# State-Dependent Implication and   Equivalence in Quantum Logic

Authors: Fedor Herbut
Journal: Advances in Mathematical Physics
 (2012)

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## Abstract

Ideal occurrence of an event (projector)E leads to the known change of a state (density operator) ρ into EρE/[tr(Eρ)] (the Lüders state). It is shown that two events E and F give the same Lüders  state if and only if the equivalence relation  Eρ=Fρ is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector Q- of the Lüders state can be evaluated as  Q-=E-(E⋀Q0), where ⋀ denotes the greatest lower bound, and Q0 is the null projector of  ρ.  State-dependent implication ≤ρ extends absolute implication (which, in turn, determines the entire structure of quantum logic). Q- and ≤ρ are investigated in a closely related way to mutual benefit. Inherent in the preorder ≤ρ is the state-dependent equivalence ~ρ, defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the  elements,  has itself a partially ordered structure, and so has each class. In a  complete Boolean subalgebra, both structures are complete lattices. Physical meanings are  discussed.


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## Body

## 1. Introduction

The basic object of this study is the concept of a  quantum-mechanical state (density operator)ρ and its change when an event (projector) E with positive probability tr(ρE)>0 occurs (the result 1 is obtained) in  ideal measurement. The terms “state” and “density operator” as well as “event,” and “projector” will be used interchangeably.As it is well known, ideal  measurement is the simplest special case of  measurement of the first kind (synonyms: predictive, repeatable, and nondemolition  measurement). It causes a change of state according to the Lüders selective (or definite-result) formula(1.1)ρ→ρ-≡EρE[tr(ρE)].
See [1–3]. We call ρ- the  Lüders state, and E a  state-determining projector (it is all meant with respect to the initially given state ρ).Usually one treats the special case of a pure-stateρ=|ψ〉〈ψ|. Then, as easily seen, (1.1) takes the simple form
(1.2)|ψ〉→E|ψ〉(〈ψ|E|ψ〉)1/2,
which is sometimes called the von Neumann-Lüders projection (it is actually a normalized projection). Von Neumann treated the even more special case when the event is elementary (an atom) E=|ϕ〉〈ϕ| [4]. The change of state then is
(1.3)|ψ〉→|ϕ〉.The Lüders state was postulated by Lüders. It was derived by several authors including the present one [5, 6] (and the first derivation was repeated in different context in [7]; see also references in these articles).For a different approach, see references [8–10]. In Khrennikovʼs terminology, one deals with the postulates of Lüders and von Neumann (and he carefully examines their effect on some foundational issues).As it was mentioned, in my view, ideal  measurement is the simplest kind of  measurement, and of which kind a  measurement is depends on the interaction of object and measuring instrument (an elaboration of this will be presented in a followup). Both Lüders and von Neumann treat ideal  measurement, but the former allows degenerate eigenvalues of the measured observable, whereas the latter is confined to the special case of complete observables, that is, to ones with all eigenvalues nondegenerate.Incidentally, I have shown in my mentioned article [6] (see Section  3 there) that the Lüders change of state can be obtained by measuring a suitable complete observable as its refinement (in the way of von Neumann), but the choice of the latter depends on the state in which the  measurement is performed. Khrennikov has rediscovered essentially the same result independently in [9].Change of state (1.1) is made use of in the concept of  state-dependent (SD) implication E≤ρF, where E and F are events, and ρ is a given state [11–13]. Absolute (or state independent) implication E≤F⇔EF=E in quantum logic 𝒫(ℋ) (the complete lattice of all projectors in the Hilbert space ℋ of the given  quantum system) is such that if an event E occurs in a state ρ, that is, if tr(ρE)=1, then also the implied event F occurs; and this is so for every state.SD implication≤ρ, which is a generalization if the absolute implication (cf. Corollary 2.3 below), is, on the other hand, based on the following reasoning. If an event E occurs in an ideal way in a given state ρ and, after the occurrence, another event Fipso facto also occurs, then one says that E implies state-dependently F and one writes E≤ρF.When one takes into account that ideal occurrence gives rise to (1.1), one obtains the equivalence relation
(1.4)E≤ρF⟺tr[(EρE[tr(ρE)])F]=1.
Evidently, the SD implication relation (1.4) is restricted to nonzero-probability state-dependently implying events E.SD implication is defined on a given  Boolean subalgebraℬ of 𝒫(ℋ) (to achieve transitivity) [11]. SD implication on ℬ is a preorder [11, 14]. As every preorder, being reflexive and transitive but not antisymmetric, SD implication defines an equivalence relation on ℬ,  SD equivalence   ~ρ, as follows:
(1.5)E~ρF⟺E≤ρF,F≤ρE.
SD equivalence breaks up ℬ into equivalence classes, the elements of the  SD quotient set ℬ/~ρ. By this all zero-probability projectors {E:tr(ρE)=0} are taken to form one class. (This will be seen to be correct, cf. Remark 2.2.) The SD  equivalence class in ℬ to which a given projector E belongs will be denoted by [E]ℬ.The SD implication concept is further elaborated in the next section. But, unlike in [11, 12], this time the powerful structure of Boolean algebras is not made use of. Only poset (partially ordered set) theory is utilized, and poset structure is introduced in the SD quotient set ℬ/~ρ.


## 2. SD Implication Further Elaborated

Simple algebraic relations characterizing SD implication and SD  equivalence are now derived.Proposition 2.1.
Iftr(Eρ)>0, then E≤ρF  if and only if Eρ=EFρ is valid.Proof.
On account of commutation under the trace, idempotency of projectors, and commutation inℬ, the mentioned former definition (1.4) of  state-dependent  implication boils down to tr(Eρ)=tr(EFρ), or equivalently to tr[(E-EF)ρ]=0. Since (E-EF) is a projector, one can further equivalently write tr[(E-EF)ρ(E-EF)]=0, and (E-EF)ρ(E-EF)=0 (no other positive operator then zero gives trace zero). Finally the Lüders lemma (page 326 in [1], cf. also Remark 4.3) gives (E-EF)ρ=0.Remark 2.2.
Relation (1.4) as the definition of SD implication cannot be extended to projectors E for which Eρ=0, but the equivalent definition Eρ=EFρ for Eρ≠0 can. Hence, we extend  formally SD implication to these projectors. If an event E is such that tr(ρE)=0, then it (formally) state-dependently implies every other event in ℬ. This statement cannot be falsified because E cannot occur in ρ. Hence it is formally true though trivial (more on this in Lemma 4.7).
One should note the equality of sets{E:Eρ=0}={E:tr(ρE)=0} due to 0=tr(ρE)=tr(EρE)⇒0=EρE=Eρ (cf. for analogy proof of Proposition 2.1).Corollary 2.3.
SD implication generalizes and extends absolute implication, that is, the latter is a special case of the former, and the former can be valid for pairs of events for which the latter is not valid.Proof.
IfE=EF, then relation (1.4) is obviously valid, and E≤ρF ensues.SD implication≤ρ, being reflexive, transitive but not antisymmetric, is a partial order [14]. As for every partial order, one can take the  SD (equivalence) quotient setℬ/~ρ, the elements of which are the SD  equivalence classes [E]ℬ in ℬ. Next, one can induce the corresponding  partial-order relation in the quotient set and thus obtain a proper binary relation for  quantum logic.We proceed by acquiring a simple insight on which the poset structure ofℬ/~ρ is based.Proposition 2.4.
Let[E]ℬ and [F]ℬ be distinct SD  equivalence classes, that is, elements of ℬ/~ρ, such that ∃E'∈[E]ℬ and ∃F'∈[F]ℬ so that E'≤ρF'. Let further E′′ and F′′ be arbitrary elements in the same respective classes. Then also E′′≤ρF′′. In other words, if one projector from one class implies state dependently a projector from another class, then every projector from the former class implies state dependently every projector from the latter one.Proof.
The very definition of the classes impliesE′′≤ρE' and F'≤ρF′′. In view of the assumed relation E'≤ρF', and due to the validity of transitivity of SD implication in ℬ, E′′≤ρF′′ follows.On ground of Proposition2.4 we can define a poset structure in ℬ/~ρ.Definition 2.5.
One has[E]ℬ≤ρ[F]ℬ if for all E'∈[E]ℬ and for all F'∈[F]ℬ the relation E'≤ρF' is valid.Lemma 2.6.
SD implication thus defined in the SD quotient setℬ/~ρ is a partial order relation, that is, it has the properties of reflexivity, transitivity, and antisymmetry.Proof.
Reflexivity. If [E]ℬ is an arbitrary SD  equivalence class in ℬ, and E is its arbitrary element, one has E≤ρE and E≥ρE, so that, according to Definition 2.5, also [E]ℬ≤ρ[E]ℬ and [E]ℬ≥ρEℬ are valid.
Transitivity. If [E]ℬ≤ρ[F]ℬ and [F]ℬ≤ρ[G]ℬ, and E',F',G' are arbitrary elements from the respective classes, then, according to Proposition 2.4, the relations E'≤ρF' and F'≤ρG' hold true. Transitivity of SD implication in ℬ then implies the same for the SD quotient set: from E'≤ρG' ensues [E]ℬ≤ρ[G]ℬ.
Antisymmetry. Let [E]ℬ≤ρ[F]ℬ and [F]ℬ≤ρ[E]ℬ be valid. Then  Definition 2.5 implies E≤ρF and F≤ρE, having E~ρF as its consequence, that is, one obtains [E]ℬ=[F]ℬ.Remark 2.7.
Having in mind thatℬ can be arbitrarily chosen in 𝒫(ℋ), we may call the SD quotient set ℬ/~ρ  SD quantum logic paralleling the set of all projectors, quantum logic 𝒫(ℋ). Since the term “quantum logic” has also another meaning as a branch of mathematical physics, one might use the term “SD  quantum logic” also to designate a part of  quantum logic (meant as the mentioned branch), the part treated in [11–13], and in this paper (cf. also the last passage in Section 11).Proposition 2.8.
Two projectorsE,F∈ℬ are  state-dependently equivalent E~ρF  if and only if Eρ=Fρ is satisfied.Proof.
Relation (1.5) and Proposition 2.1 imply Eρ=EFρ and Fρ=FEρ. Since all projectors in ℬ commute, necessity of the claimed condition obviously follows. As to sufficiency, the claimed condition and idempotency obviously imply Eρ=EFρ and Fρ=FEρ, which, in view of Proposition 2.1 and relation (1.5), is tantamount to E~ρF.Remark 2.9.
The SD quotient setℬ/~ρ is complemented because orthocomplementation in ℬ takes each SD  equivalence class onto another, and none is self-complemented. Finite Boolean subalgebras thus consist of an even number of elements.Evidently, SD  equivalenceE~ρF(⇔Eρ=Fρ) can be extended to the entire  quantum logic 𝒫(ℋ). In what immediately follows we study SD equivalence in the entire 𝒫(ℋ), and we write the  equivalence classes as [E] (without index). Subsequently, beginning with Section 8, we shall return to restricting our investigation to ℬ to obtain additional insight in SD  quantum logic.


## 3. Lüders States

We now express SD implication in terms of a (known but, perhaps, not well known) projector relation.Lemma 3.1.
Denoting byQ the range projector of a given density operator ρ, the general claim is valid that an event E has probability 1 in the state ρ  if and only if Q≤E.Proof.
The proof is given for the reader's convenience in AppendixA.Corollary 3.2.
Denoting the range projector of the Lüders stateρ- (cf. (1.1)) by Q-, and calling it  Lüders projector, relation (1.4) and Lemma 3.1 entail the following necessary and sufficient condition for  SD implication  in terms of it:
(3.1)E≤ρF⟺Q-≤F.Evidently, the projectorE in Corollary 3.2 is somehow inherent in ρ-. We want to find out how, and proceed by evaluating the Lüders projector Q-. One needs, not Q, the  initial range projector (of the initially given state ρ), but its ortho-complementary projector, the null-space projector, which one denotes by Q0 and call  the initial null projector. Besides, one shall need more insight into the properties of E that appears in (1.1). As stated, one calls such a projector a  state determining one.Lemma 3.3.
LetE and ρ be an arbitrary projector and an arbitrary density operator, respectively. Then the following  equivalence relation  is valid
(3.2)tr(Eρ)=1⟺Eρ=ρ.This general equivalence is known, but perhaps not well known. For the reader's convenience it is proved in AppendixB.Corollary 3.4.
LetE be a  state determining projector. Then, obviously Eρ-=ρ- (cf. (1.1)), and, on account of Lemmas 3.3 and 3.1, one has
(3.3)Q-≤E.Definition 3.5.
On account of Corollary3.4, we can define a new projector E- as follows:
(3.4)E-≡E-Q-.
We call E- the  redundancy projector (for reasons that will become clear below).Proposition 3.6.
The redundancy projectorE- of a state-determining projector E in (1.1) satisfies both E-≤Q-0 and E-≤Q0, where Q-0≡I-Q-,  I being the identity operator, is the Lüders null projector and Q0  is the initial null projector (that of ρ).Proof.
Definition3.5 makes E- orthogonal to Q-, hence  E-≤Q-0 it must be a subprojector of the null projector Q-0, E-≤Q-0, because Q-0=I-Q-, where I is the identity operator.
Let|ϕ〉∈ℛ(Q-0), where ℛ(Q-0) is the range of Q-0, that is, the null space of the Lüders state (cf. (1.1)). Further, let ρ=∑i‍ri|i〉〈i| be a spectral decomposition of the initially given density operator ρ into eigen-ray-projectors with all eigenvalues {ri:∀i} positive. This is always possible. Then, in view of (1.1),
(3.5)0=〈ϕ|EρE|ϕ〉=∑i‍ri〈ϕ|E|i〉〈i|E|ϕ〉=∑i‍ri|〈i|E|ϕ〉|2,
implying
(3.6)∀i:〈i|E|ϕ〉=0.
Since|ϕ〉 is any state vector in ℛ(Q-0), and the vectors {|i〉:∀i} span the range of the initial state ρ, the last relations mean that the image subspace E[ℛ(Q-0)] is orthogonal to ℛ(ρ), that is, that E maps the null space ℛ(Q-0) into the null space ℛ(Q0):
(3.7)E[ℛ(Q-0)]⊆ℛ(Q0).
Further, E- being a subprojector of Q-0, one has ℛ(E-)⊆ℛ(Q-0). Finally, since E- is also a sub-projector of the idempotent operator E (cf. Definition 3.5), it follows that
(3.8)E[ℛ(E-)]=ℛ(E-)⊆E[ℛ(Q-0)].
Therefore, transitivity of the subset relation implies ℛ(E-)⊆ℛ(Q0), or equivalently E-≤Q0 as claimed.Theorem 3.7.
The Lüders projectorQ- can be evaluated in terms of a state determining projector E (cf. (1.1)) and the initial null projector Q0 as follows:
(3.9)Q-=E-(E∧Q0),
where E∧Q0 is the greatest lower bound (glb) or largest common subprojector of E and Q0.Proof.
Let us introduceG≡E-(E∧Q0) or equivalently
(3.10)E=(E∧Q0)+G.
Since (E∧Q0) is a sub-projector of the initial null projector Q0, one has (E∧Q0)ρ=0=ρ(E∧Q0) implying
(3.11)GρG=EρE.
Hence, alsoG is a determining projector for ρ- (cf. (1.1)), and all that has been proved for the determining projector E in (1.1) is valid also for G. In particular, Q-≤G, G≡Q-+G- (definition of G-).
The definition ofG- implies G-≤G. On the other hand, it is seen from (3.10) that G≤E. Hence, due to transitivity of absolute implication, G-≤E. Since also Proposition 3.6 is valid for G and its redundancy projector G-, one has G-≤Q0, and
(3.12)G-≤(E∧Q0)
follows.
On can see from (3.10) that G is orthogonal to (E∧Q0): G(E∧Q0)=0, and so is its subprojector G-: G-(E∧Q0)=(G-G)(E∧Q0)=0. The projector orthogonality amounts to G-≤(E∧Q0)c, where the suffix c denotes taking the ortho-complementary projector. Comparing this with (3.12), we conclude that G-=0 because only zero is the common subprojector of a projector and its ortho-complementary projector.
Returning to the definition ofG-(≡G-Q-), we see that we are led to the equality G=Q-, which, due to the definition (3.10) of G, completes the proof.Theorem3.7 immediately implies the following conclusions.Corollary 3.8.
The redundancy projectorE- of any state-determining projector E∈𝒫(ℋ) is given by E-=E∧Q0.Corollary 3.9.
A projectorE is a Lüders projector Q-  if and only if its redundancy projector is zero, and  if and only if the only common subprojector of E and Q0 is zero:
(3.13)E=Q-⟺E-=0⟺E∧Q0=0.Corollary 3.10.
If[E,ρ]=0, or at least [E,Q0]=0, then Q-=E-EQ0.Proof.
The first commutation is equivalent, as well known, to reduction ofE in each eigen-subspace of ρ. Since the null-space of ρ is its eigen-subspace corresponding to the eigenvalue zero, E reduces in it, hence it commutes with Q0 that projects onto the latter. Then the glb equals the product as claimed.Remark 3.11.
WheneverFρ=Eρ is valid, one can replace E by F in the evaluation formula of the Lüders projector (cf. (1.1) and Theorem 3.7), that is, also Q-=F-(F∧Q0) is satisfied.Remark 3.12.
If the initial density operatorρ is nonsingular, that is, when Q0=0, then the Lüders projector Q- is the only state determining projector as obvious from Theorem 3.7.


## 4. Generalized Lüders Lemma

The SD equivalence relationEρ=Fρ implies that both E and F are state determining projectors for the same Lüders state ρ- (cf. (1.1)). We want to clarify if the converse implication is also valid.Theorem 4.1.
Generalized Lüders lemma.  IfE,F are two projectors and ρ is a density operator, then EρE=FρF implies Eρ=Fρ.Proof.
One can writeE=Q-+E- and F=Q-+F- (cf.  Definition 3.5). Further, on account of Proposition 3.6, both E- and F- are subprojectors of Q0. Hence, Eρ=(Q-+E-)ρ=Q-ρ. On the other hand, symmetrically Fρ=(Q-+F-)ρ=Q-ρ.Corollary 4.2.
IfEρE=0, then Eρ=0.Proof.
One can writeEρE=0ρ0. Then Theorem 4.1 implies the claim.Remark 4.3.
The claim of Corollary4.2 was proved by Lüders [1] (page 326) in the following more general form: If B is a positive operator, and C†BC=0, where the dagger denotes adjoining (of the linear operator C), then BC=0 is valid. This auxiliary result is usually called "the Lüders lemma."Lemma 4.4.
The equivalence relationE~ρF gives, in general, finer  equivalence classes in 𝒫(ℋ) then the  equivalence relation E~trρF≡(tr(Eρ)=tr(Fρ)).Proof.
The proof is given by a simple example. We take the two-dimensional spin-1/2 space and we define: ρ≡1/2=(1/2)|+〉〈+|+(1/2)|-〉〈-|, E≡|+〉〈+|, F≡|-〉〈-|. Then Eρ=(1/2)|+〉〈+|≠Fρ=(1/2)|-〉〈-|, though tr(Eρ)=1/2=tr(Fρ). Thus, E~trρF though E~ρF is not valid. The latter, finer,  equivalence relation breaks up the coarser class [E,F]trρ, so that the two distinct finer classes [E]ℬ and [F]ℬ are within it.Remark 4.5.
Theorem4.1 can be put as the claim that EρE=FρF and Eρ=Fρ are equivalent, that is, they both define the same equivalence relation ~ρ in 𝒫(ℋ).Lemma 4.6.
RelationEρ=Fρ is valid  if and only if E and F act equally on each element of the range of ρ. In particular, SD equivalence is equivalent to EQ=FQ.Proof.
The general element of the range is of the formρ|ψ〉, and SD  equivalence in  Proposition 2.8 amounts to Eρ|ψ〉=Fρ|ψ〉. This bears out the first claim of the lemma. Since the last relation is obviously equivalent to EQ|ψ〉=FQ|ψ〉, also the second claim is shown to hold true.Lemma 4.7.
One hasEρ=0, or equivalently E~ρ0,   if and only if  EQ0=E. The last relation means that E is a subprojector of the initial null projector Q0, that is, that E≤Q0 is valid. Thus, the SD  equivalence class [0]=[Q0] consists of all subprojectors of Q0. The projectors 0 and Q0 are the minimal and the maximal elements in the class, respectively, (cf. Remark 2.2).Proof.
Sufficiency. Let the subprojector condition EQ0=E be valid. Then Eρ=(EQ0)(Qρ)=0.
Necessity. Let Eρ=0=0ρ hold true. Then, according to Lemma 4.6, also EQ=0Q=0 is true. Finally, E=EQ+EQ0=EQ0 as claimed.Lemma 4.8.
One hasEρ=ρ, or equivalently E~ρI   if and only if  EQ=Q. The last relation means that Q is a subprojector of E, that is, that Q≤E is valid. Thus, the class [Q]=[I] consists of all projectors of which Q is a subprojector. The projectors Q and the identity operator I are the minimal and the maximal elements of the class, respectively.Proof.
Follows immediately from  Lemma4.6.Lemma 4.9.
The  equivalenceE~ρF implies E~ρG~ρF, where G is defined as the common glb of E and F.Proof.
It is known thatG≡E∧F=limn→∞(EFE)n (cf. solution 96 in [15]). On account of  Proposition 2.8, the assumed  equivalence relation (with the idempotencies of the projectors) implies (EFE)ρ=Eρ. Further, (EFE)nρ=Eρ ensues. Finally, Gρ=[limn→∞(EFE)n]ρ=limn→∞Eρ=Eρ.Proposition 4.10.
LetE~ρF, and let by definition H≡E-G, J≡F-G (cf. Lemma 4.9). Then E~ρF is equivalent to the simultaneous validity of the following two sub-projector relations
(4.1)H≤Q0,J≤Q0.Proof.
Sufficiency. Let the two subprojector relations be valid. Then one can write E=G+HQ0 and F=G+JQ0. SD equivalence then immediately follows.
Necessity.
E
~
ρ
F and Lemma 4.9 imply Eρ=Gρ=Fρ, and one has
(4.2)Hρ=(E-G)ρ=0=(F-G)ρ=Jρ.Proposition4.10 has introduced the entities H and J in the general case. Incidentally, there is a general lemma that says: Two arbitrary projectors E and F commute  if and only if H and J are orthogonal (since we actually do not need this claim, let its proof be left to the reader).


## 5. Complete-Lattice Structure of the SD  Equivalence Classes in𝒫(ℋ)

We assume that an arbitrary density operatorρ is given and subject to scrutiny the SD equivalence classes. On account of Remark 4.5 we know that each such class contains precisely all projectors that are state determining for one and the same Lüders state ρ-. We begin by paying attention to the Lüders projectors in the classes.Proposition 5.1.
(A) Each SD  equivalence class[E]  contains a Lüders projector Q-, and it can be evaluated from any element E of the class according to Theorem 3.7.
(B) Each Lüders projectorQ- is the  unique minimal element in its class [Q-]: E~ρQ- implies Q-≤E.
(C) The following  bijectionb^ maps the SD quotient set 𝒫(ℋ)/~ρ onto the set of all Lüders projectors in 𝒫(ℋ), which we denote by ℒ:
(5.1)b^([E])=Q-≡E-(E∧Q0),[E]∈𝒫(ℋ)~ρ,b^-1(Q-)=[Q-],Q-∈ℒ.Proof.
(A) We assume that[E] is any  state-dependent  equivalence class, and that E is any of its elements. We take Q- as defined by the relation in Theorem 3.7. Since (E∧Q0) is a sub-projector of Q0, it follows that we have (E∧Q0)ρ=0, and then, according to the relation in Theorem 3.7, Q-ρ=Eρ. Thus, each class does contain at least one Lüders projector.
(B) ThatQ-≤E is immediately seen from Corollary 3.4. This makes Q- the unique minimal element in [E] (there cannot be two distinct minimal elements, because they would have to be subprojectors of each other, and ≤ is a partial order, the anti-symmetric property of which implies the equality).
(C) is now obvious.It is now useful to be reminded of a general property of the complete lattice𝒫(ℋ).Lemma 5.2.
LetE be any nonzero projector. The set of all lower bounds {E':E'≤E} is a complete lattice.Though this a known and simple property of complete lattices, we prove it for the reader's convenience in AppendixC.Theorem 5.3.
An arbitrary SD equivalence class[E], as a poset subset of 𝒫(ℋ), is  isomorphic  to the complete lattice {E':E'≤(Q0∧Q-0)}. The isomorphism i^ acts as follows:
(5.2)i^(E)≡E-=E-Q-,E∈[E];i^-1(E')=Q-+E',E'≤(Q0∧Q-0).Proof.
LetE',F'≤(Qc-∧Q0). Then, on account of the assumed orthogonalities Q-E'=0=Q-F', one has
(5.3)(Q-+E')(Q-+F')=Q-+E'F'.
Thus, one has (Q-+E')(Q-+F')=(Q-+E'), symbolically (Q-+E')≤(Q-+F')  if and only if E'F'=E', symbolically E'≤F'.We have the immediate consequence.Corollary 5.4.
Each SD  equivalence class is a  complete lattice with the Lüders projectorQ- as the minimal element and Q-+(Q0∧Q-0) as the maximal element.The set of all Lüders projectorsℒ, as any subset of 𝒫(ℋ), is a poset. It has a lattice structure and it is easily connected with the ortho-complemented SD quotient set.Theorem 5.5.
(A) The posetℒ has the structure of a  complete glb-semilattice, the operation of glb (greatest lower bound) being the same as in the entire  quantum logic 𝒫(ℋ). Its minimal element is the zero projector.
(B) There is a simple bijectioni'^ mapping ℒ onto the SD quotient set 𝒫(ℋ)/~ρ that associates with each Lüders projector its SD  equivalence class, and the inverse of which singles out the minimal element in the SD  equivalence class:
(5.4)i'^(Q-)≡[Q-],(i′^)-1([E])=Q-,Q-~ρE.
The term "glb-semilattice" is consistent with the terminology of Birkhoff, who used the synonymous term "meet-semilattice" (cf. the definition on page 22 in [14]).Proof of Theorem5.5.
Let{Em:m∈M} be any subset of ℒ, and let Eglb≡∧m∈MEm  be the greatest lower bound of the projectors in the subset taken in 𝒫(ℋ). We show now that it is a Lüders projector. Corollary 3.9 implies that a projector is a Lüders projector  if and only if only the zero projector is a common lower bound of it and Q0.
We takeP as a common sub-projector of Eglb and of Q0. On account of Eglb being a lower bound (a sub-projector) of each Em in the subset, P is a common sub-projector of each Em and of Q0. Since each Em is a Lüders projector, P must be zero.


## 6. Physical Meaning of Complete-Lattice Structure

The physical meaning of an event (projector)E consists in the fact that it can occur in a given state (density operator) ρ, when tr(Eρ)=1, or “not occur” when tr(Eρ)=0. In the latter case the opposite event (ortho-complementary event) Ec≡I-E occurs. (There is a third possibility, E can de indeterminate, that is, 0<tr(Eρ)<1. Then it is made to occur or not occur in an ensemble of measurements giving relative frequencies.)Let{Em:m∈M} be an arbitrary set of projectors in 𝒫(ℋ). One may wonder if the set can, as a whole, occur? The answer is of course, that it can. This is the case when all the events in it occur. One may further wonder if this occurrence of the set of events can be equivalently replaced by occurrence of a single event. Affirmative answer is given by the following claim.Proposition 6.1.
The system of relations(6.1)∀m∈M:tr(Emρ)=1
is  equivalent  to the single relation tr[(∧m∈MEm)ρ]=1.Proof.
Necessity. On account of Lemma 3.1, one can rewrite the system of relations in Proposition 6.1 equivalently as follows:
(6.2)∀m∈M:Q≤Em.
Being a common lower bound of the set {Em:m∈M}, Q must satisfy Q≤∧m∈MEm. This is equivalent, due to Lemma 3.1, to tr[(∧m∈MEm)ρ]=1.
Sufficiency. Assuming tr[(∧m∈MEm)ρ]=1, or equivalently Q≤(∧m∈MEm), and taking into account that ∀m∈M:(∧m'∈MEm')≤Em, transitivity implies ∀m∈M:Q≤Em. This is equivalent to the system of relations in Proposition 6.1.Let us turn to the other extreme case. Nonoccurrence of the set of events{Em:m∈M} also amounts to that of a single event.Proposition 6.2.
Non-occurrence of a set of events{Em:m∈M} is equivalent to  non-occurrence of the single event ∨m∈MEm, where ∨ denotes the least upper bound.Proof.
The non-occurrence in question is equivalent to the occurrence of the set{Emc:m∈M} of opposite events. This, in turn, is, according to Proposition 6.1, equivalent to the occurrence of ∧m∈MEmc. On account of the corresponding de Morgan rule, this projector can be rewritten as (∨m∈MEm)c. Occurrence of this event is equivalent to non-occurrence of the opposite event (∨m∈MEm).Propositions6.1 and 6.2 have made it clear that the physical meaning of the greatest lower bound and the least upper bound in a complete lattice is occurrence and non-occurrence of a set of events, respectively.


## 7. Physical Meaning of the Complete-Lattice Structure of the SD  Equivalence Classes

We have seen that every SD  equivalence class contains precisely all those events, that, as projectors, are the state-determining ones for one and the same Lüders state. These events have the same probability in the initially given stateρ, and, more importantly, they give rise to the same change of state in ideal measurement. In this sense, they are indistinguishable, and the SD  equivalence class that they constitute, as a whole, correctly replaces its individual elements.According to Proposition6.1, the minimal element in the class, which is its glb, and which is the Lüders projector Q-, determines via its occurrence the occurrence of the whole class. Its maximal element Q-+(Q0∧Q-0), via its non-occurrence determines that of the entire class. Thus, the two extreme elements in the class make possible the occurrence or non-occurrence of the class as a set of indistinguishable events  (as far as probability and ideal measurement go).One wonders if the elements of the class that are intermediate between the extreme ones do have any physical meaning. They are distinguished via the redundancy projectors(7.1)E-∈{E′:E′≤(Q0∧Q-0)},
(cf. Theorem 5.3). They have a subtle physical meaning of redundancy. We elaborate this point.(A) The Lüders projectorQ- of any class [E] has no redundancy as far as occurrence goes in the sense that they occur (have probability one) in the corresponding Lüders state ρ- and there exists no other event that occurs in ρ- and that implies the event Q-.(B) All other eventsE that are  state-dependent ly equivalent to Q- do have redundancy, and a kind of its measure is uniquely expressed by the redundancy projector E- because E=Q-+E- (cf. Definition 3.5). They occur in ρ-, but there exists another event that also occurs in this state and it implies E (it is, of course, the Lüders state).(C) As to the eventsE' that are implied by Q- but belong to another SD-class, one must distinguish two distinct sets of Lüders entities, besides ρ-,Q-,Q-0 that were in focus so far, also another set ρ'-,Q'-,Q0′¯ connected to the SD-class to which E' belongs.For such eventsE' the redundancy projector E'- satisfies formally the same last relation E'-∈{E′′:E′′≤(Q0∧Q0′¯)}, but this time with Q0′¯.Next we return to restricting𝒫(ℋ) to one of its Boolean subalgebras.


## 8. SD  Equivalence Classes inℬ and Their Minimal Elements

After having restricted𝒫(ℋ) to an arbitrary Boolean subalgebra ℬ, “shrinking” further ℬ into ℬ/~ρ is mathematically necessary to upgrade the preorder into an order relation (cf.  Lemma 2.6). In other words, the SD  implication "≤ρ" is actually valid in the quotient set. Physically, the events are now the classes, and the question arises how do they “occur” or “not occur.”The subalgebraℬ can be a  finite lattice, if it is closed to taking the greatest lower bound (glb) and the least upper bound (lub) of at most finite subsets; it can be a  σ-lattice if it contains the glb and lub of its subsets with at most countably infinite number of elements, and, finally, it can be a  complete lattice if the glb and lub of any subset is in it. We will call this the  lattice type of ℬ.In restricting𝒫(ℋ) to ℬ each SD  equivalence class [E]ℬ is part of the corresponding SD  equivalence class in 𝒫(ℋ):
(8.1)[E]ℬ≡[E]∩ℬ.
The importance of the intersection relation (8.1) is in the fact that if two projectors E,F are in ℬ, then they belong to one and the same SD  equivalence class [E]ℬ  if and only if they are equivalent in 𝒫(ℋ). For the latter, we have, besides Eρ=Fρ, additional necessary and sufficient conditions like in  Lemma 4.6 and in  Proposition 4.10.It was seen in  Corollary5.4 that each SD  equivalence class [E] in 𝒫(ℋ) is a complete lattice. Hence, it follows from (8.1) that each SD  equivalence class [E]ℬ in ℬ is also a lattice of the same lattice type as that of ℬ.To acquire a feeling for the SD  equivalence classes[E]ℬ defined by (8.1), we consider two extreme cases. Firstly, we assume that ρ is nonsingular. Then Q=I and Q0=0. According to  Remark 3.12, each projector in 𝒫(ℋ) equals the minimal element Q- in its SD  equivalence class. Hence all SD  equivalence classes in 𝒫(ℋ) have only one element. Relation (8.1) then tells us that also the SD  equivalence classes [E]ℬ all consist of one element and [E]ℬ=[E] if E∈ℬ.Secondly, let us take a singular density operator, but letℬ consist of only two elements. They must, of course, be 0 and I. Then, only [Q0](=[0]) and [Q](=[I]) (cf.  Lemmas 4.7 and  4.8) have nonempty intersections with ℬ. The former intersection consists of only the maximal element in the 𝒫(ℋ) class, whereas the latter intersection contains only the minimal element of the corresponding class. We proceed by investigating the general case of [E]ℬ in detail.If not stated otherwise, we assume thatℬ is a complete lattice in what follows. We introduce two key entities for further study.Definition 8.1.
LetQ0ℬ be the greatest subprojector of Q0 that is in ℬ: Q0ℬ≡∨({E:E≤Q0}∩ℬ). We call Q0ℬ  the Boolean initial null projector. Further, let Qℬ be the least among the projectors that are greater than Q and belong to ℬ: Qℬ≡∧({E:E≥Q}∩ℬ). We call Qℬ  the Boolean initial range projector.
Note that, sinceℬ, {E:E≤Q0}, and {E:E≥Q} are complete lattices (cf. Lemma 5.2 and its proof in Appendix C), so are the above intersections.Proposition 8.2.
The Boolean null projector and the Boolean range projector are dual entities, that is, they are the ortho-complementary projectors of each other:Q0ℬ=Qℬc.Proof.
The SD  equivalence classes{E:E≤Q0}∩ℬ and {E:E≥Q}∩ℬ are dual to each other. Hence, so are the maximal element in the former and the minimal element in the latter.Remark 8.3.
Evidently,Qℬ=Q, or equivalently Q0ℬ=Q0  if and only if Q∈ℬ, or equivalently Q0∈ℬ.Proposition 8.4.
One has[0]ℬ=[Q0ℬ]ℬ and [Qℬ]ℬ=[I]ℬ. The zero projector and Q0ℬ are the least and the greatest elements, respectively, in [0]ℬ: E∈[0]ℬ⇒0≤E≤Q0ℬ, whereas Qℬ and the identity operator I are the least and the greatest elements, respectively, in [I]ℬ: E∈[I]ℬ⇒Qℬ≤E≤I.Proof.
The projectorQ0ℬ is by Definition 8.1 a subprojector of Q0. Hence, Q0ℬρ=0ρ, or Q0ℬ~ρ0. The projector Q0ℬ is, by definition, the greatest subprojector of Q0 that belongs to ℬ. Hence it is the greatest element in [0]ℬ. Further, as we know (cf.  lemma 2 and  lemma 3), one has Eρ=ρ, or equivalently E~ρI,  if and only if E≥Q. Hence, Qℬ is the minimal element of [I]ℬ. The rest is obvious.Definition 8.5.
We call a projectorE0ℬ a  core element in ℬ if the relation E0ℬQ0ℬ=0 is satisfied. The set of all core elements in ℬ we call the core set and denote it by Cℬ.Proposition 8.6.
The core set isCℬ={E:E≤Qℬ}∩ℬ. It is a complete lattice.Proof.
The definitionE0ℬQ0ℬ=0 means that each core element is orthogonal to Q0ℬ, or equivalently that it is a subprojector of the ortho-complementary projector Qℬ of Q0ℬ (cf.  Proposition 8.2). The poset Cℬ is the intersection of two complete lattices, hence it is itself a complete latticeTheorem 8.7.
(A) Every  state-dependent  equivalence class[E]ℬ contains a core element E0ℬ. It can be evaluated from an arbitrary element E of the class by the relation
(8.2)E0ℬ=E-EQ0ℬ,
(cf. Definitions 8.1 and 8.5).(B) Every core elementE0ℬ is the least element of its class [E0ℬ]ℬ: E~ρE0ℬ,E∈ℬ⇒E≥E0ℬ. Hence, each class contains a unique core element.Proof.
(A) Let[E]ℬ be any  state-dependent  equivalence class in ℬ, and let E be any of its elements. The rhs of (8.2) is state-dependently equivalent to E because Q0ℬ is equivalent to 0. In other words, the lhs belongs to [E]ℬ. We prove that it is a core element: (E-EQ0ℬ)Q0ℬ=0. Hence, according to  Definition 8.5, it is a core element, and therefore, each class does contain at least one core element.
(B) LetE0ℬ be a core element of [E]ℬ. Then, according to  Theorem 8.7(A), E0ℬ=E-EQ0ℬ. Multiplication gives EE0ℬ=E0ℬ, that is, E0ℬ≤E as claimed. This makes E0ℬ the unique least element in the class [E0ℬ]ℬ to which it belongs.The core elementsE0ℬ in the SD  equivalence classes in ℬ play the same role as the Lüders range projectors Q- in the SD  equivalence classes in 𝒫(ℋ) (cf.  Corollary 5.4).


## 9. Further Investigation of the SD  Equivalence Classes inℬ

Some SD  equivalence classes[E] in 𝒫(ℋ) do and others do not contain an SD  equivalence class [E]ℬ, that is, they may be vacant in this respect. We now derive some necessary conditions for nonvacant classes [E].Proposition 9.1.
Each SD  equivalence class[E] is vacant unless Qℬ is an upper bound for the minimal element, the Lüders range projector Q-, in the class.Proof.
The core elementE0ℬ, if it exists in a class [E0ℬ], must be an upper bound of the minimal element Q- in the class. Since, further, E0ℬ≤Qℬ (cf.  Proposition 8.6), transitivity establishes the claim.Corollary 9.2.
In the special case whenQℬ=Q, or equivalently, Q∈ℬ, SD  equivalence classes in ℬ can appear only in those SD  equivalence classes in 𝒫(ℋ) in which Q-≤Q, and then not necessarily in all (take, e.g., a non-singular ρ and the minimal Boolean subalgebra).We now investigate  the properties of theintermediate elements of the SD  equivalence classes in ℬ analogously as one did in 𝒫(ℋ).Theorem 9.3.
(A) Each elementE of each  state-dependent  equivalence class [E]ℬ can (uniquely) be written as follows:
(9.1)E=E0ℬ+E',
so that
(9.2)E0ℬ~ρE,E0ℬ∈Cℬ,E'∈[0]ℬ.
Conversely, for each core elementE0ℬ, each projector E'∈[0]ℬ determines an element E≡E0ℬ+E' that is  state-dependently equivalent to E0ℬ. In this way one has a one-to-one map of {E':E'≤Q0ℬ}∩ℬ(=[0]ℬ=[Q0ℬ]ℬ) onto any class [E0ℬ]ℬ.
(B) The map associating with eachE'∈[0]ℬ the image E0ℬ+E', where E0ℬ is the core element of any given SD  equivalence class in ℬ, is an  isomorphism of the poset [0]ℬ onto the poset [E0ℬ]B.Proof.
(A)  Theorem8.7 has established that each SD  equivalence class [E]ℬ contains a unique core element E0ℬ and that the latter is the least element of the class. The projector E'≡E-E0ℬ is obviously in ℬ since one can write E'=(E0ℬ)cE. That E' is a subprojector of Q0ℬ is seen as follows. We take into account that E0ℬρ=Eρ, which implies E'ρ=0, E'~ρQ0ℬ and E'≤Q0ℬ (cf.  Proposition 8.4). Thus, E'∈[0]ℬ.
Conversely, we assume that(9.3)E'≤Q0ℬ,E'∈ℬ.
One has E0ℬ≤Qℬ (cf.  Definition 8.5 and  Proposition 8.4), and, taking the ortho-complement of this inequality, Q0ℬ≤(E0ℬ)c ensues. Hence, transitivity and (9.3) give E'≤(E0ℬ)c, implying E'E0ℬ=0. This orthogonality makes it possible to add E' to E0ℬ.
Finally, as we know from  Proposition8.4, E'ρ=0. Hence, (E0ℬ+E')~ρE0ℬ as claimed. The map at issue is clearly a one-to-one map because it can be inverted beginning with any element of the image set [E]ℬ.
(B) To prove isomorphism, letE′≤E′′, E′,E′′∈[0]ℬ. Let us, further, multiply out the product
(9.4)(E0ℬ+E')(E0ℬ+E′′)=E0ℬ+E0ℬE′′+E'E0ℬ+E'E′′.
One obtains
(9.5)(E0ℬ+E′)(E0ℬ+E′′)=E0ℬ+E′E′′,
if one utilizes the orthogonalities stemming from E0ℬE′′=E0ℬ(Q0ℬE′′)=0 (cf.  Definition 8.5), and E'E0ℬ=(E'Q0ℬ)E0ℬ=0. Obviously,  if and only if E'E′′=E', then (E0ℬ+E')(E0ℬ+E′′)=E0ℬ+E' as claimed.Corollary 9.4.
Each SD  equivalence class[E]ℬ has, as a poset, the same structure as the poset [Q0ℬ]ℬ. In particular, in the former, its core element E0ℬ is the least and E0ℬ+Q0ℬ is the greatest element.Corollary 9.5.
Each SD  equivalence class[E]ℬ has precisely one element, the core element E0ℬ,  if and only if Q0ℬ=0. This is the case, for example, when ρ is non-singular, because then Q0=0, and Q0ℬ≤Q0. But Q0 may be non-zero, and Q0ℬ=0.Corollary 9.6.
Theorem9.3 establishes an isomorphism between any two SD  equivalence classes in ℬ (in an obvious way).The reader who remembers well the results in the decomposition of entire  quantum logic𝒫(ℋ) into SD  equivalence classes is rightly perplexed. He wonders how did it happen that restriction to a Boolean subalgebra made all the SD  equivalence classes [E]ℬ mutually isomorphic, whereas the larger SD  equivalence classes [E] (cf. (1.1)) are not isomorphic due to the fact that the Lüders null projectors Q-0 are, in general, different for different classes.To throw additional light on this puzzling fact, we shortly repeat the relevant parts of the above arguments. The (minimal) core elementsE0ℬ in the classes [E]ℬ are obtained from an arbitrary element in the class in a completely parallel way as in [E] (cf.  Theorem 8.7(A) above and  Theorem 3.7). As a consequence, the core elements satisfy E0ℬQ0ℬ=0 parallelly to the Lüders range projectors for which Q-∧Q0=0 is valid (cf.  Definition 8.5 above and  Corollary 3.9).Also  Theorem9.3(A) above could have been put in the form of a map of {E':E'≤Q0ℬ∧(E0ℬ)c}∩ℬ (see below) onto [E]ℬ, to display further parallelism with 𝒫(ℋ), where the counterpart is {E'≤Q0∧Q-c}. So far the classes in ℬ and in 𝒫(ℋ) go parallelly.What makes a difference is the fact that in a Boolean subalgebra any two projectors commute. Hence, glb takes the form of multiplication. If the glb is zero, as it is in our case (cf.  Definition8.5), we have orthogonality, leading to E0ℬ≤Qℬ and, taking the ortho-complement of this inequality, to Q0ℬ≤(E0ℬ)c. Since in a Boolean subalgebra the last inequality means Q0ℬ(E0ℬ)c=Q0ℬ, we have Q0ℬ∧(E0ℬ)c=Q0ℬ(E0ℬ)c=Q0ℬ as a consequence.Thus,(E0ℬ)c, the counterpart of Q-0, has disappeared from the scene (whereas Q-0(=Q-c) in [E] does remain in general).Remark 9.7.
IfQ-≤Q, then, taking the ortho-complementary projectors, we obtain Q0≤Q-0 implying Q0∧Q-0=Q0. In other words, in the classes considered, Q-0 disappears from the maximal element in the class, which is now Q-+Q0. Thus, the considered classes are mutually isomorphic just like the SD  equivalence classes in ℬ that they may contain.


## 10. The Core Decomposition

The core set𝒞ℬ in ℬ (cf.  Definition 8.5) runs parallel to the Lüders set ℒ in 𝒫(ℋ) (cf.  Proposition 5.1(C)). We investigate now if also the lattice structure of the former parallels that of ℒ.Theorem 10.1.
The core set𝒞ℬ has the structure of a  complete lattice with the greatest lower bound operation being the same as in ℬ, but the least upper bound operation being, in general, different. The least element in 𝒞ℬ is the zero projector 0, and the greatest element is Qℬ.Proof.
Let𝒮 be any subset of 𝒞ℬ, and let ∧𝒮 be the glb of the projectors in 𝒮 taken in 𝒫(ℋ). We show that it is a core element.
Arguing  ab contrario, we assume that there exists a non-zero projectorE in ℬ that is a common subprojector of ∧𝒮 and of Q0ℬ. But then, on account of ∧𝒮 being a lower bound (a subprojector) of each E0ℬ∈𝒮 and of Q0ℬ, it cannot be non-zero (cf.  Definition 8.5), and ∧𝒮∈𝒞ℬ follows. It is obvious that zero is the least element in 𝒞ℬ.
The projectorQℬ is a core element (cf.  Definition 8.5) since it is orthogonal to Q0ℬ as its ortho-complementary projector. That it is the largest element in 𝒞ℬ is evident from  Proposition 8.6.
It is an elementary fact that every complete glb-semilattice with a greatest element is a complete lattice because the lub of any subset is the glb of the subset of all upper bounds of the given subset (for the reader's convenience the latter claim is proved in AppendixD).
The lub∨𝒮 in 𝒞ℬ need not be the same as the lub of the same subset in ℬ, in general, because, in the latter case, it equals the glb of the subset of all upper bounds of the given subset in ℬ. This is the glb of a larger set, hence it is, in general, less than the mentioned glb in 𝒞ℬ.Unlike in𝒫(ℋ), the core set has isomorphic  complete lattices.Theorem 10.2.
For eachE'∈[0]ℬ(=[Q0ℬ]ℬ), the poset E'+𝒞ℬ, which we call the  core class determined by E', with the absolute implication ≤ defined in 𝒫(ℋ), is  isomorphic to 𝒞ℬ; hence, it is a complete lattice like 𝒞ℬ.One should note thatE'+𝒞ℬ≡{E'+E:E∈𝒞ℬ} is the set of all projector sums of E' and an element of 𝒞ℬ. A sum of two projectors is a projector if the terms are orthogonal as well known. It is the case here because E'≤Q0ℬ (cf.  Theorem 9.3(A) and  Proposition 8.4), E0ℬ≤Qℬ (cf.  Proposition 8.6), and Q0ℬ and Qℬ are orthogonal to each other (cf.  Proposition 8.2).Proof of Theorem10.2.
LetE'∈[0]ℬ, E0ℬ,E0′ℬ∈𝒞ℬ. On account of the orthogonalities due to  Definition 8.5 and the fact that E'≤Q0ℬ, one has
(10.1)(E'+E0ℬ)(E'+E0′ℬ)=E'+E0ℬE0′ℬ.
This is (E'+E0ℬ)(E'+E0′ℬ)=E'+E0ℬ, that is, (E'+E0ℬ)≤(E'+E0′ℬ)  if and only if E0ℬE0′ℬ=E0ℬ, that is, E0ℬ≤E0′ℬ.Definition 10.3.
We call the (nonoverlapping) decomposition ofℬ(10.2)ℬ=∑E′∈[0]ℬ‍{E′+𝒞ℬ},
the  core decomposition of the given Boolean subalgebra ℬ of 𝒫(ℋ) into  core classes.We can pictureℬ as a matrix of elements such that the rows are the SD  equivalence classes and the columns are the core classes. Then, the rows are enumerated by the core elements (elements of 𝒞ℬ), and the columns by the elements of [0]ℬ.One should pay attention to the fact that the same notation is used for summation (+ and∑‍) in two different senses: as a summation of projectors like in the core class E'+𝒞ℬ, and as a union of non-overlapping subsets as in  Definition 10.3.Remark 10.4.
In view of  Remark9.7, we see that if we restrict ourselves to 𝒩𝒱≡∑Q-≤Q‍[Q-] (a subset of 𝒫(ℋ)), that is, if we consider only those SD  equivalence classes in 𝒫(ℋ) that may contain a SD  equivalence class in ℬ (that are nonvacant in this respect), then we have a decomposition of 𝒩𝒱 parallelling the core decomposition of ℬ. But in 𝒫(ℋ), when no ℬ is chosen, there is no motivation for this.


## 11. Back to the SD Quotient Set

There is more insight to be gained about the poset structure of the SD quotient setℬ/~ρ.A remarkable property ofℬ/~ρ is the following fact [11] (Lemma  1 there). A SD equivalence class implies state-dependently another such class  if and only if there is an event E in the former and an event F in the latter so that E≤F, that is, the former event implies the latter one in the absolute sense. Since this result is of great importance for the aims of this study, we re derive it (with new insight) to make the present text self-contained.Proposition 11.1.
IfE≤F, then [E]ℬ≤ρ[F]ℬ. Conversely, if [E]ℬ≤ρ[F]ℬ, then ∃E'∈[E]ℬ and F'∈[F]ℬ such that E'≤F'.Proof.
The relationE≤F can be rewritten as E≤ρF (cf. Corollary 2.3), then the claimed relation [E]ℬ≤ρ[F]ℬ immediately follows (cf. Definition 2.5). If [E]ℬ≤ρ[F]ℬ, then Eρ=EFρ (cf.  Propositions 2.4 and 2.1). Hence, EF~ρE, that is, EF∈[E]ℬ, and, taking E'≡EF and F'≡F one has E'≤F'.Remark 11.2.
In the partially ordered setℬ/~ρ one can replace ≤ρ by ≤ because Proposition 11.1 allows one to define [E]ℬ≤[F]ℬ via E≤F that the respective CD  equivalence classes contain.
For the sake of generality, let us assume now that the Boolean subalgebraℬ is of any lattice type (finite, σ- or complete, cf. the second passage in section 8).Theorem 11.3.
(A) The map associating with each projectorE∈ℬ the SD  equivalence class [E]ℬ to which it belongs is a poset  homomorphism of the Boolean subalgebra ℬ onto the Boolean algebra ℬ/~ρ.
(B)ℬ/~ρ is of the same lattice type as ℬ.Proof.
(A) In view of  Proposition11.1, it is obvious that the map taking ℬ onto ℬ/~ρ by associating with each element E∈ℬ the SD  equivalence class [E]ℬ to which it belongs is a homomorphism of a poset with absolute implication "≤" onto the poset ℬ/~ρ with the same implication "≤." The map evidently preserves the ortho-complementation operation. We prove now that it preserves also the glb and lub operations.
Let{Em:m∈M} be an arbitrary subset of ℬ of a finite number of elements if ℬ is a finite lattice, of at most a countably infinite number of elements if ℬ is a σ-lattice, and a completely arbitrary subset if ℬ is a complete lattice.
The SD  equivalence class[∧m∈MEm]ℬ is a lower bound of the set {[Em]ℬ:m∈M} of classes because so is ∧m∈MEm for the set of projectors {Em:m∈M}. If [F]ℬ is an arbitrary lower bound of {[Em]ℬ:m∈M}, then ∀m∈M:∃Fm∈[F]ℬ such that Fm≤Em (cf.  Proposition 11.1 and its proof). Let F'≡∧m∈MFm. Since ∀m∈M:F'≤Fm≤Em, one has F'≤∧m∈MEm.
On the other hand,F'∈[F]ℬ because also [F]ℬ is a lattice of the specified lattice type (cf. (8.1)). Hence, on account of the last inequality, [F]ℬ≤[∧m∈MEm]ℬ, and [∧m∈MEm]ℬ is seen to be the glb of {[Em]ℬ:m∈M}. Thus the map in question preserves the glb operation. Symmetrically (or dually) one can prove preservation of the lub operation.
To prove that the quotient set is an ortho-complemented lattice, let again{[Em]ℬ:m∈M} be an arbitrary subset of ℬ (of cardinality corresponding to the lattice type of ℬ). Since the first Morgan rule (∧m∈MEm)c=∨m∈MEmc is valid in ℬ, and the surjection is such that [E]ℬc=[Ec]ℬ, one obtains the validity of the first Morgan rule (∧m∈M[Em]ℬ)c=∨m∈M[Em]ℬc) in ℬ/~ρ. Symmetrically one proves that also the second Morgan rule is true.
To show thatℬ/~ρ is a Boolean algebra, we have to prove that the glb and the lub operations are mutually distributive. It is, actually, a straightforward consequence of the preservation of these operations in the map E→[E]ℬ:
(11.1)[E]ℬ∧([F]ℬ∨[G]ℬ)=[E]ℬ∧[F∨G]ℬ=[E∧(F∨G)]ℬ.
Since ℬ is a Boolean subalgebra, it is distributive, and one further has
(11.2)lhs=[(E∧F)∨(E∧G)]ℬ=[(E∧F)]ℬ∨[(E∧G)]ℬ=([E]ℬ∧[F]ℬ)∨([E]ℬ∧[G]ℬ).
The converse distributivity of lub with respect to glb is proved analogously.
(B) We prove now thatℬ/~ρ is a lattice of the same lattice type as ℬ. Let {[Em]ℬ:m∈M} be an arbitrary subset of the former (of cardinality corresponding to the lattice type of ℬ). Then ∧m∈MEm, being the glb of {Em:m∈M}, makes (according to the preceding passage) [∧m∈MEm]ℬ the glb of {[Em]ℬ:m∈M}. Symmetrically one deals with the lub operation.The results of this and the next section, in conjunction with all the results of the four articles on the topic ([11–13], and this paper) make it, hopefully, desirable to accept the proposed term “SD  quantum logic’’ in its two meanings (cf.  Remark 2.7).


## 12. Physical Meaning of the Quotient Set

We have seen that SD implication extends absolute implication in a given Boolean subalgebraℬ of quantum logic 𝒫(ℋ), but there is a price to be paid: SD implication is a preorder. Inherent in it is an equivalence relation that makes equivalent events indistinguishable. These are the events that have same probability in the given state ρ, and change this state in the same way when ideally measured.When one makes the transition to the SD quotient setℬ/~ρ (by a homomorphism of a Boolean subalgebra to a Boolean algebra of the same kind, cf.  Theorem 11.3), then one can even forget about the SD implication, because ℬ/~ρ can be characterized by the equivalence relation in ℬ and absolute implication (cf.  Remark 11.2).This throws  new light on SD implication. The SD quotient set, that is, the SD  quantum logic (cf.  Remark2.7), can be viewed, instead of as extending absolute implication in ℬ, as a contraction, a  shrinking of sets of indistinguishable events with respect to ρ into, effectively, single events (the corresponding SD  equivalence classes in ℬ).This is where, whenℬ is a complete lattice, the extreme elements of the SD  equivalence classes acquire physical meaning. As it has been shown in Propositions 6.1 and 6.2 (cf. also  Corollary 9.4), occurrence of a SD  equivalence class amounts to the occurrence of its minimal element, the core element E0ℬ, and its non-occurrence to that of its maximal element E0ℬ+Q0ℬ.If for a set of SD  equivalence classes, that is, elements of the SD  quantum logicℬ/~ρ, the question of occurrence or non-occurrence appears, then the physical answer is given by the corresponding glb and lub operation in the core set 𝒞ℬ (cf.  Definition 8.5 and  Proposition 8.6).


## 13. An Illustration

Let the state space be a separable, that is, a countably infinite dimensional, complex Hilbert spaceℋ, and let {|n〉:n=1,2,…,∞} be a given complete orthonormal basis in ℋ. We consider an observable A defined by its spectral form
(13.1)A=∑n‍an|n〉〈n|,n≠n'⇒an≠an',
and a state vector
(13.2)|Ψ〉≡∑n=2,4,…,∞‍ψn|n〉;∑n=2,4,…,∞‍|ψn|2=1.
The last two sums are over all even integers, and all ψn  in them are non-zero.We define the Boolean subalgebraℬ as the set of all spectral projectors of the observable A. Let 𝒮 be an arbitrary  set of integers including the vacant set, and let 𝒮=𝒮e+𝒮o be the unique decomposition of 𝒮 into the subset of all even integers that it contains and the subset of all odd integers in it. Let us denote by 𝒫 the power set, that is, the set of all subsets (the vacant set included) of the set 𝒩 of all integers.It is easily seen that𝒫 can be used as the index set of the spectral projectors, that is, that ℬ≡{E𝒮:𝒮∈𝒫}, E𝒮≡∑n∈𝒮‍|n〉〈n|. ℬ and 𝒫 are isomorphic posets with respect to the map 𝒫∋𝒮→E𝒮∈ℬ. In ℬ absolute implication "≤" defined in 𝒫(ℋ), and in 𝒫 the set-theoretical inclusion relation "⊆" are the partial orders. Evidently, 𝒫 and ℬ are complete lattices.The SD  equivalence classes are defined by the SD  equivalence relation(13.3)E𝒮~|Ψ〉E𝒮'⟺E𝒮|Ψ〉=E𝒮′|Ψ〉.
Evidently, E𝒮~ρE𝒮'  if and only if 𝒮e=(𝒮')e. Hence. the SD  equivalence classes are the sets {E𝒮:∀𝒮o,𝒮efixed} with E𝒮e as the core elements.To find the greatest elements in the SD  equivalence classes, we note thatE𝒮M≡∑n=1,3,…,∞‍|n〉〈n|, where the summation goes over all odd integers, is the greatest spectral projector that takes |Ψ〉 into zero. It is the Boolean null projector Q0ℬ=E𝒮M. The greatest elements in the SD  equivalence classes are E𝒮e+E𝒮M. The Boolean range projector is Qℬ=(Q0ℬ)c=E𝒮Mc=∑n=2,4,…,∞‍|n〉〈n|, where the summation goes over all even integers.In particular, the projectors0 and E𝒮M are the extreme elements of [E]ℬ when E|Ψ〉=0, and E𝒮Mc and I are the extreme elements of [E]ℬ if Eρ=ρ (cf.  Proposition 8.4).Incidentally, the initial range projector isQ=|Ψ〉〈Ψ|, and the initial null projector is Q0=I-|Ψ〉〈Ψ|. In a given SD  equivalence class [E0ℬ≡E𝒮e] in 𝒫(ℋ) the Lüders range projector is Q-=E0ℬ-(E0ℬ∧Q0).The core set is𝒞ℬ={E𝒮e:∀𝒮e}. The core classes are
(13.4){E𝒮o+E𝒮e:∀E𝒮e,E𝒮e∈𝒞ℬ;E𝒮ofixed}.
Thus an arbitrary element E in ℬ is uniquely decomposed as E=E0ℬ+E', where E0ℬ≡E𝒮e, and E'≡E𝒮o. If one pictures ℬ as a matrix of elements (projectors) so that the SD  equivalence classes are the rows and the core classes are the columns, then E0ℬ shows in which row, and E' displays in which column the element is.One could replace the above density operator|Ψ〉〈Ψ| by a mixed state, and make the observable incomplete with arbitrary multiplicities of the eigenvalues, but still with a purely discrete spectrum. Such an observable would be the general one that is measured exactly. But the additional intricacies would then diminish the transparency of the illustration.



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*Source: 385341-2012-11-11.xml*