langle x || \Psi_T \rangle_n|^2 dx = \int_{|L-x| \le l/2} |c_2^2 \Psi_I^{\#2}(x, T)|^2 dx = |c_2|^2 \int_{|L-x| \le l/2} |\Psi_I^{\#}(x, T)|^2 dx = |c_2|^2. \quad (3.1.31) $$

Note that. When Schrödinger's cat has permormed this measurement and the result $x \approx L \pm l$ is observed, then the immediate post measurement state of $\alpha$-particle (by von Neumann measurement postulate **Q.III.5**, see subsection **I.7.1.**.) will end up in the state

$$ \begin{aligned} |\Psi_{T_{\text{col}}}\rangle_n = & \frac{\int_{|L-x| \le l/2} |x\rangle\langle x || \Psi_{T_{\text{col}}} \rangle_n dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x || \Psi_{T_{\text{col}}} \rangle_n|^2 dx}} = \frac{\int_{|L-x| \le l/2} |x\rangle\langle x | (c_1|s_1(T_{\text{col}})\rangle + c_2|s_2(T_{\text{col}})\rangle) dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x || \Psi_{T_{\text{col}}} \rangle_n|^2 dx}} = \\ & \frac{c_1 \int_{|L-x| \le l/2} |x\rangle\langle x || s_1(T_{\text{col}}) \rangle + c_2 \int_{|L-x| \le l/2} |x\rangle\langle x || s_2(T_{\text{col}}) \rangle dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x || \Psi_{T_{\text{col}}} \rangle_n|^2 dx}} \in S_{\Theta}, \Theta = \{x || L - x \le l/2\}. \end{aligned} \quad (3.1.32) $$

From Eq.(3.1.32) by Eq.(3.1.31) and by Remark 3.1.7 one obtains

$$ \begin{aligned} |\Psi_{T_{\text{col}}}\rangle_n = & \frac{\int_{|L-x| \le l/2} |x\rangle\langle x || \Psi_{T_{\text{col}}} \rangle_n dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x || \Psi_{T_{\text{col}}} \rangle_n|^2 dx}} = \frac{\int_{|L-x| \le l/2} |x\rangle\langle x | (c_1|s_1(T_{\text{col}})\rangle + c_2|s_2(T_{\text{col}})\rangle) dx}{\sqrt{\int_{|L-x| \le l/2} |\langle x || \Psi_{T_{\text{col}}} \rangle_n|^2 dx}} = \\ & \frac{c_2}{|c_2|} \int_{|L-x| \le l/2} |x\rangle\langle x || s_2(T_{\text{col}}) \rangle dx. \end{aligned} \quad (3.1.32) $$

Obviously by Remark 3.1.4 the state $|\Psi_{T_{\text{col}}}\rangle_n$ kills Schrödinger's cat with a probability $P_{T_{\text{col}}}(|\text{death cat}\rangle) = 1$.

**Proof. (ii)** The probability of getting a result $L$ at any instant $T > T_{\text{col}}$ with

accuracy of $\delta x = l$ by Eq.(3.1.31) and Eq.(3.1.13) given by

$$ \begin{aligned} \int_{|L-x| \le l/2} |\langle x || \Psi_T \rangle_n|^2 dx &= \int_{|L-x| \le l/2} |c_2^2 \Psi_I^{\#2}(x, T)|^2 dx = |c_2|^2 \int_{|L-x| \le l/2} |\Psi_I^{\#}(x, T)|^2 dx = \\ &= (\pi r_c^2)^{-1/2} \int_{|L-x| \le l/2} \theta_1(x + \pi T \sqrt{8E/m}, l) \exp\left[-\frac{(x + \pi T \sqrt{8E/m})^2}{r_c^2}\right] \times 0. \end{aligned} \quad (3.1.33) $$

Thus standard formalism of continuous quantum measurements [2],[3],[4],[5] leads to a definite but unpredictable measurement outcomes, and therefore $|\Psi_I\rangle_n$ suddenly “collapses” at unpredictable instant $t'$ into one of the states $|s_i(t')\rangle$, $i = 1,2$.

### III.2. Resolution of the Schrödinger's cat paradox using generalized von Neumann postulate.

**Proposition 3.2.1.** Suppose that a nucleus **n** is in the superposition state given by Eq.(3.1.19). Then the collaps: $|\text{live cat}\rangle \rightarrow |\text{death cat}\rangle$ arises at instant
$$ T = \frac{L}{|c_2|^2 \sqrt{8\pi^2 E/m}}. \qquad (3.2.1) $$

with a probability $P_T(|\text{death cat}\rangle)$ to observe a state $|\text{death cat}\rangle$ at instant $T$ is $P_T(|\text{death cat}\rangle) = 1$.

**Proof.** Let us consider now a state $|\Psi_t\rangle_n$ given by Eq.(3.1.19). This state consists of a summ of two wave packets $c_1\Psi_{II}^\#(x,t)$ and $c_2\Psi_I^\#(x,t)$. Wave packet $c_1\Psi_{II}^\#(x,t)$ present an $\alpha_{II}$-particle which lives in region II with a probability $|c_1|^2$ (see Pic.2.1.1). Wave packet $c_2\Psi_I^\#(x,t)$ present an $\alpha_I$-particle which lives in region I with a probability $|c_2|^2$ (see Pic.2.1.1) and moves from the right to the left. Note that $I \cap II = \emptyset$. From Eq.(3.1.28) follows that $\alpha_I$-particle at each instant $t \ge 0$ moves quasiclassically from right to left by the law

$$ x(t) = -|c_2|^2 \pi t \sqrt{8E/m}, \qquad (3.2.2) $$

From Eq.(3.2.2) one obtains

$$ T = T_{\text{col}} \approx \frac{L}{|c_2| \sqrt{8\pi^2 E/m}}. \qquad (3.2.3) $$

Note that. In this case Schrödinger's cat in fact permorm a single measurement of $|\Psi_t\rangle_n$-particle position with accuracy of $\delta x = l$ at instant $t = T = T_{\text{col}}$ given by Eq.(3.2.3), by internal monitor (see Pic.1.1.1). The probability $P(L, l, T_{\text{col}})$ of getting

the result $L$ at instant $t = T_{\text{col}}$ with accuracy of $\delta x = l$ [by Remark 3.1.7 and by Generalized von Neumann measurement postulate Q.IV.4 and by postulate Q.IV.3 (see subsection 1.7.1) and by reconcile Bohr rule, see section 1.6, Eq.(1.6.19)] is given by formula

$$ \begin{aligned} \mathbf{P}(L, l, T_{\text{col}}) &= \int_{|L-x| \le l/2} \left[ |\langle x | c_1 | s_1(T_{\text{col}}) \rangle|^2 * |\langle x | c_2 | s_2(T_{\text{col}}) \rangle|^2 \right] dx = \\ &\quad \int_{|L-x| \le l/2} |c_2|^{-2} |c_1|^{-2} \left[ |\langle x | c_1 |^{-2} | s_1(T_{\text{col}}) \rangle|^2 * |\langle x | c_2 |^{-2} | s_2(T_{\text{col}}) \rangle|^2 \right] dx = \\ &\quad \int_{|L-x| \le l/2} |c_2|^{-2} |c_1|^{-2} \left[ |\Psi_I^\#(x|c_2|^{-2}, T_{\text{col}})|^2 * |\Psi_{II}^\#(x|c_1|^{-2}, T_{\text{col}})|^2 \right] dx = 1. \end{aligned} \qquad (3.2.4) $$

Note that. When Schrödinger's cat has permormed this measurement and the result $x \approx L \pm l$ is observed, then the immediate post measurement state of $\alpha$-particle (by generalized von Neumann postulate Q.IV.4, see subsection 1.7.1) will end up in the state

$$ \begin{aligned} |\Psi_{T_{\text{col}}}\rangle_n &= \frac{\int_{|L-x| \le l/2} |x\rangle\langle x| \Psi_{T_{\text{col}}}\rangle_n dx}{\sqrt{\int_{|L-x| \le l/2} \left[ |\langle x||s_1(T_{\text{col}})\rangle|^2 + |\langle x||s_2(T_{\text{col}})\rangle|^2 \right] dx}} = \frac{|c_2|^{-2} \int_{|L-x| \le l/2} |x\rangle\langle x| s_2(T_{\text{col}})\rangle dx}{\sqrt{\int_{|L-x| \le l/2} \left[ |\langle x||s_2(T_{\text{col}})\rangle|^2 \right] dx}} \in \mathbf{H}_{\Theta}, \qquad (3.2.5) \\ \Theta &= \{x \mid |L-x| \le l/2\}. \end{aligned} $$

The state $|\Psi_{T_{\text{col}}}\rangle_n$ again kills Schrödinger's cat with a probability
$$P_{T_{\text{col}}}(|\text{death cat}\rangle) = 1.$$

Thus is the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition:

$$|\text{cat}\rangle = c_1|\text{live cat}\rangle + c_2|\text{death cat}\rangle.$$

Contrary to van Kampen's [10] and some others' opinions, "looking" at the outcome changes nothing, beyond informing the observer of what has already happened.

We remain: there are widespread claims that Schrödinger's cat is not in a definite alive or dead state but is, instead, in a superposition of the two. van Kampen, for example, writes "The whole system is in a superposition of two states: one in which no decay has occurred and one in which it has occurred. Hence, the state of the cat also consists of a superposition:

$$|\text{cat}\rangle = c_1|\text{live cat}\rangle + c_2|\text{death cat}\rangle.$$ The state remains a superposition until an observer looks at the cat" [10].

### III.3. Schrödinger's cat does not a live cat and dead cat samultaniously.

In this subsection we prove that Schrödinger's cat does not a live cat and dead cat samultaniously. Being interested only in distinction in changes of macroscopical and microscopic variables, it is easy to show that the macroscopical variable under suitable conditions can be described own wave function and the own Schrödinger equation i.e. if $X, Y = (y_1, ..., y_n)$ designates an macroscopic coordinates, and $x = (x_1, ..., x_m)$ is a set of microscopic variables then own functions of system corresponding to an value of energy, contain functions of the form [6]

$$\Phi(X, Y, x, t) = \Psi(X, Y, t)\psi(x, t), \qquad (3.3.1)$$

Here $\Psi(X, Y, t)$ is a "macroscopic" wave function such that

$$i\hbar \frac{\partial \Psi(X, Y, t)}{\partial t} = -\frac{\hbar^2}{2M} \nabla^2 \Psi(X, Y, t) + V(X, Y) \Psi(X, Y, t), \qquad (3.3.2)$$

Where the variable $X$ unequivocally corresponds to live or dead cat, $V(X, Y)$ is a conservative potential. Further, for the purpose of simplifications, we will assume that macroscopic wave function $\Psi(X, Y, t)$ depends only on one variable $X$. By variable $X$ it is possible to choose body temperature of a cat, i.e. $X = \theta$ if it is a question of the real cat. Then we obtain Schrödinger equation

$$i\hbar \frac{\partial \Psi(\theta, Y, t)}{\partial t} = -\frac{\hbar^2}{2M} \nabla^2 \Psi(\theta, Y, t) + V(\theta, Y) \Psi(\theta, Y, t), \qquad (3.3.3)$$
$$\Psi(\theta, Y, 0) = \Psi_0(\theta, Y).$$

**Definition 3.3.1.** Let $\theta$-supp[$\Psi(\theta, Y, t)$] be a set such that

$$\forall \theta \{\theta \in \theta\text{-supp}[\Psi(\theta, Y, t)] \text{ iff } \Psi(\theta, Y, t) \neq 0\}. \qquad (3.3.4)$$

**Assumption 3.3.1.** We assume now that:
(i) for a live cat any observer which measured body temperature of a cat always obtains result $\hat{\theta}$ such that $\hat{\theta} \in [\theta_1^{\text{live}}, \theta_2^{\text{live}}]$, where $0 < \theta_1^{\text{live}} < \theta_2^{\text{live}}$,

(ii) for a dead cat any observer which measured body temperature of a cat always obtains result $\hat{\theta}$ such that $\hat{\theta} \in [\theta_1^{\text{dead}}, \theta_2^{\text{dead}}]$, where $\theta_2^{\text{dead}} \le 0$ and

(iii) for a sick but a live cat any observer which measured body temperature of a cat always obtains result $\hat{\theta}$ such that $\hat{\theta} \in (\theta_2^{\text{dead}}, \theta_1^{\text{live}})$.

Let $\Psi^{\text{live}}(\theta, \mathbf{Y}, 0) = \Psi_0^{\text{live}}(\theta, \mathbf{Y})$ be a wave function of a live cat at instant $t = 0$. Then obviously

$$ \int_{\theta_1^{\text{live}}}^{\theta_2^{\text{live}}} \int_{\mathbb{R}^n} \theta \|\Psi_0^{\text{live}}(\theta, \mathbf{Y})\|^2 d\theta d^n Y = \bar{\theta}_1 \in [\theta_1^{\text{live}}, \theta_2^{\text{live}}] \quad (3.3.5) $$

and $\theta$-supp[$\Psi(\theta, \mathbf{Y}, t)$] = $[\theta_1^{\text{live}}, \theta_2^{\text{live}}]$.

Let $\Psi^{\text{dead}}(\theta, \mathbf{Y}, 0) = \Psi_0^{\text{dead}}(\theta, \mathbf{Y})$ be a wave function of a dead cat at instant $t = 0$. Then obviously

$$ \int_{\theta_1^{\text{dead}}}^{\theta_2^{\text{dead}}} \int_{\mathbb{R}^n} \theta \|\Psi_0^{\text{dead}}(\theta, \mathbf{Y})\|^2 d\theta d^n Y = \bar{\theta}_2 \in [\theta_1^{\text{dead}}, \theta_2^{\text{dead}}] \quad (3.3.6) $$

We assume now that

$$ \begin{aligned} & \int_{\mathbb{R}^n} \|\Psi_0^{\text{live}}(\theta, \mathbf{Y})\|^2 d^n Y = \psi_1(\theta), \\ & \int_{\mathbb{R}^n} \|\Psi_0^{\text{dead}}(\theta, \mathbf{Y})\|^2 d^n Y = \psi_2(\theta), \end{aligned} \quad (3.3.7) $$

where

$$ \psi_1(\theta) = \begin{cases} \left(\frac{1}{2\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{4\delta}(\theta - \bar{\theta}_1)^2\right] & \text{iff } |\theta - \bar{\theta}_1| \le \varepsilon \ll 1, \\ 0 & \text{iff } |\theta - \bar{\theta}_1| > \varepsilon \end{cases} \quad (3.3.8) $$

and

$$ \psi_2(\theta) = \begin{cases} \left(\frac{1}{2\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{4\delta}(\theta - \bar{\theta}_2)^2\right] & \text{iff } |\theta - \bar{\theta}_2| \le \varepsilon \ll 1, \\ 0 & \text{iff } |\theta - \bar{\theta}_2| > \varepsilon \end{cases} \quad (3.3.9) $$

We assume now that at instant $t = 0$ cat consist of superposition

$$ \Psi_0(\theta, \mathbf{Y}) = c_1 \Psi_0^{\text{live}}(\theta, \mathbf{Y}) + c_2 \Psi_0^{\text{dead}}(\theta, \mathbf{Y}), \quad (3.3.10) \\ |c_1|^2 + |c_2|^2 = 1. $$

Let $\psi(\theta)$ be a function

$$ \psi(\theta) = \int_{\mathbb{R}^n} \Psi_0(\theta, \mathbf{Y}) d^n Y. \quad (3.3.11) $$

Thus

$$ \psi(\theta) = c_1 \psi_1(\theta) + c_2 \psi_2(\theta). \quad (3.3.12) $$
Let $p(\theta, \alpha_1, \alpha_2)$, $\alpha_1 = |c_1|^2, \alpha_2 = |c_2|^2$ be a probability density to observe at instant $t = 0$ a body temperature of a cat with result $\theta$. By reconcile Bohr rule, [see section 1.6, Eq.(1.6.19)]

$$ 
\begin{gathered}
p(\theta, \alpha_1, \alpha_2) = [p_1(\theta, \alpha_1)] * [p_2(\theta, \alpha_2)], \\
p(\theta, \alpha_1) = \alpha_1^{-1} |\psi_1(\frac{\theta}{\alpha_1})|^2, p_2(\theta, \alpha_2) = \alpha_2^{-1} |\psi_2(\frac{\theta}{\alpha_2})|^2.
\end{gathered}
\qquad (3.3.13) $$

Thus from Eq.(3.3.8)-Eq.(3.3.9) we obtain

$$ 
\begin{cases}
p_1(\theta, \alpha_1) = \\
\quad \alpha_1^{-1} \left(\frac{1}{2\delta\pi\delta}\right)^{1/2} \exp\left[-\frac{1}{2\alpha_1^2\delta}(\theta - \alpha_1\bar{\theta}_1)^2\right] \quad \text{iff} \quad \left|\frac{\theta}{\alpha_1} - \bar{\theta}_1\right| \le \varepsilon \ll 1, \\
\quad 0 \quad \text{iff} \quad \left|\frac{\theta}{\alpha_1} - \bar{\theta}_1\right| > \varepsilon
\end{cases}
\qquad (3.3.14) $$

and

$$ 
\begin{cases}
p_2(\theta, \alpha_2) = \\
\quad \alpha_2^{-1} \left(\frac{1}{2\delta\pi\delta}\right)^{1/2} \exp\left[-\frac{1}{2\alpha_2^2\delta}(\theta - \alpha_2\bar{\theta}_2)^2\right] \quad \text{iff} \quad \left|\frac{\theta}{\alpha_2} - \bar{\theta}_2\right| \le \varepsilon \ll 1, \\
\quad 0 \quad \text{iff} \quad \left|\frac{\theta}{\alpha_2} - \bar{\theta}_2\right| > \varepsilon.
\end{cases}
\qquad (3.3.15) $$

From Eq.(3.3.13)-Eq.(3.3.15) by using formulae

$$ 
\begin{gathered}
f_1(x) = \frac{1}{\sigma_1 \sqrt{2\pi}} \exp\left[-\frac{(x-m_1)}{2\sigma_1^2}\right], f_2(x) = \frac{1}{\sigma_2 \sqrt{2\pi}} \exp\left[-\frac{(x-m_2)^2}{2\sigma_2^2}\right], \\
f_1(x) * f_2(x) = \frac{1}{\sqrt{2\pi(\sigma_1^2 + \sigma_2^2)}} \exp\left\{-\frac{[x-(m_1+m_2)]^2}{2(\sigma_1^2 + \sigma_2^2)}\right\}
\end{gathered}
\qquad (3.3.16) $$

we obtain

$$ 
\begin{gathered}
p(\theta, \alpha_1, \alpha_2) \asymp \\
\frac{1}{\delta \sqrt{2\pi \delta (\alpha_1^2 + \alpha_2^2)}} \exp\left\{-\frac{[\theta - (\alpha_1 \bar{\theta}_1 + \alpha_2 \bar{\theta}_2)]^2}{2\delta^2 (\alpha_1^2 + \alpha_2^2)}\right\} \quad \text{iff} \quad |\theta - (\alpha_1 \bar{\theta}_1 + \alpha_2 \bar{\theta}_2)| \le \varepsilon \ll 1, \\
p(\theta, \alpha_1, \alpha_2) = 0 \quad \text{iff} \quad |\theta - (\alpha_1 \bar{\theta}_1 + \alpha_2 \bar{\theta}_2)| > \varepsilon
\end{gathered}
\qquad (3.3.17) $$

We assume now that potential $V(\theta, \mathbf{Y})$ is a polynomial function of variables $z = (z_1, \dots, z_{n+1}) = (\theta, \mathbf{Y}) = (\theta, y_1, \dots, y_n)$. Using replacement

$$ 
\begin{gathered}
z_i \rightarrow \frac{y_i}{(1 + \varepsilon^{2k}|x|^2 k)}, \quad i = 1, \dots, n+1, \\
\varepsilon \in (0, 1], k \ge 1,
\end{gathered}
\qquad (3.3.18) $$

we obtain from potential $V(z) = V(\theta, \mathbf{Y})$ regularized potential $V_\epsilon(z), \epsilon \in (0, 1]$, such that $V_{\epsilon=0}(z) = V(z)$ and
$$ \sup_{z \in \mathbb{R}^{n+1}} (|V_{\varepsilon}(z)|) < +\infty, \varepsilon \in (0,1]. \quad (3.3.19) $$

Finally we obtain from Schrödinger equation (3.3.3) regularized Schrödinger equation of the Colombeau form

$$ \begin{gathered} i\hbar \left( \frac{\partial \Psi_{\varepsilon}(z,t)}{\partial t} \right)_{\varepsilon} = -\frac{\hbar^2}{2M} (\nabla^2 \Psi_{\varepsilon}(z,t))_{\varepsilon} + (V_{\varepsilon}(z) \Psi_{\varepsilon}(z,t))_{\varepsilon}, \quad (3.3.20) \\ (\Psi_{\varepsilon}(z,0))_{\varepsilon} = \Psi_0(z). \end{gathered} $$

**Theorem 3.3.1.** Let us consider Cauchy problem (3.3.20) with $\Psi_0(z)$ given by formula

$$ \Psi_0(z) = \begin{cases} \left(\frac{1}{2\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{4\delta}(z-z_0)^2\right] & \text{iff } \|z-z_0\| \le \varepsilon_1 \ll 1, \\ 0 & \text{iff } \|z-z_0\| > \varepsilon_1, \end{cases} \quad (3.3.21) $$

1. We assume that:

(i) $(V_{\varepsilon}(z))_{\varepsilon} \in G(\mathbb{R}^{n+1})$,

(ii) function $V(x)$ is a polynomial on variable $z = (z_1, \dots, z_{n+1})$, i.e.

$$ V(z) = \sum_{\|a\| \le m} g_a z^a, \quad \alpha = (i_1, \dots, i_{n+1}), \quad z^a = z_1^{i_1} \times \dots \times z_{n+1}^{i_{n+1}}, \quad \|a\| = \sum_{r=1}^{n+1} i_r $$

2. Let $u(\tau, t, \lambda, z, y) = (u_1(\tau, t, \lambda, z, y), \dots, u_{n+1}(\tau, t, \lambda, z, y))$ be the solution of the boundary problem:

$$ \begin{gathered} \frac{\partial^2 u^T(\tau, t, \lambda, z, y)}{\partial \tau^2} = \text{Hess}[V(\lambda, \tau)]u^T(\tau, t, \lambda, x, y) + [V'(\lambda, \tau)]^T, \quad (3.3.22) \\ u(0, t, \lambda, z, y) = y, u(t, t, \lambda, z, y) = z. \end{gathered} $$

Here

$$ \begin{gathered} \lambda = (\lambda_1, \dots, \lambda_{n+1}) \in \mathbb{R}^{n+1}, u^T(\tau, t, \lambda, z, y) = (u_1(\tau, t, \lambda, z, y), \dots, u_{n+1}(\tau, t, \lambda, z, y))^T, \\ V'(\lambda, \tau) = ([\partial V(z,t)/\partial z_1]_{z=\lambda}, \dots, [\partial V(z,t)/\partial z_{n+1}]_{z=\lambda}), \quad (3.3.23) \\ \text{Hess}[V(\lambda, \tau)] = [(\partial^2 V(z,t))/\partial x_i \partial x_j]_{x=\lambda} \end{gathered} $$

3. Let $S(t, \lambda, z, y)$ be the function given by formula

$$ S(t, \lambda, z, y) = \int_{0}^{t} \mathcal{L}(\dot{u}(\tau, t, \lambda, z, y), u(\tau, t, \lambda, z, y), \tau)d\tau, \quad (3.3.24) $$

where master Lagrangian $\mathcal{L}(\dot{u}, u, \tau)$ is

$$ \begin{gathered} \mathcal{L}(\dot{u}, u, \tau) = m/2 \dot{u}^2(\tau, t, \lambda, z, y) - \hat{V}(u(\tau, t, \lambda, z, y), \tau), \dot{u} = ((\partial u_1)/\partial \tau, \dots, (\partial u_{n+1})/\partial \tau), \dot{u}^2 = \langle \dot{u}, \dot{u} \rangle, \quad (3.3.25) \\ \hat{V}(u(\tau, t, \lambda, z, y), \tau) = u(\tau, t, \lambda, z, y) \text{Hess}[V(\lambda, \tau)] u^T(\tau, t, \lambda, z, y) + V'(\lambda, \tau) u^T(\tau, t, \lambda, z, y). \end{gathered} $$

4. Let $y_{cr} = y_{cr}(t, \lambda, z) \in \mathbb{R}^{n+1}$ be solution of the linear system of the algebraic equations

$$ [\partial S(t, \lambda, z, y) / (\partial y_i)]_{y=y_{cr}} = 0, i = 1, \dots, n+1. \quad (3.3.26) $$
5. Let $\hat{z} = \hat{z}(t, \lambda, z_0) \in \mathbb{R}^{n+1}$ be solution of the linear system of the algebraic equations

$$y_{cr}(t, \lambda, \hat{z}) + \lambda - z_0 = 0. \tag{3.3.27}$$

6. Assume that: for a given values of the parameters $t, \lambda, z_0$ the point $\hat{z} = \hat{z}(t, \lambda, z_0)$ is not a focal point on a corresponding trajectory is given by corresponding solution of the boundary problem (3.3.22). Then:

(i) for the limiting quantum average given by formula

$$(\lim_{h \to 0} \langle i, t, z_0, \varepsilon; \hbar \rangle)_\varepsilon = (\lim_{h \to 0} \int z_i |\Psi_\varepsilon(z, t; \hbar)|^2 dz)_\varepsilon, \varepsilon \in (0, 1], z \in \mathbb{R}^{n+1}, i = 1, \dots, n+1. \tag{3.3.28}$$

the following inequalities are satisfied

$$\begin{align}
& \lim_{\varepsilon \to 0} |\langle i, t, z_0, \varepsilon; \hbar \rangle - \lambda_i(t, z_0)| \le \nonumber \\
& 2[|\det(S_{y_{cr}y_{cr}}(t, \lambda, \hat{z}(t, \lambda, z_0), y_{cr}(t, \lambda, \hat{z}(t, \lambda, z_0)))|]^{-1} |\hat{z}_i(t, \lambda, z_0)|, \tag{3.3.29} \\
& (S_{y_{i,cr}y_{j,cr}}) = [\partial S(t, \lambda, z, y)/\partial y_i \partial y_j]_{y_i=y_{i,cr}, y_j=y_{j,cr}} \nonumber \\
& i, j = 1, \dots, n+1. \nonumber
\end{align}$$

(ii) therefore one can to calculate the limiting quantum trajectories $z_i^*(t, z_0, t_0)$

$$z_i^*(t, z_0, t_0) \triangleq \lambda_i(t, z_0), i = 1, \dots, n+1 \tag{3.3.30}$$

corresponding to potential $V(z, t)$ by using following system of the transcendental master equations

$$\hat{z}_i(t, \lambda, z_0) = 0, i = 1, \dots, n+1. \tag{3.3.31}$$

(iii) In the limit $\hbar \to 0, \varepsilon \to 0$ Schrödinger equation Eq.(3.3.20) completely evolve quasiclassically i.e. for expectation value of the position $\langle\langle z_i \rangle\rangle(t, x_0, t_0, \varepsilon; \hbar)\rangle_{i=1}^{n+1}$ at each instant $t$ the inequality

$$\lim_{\varepsilon \to 0} |\langle z_i \rangle(t, z_0, t_0, \varepsilon; \hbar) - z_i^*(t, z_0, t_0)| \le \delta \ll 1, i = 1, \dots, n+1, \tag{3.3.32}$$

is satisfied with a probability 1, e.g.,

$$\lim_{\varepsilon \to 0} \mathbf{P}\{|\langle z_i \rangle(t, z_0, t_0, \varepsilon; \hbar) - z_i^*(t, z_0, t_0)| \le \delta\} = 1. \tag{3.3.33}$$

**Proof.** The proof completely similarly as the proof of the theorem 3.1 from paper [19].

**Theorem 3.3.2.** Let us consider Cauchy problem (3.3.20) with $\Psi_0(z)$ given by formula

$$\begin{align}
\Psi_0(z) &= c_1 \Psi_{1,0}(z) + c_2 \Psi_{2,0}(z), \tag{3.3.34} \\
|c_1|^2 + |c_2|^2 &= 1, \nonumber
\end{align}$$

where
$$ \Psi_{1,0}(z) = \begin{cases} \left(\frac{1}{2\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{4\delta}(z-z_{1,0})^2\right] \quad \text{iff } \|z-z_0\| \le \varepsilon_1 \ll 1, & (3.3.35) \\ 0 \quad \text{iff } \|z-z_0\| > \varepsilon_1, & \end{cases} $$

and

$$ \Psi_{2,0}(z) = \begin{cases} \left(\frac{1}{2\delta\pi\delta}\right)^{1/4} \exp\left[-\frac{1}{4\delta}(z-z_{2,0})^2\right] \quad \text{iff } \|z-z_{2,0}\| \le \varepsilon_1 \ll 1, & (3.3.36) \\ 0 \quad \text{iff } \|z-z_{2,0}\| > \varepsilon_1, & \end{cases} $$

Then

(i) in the limit $\hbar \to 0, \varepsilon \to 0$ Schrödinger equation Eq.(3.3.33)-Eq.(3.3.35) completely evolve quasiclassically i.e. for expectation value of the observable $\langle\langle z_i \rangle\rangle(t, x_0, t_0, \varepsilon; \hbar)\rangle_{i=1}^{n+1}$ at each instant $t$, the inequality

$$ \begin{gathered} \lim_{\substack{\hbar \to 0 \\ \varepsilon \to 0}} |\langle z_i \rangle(t, z_0, t_0, \varepsilon; \hbar) - z_i^*(t, z_0, t_0)| \le \delta \ll 1, \quad i = 1, \dots, n+1, \\ z_0 = \alpha_1 z_{1,0} + \alpha_2 z_{2,0}, \\ \alpha_1 = |c_1|^2, \quad \alpha_2 = |c_2|^2 \end{gathered} \qquad (3.3.37) $$

is satisfied with a probability 1, e.g.,

$$ \lim_{\substack{\hbar \to 0 \\ \varepsilon \to 0}} \mathbf{P}\{|\langle z_i \rangle(t, z_0, t_0, \varepsilon; \hbar) - z_i^*(t, z_0, t_0)| \le \delta\} = 1. \qquad (3.3.38) $$

(ii) here $z_i^*(t, z_0, t_0), i = 1, \dots, n+1$ given by Eq.(3.3.30)-Eq.(3.3.31) with
$$ z_0 = \alpha_1 z_{1,0} + \alpha_2 z_{2,0}, $$
$$ \alpha_1 = |c_1|^2, \quad \alpha_2 = |c_2|^2. $$

**Proof.** Let $\Psi_1(z,t)$ be a solution of the Cauchy problem (3.3.20) with $\Psi_0(z) = \Psi_{1,0}(z)$ given by Eq.(3.3.35) and let $\Psi_2(z,t)$ be a solution of the Cauchy problem (3.3.20) with $\Psi_0(z) = \Psi_{2,0}(z)$ given by Eq.(3.3.36). From Theorem 3.3.2 follows that

$$ \forall t(t \ge 0) \left[ (\text{supp}(\Psi_1(z,t))) \cap (\text{supp}(\Psi_2(z,t))) = \emptyset \right] \qquad (3.3.39) $$

Let $p(z, t, z_{1,0}, z_{2,0}, \alpha_1, \alpha_2), \alpha_1 = |c_1|^2, \alpha_2 = |c_2|^2$ be a probability density to observe at instant $t$ vector $z$ with result $z$. By reconcile Bohr rule, [see section 1.6, Eq.(1.6.19)] we obtain

$$ \begin{align} p(z,t,z_{1,0},z_{2,0},\alpha_1,\alpha_2) &= [p_1(z,t,z_{1,0},\alpha_1)] * [p_2(z,t,z_{2,0},\alpha_2)], \notag \\ p(z,t,z_{1,0},\alpha_1) &= \alpha_1^{-(n+1)} |\Psi_1(\alpha_1^{-1}z,t,z_{1,0})|^2, \tag{3.3.40} \\ p_2(z,t,z_{2,0},\alpha_2) &= \alpha_2^{-(n+1)} |\Psi_2(\alpha_2^{-1}z,t,z_{2,0})|^2. \notag \end{align} $$

In order to obtain the inequality (3.3.37) we need to estimate the quantities

$$ \langle z_i \rangle(t, z_{1,0}, z_{2,0}, t_0, \alpha_1, \alpha_2, \varepsilon; \hbar) = \int_{\mathbb{R}^{n+1}} |z_i - \lambda_i| p(z, t, z_{1,0}, z_{2,0}, \alpha_1, \alpha_2) dz^{n+1}. \qquad (3.3.41) $$

In order to estimate the quantities we dealing completely similarly as in the proof of the theorem 3.1 from paper [19].
Quasiclassical quantum "cat" with a cubic potential
supplemented by additive sinusoidal driving.

As an example we calculate now exact quasi-classical asymptotic for quantum
cat with a cubic potential supplemented by additive sinusoidal driving. Using
Theorem3.3.2 we obtain limiting quantum trajectories given given by
Eq.(3.3.30)-Eq.(3.3.31) with $z_0 = \alpha_1 z_{1,0} + \alpha_2 z_{2,0}$, $\alpha_1 = |c_1|^2$, $\alpha_2 = |c_2|^2$. Let us
consider quantum cat with a cubic potential

$$
V(\theta) = (m\omega^2)/2\theta^2 - a\theta^3 + b\theta, \theta \in [\theta_1, \theta_2], a, b > 0 \quad (3.3.42)
$$

supplemented by additive sinusoidal driving

$$
V(\theta) = \frac{m\omega^2}{2}\theta^2 - a\theta^3 + b\theta - [A \sin(\Omega t)]\theta. \quad (3.3.43)
$$

The corresponding master Lagrangian given by Eq.(3.3.25),is

$$
L(u, \dot{u}, \tau) = (m/2)\dot{u}^2 - m\left(\left(\frac{\omega^2}{2}\right) + \left(\frac{3a\lambda}{m}\right)\right)u^2 - \left(m\omega^2\lambda + \frac{3a\lambda^2}{2} - b - A\sin(\Omega t)\right)u. \quad (3.3.44)
$$

We assume now that: $\omega^2/2 + 3a\lambda/m \ge 0$ and rewrite (3.3.44) in the following form

$$
L(\dot{u}, u, \tau) = (m/2)\dot{u}^2 - (m\omega^2\lambda/2)u^2 + g(\lambda, t)u, \quad (3.3.45)
$$

where $\omega(\lambda) = \sqrt{2|\omega^2/2 + 3a\lambda/m|}$ and $g(\lambda, t) = -[m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega t)]$. The corresponding master action $S(t, \lambda, x, y)$ given by Eq.(3.3.24), is

$$
S(t, \lambda, x, y) = \frac{m\omega}{2\sin\omega t} \left[ \begin{aligned}
& (\cos\omega t)(y^2 + x^2) - 2xy + \frac{2x}{m\omega} \int_0^t g(\lambda, \tau) \sin(\omega\tau) d\tau + \\
& + \frac{2y}{m\omega} \int_0^t g(\lambda, \tau) \sin(\omega(t-\tau)) d\tau - \frac{2}{m^2\omega^2} \int_0^t \int_0^\tau g(\lambda, \tau) g(\lambda, s) \sin\omega(t-\tau) \sin(\omega s) ds d\tau
\end{aligned} \right]. \quad (3.3.46)
$$

Therefore a linear system of the algebraic equations(3.3.) is

$$
\frac{\partial S(t, \lambda, x, y)}{\partial y} = 2y \cos \omega t - 2x + \frac{2}{m\bar{\omega}} \int_{0}^{t} g(\lambda, t) \sin((\omega(t - \tau))d\tau = 0. \quad (3.3.47)
$$

Therefore

$$
y_{cr}(t, \lambda, x) = \frac{x}{\cos \omega t} - \frac{1}{m\omega \cos \omega t} \int_{0}^{t} g(\lambda, t) \sin((\omega(t - \tau))d\tau. \quad (3.3.48)
$$

The linear system of the algebraic equations (3.3.26) is

$$
\frac{x}{\cos \omega t} - \frac{1}{m\omega \cos \omega t} \int_{0}^{t} g(\lambda, t) \sin((\omega(t - \tau))d\tau + \lambda - \theta_{0} = 0. \quad (3.3.49)
$$

Therefore solution of the linear system of the algebraic equations (3.3.49) is

$$
x(t, \lambda, \theta_0) = \frac{1}{m\bar{\omega}} \int_0^t g(\lambda, t) \sin((\omega(t - \tau))d\tau + (\lambda - \theta_0) \cos \omega t. \quad (3.3.50)
$$

Transcendental master equation (3.3.31) is

$$
\int_{0}^{t} g(\lambda, t) \sin((\omega(t - \tau))d\tau + m\omega(\lambda - \theta_{0}) \cos \omega t = 0 \quad (3.3.51)
$$

Finally from Eq.(3.3.51) we obtain

$$
\begin{equation}
\begin{split}
d(\lambda(t))(\cos(\omega t)/\omega - 1/\omega) + A(\omega\sin(\Omega t) - \Omega\sin(\omega t))/(\omega^2 - \Omega^2) - \\
(\lambda(t) - x_0)m\omega\cos(\omega t) = 0,
\end{split}
\tag{3.3.52}
\end{equation}
$$
where $d(\lambda) = m\omega^2\lambda + 3a\lambda^2 - b$.

Numerical Example. $\theta_0 = 0, m = 1, \Omega = 0, \omega = 9, a = 3, b = 10, A = 0$.

<figure><img src="image_28.png" /><figcaption>Pic. 1. Limiting quantum trajectory $\theta(t) = \lambda(t)$.</figcaption></figure>

### III.4. Stern-Gerlach experiment revisited and Schrödinger's cat paradox resolution.

#### III.4.1. Stern-Gerlach experiment revisited

In 1922, by studying the deflection of a beam of silver atoms in a strongly inhomogeneous magnetic field (cf. Fig. 3.4.1) Otto Stern and Walter Gerlach obtained an experimental result that contradicts the common sense prediction: the beam, instead of expanding, splits into two separate beams giving two spots of equal intensity $N^+$ and $N^-$ on a detector, at equal distances from the axis of the original beam. Historically, this is the experiment which helped establish spin quantization. Theoretically, it is the seminal experiment posing the problem of measurement in quantum mechanics.

<figure><img src="image_29.png" /><figcaption>Fig.3.4.1.Schematic configuration of the Stern-Gerlach experiment. Adapted from [55].</figcaption></figure>

$$ \Psi(z,y,t)|_{t=0} = \Psi^0(z,y) = \Psi_1^0(z)\Psi_2^0(y). \qquad (3.4.1) $$
We assume now that both density $\Psi^0(z)$ and $\Psi^0(y)$ is very narrow, in fact constrained such that

$$ 
\begin{aligned}
\Psi_1^0(z) &= \Psi_1^0(z, \delta) = 0 \quad \text{iff} \quad |x| > \delta, \\
\Psi_2^0(y) &= \Psi_2^0(y, \delta) = 0 \quad \text{iff} \quad |y| > \delta,
\end{aligned}
\qquad (3.4.2) 
$$

and

$$ 
\begin{aligned}
\Psi_1^0(z) &= \Psi_1^0(z, \delta) = (2\pi\sigma_0\sigma_0^2)^{-\frac{1}{4}} e^{-\frac{z^2}{4\sigma_0^2}} \left( \begin{array}{c} \cos \frac{\theta_0}{2} e^{-i\frac{\varphi_0}{2}} \\ \sin \frac{\theta_0}{2} e^{i\frac{\varphi_0}{2}} \end{array} \right) \quad \text{iff} \quad |z| \le \delta, \\
\| \Psi_1^0(z, \delta) \|_2^2 &= 1; \qquad (3.4.3)
\end{aligned}
$$

$$ 
\begin{aligned}
\Psi_2^0(y) &= \Psi_2^0(y, \delta) = (2\pi\sigma_0\sigma_0^2)^{-\frac{1}{4}} e^{-\frac{y^2}{4\sigma_0^2}} \quad \text{iff} \quad |y| \le \delta, \\
\| \Psi_2^0(y, \delta) \|_2^2 &= 1 \\
\sigma_0 &\ll 1.
\end{aligned}
$$

Silver atoms contained in the oven E (Fig.3.4.1.) are heated to a high temperature and escape through a narrow opening. A second aperture, T, selects those atoms whose velocity, $v_0$, is parallel to the y-axis. The atomic beam crosses the gap of the electromagnet $A_1$ before condensing on the detector, $P_1$. Before crossing the electromagnet, the magnetic moment of each silver atom is oriented randomly (isotropically). In the beam, we represent each atom by its wave function; one can assume that at the entrance to the electromagnet, $A_1$, and at the initial time $t = 0$, each atom can be approximatively described by a quasi-Gaussian spinor in plain $(z,y)$ given by Eqs.(3.4.1-3.4.3) corresponding to a pure state. As will be it is proved later the variable $y$ will be treated strictly quasiclassically, i.e. almost classically, with

$$ 
\begin{aligned}
\mathbf{P}\{|y - \mathbf{v}_y^{\pm}(\mathbf{v}_0, \theta_0)t| < \delta\} &= 1, \\
\mathbf{P}\{|y - \mathbf{v}_y^{\pm}(\mathbf{v}_0, \theta_0)t| \ge \delta\} &= 0
\end{aligned}
\qquad (3.4.4) 
$$

and $\sigma_0 \le \sigma'_0 = 10^{-4}m$, where $\sigma'_0$ corresponds to the size of the slot T along the z-axis and where the expression of the functions $v_y^+(v_0, \theta_0)$ and $v_y^-(v_0, \theta_0)$ will be given later.

The approximation by a quasi-Gaussian initial spinor will allow explicit calculations. Because the slot is much wider along the x-axis, the variable $z$ will be also treated strictly quasiclassically with

$$ 
\begin{aligned}
\mathbf{P}\{|z - z_{\Delta}^{\pm} - \mathbf{v}_{z}^{\pm}(u, \theta_0)t| \le \delta\} &= 1, \\
\mathbf{P}\{|z - \mathbf{v}_{z}^{\pm}(u, \theta_0)t| > \delta\} &= 0,
\end{aligned}
\qquad (3.4.5) 
$$

where the expression of the functions $v_z^{\pm}(u, \theta_0)$, $u = \frac{\mu_B B_0' (\Delta t)}{m}$ will be given later. In order to obtain an explicit solution of the Stern-Gerlach experiment, we take for the silver atom, we have $m = 1.8 \times 10^{-25} \text{kg}$, $v_0 = 500 \text{ m/s}$ (corresponding to the
temperature of $T = 1000°K)$. In equation (3.4.3.) and in figure 3.4.2., $\theta_0$ and $\phi_0$ are the polar angles characterizing the initial orientation of the magnetic moment, $\theta_0$ corresponds to the angle with the z-axis. The experiment is a statistical mixture of pure states where the $\theta_0$ and the $\phi_0$ are randomly chosen: $\theta_0$ is drawn in a uniform way from $[0, \pi]$ and that $\phi_0$ is drawn in a uniform way from $[0, 2\pi]$.

<figure><img src="image_30.png" /><figcaption>Fig.3.4.2. Orientation of the magnetic moment $\theta_0$ and $\phi_0$ are the polar angles characterizing the spin vector in the de Broglie-Bohm interpretation. Adapted from [55].</figcaption></figure>

**Assumption 3.4.1.** We assume that a particle collapses in a magnetic field **B** at some instant $t'$ by two particle, i.e. the spinor $\Psi(z, y, t)$ collapses in a magnetic field **B**

at some instant $t'$ by two spinors $\Psi_+(z, y, t, t', \delta)$ and $\Psi_-(z, y, t, t', \delta)$ given by Eq.(3.4.9.a)-Eq.(3.4.9.b). Note that such collapse obviously occurs except spinors such

that: $\sqrt{2^{-1}}(\psi_+^{(z)} + \psi_-^{(z)}) = \psi_+^{(x)}$, etc.

**Remark 3.4.1.** Note that standard assumption consist that spinor collapses on detector P1

with respect of the Born rule.

Thus the evolution of the spinor

$$ \Psi(z, y, t, t') = \begin{pmatrix} \Psi_+(z, y, t, t') \\ \Psi_-(z, y, t, t') \end{pmatrix} $$

in a magnetic field **B** is then given by the nonlocal Pauli equation:
$$ i\hbar \left( \begin{array}{c} \int dz dy \int dt \frac{\partial \Psi_+(z,y,t,t')}{\partial t} \\ \int dz dy \int dt \frac{\partial \Psi_-(z,y,t,t')}{\partial t} \end{array} \right) = -\frac{\hbar^2}{2m} \int dt \int dz dy \Delta \left( \begin{array}{c} \Psi_+(z,y,t,t') \\ \Psi_-(z,y,t,t') \end{array} \right) + \mu_B \int dt \int dz dy \mathbf{B} \sigma \left( \begin{array}{c} \Psi_+(z,y,t,t') \\ \Psi_-(z,y,t,t') \end{array} \right) \quad (3.4.6) $$

where $\mu_B = \frac{e\hbar}{2m_e}$ is the Bohr magneton and where $\sigma = (\sigma_x, \sigma_y, \sigma_z)$ corresponds to the three Pauli matrices.

**Remark 3.4.2.** The particle first enters an electromagnetic field **B** directed along the z-axis, $B_x = B'_0 x, B_y = 0, B_z = B_0 - B'_0 z$, with $B_0 = 5$ Tesla, $B'_0 = \left| \frac{\partial B}{\partial z} \right| = 10^3$ Tesla/m over a length $\Delta l = 1$ cm.

**Remark 3.4.3.** On exiting the magnetic field, the both particles is free until it reaches the detector $P_1$ placed at a $D = 20$ cm distance.

The particles stays within the magnetic field for a time $\Delta t$ with

$$ \Delta t = \frac{\Delta l}{v_0}. \quad (3.4.7) $$

**Assumption 3.4.2.** We assume now for simplification that

$$ t' \approx \Delta t. \quad (3.4.8) $$

Thus during this time $t \in [0, t') \approx [0, \Delta t)$, the spinor $\Psi(z, y, t, t', \delta)$ is:

$$ \Psi(z, y, t, t', \delta) = \begin{pmatrix} \Psi_+(z, y, t, t', \delta) \\ \Psi_-(z, y, t, t', \delta) \end{pmatrix} = \begin{pmatrix} \Psi_+(z, t, t', \delta) \Psi(y, t, t', \delta) \\ \Psi_-(z, t, t', \delta) \Psi(y, t, t', \delta) \end{pmatrix}, \quad (3.4.9.a) $$

where
$$
\begin{align*}
\Psi_+(z,t,t',\delta) &= \cos \frac{\theta_0}{2} e^{-i \frac{\varphi_0}{2}} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp \left[ -\frac{\left(z - \frac{\mu_B B_0'}{2m} t^2\right)^2}{4\sigma_0^2} \right] \times \\
&\quad \exp \left[ i \frac{\mu_B B_0' tz - \frac{\mu_B^2 B_0'^2}{6m} t^3 + \mu_B B_0 t + 0.5h\varphi_0}{\hbar} \right] \text{iff } \left|z - \frac{\mu_B B_0'}{2m} t^2\right| \le \delta, \\
\Psi_+(z,t,\delta) &= 0 \text{ iff } \left|z - \frac{\mu_B B_0'}{2m} t^2\right| > \delta,
\end{align*}
$$

$$
\begin{align}
\Psi_-(z,t,t',\delta) &= i \sin \frac{\theta_0}{2} e^{i \frac{\varphi_0}{2}} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp \left[ - \frac{\left(z + \frac{\mu_B B_0'}{2m} t^2\right)^2}{4\sigma_0^2} \right] \times \tag{3.4.9.b} \\
&\quad \exp \left[ i \frac{-\mu_B B_0' tz - \frac{\mu_B^2 B_0'^2}{6m} t^3 - \mu_B B_0 t - 0.5h\varphi_0}{\hbar} \right] \text{iff } \left|z + \frac{\mu_B B_0'}{2m} t^2\right| \le \delta, \nonumber \\
\Psi_-(z,t,t',\delta) &= 0 \text{ iff } \left|z + \frac{\mu_B B_0'}{2m} t^2\right| > \delta; \nonumber \\
\Psi(y,t,t',\delta) &= (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp \left[ - \frac{(y - v_0 t)^2}{4\sigma_0^2} \right] \text{iff } |y - v_0 t| \le \delta, \nonumber \\
\Psi(y,t,t',\delta) &= 0 \text{ iff } |y - v_0 t| > \delta. \nonumber
\end{align*}
$$

After the magnetic field, at time $t + \Delta t$ ($t > 0$) in the free space, the both spinors
becomes:

$$
\Psi_+(z,y,t+t',\delta) \approx \Psi_+(z,y,t+\Delta t,\delta) = \Psi_+(z,t+\Delta t,\delta)\Psi(y,t+\Delta t,\delta) \quad (3.4.10)
$$

and

$$
\Psi_-(z,y,t+t',\delta) \approx \Psi_-(z,y,t+\Delta t,\delta) = \Psi_-(z,t+\Delta t,\delta)\Psi(y,t+\Delta t,\delta). \quad (3.4.11)
$$

Here

$$
\Psi_+(z, t + \Delta t, \delta) \approx \cos \frac{\theta_0}{2} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp \left[ - \frac{(z - z_\Delta - ut)^2}{4\sigma_0^2} \right] e^{i \frac{muz+h\varphi_+}{\hbar}} \text{ iff } |z - z_\Delta - ut| \le \delta, \quad (3.4.12) \\
0 \quad \text{iff } |z - z_\Delta - ut| > \delta
$$

and
$$
\Psi_-(z, t + \Delta t, \delta) \approx \\
\left\{
\begin{array}{ll}
\sin \frac{\theta_0}{2} (2\pi\sigma_0^2)^{-\frac{1}{4}} \exp\left[-\frac{(z+z_\Delta+ut)^2}{4\sigma_0^2}\right] e^{i\frac{-muz+h\phi_-}{\hbar}} & \text{iff } |z+z_\Delta+ut| \le \delta, \\
0 & \text{iff } |z+z_\Delta+ut| > \delta,
\end{array}
\right.
\quad (3.4.13)
$$

and

$$
\begin{gather}
\Psi(y,t,\delta) = (2\pi\sigma_0^2)^{-\frac{1}{4}} \exp\left[-\frac{(y-\mathbf{v}_0(t+\Delta t))^2}{4\sigma_0^2}\right] \quad \text{iff } |y-\mathbf{v}_0(t+\Delta\theta_0 t)|\le\delta, \tag{3.4.14} \\
\Psi(y,t,\delta) = 0 \quad \text{iff } |y-\mathbf{v}_0(t+\Delta t)|>\delta. \nonumber
\end{gather}
$$

where

$$
z_{\Delta} = \frac{\mu_B B'_0 (\Delta t)^2}{2m}, \quad u = \frac{\mu_B B'_0 (\Delta t)}{m}. \qquad (3.4.15)
$$

From Eq.(3.4.10),Eq.(3.4.12) and Eq.(3.4.14) we obtain

$$
\Psi_+(z,y,t+\Delta t,\delta) =
\left\{
\begin{array}{ll}
\cos \frac{\theta_0}{2} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp\left[-\frac{(z-z_\Delta-ut)^2}{4\sigma_0^2}\right] e^{i\frac{muz+h\phi_+}{\hbar}} & \text{iff} \\
\exp\left[-\frac{(y-\mathbf{v}_0(t+\Delta t))^2}{4\sigma_0^2}\right] & \text{and} \\
0 & \text{otherwise}
\end{array}
\right.
\quad |z-z_\Delta-ut| \le \delta \tag{3.4.16}
$$

From Eq.(3.4.16) by postulate Q.IV.3 for the probability density with respect to
observable z we obtain the expression

$$
\begin{equation}
\begin{split}
c_{|\Psi_+\rangle}(z,t) = \left\{ \right. & \left(2\pi\sigma_0^2\right)^{-\frac{1}{2}} \exp\left[-\frac{\left(\frac{z}{\eta_{\theta_0}^+} - z_\Delta - ut\right)^2}{2\sigma_0^2}\right] \quad \text{iff} \quad \left. \left|\frac{z}{\eta_{\theta_0}^+} - z_\Delta - ut\right| \le \delta \right. \\
& \left. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{otherwise} \right]
\end{split}
\tag{3.4.17}
\end{equation}
$$

$$
\eta_{\theta_0}^{+} = \cos^2 \frac{\theta_0}{2}
$$

and with respect to observable y we obtain the expression
$$
\begin{equation} \tag{3.4.18}
\begin{split}
c_{|\Psi_+\rangle}(y,t) = & \left\{ (2\pi\sigma_0^2)^{-1/2} \exp \left[ -\frac{\left( \frac{y}{\eta_{\theta_0}^+} - \mathbf{v}_0(t+\Delta t) \right)^2}{4\sigma_0^2} \right] \right\} \text{iff} \quad \left| \frac{y}{\eta_{\theta_0}^+} - \mathbf{v}_0(t+\Delta t) \right| \le \delta \\
& \qquad \text{otherwise} \\
\eta_{\theta_0}^+ &= \cos^2 \frac{\theta_0}{2}
\end{split}
\end{equation}
$$

and therefore corresponding particle movin by strictly quasiclassical law

$$
\begin{align}
\mathbf{P}\{|z^+(t) - z_\Delta^+ - \mathbf{v}_z^+(u, \theta_0)t| \le \delta\} &= 1, \nonumber \\
\mathbf{P}\{|z^+(t) - z_\Delta^+ - \mathbf{v}_z^+(u, \theta_0)t| > \delta\} &= 0, \nonumber \\
\mathbf{P}\{|y^+(t) - \mathbf{v}_y^+(\mathbf{v}_0, \theta_0)(t+\Delta t)| \le \delta\} &= 1, \tag{3.4.19} \\
\mathbf{P}\{|y^+(t) - \mathbf{v}_y^+(\mathbf{v}_0, \theta_0)(t+\Delta t)| > \delta\} &= 0, \nonumber \\
z_\Delta^+ &= \eta_{\theta_0}^+ z_\Delta, \nonumber \\
\mathbf{v}_z^+(\theta_0) &= \eta_{\theta_0}^+ u, \quad \mathbf{v}_y^+(\theta_0) = \eta_{\theta_0}^+ \mathbf{v}_0. \nonumber
\end{align}
$$

From Eq.(3.4.11),Eq.(3.4.13) and Eq.(3.4.14) we obtain

$$
\begin{equation}
\begin{split}
\Psi_-(z,y,t+\Delta t,\delta) ={}& \sin\frac{\theta_0}{2}(2\pi\sigma_0^2)^{-1/4} \\
& \exp\left[-\frac{(z+z_\Delta+ut)^2}{4\sigma_0^2}\right]e^{\frac{-muz+h\varphi_-}{\hbar}} \\
& \times \begin{array}{@{}l@{\quad\text{iff}\quad}l@{}}
  & |z+z_\Delta+ut| \le \delta \\
  \text{and} & \\
  & |y-\mathbf{v}_0(t+\Delta\theta_0 t)| \le \delta
\end{array}
\end{split}
\tag{3.4.20}
\end{equation}
$$

From Eq.(3.4.20) by postulate Q.IV.3 for the probability density with respect to
observable z we obtain the expression

$$
\begin{equation}
\begin{split}
c_{|\Psi_-\rangle}(z,t) = & \left\{ (2\pi\sigma_0^2)^{-1/2} \exp\left[ -\frac{\left(\frac{z}{\eta_{\bar{\theta}_0}} + z_\Delta + ut\right)^2}{2\sigma_0^2} \right] \right\} \text{iff} \quad \left|\frac{z}{\eta_{\bar{\theta}_0}} + z_\Delta + ut\right| \le \delta \tag{3.4.21} \\
& \qquad \text{otherwise}
\end{split}
\end{equation}
$$
$$
\eta_{\bar{\theta}_0} = \sin^2 \frac{\theta_0}{2}
$$

and with respect to observable y we obtain the expression
$$
\begin{equation}
\begin{split}
c_{|\Psi_-\rangle}(y,t) = & \left\{ (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp\left[ -\frac{\left( \frac{y}{\eta_{\bar{\theta}_0}} - \mathbf{v}_0(t+\Delta t) \right)^2}{4\sigma_0^2} \right] \right\} \text{ iff } \left| \frac{y}{\eta_{\bar{\theta}_0}} - \mathbf{v}_0(t+\Delta t_0 t) \right| \le \delta \\
& \qquad \text{otherwise} \\
\eta_{\bar{\theta}_0} &= \sin^2 \frac{\theta_0}{2}
\end{split}
\tag{3.4.22}
\end{equation}
$$

and therefore corresponding particle movin by strictly quasiclassical law

$$
\begin{align}
\mathbf{P}\{|z^{-}(t) + z_{\Delta}^{-} + \mathbf{v}_{z}^{-}(u, \theta_{0})t| \le \delta\} &= 1, \nonumber \\
\mathbf{P}\{|z^{-}(t) + z_{\Delta}^{-} + \mathbf{v}_{z}^{-}(u, \theta_{0})t| > \delta\} &= 0, \nonumber \\
\mathbf{P}\{|y^{-}(t) - \mathbf{v}_{y}^{-}(\mathbf{v}_{0}, \theta_{0})(t + \Delta t)| \le \delta\} &= 1, \tag{3.4.23} \\
\mathbf{P}\{|y^{-}(t) - \mathbf{v}_{y}^{-}(\mathbf{v}_{0}, \theta_{0})(t + \Delta t)| > \delta\} &= 0, \nonumber \\
z_{\Delta}^{-} &= \eta_{\bar{\theta}_{0}}z_{\Delta}, \nonumber \\
\mathbf{v}_{z}^{-}(u, \theta_{0}) &= \eta_{\bar{\theta}_{0}}u, \mathbf{v}_{y}^{-}(\mathbf{v}_{0}, \theta_{0}) = \eta_{\bar{\theta}_{0}}\mathbf{v}_{0}. \nonumber
\end{align}
$$

All interpretations are based on the equations (3.4.18)-(3.4.21).One deduce from Eq.(3.4.18)-Eq.(3.4.21) the probability density of a pure state in the free space after the electromagnet:

$$
\begin{align}
\rho_{\theta_0}(z, y, t + \Delta t) &= (2\pi\sigma_0^2)^{-\frac{1}{2}} \rho_{\theta_0}(z, t + \Delta_{\theta_0} t) \sum_{\pm} \exp\left[ -\frac{\left( \frac{y}{\eta_{\theta_0}^{\pm}} - \mathbf{v}_0(t + \Delta t) \right)^2}{4\sigma_0^2} \right]; \nonumber \\
\rho_{\theta_0}(z, t + \Delta t) &= (2\pi\sigma_0^2)^{-\frac{1}{2}} \left\{ \begin{aligned}[t]
& \cos^{-2} \frac{\theta_0}{2} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp\left[ -\frac{\left( \frac{z}{\eta_{\theta_0}^{+}} - z_{\Delta} - ut \right)^2}{2\sigma_0^2} \right] + \\
& \qquad \sin^{-2} \frac{\theta_0}{2} (2\pi\sigma_0^2)^{-\frac{1}{2}} \exp\left[ -\frac{\left( \frac{z}{\eta_{\theta_0}^{-}} + z_{\Delta} + ut \right)^2}{4\sigma_0^2} \right]
\end{aligned} \right\}. \tag{3.4.24}
\end{align}
$$

The decoherence time $t_{\text{dec}}$, where the two spots $N^+$ and $N^-$ are separated, is then given by the equation:

$$
t_{\text{dec}} \approx \frac{3\sigma_0 - z_{\Delta}}{u(\eta_{\theta_0}^{+} + \eta_{\bar{\theta}_0})} = \frac{3\sigma_0 - z_{\Delta}}{u}. \quad (3.4.25)
$$

This decoherence time is usually the time required to diagonalize the marginal
density matrix $\rho_{\theta_0}^S(t, \delta)$ of spin variables associated with a pure state
$$ \rho_{\theta_0}^S(t, \delta) = \left( \begin{array}{cc} \int |\Psi_+(z, y, t + \Delta t, \delta)|^2 dz dy & \int \Psi_-^*(z, y, t + \Delta t, \delta) \Psi_+(z, y, t + \Delta t, \delta) dz dy \\ \int \Psi_-(z, y, t + \Delta t, \delta) \Psi_+^*(z, y, t + \Delta t, \delta) dz dy & \int |\Psi_+(z, y, t + \Delta t, \delta)|^2 dz dy \end{array} \right) \quad (3.4.26) $$

For $t \ge t_{\text{dec}}$, the product $\Psi_-(z, y, t + \Delta t, \delta) \Psi_+^*(z, y, t + \Delta t, \delta)$ is null and the density matrix $\rho_{\theta_0}^S(t, \delta)$ is diagonal. We then obtain atoms with a spin oriented only along the z-axis (positively or negatively). Let us consider the spinor $\Psi(z, y, t + \Delta t, \delta)$ given by equations (3.4.10)-(3.4.15).

**Remark 3.4.4.** Experimentally, we do not measure the spin directly, but the z position of the particle impact on the detector P1 (Fig.3.4.3.).

<figure><img src="image_31.png" /><figcaption>Fig.3.4.3. Silver atom impacts on the detector P1.</figcaption></figure>

Adapted from [55].

**Remark 3.4.5.** Note that if we measure the z-position of the particle at instant $t$, we also measure the y-position of the particle at the same instant $t$.

**Remark 3.4.6.** Let $P_t(D - \delta, D, y_t^+)$ be the probability of obtaining the result $y_t^+$ at instant $t$, lying in the range $(D - \delta, D)$ on measuring observable $y$ in respect to spinor $\Psi_+(z, y, t + \Delta t, \delta)$. From Eq.(3.4.19) we obtain

$$ \begin{gathered} P_t(D - \delta, D + \delta, y_t^+) = 1 \quad \text{iff} \quad (3.4.27) \\ y^+(t) = D \quad \text{and} \quad |y^+(t) - v_y^+(v_0, \theta_0)(t + \Delta t)| \le \delta. \end{gathered} $$

From Eq.(3.4.27) follows that:

$$ \begin{gathered} P_t(D - \delta, D + \delta, y_t^+) = 1 \quad \text{if} \quad (3.4.28) \\ t \triangleq t_+(D) \approx \frac{D}{v_y^+(v_0, \theta_0)} = \frac{D}{v_0 \cos^2 \frac{\theta_0}{2}}. \end{gathered} $$

**Remark 3.4.7.** Let $P_t(\tilde{z}_+ - \delta, \tilde{z}_+ + \delta, z_t^+)$ be the probability of obtaining the result $z_t^+$ at instant $t$, lying in the range $(\tilde{z}_+ - \delta, \tilde{z}_+ + \delta)$, $\tilde{z}_+ \in N^+$ on measuring observable $z$ in respect to spinor $\Psi_+(z, y, t + \Delta t, \delta)$. From Eq.(3.4.19) we obtain

$$ \begin{gathered} P_t(\tilde{z}_+ - \delta, \tilde{z}_+ + \delta, z_t^+) = 1 \quad \text{iff} \quad (3.4.29) \\ z^+(t) = \tilde{z}_+ \quad \text{and} \quad |z^+(t) - z_\Delta^+ - v_z^+(u, \theta_0)t| \le \delta. \end{gathered} $$
From Eq.(3.4.29) follows that:

$$ 
\begin{gathered}
\mathbf{P}_t(\tilde{z}_+ - \delta, \tilde{z}_+ + \delta, z_t^+) = 1 \\
\text{if} \\
t \triangleq t(\tilde{z}_+) \approx \frac{\tilde{z}_+}{\mathbf{v}_z^+(u, \theta_0)} = \frac{\tilde{z}_+}{u \cos^2 \frac{\theta_0}{2}}.
\end{gathered}
\quad (3.4.30) $$

**Remark 3.4.8.** Note that from Remark 3.4.5 it follows that $t(\tilde{z}_+) \approx t(D)$ and therefore from Eq.(3.4.28) and Eq.(3.4.30) one obtains

$$ \frac{\tilde{z}_+}{u \cos^2 \frac{\theta_0}{2}} \approx \frac{D}{\mathbf{v}_0 \cos^2 \frac{\theta_0}{2}} \Rightarrow \frac{\tilde{z}_+}{u} \approx \frac{D}{\mathbf{v}_0} \quad (3.4.31) $$

as it should be, because the equality $\frac{\tilde{z}_+}{u} \approx \frac{D}{\mathbf{v}_0}$ is required by the condition of the

Stern-Gerlach experiment.

**Remark 3.4.9.** Let $\mathbf{P}_t(D - \delta, D, y_t^-)$ be the probability of obtaining the result $y_t^-$ at instant $t$, lying in the range $(D - \delta, D)$ on measuring observable $y$ in respect to spinor $\Psi_-(z, y, t + \Delta t, \delta)$. From Eq.(3.4.23) we obtain

$$ 
\begin{gathered}
\mathbf{P}_t(D - \delta, D + \delta, y_t^-) = 1 \quad \text{iff} \\
y_t^-(t) = D \quad \text{and} \quad |y_t^-(t) - \mathbf{v}_y^-(\mathbf{v}_0, \theta_0)(t + \Delta t)| \le \delta.
\end{gathered}
\quad (3.4.32) $$

From Eq.(3.4.32) follows that:

$$ 
\begin{gathered}
\mathbf{P}_t(D - \delta, D + \delta, y_t^-) = 1 \\
\text{if} \\
t \triangleq t_-(D) \approx \frac{D}{\mathbf{v}_y^-(\mathbf{v}_0, \theta_0)} = \frac{D}{\mathbf{v}_0 \sin^2 \frac{\theta_0}{2}}.
\end{gathered}
\quad (3.4.33) $$

**Remark 3.4.10.** Let $\mathbf{P}_t(\tilde{z}_- - \delta, \tilde{z}_- + \delta, z_t^-)$ be the probability of obtaining the result $z_t^-$ at instant $t$, lying in the range $(\tilde{z}_- - \delta, \tilde{z}_- + \delta)$, $\tilde{z}_- \in \mathcal{N}^-$ on measuring observable $z$ in respect to spinor $\Psi_-(z, y, t + \Delta t, \delta)$. From Eq.(3.4.32) we obtain

$$ 
\begin{gathered}
\mathbf{P}_t(\tilde{z}_- - \delta, \tilde{z}_- + \delta, z_t^-) = 1 \quad \text{iff} \\
z_t^-(t) = \tilde{z}_- \quad \text{and} \quad |z_t^-(t) - z_{\Delta} - \mathbf{v}_z^-(u, \theta_0)t| \le \delta.
\end{gathered}
\quad (3.4.34) $$

From Eq.(3.4.29) follows that:

$$ 
\begin{gathered}
\mathbf{P}_t(\tilde{z}_- - \delta, \tilde{z}_- + \delta, z_t^-) = 1 \\
\text{if} \\
t \triangleq t(\tilde{z}_-) \approx \frac{|\tilde{z}_-|}{\mathbf{v}_z^-(u, \theta_0)} = \frac{|\tilde{z}_-|}{u \sin^2 \frac{\theta_0}{2}}.
\end{gathered}
\quad (3.4.35) $$
**Remark 3.4.11.** Note that from Remark 3.4.5 it follows that $t(\tilde{z}_+) \approx t(D)$ and therefore
from Eq.(3.4.33) and Eq.(3.4.35) one obtains

$$ \frac{|\tilde{z}_-|}{u \sin^2 \frac{\theta_0}{2}} \approx \frac{D}{v_0 \sin^2 \frac{\theta_0}{2}} \Rightarrow \frac{\tilde{z}_-}{u} \approx \frac{D}{v_0} \qquad (3.4.36) $$

as it should be, because the equality $\frac{|\tilde{z}_-|}{u} \approx \frac{D}{v_0}$ is required by the condition of the Stern-Gerlach experiment.

### III.4.2. Schrödinger's cat which measure spin. Schrödinger's cat paradox resolution.

Let us consider again the Schrödinger's cat which measure spin by using the Stern-Gerlach apparatus, see subsection 1.6, (Fig.1.6.2). When a measurement is made, with the “up” outcome Schrödinger's cat is dead. When a measurement is made, with a “down” outcome Schrödinger's cat is alive. It is known many years that conventional QM with canonical explanation of the Stern-Gerlach experiment cannot give predictable and [27] As pointed out in subsection 1.6

**Theorem 3.4.1.** Any spinor

$$ \left( \begin{array}{c} \Psi_{+}(z, y, t, t') \\ \Psi_{-}(z, y, t, t') \end{array} \right) \qquad (3.4.37) $$

given by Eq.(3.4.9.a)-Eq.(3.4.9.b) with $\theta_0$ such that $\cos \frac{\theta_0}{2} \neq 0$ always kills the Schrödinger's cat at instant $t$:

$$ t \approx \frac{D}{v_0 \cos^2 \frac{\theta_0}{2}}. \qquad (3.4.38) $$

## IV. EPR Paradox Resolution

### IV.1. The relaxed locality principle.

The Special Theory of Relativity limits the speed at which any physical influences and any

real information can travel to the speed of light, c.

**The Einstein's principle of locality (EPL):** any effects do not propagate faster than the

speed of light, i.e. speed of light is a limiting factor.

The principle of locality claimed that:
(i) Any physical event $A(t_1, \mathbf{r}_1)$ which has occured at point $A(t_1, \mathbf{r}_1) \in M_4$ (see Definition

2.2.8) cannot cause (by physical interection) a physical event $B(t_2, \mathbf{r}_2)$ (result) which has

occured at point $B(t_2, \mathbf{r}_2) \in M_4$ in a time less than $T = D/c$, where D, is the distance

between the points.

(ii) An physical event $A(t, \mathbf{r}_1)$ which has occured at point $A(t, \mathbf{r}_1) \in M_4$ cannot cause a

simultaneous physical event $B(t, \mathbf{r}_2)$ (result) which has occured at another point $B(t, \mathbf{r}_2) \in M_4$.

(iii) Any real physical information about physical event $A(t_1, \mathbf{r}_1)$ at point $A(t_1, \mathbf{r}_1)$ cannot be

obtained by observer at point $B(t_2, \mathbf{r}_2)$ in a time less than $T = D/c$, where D, is the distance

between the points.

**Definition 4.1.1.** Let $[\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{t.l.s.}} \stackrel{\rightarrow}{\}$ be a set of the all timelike separated

pairs of events $\{A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)\}_{\text{t.l.s.}} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{t.l.s.}}$, (see Definition 2.2.10.a)

such that $t_2 > t_1$ and $A^{oc}(t_1, \mathbf{r}_1) \Rightarrow B^{oc}(t_2, \mathbf{r}_2)$.

Note that $[\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{t.l.s.}} \stackrel{\rightarrow}{\} \subset [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{t.l.s.}}$.

**Remark 4.1.1.** Note that the claim (i) obviously meant that

$$ \forall (t_1 > t_2) \forall A(t_1, \mathbf{r}_1) \forall B(t_2, \mathbf{r}_2) \{ [A^{oc}(t_1, \mathbf{r}_1) \Rightarrow B^{oc}(t_2, \mathbf{r}_2)] \iff \\ \{A(t_1, \mathbf{r}_1), B(t_2, \mathbf{r}_2)\} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{t.l.s.}} \}. \quad (4.1.1) $$

**Remark 4.1.2.** In spacetime diagram, see FIG.4.1.1, the interval $s_{AB}^2$ is "time-like" i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).
<figure><img src="image_32.png" /><figcaption>FIG.4.1.1. Spacetime diagram.</figcaption></figure>

**Remark 4.1.3.** Note that:

(i) the interval $s_{AC}^2$ in the diagram, see FIG.4.1.1, is "space-like"; i.e., there is a frame $\mathcal{F}_t$ of reference in which events $A(t, \mathbf{r}_1)$ and $C(t, \mathbf{r}_2)$ occur simultaneously at instant $t$, separated only in space. There are also frames in which **A** precedes **C** and frames in which **C** precedes **A**.

(ii) If it were possible for a cause-and-effect relationship to exist between events **A** and **C**, then paradoxes of causality would result. For example, if **A** was the cause, and **C** the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.

(iii) Obviously there exist space-like separated pairs of physical events $\{A(t, \mathbf{r}_1), B(t, \mathbf{r}_2)\}_{s.i.s.}$ such that the events $A(t, \mathbf{r}_1)$ and $C(t, \mathbf{r}_2)$ always occur only simultaneously at any instant $t$ i.e.,

$$A^{Oc}(t, \mathbf{r}_1) \Leftrightarrow C^{Oc}(t, \mathbf{r}_2). \qquad (4.1.2)$$

**Example 4.1.1.** Let us consider two synchronized clock A and B which at rest on given inertial frame $\mathcal{F}_t$. Assume that clock A at rest in point $\mathbf{r}_1$ and clock B at rest in point $\mathbf{r}_2$ correspondingly.
<figure><img src="image_33.png" /><figcaption>FIG.4.1.2. Clock A and clock B which at rest on given inertial frame $\mathcal{F}_I$.</figcaption></figure>

Let $\mathbf{A}(t, \mathbf{r}_1)$ be event which consist that time on clock A is $t$ at time $t$ according to clock A and let $\mathbf{B}(t, \mathbf{r}_1)$ be event which consist that time on clock B is $t$ at time $t$ according to clock B.It is clear that $\mathbf{A}^{\text{Oc}}(t, \mathbf{r}_1) \Leftrightarrow \mathbf{B}^{\text{Oc}}(t, \mathbf{r}_2)$.

**Definition 4.1.2.** Let $[\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}}^{\leftrightarrow}$ be a set of the all spacelike separated

pairs of events $\{\mathbf{A}(t_1, \mathbf{r}_1), \mathbf{B}(t_2, \mathbf{r}_2)\}_{\text{s.l.s.}} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}}$, (see Definition 2.2.10.b) such that

$$ \mathbf{A}^{\text{Oc}}(t_1, \mathbf{r}_1) \Leftrightarrow \mathbf{B}^{\text{Oc}}(t_2, \mathbf{r}_2). \qquad (4.1.3) $$

**Remark 4.1.3.** Note that the conditions (4.1.3) does not violated the Einstein's principle of locality and gives only an additional properties of the algebra $\mathcal{F}_{M_4}^{\#}$.

**Remark 4.1.4.** Note that from (4.1.3) folows that

$$ [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}}^{\leftrightarrow} \subset [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}]_{\text{s.l.s.}}. $$

On the basis of this Gedankenexperiment, which is also realized by photons, the EPR-paradox can be derived if the following two principles are taken as postulates.

**1. The principle of reality $R$ :**

If the value $A_i$ of an observable $A$ can be determined without altering the quantum system $S$, then any property $P(A_i)$ which corresponds to this value of $A$ pertains

to the system $S$.

**2. The principle of locality $L$ :**
2.1. If two quantum systems $S_1$ and $S_2$ cannot interact with each other, then a measurement with respect to one system cannot alter the other system and therefore we

can assume the existence of state vectors $|S_1\rangle$ and $|S_2\rangle$.

2.2. Let $\hat{x}_1$ and $\hat{x}_2$ be two observables measured with respect to systems $S_1$ and $S_2$

mentioned above. Then by result of measurement of the quantity $\bar{x}_2 = \langle S_2 | \hat{x}_2 | S_2 \rangle$ at instant $t$, impossible to get any information on result of measurement of the quantity

$\bar{x}_1 = \langle S_1 | \hat{x}_1 | S_1 \rangle$ at the same instant $t$.

We assume now the relaxed principle of locality. Intuitively this principle says that for even spacelike separated entangled quantum systems $S_1$ and $S_2$ any measurement at instant $t$ with respect to system $S_1$ always immediately alter the other system $S_2$ at the same instant $t$. But no additional information about the system $S_1$ can be found out upon measurement on the system $S_2$ except the canonical information which can be predicted by using correlation relations which follows from concrete type of entanglement.

3. The relaxed principle of locality $L_{rel}$:

3.1. Any spacelike separated quantum systems $S_1$ and $S_2$ cannot interact with each other

and therefore we can assume the existence of state vectors $|S_1\rangle$ and $|S_2\rangle$ correspondingly.

3.2. Let $S_{1(2)}(t, \mathbf{r}_1)$ and $S_{2(1)}(t, \mathbf{r}_2)$ be two spacelike separated entangled quantum systems

located in points $(t, \mathbf{r}_1)$ and $(t, \mathbf{r}_2)$ correspondingly.

(i) Assume that a state vector $|S_{1(2)}(t, \mathbf{r}_1)\rangle$ suddenly collapses at instant $t$ to state vector

$|S_{1(2)}^{\mathbf{s-col}}(t, \mathbf{r}_1)\rangle$:

$$|S_{1(2)}(t, \mathbf{r}_1)\rangle \xrightarrow{\mathbf{s-collapse}} |S_{1(2)}^{\mathbf{s-col}}(t, \mathbf{r}_1)\rangle, \qquad (4.1.4)$$

then a state vector $|S_{2(1)}(t, \mathbf{r}_2)\rangle$ immediately collapses to state vector
$|S_{2(1)}^{\mathbf{col}}(t, \mathbf{r}_2)\rangle$:

$$|S_{2(1)}(t, \mathbf{r}_2)\rangle \xrightarrow{\mathbf{col collapse}} |S_{2(1)}^{\mathbf{col}}(t, \mathbf{r}_2)\rangle \qquad (4.1.5)$$

(ii) Assume that a state vector $|S_{1(2)}(t, \mathbf{r}_1)\rangle$ after measurement immediately collapses at

instant $t$ to state vector $|S_{1(2)}^{\mathbf{m-col}}(t, \mathbf{r}_1)\rangle$:

$$|S_{1(2)}(t, \mathbf{r}_1)\rangle \xrightarrow{\mathbf{m-coll collapse}} |S_{1(2)}^{\mathbf{m-col}}(t, \mathbf{r}_1)\rangle, \qquad (4.1.6)$$

then a state vector $|S_{2(1)}(t, \mathbf{r}_2)\rangle$ immediately collapses to state vector
$|S_{2(1)}^{\mathbf{col}}(t, \mathbf{r}_2)\rangle$:

$$|S_{2(1)}(t, \mathbf{r}_2)\rangle \xrightarrow{\mathbf{coll collapse}} |S_{2(1)}^{\mathbf{col}}(t, \mathbf{r}_2)\rangle \qquad (4.1.7)$$
(iii) Let $S_{1(2)}^{\text{s-col}}(t, \mathbf{r}_1)$ and $S_{2(1)}^{\text{col}}(t, \mathbf{r}_2)$ be a physical events defined by formulae (4.1.4) and

(4.1.5) correspondingly, then

$$ \mathbf{Occ}[\mathbf{S}_{1(2)}^{\text{s-col}}(t, \mathbf{r}_1)] \Leftrightarrow \mathbf{Occ}[\mathbf{S}_{2(1)}^{\text{col}}(t, \mathbf{r}_2)], \qquad (4.1.8) $$

see Definition 2.2.8 (ii.9).

(iv) Let $S_{1(2)}^{\text{m-col}}(t, \mathbf{r}_1)$ and $S_{2(1)}^{\text{col}}(t, \mathbf{r}_2)$ be a physical events defined by formulae (4.1.6) and

(4.1.7) correspondingly, then

$$ \mathbf{Occ}[\mathbf{S}_{1(2)}^{\text{m-col}}(t, \mathbf{r}_1)] \Leftrightarrow \mathbf{Occ}[\mathbf{S}_{2(1)}^{\text{col}}(t, \mathbf{r}_2)], \qquad (4.1.9) $$

3.3. No any additional information about the system $S_1$ upon measurement at instant $t$ can

be found out upon measurement on the system $S_2$ upon measurement at instant $t$ except

the canonical information which can be predicted by using correlation relations which

follows from concrete type of entanglement.

**Remark 4.1.5.** Note that conditions (4.1.8)-(4.1.9) very similarly to the condition (4.1.3)

and gives only an additional properties of the algebra $\mathcal{F}_{M_4}^{\#}$.

**Remark 4.1.6.** Note that from (4.1.8) follows that

$$ \{\mathbf{S}_{1(2)}^{\text{s-col}}(t, \mathbf{r}_1), \mathbf{S}_{2(1)}^{\text{col}}(t, \mathbf{r}_2)\} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}] \stackrel{\leftrightarrow}{\mathrm{s.l.s.}} \qquad (4.1.10) $$

from (4.1.9) follows that

$$ \{\mathbf{S}_{1(2)}^{\text{m-col}}(t, \mathbf{r}_1), \mathbf{S}_{2(1)}^{\text{col}}(t, \mathbf{r}_2)\} \in [\mathcal{F}_{M_4}^{\#}, \{(t_1, \mathbf{r}_1), (t_2, \mathbf{r}_2)\}] \stackrel{\leftrightarrow}{\mathrm{s.l.s.}} \qquad (4.1.11) $$

**Remark 4.1.7.** Note that:

(i) collapse of a state vector $|S_{2(1)}(t, \mathbf{r}_2)\rangle$ given by (4.1.5) occurs without any interaction

between quantum systems $S_{1(2)}$ and $S_{2(1)}$ but only by property given by formulae (4.1.8);

(ii) collapse of a state vector $|S_{2(1)}(t, \mathbf{r}_2)\rangle$ given by (4.1.7) occurs without any interaction

between quantum systems $S_{1(2)}$ and $S_{2(1)}$ but only by property given by formulae (4.1.9);

**Remark.4.1.8.** We find that the EPR-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle. However it follows also, that the nonlocalities which are introduced above cannot be explained within the conventional quantum theory.

## IV.2. Generalized EPR argument and Postulate of Nonlocality

Entanglement is one of the most interesting properties of quantum mechanics,
and is an important ingredient of quantum information protocols such as quantum densecoding and quantum computation. In the Schrödinger picture, a necessary and sufficient criterion for the emergence of entanglement is that the state describing the entire system is inseparable, i.e. the wavefunction of the total system cannot be factored into a product of separate contributions from each sub-system. Using the Heisenberg approach, a sufficient criterion for the presence of entanglement is that correlations between conjugate observables of two subsystems allow the statistical inference of either observable in one sub-system, upon a measurement in the other, to be smaller than the standard quantum limit, i.e. the presence of non-classical correlations. The latter approach was originally proposed in the paper of Einstein, Podolsky and Rosen [21]. These two different pictures result in two distinct methods of characterizing entanglement. One is to identify an observable signature of the mathematical criterion for wave-function entanglement, i.e. inseparability of the state. The second looks directly for the onset of non-classical correlations. For pure states these two approaches return the same result suggesting consistency of the two methods. However, when decoherence is present, causing the state to be mixed, difference scan occur.

IV.2.1. The EPR-Reid criterion

We remind now EPR-Reid criterion [22]. EPR originally argued as follows. Consider two spatially separated subsystems at A and B. EPR considered two observables $\hat{x}$ (the “position”) and $\hat{p}$ (“momentum”) for subsystem A, where $\hat{x}$ and $\hat{p}$ do not commute, so that (C is nonzero)

$$[\hat{x}, \hat{p}] = 2C. \qquad (4.2.1)$$

Suppose now that one may predict with certainty the result of measurement $\hat{x}$, based on the result of a measurement performed at B. Also, for a different choice of measurement at B, suppose one may predict the result of measurement $\hat{p}$ at A. Such correlated systems are predicted by quantum theory. Assuming “local realism” EPR deduce the existence of an “element of reality”, $\tilde{x}$, for the physical quantity $\hat{x}$ and also an element of reality, $\tilde{p}$, for $\hat{p}$. Local realism implies the existence of two hidden variables $\tilde{x}$ and $\tilde{p}$ that simultaneously predetermine, with no uncertainty, the values for the result of an $\hat{x}$ or $\hat{p}$ measurement on subsystem A, should it be performed. This hidden variable state for the subsystem A alone is not describable within quantum mechanics, since simultaneous eigenstates of $\hat{x}$ and $\hat{p}$ do not exist. Hence, EPR argued, if quantum mechanics is to be compatible with local realism, we must regard quantum mechanics to be incomplete.

We remind that in original publication [21], Einstein, Podolsky and Rosen describe two particles A and B with correlated position

$$x_B = x_A + x_0 \qquad (4.2.2)$$

and anti-correlated momentum

$$p_B = -p_A, \qquad (4.2.3)$$

(see Fig.4.2.1).
<figure><img src="image_34.png" /><figcaption>Fig.4.2.1.</figcaption></figure>

In the idealized entangled state proposed by EPR,

$$|EPR\rangle = \int_{-\infty}^{\infty} |x, x\rangle dx = \int_{-\infty}^{\infty} |p, p\rangle dp$$

the positions and momenta of the two particles are perfectly correlated. Note that: this state is non-normalizable and cannot be realized in the laboratory. When coordinates $x^A$ and $p^A$ are measured in independent realizations of the same state, the correlations allow for an exact prediction of $x^B$ and $p^B$. EPR assumed that such exact predictions necessitate an "element of reality" which predetermines the outcome of the measurement. Quantum mechanics however prohibits the exact knowledge of two noncommuting variables like $x^B$ and $p^B$, since their measurement uncertainties are subject to the Heisenberg relation

$$\Delta x^B \Delta p^B \geq \hbar/2. \qquad (4.2.4)$$

Classical notion of EPR correlations was generalized to a more realistic scenario, yielding a Reid criterion [22] for the uncertainties $\Delta x_{inf}^B$ and $\Delta p_{inf}^B$ of the inferred predictions for $x^B$ and $p^B$. The EPR criterion is met if these uncertainties violate the Heisenberg inequality for the inferred uncertainties $\Delta x_{inf}^B \Delta p_{inf}^B \geq \hbar/2$.

Reid extended classical EPR argument to situations where the result of measurement $\hat{x}$ at A cannot be predicted with absolute certainty [22]. The assumption of local realism allows us to deduce the existence of an "element of reality" of some type for $\hat{x}$ at A, since we can make a prediction of the result at A, without disturbing the subsystem at A, under the locality assumption. Let $\Psi(\hat{x}^A, \hat{x}^B)$ be a wave function of composite system $A \cup B$. Let $x_i^B$ be the result of a measurement, $\hat{x}^B$ say, performed at B, where i is used to label the possible results, discrete or otherwise, of the measurement $\hat{x}^B$. As a result of the measurement of the coordinate, we have a new wave function of composite system $A \cup B$ which is given by Eq.(4.2.3) (see Remark.4.2.1)

$$\Psi_{x_i^B}(x, \hat{x}^B) = \Psi_{x_i^B}(\hat{x}^A, \hat{x}^B) = R(\hat{x}^B - x_i^B)\Psi(\hat{x}^A, \hat{x}^B) = R(\hat{x}^B - x_i^B)\Psi(x, \hat{x}^B) \quad (4.2.5)$$

and therefore adjoint probability density $p_{x_i^B}(x, \hat{x}^B) = p(x, \hat{x}^B|x_i^B)$ at instant at once after measurement is given by

$$p_{x_i^B}(x, \hat{x}^B) = p(x, \hat{x}^B|x_i^B) = \| R(\hat{x}^B - x_i^B)\Psi(x, \hat{x}^B) \|^2 \quad (4.2.6)$$
Then the conditional probability density $p_{x_i^B}(x) = p(x|x_i^B)$ conditional on a result $x_i^B$ for QM measurement at B is given by

$$ 
\begin{aligned}
p_{x_i^B}(x) = p(x|x_i^B) &= \int_{-\infty}^{\infty} p_{x_i^B}(x, \hat{x}^B) d\hat{x}^B = \int_{-\infty}^{\infty} d\hat{x}^B \| \Psi_{x_i^B}(x, \hat{x}^B) \|^2 = \\
&\qquad \int_{-\infty}^{\infty} d\hat{x}^B \| R(\hat{x}^B - x_i^B) \Psi(x, \hat{x}^B) \|^2.
\end{aligned} 
\quad (4.2.7) 
$$

The predicted results for the measurement at A, based on the measurement at B, are however no longer a set of definite numbers with zero uncertainty, but become fuzzy, being described by a set of distributions $P(x|x_i^B)$ giving the probability of a result for the measurement at A, conditional on a result $x_i^B$ for measurement at B. We define $\Delta_i^2 x$ to be the variance of the conditional distribution $P(x|x_i^B)$. Similarly we may infer the result of measurement $\hat{p}$ at A, based on a (different) measurement, $\hat{p}^B$ say, at B. Denoting the results of the measurement $\hat{p}^B$ at B by $p_j^B$, we then define the probability distribution, $P(p|p_j^B)$ which is the predicted result of the measurement for $\hat{p}$ at A conditional on the result $p_j^B$ for the measurement $\hat{p}^B$ at B. The variance of the conditional distribution $P(p|p_j^B)$ is denoted by $\Delta_j^2 p$.

**Remark.4.2.1.** We remind now that the QM-measurement is represented by the canonical scheme [5]

$$ |\psi\rangle \xrightarrow{a'} |\psi_{a'}\rangle = \mathfrak{R}_{a'}|\psi\rangle, \quad \int da' \mathfrak{R}_{a'}^{\dagger} \mathfrak{R}_{a'} = \mathbf{1}, p_{a'} = \|\psi_{a'}\|^2 = \langle\psi|\mathfrak{R}_{a'}^{\dagger} \mathfrak{R}_{a'}|\psi\rangle, \quad (4.2.8) $$

where $p_{a'}$ is a corresponding probability density. To obtain the probability that the parameter $a'$ turns out to belong to the set $\Delta$ one has to integrate over this set:

$$ \mathbf{P}[a' \in \Delta] = \int_{\Delta} da' p_{a'}. \quad (4.2.9) $$

If the state $|\psi\rangle$ is represented by the wave function $\psi(a)$ the operator $\mathfrak{R}_{a'}$ describing the measurement giving the result $a'$ will be taken in the following form

$$ \mathfrak{R}_{a'} \psi(a) = R(a - a') \psi(a), \quad (4.2.10) $$

where $R(a)$ is a function with a support concentrated in some vicinity of zero and representing the 'fuzziness' of the measurement. It is a characteristic function of the measurement and may, for example, be (and typically is) a Gaussian function. The width of this function corresponds to the resolution of the measurement.

Normalization $\int da' \mathfrak{R}_{a'}^{\dagger} \mathfrak{R}_{a'} = \mathbf{1}$ of the operators $\mathfrak{R}_{a'}$ is provided by the corresponding normalization of the function $R(a)$ as follows:

$$ \int da |R^2(a)| = 1. \quad (4.2.11) $$

If the measurement is described by the Gaussian function

$$ R(a) = \exp\left[-\frac{(a - a')^2}{4\Delta^2}\right] \quad (4.2.12) $$
it is a minimally disturbing measurement of the coordinate $a'$ with resolution $\Delta$ [5].

**Remark.4.2.2.** Consider the momentum representation $\tilde{\psi}(p)$ of the initial wave function $\psi(q)$

$$ \tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}} \int dq\psi(q) \exp\left[-\frac{i}{\hbar}pq\right]. \qquad (4.2.13) $$

As a result of the measurement of the coordinate,

$$ \Re_{q'}\psi(q) = R(q - q')\psi(q) = \psi_{q'}(q), \qquad (4.2.14) $$

we have a new wave function and its momentum representation has the form [5]

$$ \tilde{\psi}_{q'}(p) = \int dp' \tilde{R}_{q'}(p - p') \tilde{\psi}(p'), \qquad (4.2.15) $$

where $\tilde{R}_{q'}(p)$ is a momentum representation of the function $R_{q'}(q)$. Note that

$$ \tilde{R}_{q'}(p) = \tilde{R}(p) \exp\left[-\frac{i}{\hbar}pq'\right], \qquad (4.2.16) \\ \tilde{R}(p) = \frac{1}{2\pi\hbar} \int dq R(q) \exp\left[-\frac{i}{\hbar}pq\right]. $$

**Remark.4.2.3.** Consider now a coordinate measurement having a Gaussian characteristic function of width of the order of $\Delta$

$$ R(q) \approx \exp\left[-\frac{q^2}{4\Delta^2}\right]. \qquad (4.2.17) $$

Then the momentum representation of this function (characterizing the structure of the momentum uncertainty $\hbar/\Delta$ acquired in the measurement) is also Gaussian with width of the order of $\hbar/\Delta$:

$$ \tilde{R}(p) \approx \exp\left[\frac{-p^2}{\left(\frac{\hbar}{\Delta}\right)^2}\right]. \qquad (4.2.18) $$

For a given experiment one could in principle measure the individual variances $\Delta_i^2 x$ of the conditional distributions $P(x|x_i^B)$ (and also $\Delta_j^2 p$ for the $P(p|p_j^B)$). Obviously if each of the variances $\Delta_i^2 x$ and $\Delta_j^2 p$ satisfy $\Delta_i^2 x = 0$ and $\Delta_j^2 p = 0$ this would imply the demonstration of the original EPR paradox. This situation however is not practical for continuous variable measurements [5]. Instead of considering the problem of simultaneous eigenstates as originally proposed by EPR, one can suggest an different and experimentally realizable criterion based on the Heisenberg Uncertainty Principle: $\Delta\hat{x}\Delta\hat{p} \ge C$. For the sake of notational convenience we now consider in the remainder of this subsection that appropriate scaling enables $\hat{x}$ and $\hat{p}$ to be dimensionless and $C = 1$.

EPR correlations however would be demonstrated in a convincing manner if the experimentalist could measure each of the conditional distributions $P(x|x_i^B)$ and establish that each of the distributions is very narrow, in fact constrained such that [5]:
$$ 
\begin{aligned}
P(x|x_i^B) &= 0 \quad \text{iff} \quad |x - \mu_i| > \delta, \\
P(p|p_j^B) &= 0 \quad \text{iff} \quad |p - v_j| > \delta.
\end{aligned} 
\qquad (4.2.19) 
$$

Here $\mu_i$ is the mean value of the conditional distribution $P(x|x_i^B)$ and $v_j$ is the mean value of the conditional distribution $P(p|p_j^B)$. In this case the assumption of local realism would imply, since the measurement $\hat{x}^B$ at B will always imply the result of $\hat{x}$ at A to be within the range $\mu_i \pm \delta_x$, that the result of the measurement at A is predetermined to be within a bounded range of width $2\delta$. In a straightforward extension of EPR's argument, we replace the words “predict with certainty” with “predict with certainty that the result is constrained to be within the range $\mu_i \pm \delta$”, and then define an “element of reality” with this intrinsic bounded by fuzziness $\delta$. We now consider the situation where an experimenter has demonstrated that for every outcome $x_i^B$ (and $p_j^B$) for the measurement $\hat{x}^B$ (and $\hat{p}^B$) performed at B, the variance $\Delta_i x$ (and $\Delta_j p$) of the appropriate conditional distribution satisfies

$$ \Delta_i x < 1, \Delta_j p < 1 \qquad (4.2.20) $$

for any $i,j \in \mathbb{N}$. The measurement at B always allows an inference of the result at A to a precision better than given by the uncertainty bound 1.

**Remark.4.2.4.** In this case we do not predict a result at A “with certainty”, as in EPR’s

original paradox. The measurement $\hat{x}^B$ at B however does predict by Eq.(4.2.3) [or by

Eq.(4.2.9) in general case] with a certain probability constraints on the result for $\hat{x}$ at A.

**Remark 4.2.5.** Following the EPR argument, which assumes

**no action-at-a-distance,**

so that the measurement at B does not cause any instantaneous influence to the system at

A, one can attribute a probabilistic predetermined “element of reality” to the system at A.

**Remark.4.2.6.** There is a similar predicted result for the measurement $\hat{p}$ at A based on a

result of measurement at B, and a corresponding predetermined description based again

on the

**no-action-at-a-distance**

assumption.

**Remark.4.2.7.** The important point in establishing the EPR paradox for this more general

yet practical situation is that under the EPR premises the predetermined statistics (or

generalised “elements of reality”) for the physical quantities
x̂ and p̂ are attributed simultaneously to the subsystem at A.

Assuming no action-at-a-distance, the choice of the experimenter (Bob) at B to infer

information about either x̂ or p̂ cannot actually induce the result of the
measurement at A.

As there is no disturbance created by Bob's measurement, the (appropriately extended)

EPR definition of realism is that the prediction for $x$ is something (a probabilistic "element of"

reality") that can be attributed to the subsystem at A, whether or not Bob makes his

measurement.

**Remark 4.2.8.** This is also true of the prediction for $p̂$, and therefore the two "elements of"

reality" representing the physical quantities x̂ and p̂ exist to describe the
predictions for x̂

and p̂ simultaneously.

The paradox can then be established by proving the impossibility of such a
simultaneous level of prediction for both x̂ and p̂ for any quantum description of the
subsystem *A* alone. By this we mean explicitly that there can be no procedure
allowed, within the predictions of quantum mechanics, to make *simultaneous
inferences* by measurements performed at *B* or any other location, of both the
result x̂ and p̂ at *A*, to the precision indicated by Δᵢx < 1, Δⱼp < 1.

**Remark 4.2.9.** Recall that the inference of the result at *A* by measurement at *B* is actually a measurement of x̂ performed with the accuracy determined by the Δᵢx. However simultaneous measurements of x̂ and p̂ to the accuracy (4.2.9) are not possible (predicted by quantum mechanics). The reduced density matrix describing the state at *A* after such measurements would violate the H.U.P.

A simpler quantitative, experimentally testable criterion for EPR was proposed
by Reid in 1989 see for example [22]. The 1989 inferred H.U.P. criterion is based
on the average variance of the conditional distributions for inferring the result of
measurement x̂ (and also for p̂). The EPR paradox is demonstrated when the
product of the average errors in the inferred results for x̂ and p̂ violate the
corresponding Heisenberg Uncertainty Principle. The spirit of the original EPR
paradox is present, in that one can perform a measurement on B to enable an
estimate of the result x at A (and similarly for p̂).

**Abbreviation 4.2.1.** For the sake of notational convenience we now abbreviate
in the remainder of the paper: Δ<sub>loc,i</sub>x and Δ<sub>loc,i</sub>p instead Δᵢx and Δⱼp for the variance
Δᵢx and Δⱼp which were calculated under assumption no action-at-a-distance, see
Remark 4.2.5-4.2.6.

We define now [22]:
$$ \begin{aligned} \Delta_{\text{loc.inf.}}^2 x &= \sum_i P(x_i^B) \Delta_{\text{loc.}i}^2 x, \\ \Delta_{\text{loc.inf.}}^2 p &= \sum_j P(p_j^B) \Delta_{\text{loc.}j}^2 p. \end{aligned} \qquad (4.2.21) $$

Here $\Delta_{\text{loc.inf.}}^2 \hat{x}$ is the average variance for the prediction (inference) under assumption no action-at-a-distance of the result $x$ for $\hat{x}$ at $A$, conditional on a measurement $\hat{x}^B$ at $B$. Here $i \in \mathbb{N}$ labels all outcomes of the measurement $\hat{x}$ at $A$, and $\mu_i$ and $\Delta_i x$ are the mean and standard deviation, respectively, of the conditional distribution $P(x|x_i^B)$, where $x_i^B$ is the result of the measurement $\hat{x}^B$ at $B$. We define a $\Delta_{\text{loc.inf.}}^2 \hat{p}$ similarly to represent the weighted variance for the prediction (inference) under assumption no action-at-a-distance of the result $\hat{p}$ at $A$, based on the result of the measurement at $B$. Here $P(x_i^B)$ is the probability for a result $x_i^B$ upon measurement of $\hat{x}^B$, and $P(p_j)$ is defined similarly.

The Reid's criterion to demonstrate the EPR "paradox", the Reid's local signature of the EPR paradox, is

$$ (\Delta_{\text{loc.inf.}}^2 x)(\Delta_{\text{loc.inf.}}^2 p) < 1. \qquad (4.2.22) $$

This criterion is a clear criterion for the demonstration of the EPR "paradox", by way of the argument presented above. Such a prediction (4.2.21) for $\hat{x}$ and $\hat{p}$ with the average inference variances given, cannot be achieved by any quantum description of the subsystem alone. This EPR criterion has been achieved experimentally.

## IV.2.2. The Postulate of Nonlocality and signature of the EPR "paradox"

**Remark.4.2.10.** A most critical component of the EPR argument was the principle of locality. Indeed, one may regard the EPR paradox as a statement of the mutual incompatibility of locality, entanglement, and completeness. Experimental tests of Bell's inequalities have indicated that quantum mechanics is complete by ruling out the possibility of hidden variables. Therefore it is generally agreed that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state consistent with the result of measurement, regardless of how far apart the particles are. In the situation proposed by EPR, the position or momentum of the unmeasured particle becomes a reality when, and only when, the corresponding quantity of the other particle is measured.

**Remark.4.2.11.** The assumption of nonlocality allows us to deduce the existence of a fuzzy "element of reality" of some type for $\hat{x}$ at $A$, since we can make a prediction of the result at $A$, but with some disturbing of the subsystem at $A$, under the measurement, $\hat{x}^B$ say, performed at $B$. This prediction is subject to the result $x_i^B$ of a measurement, $\hat{x}^B$ say, performed at $B$, where $i$ is used to label the possible results, discrete or otherwise, of the measurement $\hat{x}^B$.

We accept now the following postulate:

**Postulate of Nonlocality**
(i) Let A and B two entangled particles. Let $\Psi(\hat{x}^A, \hat{x}^B)$ be a wave function of composite system $A \cup B$. Let $x_i^B$ be the result of a measurement, $\hat{x}^B$ say, performed at B, where i is used to label the possible results, discrete or otherwise, of the measurement $\hat{x}^B$. As a result of the measurement of the coordinate, we have a new wave function of composite system $A \cup B$ which [in contrast with Eq.(4.2.5)] is given by

$$ \Psi_{x_i^B}(\hat{x}^A, \hat{x}^B) = R_2(\hat{x}^A - x_i^A(x_i^B))R_1(\hat{x}^B - x_i^B)\Psi(\hat{x}^A, \hat{x}^B), \quad (4.2.23) \\ x_i^A(x_i^B) + x_0 \approx x_i^B. $$

(ii) Let A and B two entangled particles. Let $\tilde{\Psi}(\hat{p}^A, \hat{p}^B)$ be a wave function of composite system $A \cup B$. Let $p_j^B$ be the result of a measurement, $\hat{p}^B$ say, performed at B, where j is used to label the possible results, discrete or otherwise, of the measurement $\hat{p}^B$. As a result of the measurement of the coordinate, we have a new wave function of composite system $A \cup B$ which is given by

$$ \tilde{\Psi}_{p_j^B}(\hat{p}^A, \hat{p}^B) = \hat{R}_2(\hat{p}^A - p_j^A(p_j^B))\hat{R}_1(\hat{p}^B - p_j^B)\tilde{\Psi}(\hat{p}^A, \hat{p}^B), \quad (4.2.24) \\ p_j^A(p_j^B) \approx -p_j^B. $$

**Remark 4.2.12.** The spirit of the original EPR paradox now is present, in that the canonical EPR correlations (4.2.2) and (4.2.3) well preserved.

**Remark 4.2.13.** Note that EPR correlations $x_i^A(x_i^B) + x_0 \approx x_i^B$ and $p_j^A(p_j^B) \approx -p_j^B$ however would be demonstrated in a convincing manner if the experimentalist could measure each of the conditional distributions $P(x|x_i^B)$ and establish that each of the distributions is very narrow, in fact constrained so that [5]

$$ \begin{aligned} p(x|x_i^B) &\approx 0 & \text{iff } |x - \mu_i| > \delta, \\ p(p|p_j^B) &\approx 0 & \text{iff } |p - v_j| > \delta, \\ P(x|x_i^B) &\approx 0 & \text{iff } |x - \mu_i| > \delta, \\ P(p|p_j^B) &\approx 0 & \text{iff } |p - v_j| > \delta. \end{aligned} \quad (4.2.25) $$

Here $\mu_i$ is the mean of the conditional distribution $P(x|x_i^B)$ and $v_j$ is the mean of the conditional distribution $P(p|p_j^B)$.

**Remark 4.2.14.** We assume now that a coordinate and momentum measurements have a Gaussian characteristic function of width of the order of $2\delta$

$$ \begin{aligned} R_1(x) &= R_2(x) = R(x) \approx \exp\left[-\frac{x^2}{4\delta^2}\right] \\ \hat{R}_1(p) &= \hat{R}_2(p) = \hat{R}(p) \approx \exp\left[-\frac{p^2}{4\delta^2}\right] \end{aligned} \quad (4.2.26) $$

In this case the Postulate of Nonlocality would imply, since the measurement $\hat{x}^B$ at B will always imply the result of $\hat{x}$ at A to be within the range $\mu_i \pm \delta_x$, that the result of the measurement at A is predetermined to be within a bounded range of width $2\delta$. In a straightforward extension of EPR's argument, we replace the words
"predict with certainty" with "predict with certainty that the result is constrained to be within the range $\mu_i \pm \delta$. We now consider the situation where an experimenter has demonstrated that for every outcome $x_i^B$ (and $p_i^B$) for the measurement $\hat{x}^B$ (and $\hat{p}^B$) performed at B, the variance $\Delta_i x$ (and $\Delta_j p$) of the appropriate conditional distribution satisfies

$$ \Delta_i x < 1, \Delta_j p < 1 \qquad (4.2.27) $$

for all $i, j$. The measurement at B always allows an inference of the result at A to a precision better than given by the uncertainty bound 1.

In this case we do not predict a result at A "with certainty", as in EPR's original paradox. The measurement $\hat{x}_B$ at B however does predict with a certain probability constraints on the result for $\hat{x}$ at A.

**Remark 4.2.15.** Note that adjoint probability density $p(\hat{x}^A, \hat{x}^B|x_i^B)$ at instant at once after measurement [in contrast with Eq.(4.1.6)] is given by

$$ \begin{aligned} p(\hat{x}^A, \hat{x}^B|x_i^B) = & \| \Psi_{x_i^B}(\hat{x}^A, \hat{x}^B) \|^2 = \\ & \| R(\hat{x}^A - x_i^A(x_i^B)) R(\hat{x}^B - x_i^B) \Psi(\hat{x}^A, \hat{x}^B) \|^2, \end{aligned} \qquad (4.2.27) \\ x_i^A(x_i^B) + x_0 \approx x^B \pm \mu_i. $$

Then the conditional probability density $p_{x_i^B}(x) = p(x|x_i^B)$ conditional on a result $x_i^B$ for QM measurement at B is given by

$$ \begin{aligned} p_{x_i^B}(x) = p(x|x_i^B) = & \int_{-\infty}^{\infty} p_{x_i^B}(x, \hat{x}^B) d\hat{x}^B = \int_{-\infty}^{\infty} d\hat{x}^B \| \Psi_{x_i^B}(x, \hat{x}^B) \|^2 = \\ & \int_{-\infty}^{\infty} d\hat{x}^B \| R(\hat{x}^A - x_i^A(x_i^B)) R(\hat{x}^B - x_i^B) \Psi(x, \hat{x}^B) \|^2. \end{aligned} \qquad (4.2.28) $$

There is a similar predicted result for the measurement $\hat{p}$ at A based on a result of measurement at B, and a corresponding predetermined description based on the QM constraints

$$ \begin{aligned} \widetilde{\Psi}_{p_j^B}(\hat{p}^A, \hat{p}^B) = & \hat{R}(\hat{p}^A - p_j^A(p_j^B)) \hat{R}(\hat{p}^B - p_j^B) \widetilde{\Psi}(\hat{p}^A, \hat{p}^B), \\ p_j^A(p_j^B) \approx & -p^B. \end{aligned} \qquad (4.2.29) $$

The spirit of the original EPR "paradox" is present, in that one can perform a measurement on B to enable an estimate of the result x at A (and similarly for $\hat{p}$).

**Abbreviation 4.2.2.** For the sake of notational convenience we now abbreviate in the remainder of the paper: $\Delta_{\text{nonloc.}i}x$ and $\Delta_{\text{nonloc.}j}p$ instead $\Delta_i x$ and $\Delta_j p$ for the variance $\Delta_i x$ and $\Delta_j p$ which were calculated under nonlocality assumption (postulate) by conditional probability density given by Eq.(4.1.28).

We define now
$$ 
\begin{aligned}
\Delta_{\text{nonloc.inf.}}^2 x &= \sum_i P(x_i^B) \Delta_{\text{nonloc.}i}^2 x, \\
\Delta_{\text{nonloc.inf.}}^2 p &= \sum_j P(p_j^B) \Delta_{\text{nonloc.}j}^2 p.
\end{aligned}
\qquad (4.2.30) $$

Here $\Delta_{\text{nonloc.inf.}}^2 x$ is the average variance for the prediction (inference) of the result $x$ for $\hat{x}$ at $A$, conditional on a measurement $\hat{x}^B$ at $B$. Here $i$ labels all outcomes of the measurement $\hat{x}$ at $A$, and $\mu_i$ and $\Delta_i x$ are the mean and standard deviation, respectively, of the conditional distribution $P(x|x_i^B)$, where $x_i^B$ is the result of the measurement $\hat{x}^B$ at $B$. We define a $\Delta_{\text{nonloc.inf.}}^2 p$ similarly to represent the weighted variance for the prediction (inference) of the result $\hat{p}$ at $A$, based on the result of the measurement at $B$. Here $P(x_i^B)$ is the probability for a result $x_i^B$ upon measurement of $\hat{x}^B$, and $P(p_j^B)$ is defined similarly. The criterion to demonstrate the EPR paradox, the signature of the EPR paradox, is The criterion to demonstrate the EPR "paradox", the nonlocal signature of the EPR paradox, is given by

$$ (\Delta_{\text{nonloc.inf.}} x)(\Delta_{\text{nonloc.inf.}} p) < 1. \qquad (4.2.31) $$

This criterion is a clear criterion for the demonstration of the EPR "paradox", by way of the argument presented above. Such a prediction for $\hat{x}$ and $\hat{p}$ with the average inference variances given, cannot be achieved by any quantum description of the subsystem alone.

### IV.2.3. The EPR-nonlocality criteria

**Remark 4.2.16.** A critical component of the EPR argument was the principle of locality. Indeed, one may regard the EPR paradox as a statement of the mutual incompatibility of locality, entanglement, and completeness. Experimental tests of Bell's inequalities have indicated that quantum mechanics is complete by ruling out the possibility of hidden variables. Therefore it is generally agreed that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state consistent with the result of measurement, regardless of how far apart the particles are. In the situation proposed by EPR, the position or momentum of the unmeasured particle becomes a reality when, and only when, the corresponding quantity of the other particle is measured. Since only one quantity or the other is measured, the position and the momentum of the unmeasured particle need not be simultaneous realities. In this way the EPR "paradox" also is resolved. From Eq.(4.2.21) and Eq.(4.2.30) we obtain the EPR-nonlocality criteria

$$ 
\begin{aligned}
|\Delta_{\text{loc.inf.}}^2 x - \Delta_{\text{nonloc.inf.}}^2 x| &= |\sum_i P(x_i^B)[\Delta_{\text{loc.}i}^2 x - \Delta_{\text{nonloc.}i}^2 x]| > 0, \\
|\Delta_{\text{loc.inf.}}^2 p - \Delta_{\text{nonloc.inf.}}^2 p| &= |\sum_j P(p_j^B)[\Delta_{\text{loc.}j}^2 p - \Delta_{\text{nonloc.}j}^2 p]| > 0,
\end{aligned}
\qquad (4.2.32) $$

and

$$ |(\Delta_{\text{nonloc.inf.}} x)(\Delta_{\text{nonloc.inf.}} p) - (\Delta_{\text{loc.inf.}} x)(\Delta_{\text{loc.inf.}} p)| > 0. \qquad (4.2.33) $$

These EPR-nonlocality criteria has been achieved experimentally [66],[67], (see subsection IV.5, Remark 4.5.3-Remark 4.5.4).
IV.3. Nonlocal Schrödinger equation implies the Postulate of Nonlocality

In this subsection we obtain nonlocal Schrödinger equation (NSE) which corresponding to position-momentum entangled pairs A and B (see Fig.4.2.1) with well correlated position

$$ \langle x_B \rangle \approx \langle x_A \rangle + x_0 \qquad (4.3.1) $$

and anti-correlated momentum

$$ \langle p_B \rangle \approx -\langle p_A \rangle. \qquad (4.3.2) $$

**Remark 4.3.1.** As pointed out in subsection IV.2 it is generally agreed that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state consistent with the result of measurement, regardless of how far apart the particles are. It allows us to use special nonlocal generalization of the canonical Schrödinger equation.

**Remark 4.3.2.** As pointed out in subsection II.2 from nonlocal Schrödinger equation (2.1.17) one obtains collapsed wave function corresponding to GRW collapse model.

It allows us to use similar nonlocal Schrödinger equation also for entangled states.

**Remark 4.3.3.** The spirit of the original EPR paradox is present, in that the canonical EPR correlations (4.3.1) and (4.3.2) gives an boundary conditions for the solutions of the nonlocal Schrödinger equation.

**Remark 4.3.4.** In this subsection we denote (i) $x_A = x_1, x_B = x_2$, (ii) $x^A = \tilde{x}_1$, $x^B = \tilde{x}_2 = \tilde{x}_1 + x_0$.

**Definition 4.3.1.** Let us consider the time-dependent canonical Schrödinger equation

$$ i\hbar \frac{\partial \Psi(x_1, x_2, t)}{\partial t} = H\Psi(x_1, x_2, t), \qquad (4.3.3) \\ t \in [0, T], (x_1, x_2) \in \mathbb{R}^2. $$

Let $\Psi(x_1, x_2, t)$ be a classical solution of the time-dependent Schrödinger equation (4.3.3). The time-dependent Schrödinger equation (4.3.3) is a weakly well preserved (in sense of Colombeau generalized functions) by corresponding to $\Psi(x_1, x_2, t)$ collapsed Colombeau generalized wave function $(\Psi_{\varepsilon}^{\#}(x_1, x_2, t))_{\varepsilon}, \varepsilon \in (0, 1]$, where
$$ (\Psi_{\varepsilon}^{\#}(x_1, x_2, t))_{\varepsilon} = (\Psi_{\varepsilon}(x_1, x_2, t; \tilde{x}_1(t), \tilde{x}_2(t)))_{\varepsilon} = $$

$$ = \left( \frac{\Re_{1,2}(x_1, \tilde{x}_1(t), x_2, \tilde{x}_2(t); \delta, \varepsilon) \Psi(x_1, x_2, t)}{\| \Re_{1,2}(\tilde{x}_1(t), \tilde{x}_2(t); \delta, \varepsilon) \Psi(x_1, x_2, t) \|_2} \right)_{\varepsilon}, $$

$$ (\Re_{1,2}(x_1, \tilde{x}_1, x_2, \tilde{x}_2; \delta, \varepsilon))_{\varepsilon} = \prod_{i=1}^{2} (\Re_i(x_i, \tilde{x}_i; \delta, \varepsilon))_{\varepsilon}, \quad (4.3.4) $$

$$ \Re_i(x, \tilde{x}_i(t); \delta, \varepsilon) = \begin{cases} (\pi \delta \delta^2)^{-1/4} \exp \left[ - \frac{(x_i - \tilde{x}_i(t))^2}{2\delta^2} \right] & \text{iff } \|x_i - \tilde{x}_i\| \le \varepsilon, \\ 0 & \text{iff } \|x_i - \tilde{x}_i\| > \varepsilon. \end{cases} \\ i = 1, 2. $$

in region $\Gamma \subseteq \mathbb{R}^2$ if there exist an solution $\Psi(x_1, x_2, t)$ of Schrödinger equation (4.2.1) such that the estimate

$$ \left( \int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi_{\varepsilon}^{\#}(x_1, x_2, t)}{\partial t} - \hat{H} \Psi_{\varepsilon}^{\#}(x_1, x_2, t) \right\} dx_1 dx_2 \right)_{\varepsilon} = (O(\hbar^{\alpha}))_{\varepsilon}, \quad (4.3.5) \\ t \in [0, T], x_1, x_2 \in \Gamma^2, $$

with $1/2 \le \alpha$, is satisfied.

**Definition 4.3.2.** Equation (4.3.5) with a following boundary conditions

$$ \begin{align} \langle x_B^t \rangle &\approx \langle x_A^t \rangle + x_0, \nonumber \\ \langle x_A^t \rangle &= \left( \int x_A |\Psi_{\varepsilon}^{\#}(x_A, x_B, t)|^2 dx_A dx_B \right)_{\varepsilon}, \tag{4.3.6} \\ \langle x_B^t \rangle &= \left( \int x_B |\Psi_{\varepsilon}^{\#}(x_A, x_B, t)|^2 dx_A dx_B \right)_{\varepsilon}, \nonumber \end{align} $$

that is time-dependent nonlocal Schrödinger equation corresponding to EPR entangled state.

**Definition 4.3.3.** (i) The time-dependent integral equation (4.3.5) with a boundary conditions (4.3.6) is colled the time-dependent nonlocal Schrödinger equation of the order $\hbar^\alpha$ corresponding to EPR entangled state.

(ii) Such collapsed wave function $\Psi^{\#}(x_1, x_2, t,)$ as mentioned in Definition 4.3.2 is colled the

$\hbar^\alpha$- solution of the nonlocal Schrödinger equation (4.3.5)-(4.3.6) of the order $\alpha$.

**Definition 4.3.4.** Let us consider the time-independent canonical Schrödinger equation

$$ \hat{H} \Psi(x_1, x_2) = 0, (x_1, x_2) \in \mathbb{R}^2. \quad (4.3.7) $$
Let $\Psi(x_1, x_2)$ be a classical solution of the time-independent Schrödinger equation (4.3.7). The time-independent Schrödinger equation (4.3.7) is a weakly well preserved (in sense of Colombeau generalized functions) by corresponding to $\Psi(x_1, x_2)$ Colombeau generalized collapsed wave function $(\Psi_{\varepsilon}^{\#}(x_1, x_2))_{\varepsilon} \in (0, 1]$, where

$$ (\Psi_{\varepsilon}^{\#}(x_1, x_2, \delta))_{\varepsilon} = (\Psi_{\varepsilon}(x_1, x_2; \tilde{x}_1, \tilde{x}_2, \delta))_{\varepsilon} = \\ = \left( \frac{\Re_{1,2}(x_1, \tilde{x}_1, x_2, \tilde{x}_2; \delta, \varepsilon) \Psi(x_1, x_2)}{\| \Re_{1,2}(\tilde{x}_1, \tilde{x}_2; \delta, \varepsilon) \Psi(x_1, x_2) \|_2} \right)_{\varepsilon}, \\ (\Re_{1,2}(x_1, \tilde{x}_1, x_2, \tilde{x}_2; \delta, \varepsilon))_{\varepsilon} = \prod_{i=1}^{2} (\Re_i(x_i, \tilde{x}_i; \delta, \varepsilon))_{\varepsilon}, \quad (4.3.8) $$

$$ \Re_i(x, \tilde{x}_i; \delta, \varepsilon) = \begin{cases} (\pi \delta \delta^2)^{-1/4} \exp \left[ - \frac{(x_i - \tilde{x}_i)^2}{2\delta^2} \right] & \text{iff } \|x_i - \tilde{x}_i\| \le \varepsilon, \\ 0 & \text{iff } \|x_i - \tilde{x}_i\| > \varepsilon. \end{cases} $$

in region $\Gamma \subseteq \mathbb{R}^2$ if there exist an solution $\Psi(x_1, x_2)$ of Schrödinger equation (4.3.7) such that the estimate

$$ \left( \int_{\Gamma} \hat{H} \Psi_{\varepsilon}^{\#}(x_1, x_2) dx_1 dx_2 \right)_{\varepsilon} = (O(h^{\alpha}))_{\varepsilon}, \quad (4.3.9) \\ (x_1, x_2) \in \Gamma^2, $$

with $1/2 \le \alpha$, is satisfied.

**Definition 4.3.5.** Equation (4.3.9) with a boundary conditions

$$ \begin{align} \langle x_B \rangle &\approx \langle x_A \rangle + x_0, \nonumber \\ \langle x_A \rangle &= \left( \int x_A |\Psi_{\varepsilon}^{\#}(x_A, x_B)|^2 dx_A dx_B \right)_{\varepsilon}, \tag{4.3.10} \\ \langle x_B \rangle &= \left( \int x_B |\Psi_{\varepsilon}^{\#}(x_A, x_B)|^2 dx_A dx_B \right)_{\varepsilon}, \nonumber \end{align} $$

that is time-independent nonlocal Schrödinger equation corresponding to EPR entangled state.

**Definition 4.3.6.** (i) The time-independent integral equation (4.3.9) with a boundary conditions (4.3.10) is colled the time-independent nonlocal Schrödinger equation of the order $h^\alpha$ corresponding to EPR entangled state.

(ii) Such collapsed wave function $\Psi^{\#}(x_1, x_2)$ as mentioned in Definition 4.3.5 is colled the

$h^\alpha$- solution of the nonlocal Schrödinger equation (4.3.9)-(4.3.10) of the order $\alpha$.
**Lemma 4.3.1.** Let $\Phi(\lambda)$ be a function

$$ \Phi(\lambda) = \int_{0}^{a} x^{\beta-1} \exp(-\lambda x^{\alpha}) f(x) dx, \qquad (4.3.11) $$

where $\lambda \gg 1, 0 < a < \infty, 0 < \beta, 0 < \alpha$. Assume that $f(x)$ is continuous on $[0, a]$. Then

$$ \Phi(\lambda) = \alpha^{-1} \Gamma\left(\frac{\beta}{\alpha}\right) [f(0) + o(1)] \lambda^{-\beta/\alpha} \qquad (4.3.12) $$

**Lemma 4.3.2.** Let $f(x)$ be a function such that $f \in C^2(x < x_0)$ and $f \in C^2(x > x_0)$. Then

$$ \begin{aligned} f'(x) &= \{f'(x)\}_{x \neq x_0} + [f]_{x_0} \delta(x - x_0), \\ f''(x) &= \{f''(x)\}_{x \neq x_0} + [f']_{x_0} \delta(x - x_0) + [f]_{x_0} \delta'(x - x_0), \end{aligned} \qquad (4.3.13) \\ [f]_{x_0} = f(x_0 + 0) - f(x_0 - 0), \\ [f']_{x_0} = f'(x_0 + 0) - f'(x_0 - 0). $$

**Theorem 4.3.1.** Assume that there exist a classical solution $\Psi(x_1, x_2)$ of the Schrödinger equation (4.3.7) such that

$$ \begin{aligned} & \sup_{(x_1,x_2) \in \Gamma} |\Psi(x_1,x_2)| = O(\hbar^{-1/2}), \\ & \sup_{(x_1,x_2) \in \Gamma} |\partial\Psi(x_1,x_2)/\partial x_1| = O(\hbar^{-3/2}), \quad \sup_{(x_1,x_2) \in \Gamma} |\partial\Psi(x_1,x_2)/\partial x_2| = O(\hbar^{-3/2}). \end{aligned} \qquad (4.3.14) $$

Then any collapsed wave function $\Psi^#(x)$ given by Eq.(4.3.8) with $\sqrt{\hbar/\delta} = \hbar^\alpha, 1/4 < \alpha < 1/2$ that is $\hbar^\alpha$-solution of the time-independent nonlocal Schrödinger equation (4.3.9)-(4.3.10) of the order $\alpha$.

**Proof.** The Schrödinger equation (4.3.7) has the following form

$$ H\Psi(x_1, x_2) = \hbar^2 \frac{\partial^2\Psi(x_1, x_2)}{\partial x_1^2} + \hbar^2 \frac{\partial^2\Psi(x_1, x_2)}{\partial x_2^2} + V(x_1, x_2)\Psi(x_1, x_2) = 0. \qquad (4.3.15) $$

Let $\Psi_\delta^#(x_1, x_2)$ be a function

$$ \Psi_{\delta}^{\#}(x_1, x_2) = R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \Psi(x_1, x_2), \qquad (4.3.16) $$

where

$$ R_{\delta}(x_i, \tilde{x}_i) = \begin{cases} (\pi_{\delta}\delta^2)^{-1/4} \exp\left[-\frac{(x_i - \tilde{x}_i)^2}{2\delta^2}\right] & \text{iff } \|x_i - \tilde{x}_i\| \le \varepsilon, \\ 0 & \text{iff } \|x_i - \tilde{x}_i\| > \varepsilon. \end{cases} \qquad (4.3.17) $$

From Eq.(4.3.17) by using Eq.(4.3.13) we obtain
$$
\begin{align}
\frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} &= -(\pi_{\delta}\delta)^{-1/4}\delta^{-1}(x_1 - \tilde{x}_1)\exp\left[-\frac{(x_1 - \tilde{x}_1)^2}{2\delta}\right] + \nonumber \\
&\quad + ([R_{\delta}(x_1, \tilde{x}_1)]_{\tilde{x}_1-\epsilon})\delta(x_1 - \tilde{x}_1 + \epsilon) + ([R_{\delta}(x_1, \tilde{x}_1)]_{\tilde{x}+\epsilon})\delta(x_1 - \tilde{x}_1 - \epsilon), \nonumber \\
\frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} &= -(\pi_{\delta}\delta)^{-1/4}\delta^{-1}\exp\left[-\frac{(x_1 - \tilde{x}_1)^2}{2\delta}\right] + \nonumber \\
&\quad (\pi_{\delta}\delta)^{-1/4}\delta^{-2}(x_1 - \tilde{x}_1)^2 \exp\left[-\frac{(x_1 - \tilde{x}_1)^2}{2\delta}\right] + \tag{4.3.18} \\
&\quad \left(\left[\frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1}\right]_{\tilde{x}-\epsilon}\right)\delta(x_1 - \tilde{x}_1 + \epsilon) + \left(\left[\frac{R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1}\right]_{\tilde{x}+\epsilon}\right)\delta(x_1 - \tilde{x}_1 - \epsilon) + \nonumber \\
&\quad ([R_{\delta}(x_1, \tilde{x}_1)]_{\tilde{x}-\epsilon})\delta'(x_1 - \tilde{x}_1 + \epsilon) + ([R_{\delta}(x_1, \tilde{x}_1)]_{\tilde{x}+\epsilon})\delta'(x_1 - \tilde{x}_1 - \epsilon) \nonumber
\end{align}
$$

and

$$
\begin{align}
\frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} = &-(\pi_{\delta}\delta)^{-1/4}\delta^{-1}(x_2 - \tilde{x}_2)\exp\left[-\frac{(x_2 - \tilde{x}_2)^2}{2\delta}\right] + \nonumber \\
&+ ([R_{\delta}(x_2, \tilde{x}_2)]_{\tilde{x}_1-\epsilon})\delta(x_2 - \tilde{x}_2 + \epsilon) + ([R_{\delta}(x_2, \tilde{x}_2)]_{\tilde{x}+\epsilon})\delta(x_2 - \tilde{x}_2 - \epsilon), \nonumber \\
\frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} = &-(\pi_{\delta}\delta)^{-1/4}\delta^{-1}\exp\left[-\frac{(x_2 - \tilde{x}_2)^2}{2\delta}\right] + \nonumber \\
& (\pi_{\delta}\delta)^{-1/4}\delta^{-2}(x_2 - \tilde{x}_2)^2 \exp\left[-\frac{(x_2 - \tilde{x}_2)^2}{2\delta}\right] + \tag{4.3.19} \\
& \left(\left[\frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2}\right]_{\tilde{x}-\epsilon}\right)\delta(x_2 - \tilde{x}_2 + \epsilon) + \left(\left[\frac{R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2}\right]_{\tilde{x}+\epsilon}\right)\delta(x_2 - \tilde{x}_2 - \epsilon) + \nonumber \\
& + ([R_{\delta}(x_2, \tilde{x}_2)]_{\tilde{x}-\epsilon})\delta'(x_2 - \tilde{x}_2 + \epsilon) + ([R_{\delta}(x_2, \tilde{x}_2)]_{\tilde{x}+\epsilon})\delta'(x_2 - \tilde{x}_2 - \epsilon). \nonumber
\end{align}
$$

From Eq.(4.3.16) by differentiation we obtain

$$
\begin{align}
\frac{\partial^2 \Psi_{\delta}^{\#}(x_1, x_2)}{\partial x_1^2} &= \frac{\partial^2 [R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \Psi(x_1, x_2)]}{\partial x_1^2} = \nonumber \\
&\quad \frac{\partial}{\partial x_1} \left[ \Psi(x_1, x_2) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} + R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial \Psi(x_1, x_2)}{\partial x_1} \right] = \nonumber \\
&\quad 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_1} R_{\delta}(x_2, \tilde{x}_2) \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} + \nonumber \\
&\quad + \Psi(x_1, x_2) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} + R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 \Psi(x_1, x_2)}{\partial x_1^2} \tag{4.3.20}
\end{align}
$$

and
$$ \frac{\partial^2 \Psi_{\delta}^{\#}(x_1, x_2)}{\partial x_2^2} = \frac{\partial^2 [R_{\delta}(x_1) R_{\delta}(x_2, \tilde{x}_2) \Psi(x_1, x_2)]}{\partial x_2^2} = \\ \frac{\partial}{\partial x_2} \left[ \Psi(x_1, x_2) R_{\delta}(x_1, \tilde{x}_1) \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} + R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial \Psi(x_1, x_2)}{\partial x_2} \right] = \qquad (4.3.21) $$

$$ 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_2} R_{\delta}(x_1, \tilde{x}_1) \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} + \\ + \Psi(x_1, x_2) R_{\delta}(x_1, \tilde{x}_1) \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} + R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 \Psi(x_1, x_2)}{\partial x_2^2}. $$

By substitution Eq.(4.3.15) and Eq.(4.3.20)-Eq.(4.3.21) into LHS of the Eq.(4.3.9) we obtain

$$ \begin{aligned} \int_{\Gamma} \hat{H} \Psi_{\delta}^{\#}(x_1, x_2) dx_1 dx_2 = & \\ \int_{\Gamma} dx_1 dx_2 R_{\delta}(x_1, \tilde{x}_1) R_{\delta}(x_2, \tilde{x}_2) \times & \\ \left[ \hbar^2 \frac{\partial^2 \Psi(x_1, x_2)}{\partial x_1^2} + \hbar^2 \frac{\partial^2 \Psi(x_1, x_2)}{\partial x_2^2} + V(x_1, x_2) \Psi(x_1, x_2) \right] + & \\ + \hbar^2 \int_{\Gamma} dx_1 dx_2 \times & \\ \left[ 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_1} R_{\delta}(x_2, \tilde{x}_2) \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} + \Psi(x_1, x_2) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} \right] + & \\ + \hbar^2 \int_{\Gamma} dx_1 dx_2 \times & \\ \left[ 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_2} R_{\delta}(x_1, \tilde{x}_1) \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} + \Psi(x_1, x_2) R_{\delta}(x_1, \tilde{x}_1) \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right] = & \\ = \Sigma_1(\hbar, \delta) + \Sigma_2(\hbar, \delta). & \end{aligned} \qquad (4.3.22) $$

Now we go to estimate the quantities

$$ \Sigma_1(\hbar, \delta) = \hbar^2 \int_{\Gamma} dx_1 dx_2 \times \\ \left[ 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_1} R_{\delta}(x_2, \tilde{x}_2) \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} + \Psi(x_1, x_2) R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} \right] \qquad (4.3.23) $$

and
$$ \Sigma_2(\hbar, \delta) = \hbar^2 \int_{\Gamma} dx_1 dx_2 \times \left[ 2 \frac{\partial \Psi(x_1, x_2)}{\partial x_2} R_{\delta}(x_1, \tilde{x}_1) \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} + \Psi(x_1, x_2) R_{\delta}(x_1, \tilde{x}_1) \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right] \quad (4.3.24) $$

From Eq.(4.3.23) using Eq.(4.3.14) we obtain

$$ \begin{aligned} |\Sigma_1(\hbar, \delta)| &\le \hbar^2 \int_{\Gamma} dx_1 dx_2 \times \\ & \left[ 2 \left| \frac{\partial \Psi(x_1, x_2)}{\partial x_1} \right| R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_1} \right| + \left| \Psi(x_1, x_2) \right| R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_1^2} \right| \right] \\ &\le 2O(\hbar^{1/2}) \int_{\Gamma} R_{\delta}(x_2, \tilde{x}_2) \left| \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} \right| dx_1 dx_2 + \\ &\quad + O(\hbar^{3/2}) \int_{\Gamma} R_{\delta}(x_2, \tilde{x}_2) \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} dx_1 dx_2 = \\ &\quad = 2O(\hbar^{1/2}) \int_{\mathbb{R}} R_{\delta}(x_2, \tilde{x}_2) dx_2 \int_{\mathbb{R}} \left| \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} \right| dx_1 + \\ &\quad O(\hbar^{3/2}) \int_{\mathbb{R}} R_{\delta}(x_2, \tilde{x}_2) dx_2 \int_{\mathbb{R}} \left| \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} \right| dx_1 = \\ &\quad = \int_{\mathbb{R}} R_{\delta}(x_2, \tilde{x}_2) dx_2 \left[ 2O(\hbar^{1/2}) \int_{\mathbb{R}} \left| \frac{\partial R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1} \right| dx_1 + O(\hbar^{3/2}) \int_{\mathbb{R}} \left| \frac{\partial^2 R_{\delta}(x_1, \tilde{x}_1)}{\partial x_1^2} \right| dx_1 \right]. \end{aligned} \quad (4.3.25) $$

From Eq.(4.3.24) using Eq.(4.3.14) we obtain

$$ \begin{aligned} |\Sigma_2(\hbar, \delta)| &\le \hbar^2 \int_{\Gamma} dx_1 dx_2 \times \\ & \left[ 2 \left| \frac{\partial \Psi(x_1, x_2)}{\partial x_2} \right| R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} \right| + \left| \Psi(x_1, x_2) \right| R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right| \right] \\ &\le 2O(\hbar^{1/2}) \int_{\Gamma} R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} \right| dx_1 dx_2 + \\ &\quad O(\hbar^{3/2}) \int_{\Gamma} R_{\delta}(x_1, \tilde{x}_1) \left| \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right| dx_1 dx_2 \\ &\quad 2O(\hbar^{1/2}) \int_{\mathbb{R}} R_{\delta}(x_1, \tilde{x}_1) dx_1 \int_{\mathbb{R}} \left| \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} \right| dx_2 + \\ &\quad O(\hbar^{3/2}) \int_{\mathbb{R}} R_{\delta}(x_1, \tilde{x}_1) dx_1 \int_{\mathbb{R}} \left| \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right| dx_2 = \\ &\quad \int_{\mathbb{R}} R_{\delta}(x_1, \tilde{x}_1) dx_1 \left[ 2O(\hbar^{1/2}) \int_{\mathbb{R}} \left| \frac{\partial R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2} \right| dx_2 + O(\hbar^{3/2}) \int_{\mathbb{R}} \left| \frac{\partial^2 R_{\delta}(x_2, \tilde{x}_2)}{\partial x_2^2} \right| dx_2 \right]. \end{aligned} \quad (4.3.26) $$
Having substituted Eq.(4.3.18) into Eq.(4.3.25) and Eq.(4.3.19) into Eq.(4.3.26) and having applied Lemma 4.3.1 we have finalized the proof of the Eq.(4.3.9).

We assume now that

$$N \int_{\mathbb{R}} \int_{\mathbb{R}} R_{\delta}^{2}(x_{1}, \tilde{x}_{1}) R_{\delta}^{2}(x_{2}, \tilde{x}_{2}) |\Psi(x_{1}, x_{2})|^{2} dx_{1} dx_{2} = 1 \quad (4.3.27)$$

From Eq.(4.3.27) and Eq.(4.3.17) by Lemma 4.3.1 we obtain

$$N \int_{\mathbb{R}} \int_{\mathbb{R}} R_{\delta}^{2}(x_{1}, \tilde{x}_{1}) R_{\delta}^{2}(x_{2}, \tilde{x}_{2}) |\Psi(x_{1}, x_{2})|^{2} dx_{1} dx_{2} = \quad (4.3.28)$$

From Eq.(4.3.16)-Eq.(4.3.17) we obtain

$$\langle x_A \rangle = \int \int x_1 \Psi_{\delta}^{\#}(x_1, x_2) dx_1 dx_2 = \int \int x_1 R_{\delta}^2(x_1, \tilde{x}_1) R_{\delta}^2(x_2, \tilde{x}_2) |\Psi(x_1, x_2)|^2 dx_1 dx_2, \quad (4.3)$$

## IV.4. Position-momentum entangled photon pairs in non-linear wave-guide

The physical system where we expect the entangled photon states to appear include: (A) a Kerr-type nonlinear single-mode wave-guide characterized by strong photon-photon coupling [56], [57], or (B) a chain of coupled non-linear resonators [58–61]. For two photons with momenta $\hbar k_1 = \hbar(k_0 - \delta k)$ and $\hbar k^2 = \hbar(k_0 + \delta k)$ and dispersion

$$\omega k_0 + \delta k \approx \omega k_0 + v \delta k + \beta \delta k^2 / 2, \quad (4.4.1)$$

where $v$ is the photon group velocity, the variation of the energy of a photon pair

$$\Delta^{(2)} \omega = \omega k_0 - \delta k + \omega k_0 + \delta k - 2 \omega k_0 \approx \beta \delta k^2. \quad (4.4.2)$$

As the photon-photon interaction conserves both energy and longitudinal momentum, the two-photon states propagating along the non-linear transmission line can be described by the Fock function

$$|\psi\rangle_{2k_0} = \int dk_1 dk_2 \delta(k_1 + k_2 - 2k_0) f(k_1 - k_2) |k_1, k_2\rangle \quad (4.4.3)$$
<figure><img src="image_35.png" /><figcaption>Fig. 4.4.1: Entangled two-photon states in non-linear wave guides.</figcaption></figure>

(a) Spectrum of a two-photon state, $\tilde{E} = (E - 2\omega k_0)|\beta|/\kappa^2$, with total momentum $2k^0$ in wave-guide with quadratic dispersion (4.3.1) for $\beta < 0, \kappa > 0$ (left) and $\beta > 0, \kappa < 0$ (right). Solid line corresponds to the continuous spectrum, while the single eigenvalue corresponding to the entangled state is shown by dashed line. (b) Wigner function of the two-photon entangled state. It takes negative values, which is a hallmark of non-Gaussian entangled states.

(A) To demonstrate the principle of position-momentum entanglement of photons in Kerr-nonlinear systems, we, first, consider the entangled photon pairs in non-linear optical wave-guides. Classically, Kerr nonlinearity in an isotopic medium manifests itself in the third-order polarisation $\mathbf{P}^{(3)(+)} = \chi^{(3)}[(\mathbf{E}^{(-)} \cdot \mathbf{E}^{(+)})\mathbf{E}^{(+)} + \alpha(\mathbf{E}^{(+)} \cdot \mathbf{E}^{(+)})\mathbf{E}^{(-)}]$, where "+" and "-" correspond to positive and negative frequency parts, $\mathbf{E}$ is electric field, $\chi^{(3)}$ is the susceptibility of the medium $\chi^{(3)} = \chi_{xyxy}^{(3)}, \alpha = \chi_{xxyy}^{(3)}/(2\chi^{(3)})$. Quantizing electromagnetic field, integrating over transverse degrees of freedom, and neglecting magneto-optical effects ($\alpha = 0$) leading to entanglement over polarization degrees of freedom, one obtains the following Hamiltonian ($\hbar = c = 1$):

$$ H = H_0 + H_{\text{int}}, H_0 = \sum_k k\omega k a_k^\dagger a_k, \\ H_{\text{int}} = \frac{k}{L} \sum_{k_1,k_2,k_3,k_4} \delta(k_1 + k_2, k_3 + k_4) a_{k_4}^\dagger a_{k_3}^\dagger a_{k_1} a_{k_2}, \quad (4.4.4) $$

where $a_k(a_k^\dagger)$ is the annihilation (creation) operator of a photon with longitudinal momentum $k$ and energy $\omega_k$, $L$ is the length of the system. The non-linear term $H_{\text{int}}$ in Eq. (4.3.4) describes photon-photon interaction with coupling $\kappa = \pi\omega^2\chi^{(3)}/2n_r^4 A\epsilon_0$, where $n_r$ is refraction index, $A$ is the area occupied by the wave-guide mode and $\epsilon_0$ is the vacuum permittivity. Hamiltonian (4.3.4) can be diagonalized exactly in the case of $\Delta^{(2)}\omega \propto \delta k^2$. We consider a sector of the Hilbert space, which consists of all the two-photon states with the total pair momentum $2k_0$ and assume the effective mass approximation for the wave-guide dispersion given by Eq. (4.3.1). In the coordinate domain, $a_x = 1/\sqrt{L} \sum_k a_k \exp[i(k-k_0)x]$, the Hamilton Eq. (4.3.4) takes the form
$$H = \int dx \left( \omega_{k_0} a_x^\dagger a_x - i v a_x^\dagger \partial_x a_x - \frac{1}{2} \beta a_x^\dagger \partial_x^2 a_x \right) + \frac{1}{2} \int dx_1 dx_2 a_{x_1}^\dagger a_{x_2}^\dagger U(x_1 - x_2) a_{x_1} a_{x_2}, \quad (4.4.5)$$

where $U(x_1 - x_2) = 2\kappa\delta(x_1 - x_2)$. For a two-photon state, described by the wave-function

$$|\psi\rangle = \int dx_1 dx_2 f(x_1, x_2) a_{x_1}^\dagger a_{x_2}^\dagger |0\rangle,$$

one obtains the following Schrödinger equation:

$$[2\omega_{k_0} - i\nu(\partial_{x_1} + \partial_{x_2}) - \frac{1}{2}\beta(\partial_{x_1}^2 + \partial_{x_2}^2) + 2\kappa\delta(x_1 - x_2)]f(x_1, x_2) = Ef(x_1, x_2), \quad (4.4.6)$$

where $E$ is the energy of a two-photon state. Equation (4.4.6) has scattering state solutions, which correspond to the continuous spectrum of non-interacting photons with energies given by Eq. (4.4.2) (See Fig.4.4.1(a)). When the curvature of the wave-guide dispersion $\beta$ and the photonphoton coupling constant $\kappa$ are of opposite signs, $\beta_\kappa < 0$, there exists a bound state solution with

$$f(x_1, x_2) = \sqrt{\frac{\xi}{2L}} \exp[-|x_1 - x_2|\xi], \quad \xi = |\kappa/\beta| \quad (4.4.7)$$

The energy of this state is split from the continuum of weakly correlated scattering states, as we show in Fig. 4.4.1(a), and it is given by

$$E_b = 2\omega_{k_0} - \kappa^2/\beta, \quad (4.4.8)$$

as expected from binding of a one-dimensional massive particle to an attractive $\delta$-functional potential well [30]. In the momentum domain, the two-photon bound state wave-function is given by Eq. (4.4.3) with

$$f(k_1 - k_2) = \frac{8\xi^{3/2}}{\sqrt{2L}[(k_1 - k_2)^2 + 4\xi^2]} \quad (4.4.9)$$

The state (4.4.9) can be characterised by the Wigner function defined as the expectation value

$$\mathbf{W}(x_1, k_1; x_2, k_2) = \pi^{-2} \langle \psi | \Pi(x_1, k_1) \otimes \Pi(x_2, k_2) | \psi \rangle$$

of the parity operator

$$\Pi(x, k) = \int d\xi e^{-2ix\xi} a_{k+\xi}^{\dagger} |0\rangle\langle 0| a_{k-\xi}.$$

After straightforward calculations, one obtains

$$\mathbf{W}(x_1, k_1; x_2, k_2) = \frac{\xi^2 e^{-2\xi|\delta x|}}{2\pi^2(\delta k^2 + \xi^2)} \cos(2\delta k|\delta x|) + \frac{\xi}{\delta k} \sin(2\delta k|\delta x|) \delta(k_1 + k_2; 2k_0), \quad (4.4.10)$$

where $\delta x = x_1 - x_2$. This function is negative for $\cos(2\delta k|\delta x|) + (\xi/\delta k)\sin(2\delta k|\delta x|) < 0$, as shown in Fig. 4.4.1(b), which implies that the state (4.4.9) is entangled in position-momentum degrees of freedom. Moreover, for $\xi \to \infty$, the two-photon wave-function approaches the ideal Einstein-Podolsky-Rosen state in which position and momenta are perfectly (anti-) correlated:

$$|\psi\rangle = \int d(\delta k) |k_0 + \delta k, k_0 - \delta k\rangle = \int dx e^{2ik_0 x} |x, x\rangle.$$

Alternatively, to demonstrate that the state (4.4.9) is entangled in
position-momentum degrees of freedom, one can find the uncertainties $\Delta(x_1 - x_2)$ and $\Delta(k_1 + k_2)$ calculated over the joint probability distributions $P(x_1, x_2)$ and $P(k_1, k_2)$ respectively, for which, the separability criterion:

$$[\Delta(x_2 - x_1)]^2[\Delta(k_2 + k)]^2 \geq 1, \qquad (4.4.11)$$

can be applied. Although, the states for which the inequality (4.4.11) is violated are inseparable, they do not necessarily lead to EPR paradox. In order for an EPR "paradox" to arise, correlations must violate a more strict inequality []:

$$[\Delta(x_2 - x_1)]^2[\Delta(k_2 + k)]^2 \geq 1/4, \qquad (4.4.12)$$

which can be accessible experimentally [].

## IV.5. Position-momentum entangled photon pairs and the experimental verification of the postulate of nonlocality.

In paper [23] is reported on a demonstration of the EPR paradox using position- and momentum-entangled photon pairs produced by spontaneous parametric down conversion. Transverse correlations from parametric down conversion have been studied both theoretically and experimentally. It was found experimentally that the position and momentum correlations are strong enough to allow the position or momentum of a photon to be inferred from that of its partner with a product of variances $\leq 0.01\hbar^2$, which violates the separability bound by 2 orders of magnitude. In the idealized entangled state proposed by EPR, the positions and momenta of the two particles are perfectly correlated. However such idealized entangled state is non-normalizable and cannot be realized in the laboratory. However, the state of the light produced in parametric down conversion can be made to approximate the EPR state under suitable conditions. In parametric down conversion, a pump photon is absorbed by a nonlinear medium and reemitted as two photons (conventionally called signal and idler photons), each with approximately half the energy of the pump photon. Considering only the transverse components, the momentum conservation of the down conversion process requires $p_1 + p_2 = p_p$, where 1, 2, and p refer to the signal, idler, and pump photons, respectively. Provided the uncertainty in the pump transverse momentum is small, the transverse momenta of the signal and idler photons are highly anticorrelated. The exact degree of correlation depends on the structure of the signal idler state. In the regime of weak generation, this state has the form

$$|\psi\rangle_{1,2} = |\text{vac}\rangle + \int d\mathbf{p}_1 d\mathbf{p}_2 A(\mathbf{p}_1, \mathbf{p}_2)|\mathbf{p}_1, \mathbf{p}_2\rangle, \qquad (4.5.1)$$

where $|\text{vac}\rangle$ denotes the vacuum state and the two-photon amplitude $A(\mathbf{p}_1, \mathbf{p}_2)$ is

$$A(\mathbf{p}_1, \mathbf{p}_2) = \chi E_p(\mathbf{p}_1, \mathbf{p}_2) \frac{\exp(i\Delta\mathbf{k}_z L) - 1}{i\Delta\mathbf{k}_z}. \qquad (4.5.2)$$

Here is the coefficient of the nonlinear interaction, $E_p$ is the amplitude of the
plane-wave component of the pump with transverse momentum $p_1p_2, L$ is the length of the nonlinear medium, and $\Delta\mathbf{k}_z = \mathbf{k}_{p,z} - \mathbf{k}_{1,z} - \mathbf{k}_{2,z}$ (where $\mathbf{k} = \mathbf{p}/\hbar$) is the longitudinal wave vector mismatch, which generally increases with transverse momentum and limits the angular spread of signal and idler photons. The vacuum component of the state makes no contribution to photon counting measurements and may be ignored. Also, there is no inherent difference between different transverse components; so without loss of generality, we consider scalar position and momentum. The narrower the angular spectrum of the pump field and the wider the angular spectrum of the generated light, the more closely the integral (4.4.1) approximates $\int dp_1 dp_2 \delta(p_1 + p_2) |p_1, p_2\rangle = |EPR\rangle$ and the stronger the correlations in position and momentum become. The experimental setup used to determine position and momentum correlations is portrayed in Fig. 4.5.1(a)-(b). The idea is to measure the positions and momenta by measuring the down converted photons in the near and far fields, respectively [34]. The source of entangled photons is spontaneous parametric down conversion generated by pumping a 2 mm thick type-II -barium-borate (BBO) crystal with a 30 mW, cw, 390 nm laser beam. A prism separates the pump light from the down converted light. The signal and idler photons have orthogonal polarizations and are separated by a polarizing beam splitter. In each arm, the light passes through a narrow 40 m vertical slit, a 10 nm spectral filter, and a microscope objective. The objective focuses the transmitted light onto a multimode fiber which is coupled to an avalanche photodiode single-photon counting module. The spectral filter ensures that only

<figure><img src="image_36.png" /><figcaption>Fig.4.4.1(a) [23]. Experimental setup for measuring position photon correlations. Position correlations are obtained by imaging the birth place of each photon of a pair onto a separate detector.</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>Fig.4.4.1(b) [23].Experimental setup for measuring correlations in transverse momentum. Correlations in transverse momentum are obtained by imaging the propagation direction of each photon of a pair onto a separate detector.</figcaption></figure>

photons with nearly equal energies are detected. To measure correlations in the positions of the photons, a lens of focal length 100 mm (placed prior to the beam splitter) is used to image the exit face of the crystal onto the planes of the two slits [Fig. 4.4.1(a)]. One slit is fixed at the location of peak signal intensity. The other slit
is mounted on a translation stage. The photon coincidence rate is then recorded as a function of the displacement of the second slit. To measure correlations in the transverse momenta of the photons, the imaging lens is replaced by two lenses of focal length 100 mm, one in each arm, a distance $f$ from the planes of the two slits [Fig.4.5.1(b)]. These lenses map transverse momenta to transverse positions, such that a photon with transverse momentum $\hbar k_{\perp}$ comes to a focus at the point $x = f k_{\perp}/k$ in the plane of the slit. Again, one slit is fixed at the location of the peak count rate while the other is translated to obtain the coincidence distribution. By normalizing the coincidence distributions, the conditional probability density functions $p(x_2|x_1)$ and $p(p_2|p_1)$ was obtained (see Fig. 4.5.2-Fig. 4.5.3). These probability densities are then used to calculate the uncertainty in the inferred position or momentum of photon 2 given the position or momentum of photon 1:

$$ \begin{aligned} \Delta x_2^2(x_1) &= \int x_2^2 p(x_2|x_1) dx_2 - \left( \int x_2 p(x_2|x_1) dx_2 \right)^2, \\ \Delta p_2^2(p_1) &= \int p_2^2 p(p_2|p_1) dx_2 - \left( \int p_2 p(p_2|p_1) dx_2 \right)^2. \end{aligned} \quad (4.5.3) $$

Because of the finite width of the slits, the raw data in Fig. 4.5.2-Fig.4.5.3 describe a slightly broader distribution than is associated with the down conversion process itself. By adjusting the computed values of $\Delta x_2$ and $\Delta p_2$ to account for this broadening (an adjustment smaller than 10%), we obtain the correlation uncertainties $\Delta x_2 = 0.027$ mm and $\Delta p_2 = 3.7\hbar mm^{-1}$.

<figure><img src="image_38.png" /><figcaption>Fig.4.5.2.[23]. The conditional probability distribution of the relative birthplace of the entangled photons. The solid line are the theoretical prediction and the dot are the experimental data.</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>Fig.4.5.3.[23]. The conditional probability distribution of the relative transverse momentum of the entangled photons. The solid line are the theoretical prediction and the dot are the experimental data.</figcaption></figure>

The widths of the distributions determine the uncertainties in inferring the position or the momentum of one photon from that of the other. The experimentally measured variance product is then [23]
$$[\Delta^{\text{exp}}x_2^2(x_1)][\Delta^{\text{exp}}p_2^2(p_1)] = 0.01\hbar^2. \quad (4.5.4)$$

Also shown in Fig.4.5.2-Fig.4.5.3 are the predicted probability densities. These curves contain no free parameters and are obtained directly from the two-photon amplitude $A(p_1, p_2)$ [23], which is determined by the optical properties of BBO and the measured profile of the pump beam. Figure 4.5.2 indicates that the correlation widths was obtained are intrinsic to the down conversion process and are limited only by the degree to which it deviates from the idealized EPR state (4.5.1). The value of $\Delta(p_2 + p_1)$ is limited by the finite width of the pump beam. The pump photons in a Gaussian beam of width $w$ have an uncertainty $\hbar/2w$ in transverse momentum which, due to conservation of momentum, is imparted to the total momentum $p_1 + p_2$ of the signal and idler photons. The value of $\Delta(x_2 - x_1)$ is limited by the range of angles over which the crystal generates signal and idler photons. If the angular width of emission is $\Delta\phi$, then the principle of diffraction indicates that the photons cannot have a smaller transverse dimension than $\sim(k_{s,i}\Delta\phi)^{-1}$. Careful analysis based on the angular distribution of emission yields

$\Delta(x_2 - x_1) = 1.88(k_{s,i}\Delta\phi)^{-1}$ [23]. With the measured beam width of $w = 0.17$mm and predicted angular width $0.012$ rad, the theory predicts [23]:

$$[\Delta^{\text{th}}x_2^2(x_1)][\Delta^{\text{th}}p_2^2(p_1)] = 0.0036\hbar^2. \quad (4.5.5)$$

**Remark 4.5.1.** This is somewhat smaller than the experimentally calculated value of $0.01\hbar^2$,

even though the data appear to closely match the theoretical curves.

$$\Delta^{\text{exp}}x_2^2(x_1)\Delta^{\text{exp}}p_2^2(p_1) - \Delta^{\text{th}}x_2^2(x_1)\Delta^{\text{th}}p_2^2(p_1) = 0.01\hbar^2 - 0.0036\hbar^2 = 0.0064\hbar^2. \quad (4.5.6)$$

**Remark 4.5.2.** The reason for this discrepancy is that the experimental distributions have

small (1% of the peak) but very broad wings.

**Remark 4.5.3.** The origin of this uncoincidence counts is unknown [23].

**Remark 4.5.4.** In paper [23] it was assumed that this counts are perhaps due to scattering

from optical components. If these counts are treated as a noise background and

subtracted, the experimentally obtained uncertainties come into somewhat better

agreement with the theoretically predicted values, yielding an uncertainty product of

$0.004\hbar^2$:

$$\delta_{\text{EPR}}^{\text{nonloc}}(x_2 - x_1, p_2 + p_1) = \Delta^{\text{exp}}x_2^2(x_1)\Delta^{\text{exp}}p_2^2(p_1) - \Delta^{\text{th}}x_2^2(x_1)\Delta^{\text{th}}p_2^2(p_1) = 0.006\hbar^2. \quad (4.5.7)$$

Thus final value of uncoincidence counts is

$$\delta_{\text{EPR}}^{\text{nonloc}}(x_2 - x_1, p_2 + p_1) = 0.006\hbar^2. \quad (4.5.8)$$

**Remark 4.5.5.** Note that the separability criterion derived by Mancini et al. [35] is more

useful here. We remind that it states that separable systems satisfy the joint uncertainty product
$$ \Delta(x_2 - x_1)\Delta(p_2 + p_1) \ge \hbar^2, \qquad (4.5.9) $$

where the uncertainties are calculated over the joint probability distributions $P(x_1, x_2)$

and $P(p_1, p_2)$, respectively.

In this experiments the widths of the conditional probability distributions P

Therefore our results constitute a 2-order-of-magnitude violation of Mancini's separability

criterion as well as a strong violation of EPR's criterion.

## IV.6. EPR-B experiment

The EPR-B, the spin version of the Einstein-Podolsky-Rosen experiment proposed by Bohm, see [63],[64]

[64]: "We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore

$$ \psi = \frac{1}{\sqrt{2}} [\psi_{+}(1)\psi_{-}(2) - \psi_{-}(1)\psi_{+}(2)] \qquad (1) $$

where $\psi_{+}(1)$ refers to the wave function of the atomic state in which one particle (A) has spin $+h/2$, etc. The two atoms are then separated by a method that does not influence the total spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle (A) is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A.

If this were a classical system, there would be no difficulty in interpreting the above results, because all components of the spin of each particle are well defined at each instant of time. Thus, in the molecule, each component of the spin of particle A has, from the very beginning, a value opposite to that of the same component of B; and this relationship does not change when the atom disintegrates. In other words, the two spin vectors are correlated. Hence, the measurement of any component of the spin of A permits us to conclude also that the same component of B is opposite in value. **The possibility of obtaining knowledge of the spin of particle B in this way evidently does not imply any interaction of the apparatus with particle B or any interaction between A and B.**

In quantum theory, a difficulty arises, in the interpretation of the above experiment, because only one component of the spin of each particle can have a definite value at a given time. Thus, if the $x$ component is definite, then the $y$ and $z$ components are indeterminate and we may regard them more or less as in a kind of random fluctuation.
In spite of the effective fluctuation described above, however, the quantum
theory still implies that no matter which component of the spin of A may be
measured the same component of the spin of B will have a definite and opposite
value when the measurement is over. Of course, the wave function then reduces to
$\psi_+(1)\psi_-(2)$ or $\psi_-(1)\psi_+(2)$, in accordance with the result of the measurement.
Hence, there will then be no correlations between the remaining components of the
spins of the two atoms. Nevertheless, before the measurement has taken place
(even while the atoms are still in flight) we are free to choose any direction as the
one in which the spin of particle A (and therefore of particle B) will become definite.

In order to bring out the difficulty of interpreting the result, let us recall that
originally, the indeterminacy principle was regarded as representing the effects of
the disturbance of the observed system by the indivisible quanta connecting it with
the measuring apparatus. This interpretation leads to no serious difhculties for the
case of a single particle. For example, we could say that on measuring the z
component of the spin of particle A, we disturb the x and y components and make
them fluctuate. This point of view more generally implies that the definiteness of
any desired component of the spin is (along with the indefiniteness of the other two
components) a potentiality' which can be realized with the aid of a suitably oriented
spinmeasuring apparatus.

In the case of complementary pairs of continuous variables, such as position
and momentum, one obtains from this point of view the well known wave-particle
duality. In other words, the electron, for example, has potentialities for mutually
incompatible wave-like and particle-like behavior, which are realized under suitable
external conditions. In the laboratory those conditions are generally determined by
the measuring apparatus although, more generally, they may be determined by any
arrangement of matter with which the electron interacts. But in any case, it is
essential that there must be an external interaction, which disturbs the observed
system in such a way as to bring about the realization of one of its various mutually
incompatible potentialities. As a result of this disturbance, when any one variable is
made definite, other (noncommuting) variables must necessarily become indefinite
and undergo fluctuation".

Evidently, the foregoing interpretation is not satisfactory when applied to the
experiment of ERP. It is of course acceptable for particle A alone (the particle
whose spin is measured directly). But it does not explain why particle B (which
does not interact with A or with the measuring apparatus) realizes its potentiality for
a definite spin in precisely the same direction as that of A. Moreover, it cannot
explain the fluctuations of the other two components of the spin of particle B as the
result of disturbances due to the measuring apparatus.

In this subsection we explain EPR-B experiment using reduction to an sort of
generic EPR correlations for two particles A and B with maximally correlated
position $z_A$ and $z_B$. This explanation avoid the EPR-Bohm paradox.
<figure><img src="image_40.png" /><figcaption>Fig.4.5.1. Einstein-Podolsky-Rosen-Bohm experiment.</figcaption></figure>

Figure 4.5.1 presents the Einstein-Podolsky-Rosen-Bohm experiment. A source $S$ created in O pairs of identical atoms A and B, but with opposite spins. The atoms A and B split following the y-axis in opposite directions, and head towards two identical Stern-Gerlach apparatus $E_A$ and $E_B$. The electromagnet $E_A$ "measures" the spin of A along the z-axis and the electromagnet $E_B$ "measures" the spin of B along the z'-axis, which is obtained after a rotation of an angle $\delta$ around the y-axis.

**Remark 4.5.1.** So far we have consistently made use of the idea that if we know something definite about the state of a such physical system, say that we know the z component of the spin of a spin half particle is $S_z = \pm\frac{1}{2}\hbar$, then we assign to the system the state $|S_z\rangle = \left|\pm\frac{1}{2}\hbar\right\rangle$, or, more simply, $|\pm\rangle$.

**Remark 4.5.2.** We can also note that these two states $|+\rangle$ and $|-\rangle$ are mutually exclusive, i.e. if atom in the state $|+\rangle$, then the result $S_z = -\frac{1}{2}\hbar$ is never observed, and furthermore, we note that the two states $|+\rangle$ and $|-\rangle$ cover all possible values for $S_z$.

**Remark 4.5.3.** When we say that we 'know' the value of some physical observable of a quantum system, we are presumably implying that some kind of measurement has been made that provided us with this knowledge. It is furthermore assumed that in the process of acquiring this knowledge, the system, after the measurement has been performed, survives the measurement, and moreover if we were to immediately remeasure the same quantity, we would get the same result. This is certainly the situation with the measurement of spin in a Stern-Gerlach experiment. If an atom emerges from one such set of apparatus in a beam that indicates that $S_z = \frac{1}{2}\hbar$ for that atom, and we were to pass the atom through a second apparatus, also with its magnetic field oriented in the z direction, we would find the atom emerging in the $S_z = \frac{1}{2}\hbar$ beam once again. Under such circumstances, we would be justified in saying that the *atom has been prepared in the state* $|S_z = \frac{1}{2}\hbar\rangle$, etc.

**Definition 4.5.1.** Assume that *atom A has been prepared in the state* $|S_z = \frac{1}{2}\hbar\rangle$,

$|S_z = -\frac{1}{2}\hbar\rangle$, etc. Then we will say that these events $|S_z = \frac{1}{2}\hbar\rangle$, $|S_z = -\frac{1}{2}\hbar\rangle$, etc. occurs. We will be denoted these events by symbols $|S_z = \frac{1}{2}\hbar\rangle^A$, $|S_z = -\frac{1}{2}\hbar\rangle^A$, etc.,
or $|\frac{1}{2}\hbar\rangle^A, |-\frac{1}{2}\hbar\rangle^A$, etc.

**Definition 4.5.2.** Assume that we know exactly that *atom A* in the state $|\frac{1}{2}\hbar\rangle, |-\frac{1}{2}\hbar\rangle$, etc.

Then we will say that these events $|\frac{1}{2}\hbar\rangle, |-\frac{1}{2}\hbar\rangle$, etc. occurs and we will be denoted these events again by symbols $|\frac{1}{2}\hbar\rangle^A, |-\frac{1}{2}\hbar\rangle^A$, etc.

**Definition 4.5.3.** Assume that these events $|\frac{1}{2}\hbar\rangle^A, |-\frac{1}{2}\hbar\rangle^A$, etc. occurs in point $\mathbf{x} = (t, x_1, x_2, x_3) = (t, \mathbf{r}) \in M_4$ of Minkowski spacetime $M_4$. Then we will be denoted these

events by symbols $|\frac{1}{2}\hbar\rangle_x^A, |-\frac{1}{2}\hbar\rangle_x^A$, etc. or $|\frac{1}{2}\hbar\rangle_{(t_A,z_A)}^A, |-\frac{1}{2}\hbar\rangle_{(t_A,z_A)}^A$, etc.

**Assumption 4.5.1.** We claim for any $\mathbf{x} \in M_4$ that:

$$|\frac{1}{2}\hbar\rangle_x^A \in \mathcal{F}_{M_4}, |-\frac{1}{2}\hbar\rangle_x^A \in \mathcal{F}_{M_4}, \text{ etc.} \qquad (4.5.1)$$

Here $\mathcal{F}_{M_4}$ is a measure algebra $\mathcal{F}_{M_4} = (\mathbf{B}_{M_4}, \mathbf{P})$ with a probability measure $\mathbf{P}$, see

subsection II.2, Definition 2.2.2.

**Remark 4.5.4.** Note that for any $\mathbf{x} \in M_4$ and for any atom **A** these events $|\frac{1}{2}\hbar\rangle_x^A, |-\frac{1}{2}\hbar\rangle_x^A$

are mutually exclusive, see Remark 4.5.2, and therefore for any $\mathbf{x} \in M_4$

$$\mathbf{P}\left(|\frac{1}{2}\hbar\rangle_x^A \wedge |-\frac{1}{2}\hbar\rangle_x^A\right) = 0. \qquad (4.5.2)$$

**Remark 4.5.5.** We remind that if an atom is prepared in an arbitrary initial state $|S\rangle$, then

the probability amplitude of finding it in some other state $|S'\rangle$ is given by

$$\langle S'|S\rangle = \langle S'|+\rangle\langle +|S\rangle + \langle S'|-\rangle\langle -|S\rangle \qquad (4.5.3)$$

which leads, by the cancellation trick to

$$|S\rangle = |+\rangle\langle +|S\rangle + |-\rangle\langle -|S\rangle \qquad (4.5.4)$$

and therefore the states $|\pm\rangle$ form a complete set of orthonormal basis states for the state space of the system.

Suppose we have an $n$-dimensional quantum system which contains only a quantum observable with discrete values such as $S_z$, etc.

## II. Then we claim the following:

**Q<sub>d</sub>.I.1.** Any given $n$-dimensional quantum system which contains only a quantum observable with discrete values such that mentioned above is identified by a set $Q_d$:

$$Q_d \triangleq \langle \mathbf{H}_d, \mathfrak{I}_d, \mathfrak{R}_d, \mathcal{L}_{2,1}^d, \mathbf{G}_d, |\psi_t\rangle \qquad (4.5.5)$$

where:

(i) $\mathbf{H}_d$ that is some finite-dimensional complex Hilbert space,

(ii) $\mathfrak{I}_d = (\Omega_d, \mathcal{F}_d, \mathbf{P}_d)$ that is complete probability space,

(iii) $\mathfrak{R}_d = (\mathbb{R}^n, \Sigma_d)$ that is measurable space,

(iv) $\mathcal{L}_{2,1}^d(\Omega_d)$ that is complete space of discrete random variables $X_d: \Omega_d \to \mathbb{R}^n$
such that

$$ \int_{\Omega_d} \|X_d(\omega)\| d\mathbf{P}_d < \infty, \int_{\Omega_d} \|X_d(\omega)\|^2 d\mathbf{P}_d < \infty \quad (4.5.6) $$

(v) $\mathbf{G}_d : \mathbf{H}_d \to \mathcal{L}_{2,1}(\Omega_d)$ that is one to one correspondence such that

$$ \langle \psi | \hat{Q}_d | \psi \rangle = \int_{\Omega_d} (\mathbf{G}_d[\hat{Q}_d|\psi\rangle](\omega)) d\mathbf{P}_d = \mathbf{E}_{\Omega_d}(\mathbf{G}_d[\hat{Q}_d|\psi\rangle](\omega)) \quad (4.5.7) $$

for any $|\psi\rangle \in \mathbf{H}_d$ and for any Hermitian operator with discrete spectrum

$\hat{Q}_d : \mathbf{H}_d \to \mathbf{H}_d,$

(vi) $|\psi_t\rangle$ is a continuous vector function $|\psi_t\rangle : \mathbb{R}_+ \to \mathbf{H}_d$ which represented the evolution of the quantum system $\mathbf{Q}_d$.

**Q_d.I.2.** For any $|\psi_1\rangle, |\psi_2\rangle \in \mathbf{H}_d$ and for any Hermitian operator $\hat{Q}_d : \mathbf{H}_d \to \mathbf{H}_d$

such that

$$ \langle \psi_1 | \hat{Q}_d | \psi_2 \rangle = \langle \psi_2 | \hat{Q}_d | \psi_1 \rangle = 0 \quad (4.5.8) $$

valid the equality

$$ \mathbf{G}_d[\hat{Q}_d(|\psi_1\rangle + |\psi_2\rangle)](\omega) = \mathbf{G}_d[\hat{Q}_d|\psi_1\rangle](\omega) + \mathbf{G}_d[\hat{Q}_d|\psi_2\rangle](\omega). \quad (4.5.9) $$

**Remark 4.5.6.** Let $S_z^+(\omega)$ and $S_z^-(\omega)$ be discrete random variables

$S_z^+ : \Omega_d \to \{1,-1\},$

$S_z^- : \Omega_d \to \{-1,1\}$ correspondingly such that:

(i) $S_z^+(\omega) = \mathbf{G}[|+\rangle],$ (ii) $\mathbf{P}_d(\Delta_+^{=1}) = 1$, where $\Delta_+^{=1} \triangleq \{\omega | S_z^+(\omega) = 1\}$,

$$ \text{(iii) } \mathbf{P}_d(\Delta_+^{-1}) = 0, \text{ where } \Delta_+^{-1} \triangleq \{\omega | S_z^+(\omega) = -1\} \qquad \text{and} \quad (4.5.10) $$

(i) $S_z^-(\omega) = \mathbf{G}[|-\rangle],$ (ii) $\mathbf{P}_d(\Delta_-^{-1}) = 1$, where $\Delta_-^{-1} \triangleq \{\omega | S_z^-(\omega) = -1\}$,

$$ \text{(iii) } \mathbf{P}_d(\Delta_-^{=1}) = 0, \text{ where } \Delta_-^{=1} \triangleq \{\omega | S_z^-(\omega) = 1\}. $$

Let $\mathbf{Q}_c$ be any $n$-dimensional quantum system which contains only a quantum observable with continuous values. We remind that such quantum system is identified by a set $\mathbf{Q}$

$$ \mathbf{Q} \triangleq \langle \mathbf{H}, \mathfrak{T}, \mathfrak{R}, \mathcal{L}_{2,1}, \mathbf{G}, |\psi_t\rangle \rangle, \quad (4.5.11) $$

see subsection I.7.1.

**Definition 4.5.4.** We define now a composite quantum system $\mathbf{Q}_{c,d}$ which contains both sort of quantum observables by a set $\mathbf{Q}_{c,d}$

$$ \mathbf{Q}_{c,d} \triangleq \langle \mathbf{H}_{c,d}, \mathfrak{T}_{c,d}, \mathfrak{R}_{c,d}, \mathcal{L}_{2,1}^{c,d}, \mathbf{G}_{c,d}, |\psi_t\rangle \rangle \quad (4.5.12) $$

where:

(i) $\mathbf{H}_{c,d} = \mathbf{H}_c \times \mathbf{H}_d$ that is composite complex Hilbert space,

(ii) $\mathfrak{T}_{c,d} = (\Omega_{c,d}, \mathcal{F}_{c,d}, \mathbf{P}_d)$ that is complete probability space,
with

$$ \Omega_{c,d} = \Omega_c \times \Omega_d, \mathcal{F}_{c,d} = \mathcal{F}_c \times \mathcal{F}_d, \Re_{c,d} = \Re_c \times \Re_d, \mathcal{L}_{2,1}^{c,d} = \mathcal{L}_{2,1}^c \times \mathcal{L}_{2,1}^d, \mathbf{G}_{c,d} = \mathbf{G}_c \times \mathbf{G}_d, $$

(iii) $\mathfrak{R}_{c,d} = (\mathbb{R}^n, \Sigma_{c,d})$ that is measurable space with $\Sigma_{c,d} = \Sigma_c \times \Sigma_d,$

(iv) $\mathcal{L}_{2,1}^{c,d}(\Omega_d)$ that is complete space of random variables $X_{c,d} : \Omega_{c,d} \to \mathbb{R}^n$ such that

$$ \int_{\Omega_{c,d}} \|X_{c,d}(\omega)\| d\mathbf{P}_c \times d\mathbf{P}_d < \infty, \quad \int_{\Omega_{c,d}} \|X_{c,d}(\omega)\|^2 d\mathbf{P}_c \times d\mathbf{P}_d < \infty, \omega \in \Omega_{c,d} \quad (4.5.13) $$

(v) $\mathbf{G}_{c,d} : \mathbf{H}_{c,d} \to \mathcal{L}_{2,1}^{c,d}(\Omega_d)$ that is one to one correspondence such that

$$ \langle \psi | \hat{Q}_{c,d} | \psi \rangle = \int_{\Omega_{c,d}} \left( \mathbf{G}_{c,d} \left[ \hat{Q}_{c,d} | \psi \rangle \right] (\omega) \right) d\mathbf{P}_c \times d\mathbf{P}_d = \mathbf{E}_{\Omega_{c,d}} \left( \mathbf{G}_{c,d} \left[ \hat{Q}_{c,d} | \psi \rangle \right] (\omega) \right) \quad (4.5.14) $$

## IV.6. EPR-B paradox resolution

The usual conclusion of EPR-B experiment is to reject the non-local realism for two reasons: the impossibility of decomposing a pair of entangled atoms into two states, one for each atom, and the impossibility of interaction faster than the speed of light.

**Remark.4.6.1.** We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle. That is the "relaxed locality principle" introduced in subsection IV.1.

**Remark 4.6.2.** The solution to the entangled state is obtained by resolving the Pauli equation from an initial singlet wave function with a spatial extension as [55]:

$$ \Psi_0(\mathbf{r}_A, \mathbf{r}_B) = \frac{1}{\sqrt{2}} f(\mathbf{r}_A) f(\mathbf{r}_B) (|+_A\rangle \otimes |-_B\rangle - |-_A\rangle \otimes |+_B\rangle), \quad (4.6.1) $$

**Remark 4.6.2.** The initial wave function of the entangled state is the singlet state (4.6.1)

with

$$ \begin{array}{l} f(\mathbf{r}) \approx \begin{cases} \frac{x^2 + y^2 + z^2}{4\sigma_0^2} & \text{iff } \|\mathbf{r}\| \le \epsilon, \\ (2\pi\sigma_0^2)^{-\frac{1}{2}} e^{-\frac{x^2+y^2+z^2}{4\sigma_0^2}} & \text{0 iff } \|\mathbf{r}\| > \epsilon \end{cases} \\ \mathbf{r} = (x,y,z), \sigma_0 \ll 1, \epsilon \ll 1 \end{array} \quad (4.6.2) $$

and where $|\pm_A\rangle$ and $|\pm_B\rangle$ are the eigenvectors of the operators $\sigma_{z_A}$ and $\sigma_{z_B}$:

$$ \sigma_{z_A}|\pm_A\rangle = \pm|\pm_A\rangle, \sigma_{z_B}|\pm_B\rangle = \pm|\pm_B\rangle. \quad (4.6.3) $$

**Remark 4.6.3.** We treat the dependence with y strictly quasiclassically as in subsection III.4, i.e. with speed $-\mathbf{v}_{y\pm}^A(\mathbf{v}_0, \theta_0^A)$ for A and $\mathbf{v}_{y\pm}^B(\mathbf{v}_0, \theta_0^B)$ for B such that