quantum measurement

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# Quantum measurement
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## Idea
 {#Idea}

Quantum measurement is [[measurement]] in [[quantum mechanics]].

The "[[projection postulate]]" of [[quantum physics]] asserts ([von Neumann 1932](#vonNeumann32); [Lüders 1951](#Lüders51)) that:

1. measurement of [[quantum states]] is *with respect* to a choice of [[orthonormal linear basis]] $\big\{\vert \psi_b \rangle \big\}_{b : B}$ of the given [[Hilbert space]] $\mathscr{H}$ [[space of quantum states|of]] [[pure quantum states]];

1. the result of measurement on [[pure quantum states]] $\vert \psi \rangle \;\in\; \mathscr{H}$ is

   1. a random value $b \in B$;

   1. the "[[collapse of the wavefunction|collapse]]" of the [[quantum state]] being measured by [[orthogonal linear basis|orthogonal]] [[projection operator|projection]] to the [[linear span]] of the $b$th basis state.

      $$
        \array{
          P_b
            &\colon&
          \mathscr{H}
            &\xrightarrow{\phantom{---}}&
          \mathscr{H}_b \hookrightarrow \mathscr{H}
          \\
          &&
          \vert \psi \rangle
          &\mapsto&
          P_b \vert \psi \rangle
          \mathrlap{
            =
            \vert \psi_b \rangle
            \langle \psi_b \vert \psi \rangle
          }
        }
      $$

In terms of [[mixed quantum states]] represented by [[density matrices]], this prescription translates into a [[quantum operation]] which is given by a [[positive-operator valued measure]] (this is what [Lüders (1951)](#Lüders51) first wrote down).

There are different ways to *[[type theory|type]]* the quantum measurement, taking into account the non-deterministic nature of its outcome:

1. Regarding the [[direct sum]] $\bigoplus_{b \colon B} \mathscr{H}$ of [[Hilbert spaces]] as the [[logical disjunction]] ("or") of [[quantum logic]], one may regard measurement as being the [[linear map]] into the direct sum whose $b$th component is $P_b$.

   > This choice of typing appears (briefly) in [Selinger 2004](#Selinger04), [p. 39](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf#page=39), in a precursor discussion that led to the formulation of the [[quantum programming language]] *[[Quipper]]*.

1. Regarding the measurement outcome $b \in B$ as the observed [[context]] of the actual quantum collapse, one may regard the collapse projection as [[dependent type theory|dependently typed]].

   > Getting from previous option back to this one is known in the the [[Quipper]]-community as *[dynamic lifting](Quipper#ReferencesDynamicLifting)* (namely "of the measured bits back into the context")

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Both of these options naturally emerge and are naturally unified in the "[Quantum Modal Logic](necessity+and+possibility#ModalQuantumLogic)" inherent to [[dependent linear type theory]]: This is discussed at *[[quantum circuits via dependent linear types]]*.

## Properties

## Various

* [[collapse of the wavefunction]]

* [[deferred measurement principle]]

[[!include environment rep of quantum measurement channel - section]]

\linebreak

## The "measurement problem"

In the context of _[[interpretation of quantum mechanics]]_ it is common to speak of the "measurement problem" when referring to the tension between regarding [[quantum physics]] as a [[probability theory|probabilistic]] theory and the idea of realism.

Namely -- by the [above](#Definition) -- a quantum measurement is formally reflected in a change of [[probabilities]]. But since in any given [[measurement]] [[experiment]] one definite outcome is observed, one may wonder how that particular outcome was actually chosen, given that the theory only gives its probability.

(...)

## Related concepts

* [[quantum measurement channel]]

* [[deferred measurement principle]]

* [[measurement-based quantum computation]]

* [[Bell's inequality]]

* [[open quantum system]]

* [[decoherence]]

* [[quantum sensing]]

* [[quantum noise]]

* [[no-cloning theorem]]

* [[quantum reader monad]]

* [[quantum state preparation]]

* [[quantum fluctuation]]

* [[quantum probability]]

* [[interpretation of quantum mechanics]]

* [[mutually unbiased bases]]

## References

### General

The original axiomatization of quantum measurement via the *projection postulate*:

* {#vonNeumann32} [[John von Neumann]], §III.3, §V.1 and §VI  of:

  *Mathematische Grundlagen der Quantenmechanik* (German) (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

* {#Lüders51} [[Gerhart Lüders]]:

  *Über die Zustandsänderung durch den Meßprozeß*, Ann. Phys. **8** (1951) 322–328  [[doi:10.1002/andp.19504430510](https://doi.org/10.1002/andp.19504430510)]

  *Concerning the state-change due to the measurement process*, Ann. Phys. **15** 9 (2006) 663-670 [[pdf](http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf), [[Lueders-StateChange.pdf:file]]]

Review and discussion:

* {#Scheibe73} [[Erhard Scheibe]], §VI of: _The logical analysis of quantum mechanics_, Pergamon Press, Oxford (1973)

* [[Roland Omnès]], §8 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

  > (with an eye towards [[interpretations of quantum mechanics]])

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], §2.4 and Part V in: *The Theory of Open Quantum Systems*, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

  > (form the perspective of [[open quantum systems]] and with focus on [[relativistic field theory]])

* {#Renes15} [[Joseph M. Renes]], §A of: *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)] De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]

* {#Landsman17} [[Klaas Landsman]], Section 11: of _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

In view of dynamical [[quantum decoherence]]:

* [[Maximilian Schlosshauer]], *Decoherence, the measurement problem, and interpretations of quantum mechanics*,  	Rev. Mod. Phys. **76** (2004) 1267-1305 [[arXiv:quant-ph/0312059](https://arxiv.org/abs/quant-ph/0312059), [doi:10.1103/RevModPhys.76.1267](https://doi.org/10.1103/RevModPhys.76.1267)]

See also:

* [[Jürg Fröhlich]], Zhou Gang, *On the Evolution of States in a Quantum-Mechanical Model of Experiments* [[arXiv:2212.02599](https://arxiv.org/abs/2212.02599)]

Discussion in the refined context of ([[local field theory|local]] [[algebraic quantum field theory|algebraic]]) [[quantum field theory]] [[AQFT on curved spacetimes|on Lorentzian manifolds]]:

* [[Christopher J. Fewster]], [[Rainer Verch]], *Measurement in Quantum Field Theory* [[arXiv:2304.13356](https://arxiv.org/abs/2304.13356)]

### Formalization

Brief mentioning of [[type|typing]] and [[categorical semantics]] of quantum measurement

in [[quantum lambda-calculus]]:

* [[Peter Selinger]], [[Benoît Valiron]], p. 143 of: *Quantum Lambda Calculus*, Semantic Techniques in Quantum Computation, Cambridge University Press (2009) 135-172 [[doi:10.1017/CBO9781139193313.005](https://doi.org/10.1017/CBO9781139193313.005), [pdf](https://www.mscs.dal.ca/~selinger/papers/qlambdabook.pdf)]

in the [[quantum programming language]] [[QPL]]/[[Quipper]]:

* {#Selinger04} [[Peter Selinger]], p. 39 of: *Towards a quantum programming language*, Mathematical Structures in Computer Science **14** 4 (2004) 527–586 [[doi:10.1017/S0960129504004256](https://doi.org/10.1017/S0960129504004256), [pdf](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf),  [web](https://www.mathstat.dal.ca/~selinger/papers.html#qpl)]

{#CoeckeEtAlClassicalStructures} Discussion of quantum measurements in terms of [[quantum information theory via dagger-compact categories]], by [[Frobenius algebra]]-structures and the [[quantum reader monad]]:

* {#CoeckePavlović08} [[Bob Coecke]], [[Duško Pavlović]], *Quantum measurements without sums*, in: [[Louis Kauffman]], [[Samuel Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Taylor & Francis (2008) 559-596 [[arXiv:quant-ph/0608035](https://arxiv.org/abs/quant-ph/0608035), [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007)]

* {#CoeckePaquette08} [[Bob Coecke]], [[Eric Oliver Paquette]], *POVMs and Naimark's theorem without sums*, Electronic Notes in Theoretical Computer Science **210** (2008) 15-31 [[arXiv:quant-ph/0608072](https://arxiv.org/abs/quant-ph/0608072), [doi:10.1016/j.entcs.2008.04.015](https://doi.org/10.1016/j.entcs.2008.04.015)]

* [[Bob Coecke]], [[Eric Paquette]], [[Duško Pavlović]], *Classical and quantum structures* (2008) [[pdf](http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf), [[Coecke-Paquette-Pavlovic-CQS.pdf:file]]]

* [[Bob Coecke]], [[Eric Oliver Paquette]], [[Duško Pavlović]], *Classical and quantum structuralism*, in: *Semantic Techniques in Quantum Computation*, Cambridge University Press (2009) 29-69 [[arXiv:0904.1997](https://arxiv.org/abs/0904.1997), [doi:10.1017/CBO9781139193313.003](https://doi.org/10.1017/CBO9781139193313.003)]

* [[Bob Coecke]], [[Duško Pavlović]], [[Jamie Vicary]],  *A new description of orthogonal bases*, Mathematical Structures in Computer Science **23** 3 (2012) 555- 567 [[arXiv:0810.0812](https://arxiv.org/abs/0810.0812), [doi:10.1017/S0960129512000047](https://doi.org/10.1017/S0960129512000047)]

* [[Chris Heunen]], [[Martti Karvonen]], Exp. 6.5 in: *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [TAC:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

and the evolution of the "classical structures"-monad into the "spider"-diagrams (terminology for [special Frobenius normal form](Frobenius+algebra#NormalFormAndSpiderTheorem), originating in [Coecke & Paquette 2008, p. 6](#CoeckePaquette08), [Coecke & Duncan 2008, Thm. 1](#CoeckeDuncan08))
of the [[ZX-calculus]]:

* {#CoeckeDuncan08} [[Bob Coecke]], [[Ross Duncan]], §3 in: *Interacting Quantum Observables*, in *Automata, Languages and Programming. ICALP 2008*, Lecture Notes in Computer Science **5126**, Springer (2008) [[doi:10.1007/978-3-540-70583-3_25](https://doi.org/10.1007/978-3-540-70583-3_25)]

* [[Aleks Kissinger]], §§2 in: *Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories*, MSc thesis, Oxford (2008) [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/Aleks.pdf), [[Kissinger-CLassicalStructures.pdf:file]]]

* [[Aleks Kissinger]], §4 in: *Exploring a Quantum Theory with Graph Rewriting and Computer Algebra*, in: *Intelligent Computer Mathematics. CICM 2009*, Lecture Notes in Computer Science **5625** (2009) 90-105 [[doi:10.1007/978-3-642-02614-0_12](https://doi.org/10.1007/978-3-642-02614-0_12)]

* {#CoeckeDuncan11} [[Bob Coecke]], [[Ross Duncan]], Def. 6.4 in: *Interacting Quantum Observables: Categorical Algebra and Diagrammatics*, New J. Phys. **13** (2011) 043016 [[arXiv:0906.4725](http://arxiv.org/abs/0906.4725), [doi:10.1088/1367-2630/13/4/043016](https://doi.org/10.1088/1367-2630/13/4/043016)]

Textbook account (though without mentioning the monad structure) in:

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], around Lem. 5.61 of: *Categories for Quantum Theory*, Oxford University Press (2019) [[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)]

Generalization to [[Hilbert bundles]]:

* [[Chris Heunen]], [[Manuel L. Reyes]], *Frobenius Structures Over Hilbert $C^\ast$-Modules*, Commun. Math. Phys. **361** (2018) 787–824 [[doi:10.1007/s00220-018-3166-0](https://doi.org/10.1007/s00220-018-3166-0), [arXiv:1704.05725](https://arxiv.org/abs/1704.05725)]

### The Measurement Problem

The article

* [[Klaas Landsman]], Robin Reuvers, _A Flea on Schrödinger's Cat_,  Found. Phys. 43, 373-407 (2013) ([arXiv:1210.2353](http://arxiv.org/abs/1210.2353), [pdf](http://www.math.ru.nl/~landsman/Catresubmission.pdf))

points out that for symmetric systems with a symmetric [[ground state]], already a tiny perturbation mixing the ground state with the first [[excited state]] causes [[spontaneous symmetry breaking]] in a suitable limit, and suggests that this already resolves the measurement problem.

See also

* {#Wallace07} [[David Wallace]], _The Quantum Measurement Problem: State of Play_ ([arXiv:0712.0149](http://arxiv.org/abs/0712.0149))

[[!redirects quantum measurements]]

[[!redirects quantum measurement]]
[[!redirects measurement problem]]


quantum measurement channel

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## Idea

By *quantum measurement channels* one means [[quantum channels]] which express the process of [[quantum measurement]] with its [[quantum state collapse]] *together* with classical uncertainty about which measurement result is obtained. Hence a quantum measurement channel applied to a (normed) [[pure state]] $\left\vert \psi \right\rangle$ produces the [[mixed state]] in which each of the possible [[quantum state collapse|collapsed states]] $P_w \left\vert \psi \right\rangle$ appears with classical [[probability]] $\left\vert\left\langle \psi \vert w \right\rangle\right\vert^2$

More in detail, consider

$$
  \mathscr{H}
     \,\equiv\,
  \underset{w \colon W}{\oplus} \mathscr{H}_w
$$

a [[Hilbert space]] exhibited as the [[direct sum]] of subspaces $\mathscr{H}_w$ [[indexed set|indexed]] by a [[finite set]] $W \,\colon\, FinSet$,
and write

$$
  P_w
  \,\colon\,
  \mathscr{H}
  \twoheadrightarrow
  \mathscr{H}_w
  \hookrightarrow
  \mathscr{H}
$$

for the corresponding [[projection operator]].

By construction this is such that

$$
  \underset{w}{\sum} P_w
  \;=\;
  \mathrm{id}_{\mathscr{H}}
  \,,
$$

whence one also refers to the [[tuple]] $( w \mapsto P_w )$ aas  *[[projection valued measure]]* (here: on the [[finite set]] $W$).

If now $W$ is a [[quantum measurement]]-[[linear basis|basis]] on $\mathscr{H}$, then the [[quantum state collapse|collapse postulate]] of [[quantum mechanics]] says that after measuring $w \,\colon\, W$ for a quantum system previously in [[pure state]] $\left\vert \psi \right\rangle\,\colon\, \mathscr{H}$, the state will have collapsed (up to normalization) according to
$$
  \left\vert \psi \right\rangle
  \;\mapsto \;
  P_w \left\vert \psi \right\rangle
  \,,
$$

hence any [[mixed state]] ([[density matrix]]) will have evolved according to

$$
  \rho \;\mapsto\; P_w \cdot \rho \cdot P_w
  \,.
$$

But if one now in addition considers classical [[probability theory|probabilistic]] uncertainty* as to which measurement result $w$ was actually found (say due to ignorance of the experimentor or imperfection of the measurement device) then all of the resulting pure states $P_w \left\vert \psi \right\rangle$ above are equivally likely and as such constitute the [[mixed state]] which is represented by the [[density matrix]]

$$
  \left\vert \psi \right\rangle
  \left\langle \psi \right\vert
  \;\;\mapsto\;\;
  \underset{w}{\sum}
  \,
  P_w \left\vert \psi \right\rangle
 \left\langle \psi \right\vert P_w
  \,.
$$

In general, if the initial state was [[mixed state|mixed]] to start with, then the stochastic quantum measurement process will be represented by

$$
  \array{
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
    &\overset{meas_W}{\longrightarrow}&
    \mathscr{H}
      \otimes
    \mathscr{H}^\ast
    \\
    \rho
    &\mapsto&
    \;\;\;\;\;
    \mathclap{
    \underset{w}{\sum}
    \,
    P_w \cdot \rho \cdot P_w
    \mathrlap{\,.}
    }
  }
$$

This is a quantum channel, and quantum channels of this form are called *quantum measurement channels*.

## Properties

[[!include environment rep of quantum measurement channel - section]]

## Related concepts

* [[quantum measurement]]

* [[positive operator valued measure]]

* [[quantum channel]]

  * [[unitary quantum channel]]

  * [[partial trace quantum channel]]

## References

The operation of quantum measurement channels on mixed states first appears (before the general concept of "[[quantum channels]]" was formulated) in:

* {#Lüders51} [[Gerhart Lüders]], equation (8) in:

  *Über die Zustandsänderung durch den Meßprozeß*, Ann. Phys. **8** (1951) 322–328  [[doi:10.1002/andp.19504430510](https://doi.org/10.1002/andp.19504430510)]

  *Concerning the state-change due to the measurement process*, Ann. Phys. **15** 9 (2006) 663-670 [[pdf](http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf), [[Lueders-StateChange.pdf:file]]]

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(In generalization of the formula restricted to [[pure state]] that was previously given by [von Neumann 1932](density+matrix#vonNeumann32)).

[[!redirects quantum measurement channels]]

[[!redirects measurement quantum channel]]
[[!redirects measurement quantum channels]]


quantum mechanics

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#Contents#
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{:toc}

## Idea

While [[classical mechanics]] considers deterministic evolution of particles and fields, quantum physics follows nondeterministic evolution where the probability of various outcomes of [[measurement]] may be predicted from the state in a Hilbert space representing the possible reality: that state undergoes a unitary evolution, what means that the generator of the evolution is $\sqrt{-1}$ times a Hermitean operator called the *quantum Hamiltonian* or the [[Hamiltonian operator]] of the system. The theoretical framework for describing this precisely is the __quantum mechanics__. It involves a constant of nature, Planck constant $h$; some quantum systems with spatial interpretation in the limit $h\to 0$ lead to classical mechanical systems (not all: some phenomena including non-integer spin are purely quantum mechanical, but the properties depending on their existence survive in the "classical" limit); in limited generality, one can motivate and find the *nonfunctorial* procedure to single out a [[right inverse]] to taking this *classical limit* under the name [[quantization]].

While quantum mechanics may be formulated for a wide range of physical systems, interpreted as particles, extended particles and fields, the quantum mechanics of fields is often called the [[quantum field theory]] and the quantum mechanics of systems of a fixed finite number of particles is often viewed as the quantum mechanics in a narrow sense.

##### nPOV

Mathematically, despite the basic formalism of quantum mechanics which is sound and clear, there are two big areas which are yet not clear. One is to understand quantization, in all cases -- of particles, fields, strings and so on. The second and possibly more central to nLab is a problem how to define rigorously a wide range of quantum field theories and some related quantum mechanical systems like the hypothetical [[superstring theory]]. Regarding that this is a central goal, we also put emphasis on the [[interpretation of quantum mechanics]] via the picture which is a special case of a FQFT, and where the time evolution functorially leads to evolution operators.

## Definition

We discuss some basic notions of quantum mechanics.

### Quantum mechanical systems

Recall the notion of a _[[classical mechanical system]]_: the formal dual of a real _commutative_ [[Poisson algebra]].

+-- {: .num_defn #QuantumSystem}
###### Definition

A **quantum mechanical system** is a [[star algebra]] $(A, (-)^\ast)$ over the [[complex number]]s. The [[category]] of of quantum mechanical systems is the [[opposite category]] of $\ast$-algebras:

$$
  QuantMechSys := {\ast}Alg_{\mathbb{C}}^{op}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

It makes sense to think of this as a deformed version of a real [[Poisson algebra]] as follows:

* the Poisson-[[Lie algebra|Lie bracket]] of a [[Poisson algebra]] corresponds to the [[commutator]] of the $\ast$-algebra:

  $$
    [a,b] := a b - b a
    \,,
  $$

* the commutative algebra structure of the Poisson algebra coresponds to the [[Jordan algebra]] structure of the $\ast$-algebra, with commutative (but non-associative!) product

  $$
    (a,b) := a b + b a
    \,.
  $$

With this interpretation the [[derivation]]-property of the Poisson bracket over the other product is preserved: for all $a,b,c \in A$ we have

$$
  [a,(b,c)] = ([a,b],c) + (b,[a,c])
  \,.
$$

We thus may regard a non-commutative star-algebra as a _[[nonassociative algebra|non-associative]] [[Poisson algebra]]_ : a [[Jordan-Lie algebra]]. See there for more details.

=--

### Observables and states

+-- {: .num_defn}
###### Definition

Given a quantum mechanical system in terms of a [[star algebra]] $A$, we say

* an **[[observable]]** is an element $a \in A$ such that $a^\ast = a$;

* a **[[state]]** is a [[linear function]] $\rho : A \to \mathbb{C}$ which is _positive_ in the sense that for all $a \in A$ we have $\rho(a a^\ast) \geq 0 \in \mathbb{R} \hookrightarrow \mathbb{C}$.

=--

+-- {: .num_remark}
###### Remark

One can formalize the idea that a quantum mechanical system is like a deformed [[classical mechanical system]] as follows:

To every ${}^\ast$-algebra $A$ is associated its [[poset of commutative subalgebras]] $Com(A)$. Then the corresponding quantum mechanical system is a [[classical mechanical system]] [[internalization|internal]] to the [[sheaf topos]] $Sh(Com(A))$:

* The $\ast$-algebra canonically induces a commuative algebra $\underline A \in Sh(Com(A))$;

* the (classical) [[state]]s of $\underline{A}$ in $Sh(Com(A))$ are in natural bijection with the quantum states externally on $A$;

* the (classical) [[observable]]s of $\underline{A}$ in $Sh(Com(A))$ correspond to the external quantum observables on $A$.

(...details...)

=--

One also says that the internal classical mechanical system $(Sh(Com(A)), \underline{A})$ is the "[[Bohrification]]" of the external quantum system $A$. See there for more details.

### Spaces of states

Given a $\ast$-algebra $A$ together with a [[state]] $\rho$ on it, the [[GNS construction]] provides an [[inner product space]] $H_\rho$ together with an [[action]] of $A$ on $H_\rho$ and a [[vector]] $\Omega = \sqrt(\rho)$ -- the [[vacuum vector]]  -- such that for all $a \in A$ the value of the state $\rho : A \to \mathbb{C}$ is obtained by applying $a$ to $\sqrt{\rho}$ and then taking the inner product with $\sqrt \rho$:

$$
  \rho(A) = \langle \sqrt\rho, a \sqrt \rho\rangle
  \,.
$$

If the [[star algebra]] $A$ happens to be a [[C-star algebra]], then this inner product space is naturally a [[Hilbert space]].

Historically and still often in the literature, such a Hilbert space is taken as a fundamental input of the definition of quantum systems.

Traditionally, [[Dirac]]'s "bra-ket" notation is used to represent vectors in such Hilbert spaces of states, where $|\psi\rangle$ represents a state and $\langle\psi|$ represents its linear adjoint. State evolutions are expressed as unitary maps. Self-adjoint operators represent physical quantities such as position and [[momentum]] and are called [[observables]]. [[measurements|Measurements]] are expressed as sets of [[projectors]] onto the eigenvectors of an observable.

In [[mixed state]] quantum mechanics, physical states are represented as [[density operators]] $\rho$, state evolution as maps of the form $\rho \mapsto U^\dagger \rho U$ for unitary maps $U$, and [[measurements]] are positive operator-valued measures (POVM's). There is a natural embedding of pure states into the space of density matrices: $|\psi\rangle \mapsto |\psi\rangle\langle\psi|$. So, one way to think of mixed states is a probabilistic mixture of pure states.

\[ \rho = \sum_i a_i |\psi_i\rangle\langle\psi_i| \]

Composite systems are formed by taking the tensor product of Hilbert spaces. If a [[pure state]] $|\Psi\rangle \in H_1 \otimes H_2$ can be written as $|\psi_1\rangle \otimes |\psi_2\rangle$ for $|\psi_i\rangle \in H_i$ it is said to be _[[separable]]_. If no such $|\psi_i\rangle$ exist, $|\Psi\rangle$ is said to be _[[entangled]]_. If a mixed state is separable if it is the sum of separable pure states. Otherwise, it is entangled.

### Flows and time evolution

As for classical mechanics, 1-parameter families of flows in a quantum mechanical system are induced from observables $a \in A$ by

$$
  \frac{d}{d \lambda} b_\lambda = \frac{1}{i \hbar}[b_\lambda, a]
  \,.
$$

In a non-relativistic system one specifies an observable $H$ -- called the [[Hamiltonian]] -- whose flow represents the time evolution of the system. (This is the [[Heisenberg picture]].)

We comment on how to interpret this from the point of view of [[FQFT]]:

Quantum mechanics of point particles may be understood as a special case of the formalism of [[quantum field theory]]. It is interpreted as the quantum analog of the [[classical mechanics]] of point particles. Of course, we can take a configuration space of a system of particles looking like the configuration space of a single particle in a higher dimensional manifold.

Remark: related query on the relation between QFT and quantum mechanics (of particles and in general) can be found [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2573&Focus=22060#Comment_22060).

One may usefully think of the quantum mechanics of a point particle propagating on a [[manifold]] $X$ as being $(0+1)$-dimensional [[quantum field theory]]:

the fields of this system are maps $\Sigma \to X$ where $\Sigma \in Riem Bord_1$ are 1-dimensional [[Riemannian manifold]] [[cobordism]]s. These are the _trajectories_ of the particle.

After [[quantization]] this yields a 1-dimensional [[FQFT]] given by a [[functor]]

$$
  U(-) : Riem Bord_1 \to Hilb
$$

from [[cobordism]]s to [[Hilbert space]]s (or some other flavor of [[vector space]]s) that assigns

* to the point the _space of states_ $\mathcal{H}$, typically the space of $L_2$-sections (with respect to a [[Riemannian metric]] on $X$) of the background [[gauge field]] on $X$ under which the particle in question is charged

* to the cobordism of Riemannian length $t$ the operator

  $$
    U(t) := \exp\left(\frac{t}{i \hbar } H \right)
    : \mathcal{H} \to \mathcal{H}
    \,,
  $$

  where $H$ is the [[Hamiltonian operator]], typically of the form $H = \nabla^\dagger \circ \nabla $ for $\nabla$ the [[covariant derivative]] of the given background [[gauge field]].

Such a setup describes the quantum mechanics of a particle that feels forces of backgound [[gravity]] encoded in the [[Riemannian metric]] on $X$ and forces of background gauge fields (such as the [[electromagnetic field]]) encoded in the covariant derivative $\nabla$.

(This is the [[Schrödinger picture]].)

### Quantum subsystems
 {#Subsystems}

+-- {: .num_defn}
###### Definition

For $\mathcal{A}$ an algebra describing a quantum system, def. \ref{QuantumSystem}, a **subsystem** is a subalgebra (a [[subobject]]) $B \hookrightarrow \mathcal{A}$.

Two subsystems $B_1, B_2 \hookrightarrow \mathcal{A}$ are called **independent subsystems** if the [[linear map]]

$$
  B_1 \otimes B_2 \to \mathcal{A}
$$

$$
  (b_1, b_2) \mapsto b_1 \cdot b_2
$$

from the [[tensor product]] of algebras (the [[composite system]]) factors as an [[isomorphism]]

$$
  B_1 \otimes B_2 \stackrel{\simeq}{\to} B_1 \vee B_2 \hookrightarrow \mathcal{A}
$$

through the algebra $B_1 \vee B_2$ that is generated by $B_1$ and $B_2$ inside $\mathcal{A}$ (the smallest subalgebra containing both).

=--

See for instance ([BrunettiFredenhagen, section 5.2.2](#BrunettiFredenhagen)).

+-- {: .num_defn}
###### Definition

Given two independent subsystems $B_1, B_2 \hookrightarrow \mathcal{A}$, and two [[states]] $\rho_1 : B_1 \to \mathbb{C}$ and $\rho_2 : B_2 \to \mathbb{C}$, then the corresponding **product state** $\rho_1 \otimes \rho_2$ on $B_1 \vee B_2$ is defined to be

$$
  (\rho_1 \otimes \rho_2) :
  (b_1 , b_2) \mapsto \rho_1(b_1) \rho_2(b_2)
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

There exist states on $B_1 \vee B_2$ that are not (convex combinations of) product states. This phenomenon is called [[entanglement]].

=--

## Quantum physics in general

More generally, _quantum physics_ is all the known physics not including [[classical physics]] in wider sense; it includes relativistic and nonrelativistic phenomena. Quantum mechanics is the standard formalism with the Hilbert space, unitary evolution etc. which explains theoretically phenomena of quantum physics: in this generality of the formalism a la von Neumann, it includes the quantum field theory. Quantum statistical mechanics in fact uses some additional assumptions not in exact quantum mechanics, which are believed to be derivable eventually (like quantum ergodicity etc.). Thus quantum statistical mechanics may or may not be included within quantum mechanics.

Remark: Another way to look at quantum processes is via [[quantum channels]] which are completely positive trace-preserving maps.

For a previous query about quantum physics (includes experimental phenomena) and quantum mechanics (formalism for such, sometimes with or without statistical principles) see [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2573&Focus=22059#Comment_22059).

## Formulations and formalization

### Order-theoretic structure in quantum mechanics

See _[[order-theoretic structure in quantum mechanics]]_.

### Quantum mechanics in terms of $\dagger$-compact categories

Many aspects of quantum mechanics and [[quantum computation]] depend only on the abstract properties of [[Hilb]] characterized by the fact that it is a [[†-compact category]].

For more on this see

* [[quantum mechanics in terms of †-compact categories]].

## Foundational theorems of quantum mechanics

The following circle of [[theorems]]

* [[Kochen-Specker theorem]]

* [[Gleason's theorem]]

* [[Alfsen-Shultz theorem]]

* [[Harding-Döring-Hamhalter theorem]]

all revolve around the phenomenon that the "[[phase space]]" in quantum mechanics and hence the [[space of quantum states]] are all determined by the [[Jordan algebra]] structure on the [[algebra of observables]], which in turn is determined by the [[poset of commutative subalgebras]] of the algebra of observables. See at _[[order-theoretic structure in quantum mechanics]]_ for more on this.

There is also

* [[Wigner's theorem]]

which says roughly that linear maps between [[spaces of quantum states]] are [[unitary operators]] (or anti-unitary) already when they preserve [[norm]], hence preserve [[probability]].

## Applications of quantum mechanics

Quantum mechanics, as opposed to [[classical mechanics]], is necessary for an accurate description of reality whenever the characteristic scale is sufficiently small. For instance

* In [[chemistry]] ("[[quantum chemistry]]") the properties of [[atoms]] and [[molecules]] are derived from quantum mechanics.

* In [[solid state physics]] the properties of [[metals]] etc. are described by quantum mechanics of electron gases.

* In [[particle physics]] of course, [[quantum field theory]] is the appropriate description.

## Examples

* [[harmonic oscillator]]

* [[hydrogen atom]]

* [[quantum lattice system]]

## Related concepts

* [[classical mechanics]]

  * [[semiclassical approximation]]

* [[quantization]]

  * [[deformation quantization]], [[geometric quantization]]

* **quantum mechanics**

  * [[Planck's constant]]

  * [[Schrödinger picture]], [[Heisenberg picture]], [[Dirac picture]]

  * [[quantum probability]]

  * [[propagator]]

  * [[quantum superposition]], [[quantum fluctuation]], [[quantum entanglement]]

  * [[quantum measurement]]

    * [[positive-operator valued probability measure]]

  * [[geometrical formulation of quantum mechanics]]

  * [[hidden variable theory]],

    * [[EPR paradox]], [[Bell's inequalities]]

    * [[Kochen-Specker theorem]]

  * [[no-cloning theorem]]

  * [[Born-Oppenheimer approximation]]

  * [[quantum tunneling]]

  * [[supersymmetric quantum mechanics]]

  * [[quantum logic]], [[quantum computing]]

    * [[linear logic]], [[linear type theory]]

  * [[finite quantum mechanics in terms of dagger-compact categories]]

  * [[JBW-algebraic quantum mechanics]]

  * [[interpretation of quantum mechanics]]

  * [[quaternionic quantum mechanics]]

* [[quantum field theory]]

  * [[FQFT]], [[AQFT]]

* [[quantum chemistry]]

* [[quantum biology]]

## References
 {#References}

### Historical origins

The seed of quantum mechanics is sown in

* [[Max Planck]] (transl. M. Martius) *The Theory of Heat Radiation* (1914) [[pdf](https://www.gutenberg.org/files/40030/40030-pdf.pdf)]

with the recognition of a quantum of "action": [[Planck's constant]] ([p. 164](https://www.gutenberg.org/files/40030/40030-pdf.pdf#page=178))

Quantum mechanics as such originates with:

* [[Werner Heisenberg]], *Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen*, Zeitschrift für Physik **33** (1925) 879–893 [[doi:10.1007/BF01328377]( https://doi.org/10.1007/BF01328377), [Engl. pdf](http://users.mat.unimi.it/users/galgani/arch/heis25ajp.pdf)]

* [[Max Born]], [[Pascual Jordan]], *Zur Quantenmechanik*, Zeitschrift für Physik **34** (1925) 858–888 [[doi:10.1007/BF01328531](https://doi.org/10.1007/BF01328531)]

* [[Paul A. M. Dirac]], *On the theory of quantum mechanics*, Proceedings of the Royal Society **112** 762 (1926) [[doi:10.1098/rspa.1926.0133](https://doi.org/10.1098/rspa.1926.0133)]

*  [[Paul A. M. Dirac]], *The physical interpretation of the quantum dynamics*, Proceedings of the Royal Society of London **113** 765 (1927) [[doi:10.1098/rspa.1927.0012](https://doi.org/10.1098/rspa.1927.0012)]

* {#HilbertvonNeumannNordheim} [[David Hilbert]], [[John von Neumann]], [[Lothar W. Nordheim]], *Über die Grundlagen der Quantenmechanik*,  Math. Ann. **98** (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

Formulating the [[Born rule]]:

* {#Born26a} [[Max Born]], *Zur Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **37** (1926) 863–867 [[doi:10.1007/BF01397477](https://doi.org/10.1007/BF01397477)]

* {#Born26b} [[Max Born]], *Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **38** (1926) 803–827 [[doi:10.1007/BF01397184](https://doi.org/10.1007/BF01397184)]

* {#Born72} [[Max Born]], *Das Adiabatenprinzip in der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 167–192 [[doi:10.1007/BF01400360](https://doi.org/10.1007/BF01400360)]

* [[Pascual Jordan]], *Über eine neue Begründung der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

Introducing the [[Hilbert space]]-formulation (and the [[projection postulate]]):

* [[John von Neumann]]:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

but see (on [[John von Neumann|von Neumann]]'s further reasoning regarding [[quantum logic]] and then of [[von Neumann algebra factors]]):

* [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

Equivalence of the [[Heisenberg picture]] and the [[Schrödinger picture]]:

* [[Erwin Schrödinger]], *Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen*, Annalen der Physil **384** 8 (1926) 734-756 [[doi:10.1002/andp.19263840804]( https://doi.org/10.1002/andp.19263840804)]

* [[Carl Eckart]], *Operator Calculus and the Solution of the Equations of Quantum Dynamics*, Phys. Rev. **28** 4  (1926) 711-726 [[doi:10.1103/PhysRev.28.711](https://doi.org/10.1103/PhysRev.28.711)]

Introducing the tool of [[group theory]] to quantum physics (cf. *[[Gruppenpest]]*):

* [[Hermann Weyl]], *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

* [[Hermann Weyl]], *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig, (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics* Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

* {#Wigner31} [[Eugene P. Wigner]]: *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), [pdf](https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf)]

* [[Eugene P. Wigner]]: *Group theory: And its application to the quantum mechanics of atomic spectra*, 5, Academic
Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]

Early discussion of [[composite quantum systems]] and their [[quantum entanglement]]:

* [[Erwin Schrödinger]], *Discussion of Probability Relations between Separated Systems*, Mathematical Proceedings of the Cambridge Philosophical Society, **31** 4 (1935) 555-563 [[doi:10.1017/S0305004100013554](https://doi.org/10.1017/S0305004100013554)]

On the historical orogin of the [[canonical commutation relations]]:

* [[Severino C. Coutinho]], *The Many Avatars of a Simple Algebra*, The American Mathematical Monthly **104** 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), [jstor:2975052](https://www.jstor.org/stable/2975052)]

### General
 {#ReferencesGeneral}

Classical textbook accounts:

* {#Dirac78} [[Paul Dirac]], *[[The Principles of Quantum Mechanics]]*, International series of monographs on physics, Oxford University Press (1930, 1935, 1947) [[ISBN:9780198520115](https://global.oup.com/academic/product/the-principles-of-quantum-mechanics-9780198520115)]

* {#Mackey63} [[George Mackey]], *The Mathematical Foundations of Quantum Mechanics: a Lecture-note Volume*, Mathematical physics monograph series, Benjamin (1963), Dover (2004) [[google books](https://books.google.de/books?id=qlpb2mWYmfYC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)]

  > (including an influential proposal for [[quantum logic]])

* Eduard Prugovecki, _Quantum mechanics in Hilbert Space_. Academic Press (1971) [[ISBN: 9780080874081](https://www.elsevier.com/books/quantum-mechanics-in-hilbert-space/prugovecki/978-0-12-566060-0)]

* {#Scheibe73} [[Erhard Scheibe]], _The logical analysis of quantum mechanics_, Pergamon Press Oxford (1973)

  > (focus on the [[interpretation of quantum mechanics]])

* {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], *Operator Algebras and Quantum Statistical Mechanics* -- vol 1: *$C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States.*, Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)]

  > ([[AQFT|algebraic]] [[quantum statistical mechanics]])

* [[James Glimm]], [[Arthur Jaffe]], *[[Glimm-Jaffe|Quantum physics: a functional integral point of view]]*, Springer (1981, 1987) [[doi:10.1007/978-1-4612-4728-9](https://doi.org/10.1007/978-1-4612-4728-9)]

  > (focus on the [[path integral]] in [[constructive quantum field theory]])

* [[Hans Primas]], *Chemistry, Quantum Mechanics and Reductionism*, Springer (1983) [[doi:10.1007/978-3-642-69365-6](https://doi.org/10.1007/978-3-642-69365-6)]

  > (with an eye towards [[quantum chemistry]] and [[interpretation of quantum mechanics]])

* [[Anthony Sudbery]], *Quantum mechanics and the particles of nature: an outline for mathematicians*, Cambridge University Press (1986) [[pdf](http://users.uoa.gr/~navidcon/sudbery.pdf), [spire:240835](https://inspirehep.net/literature/240835)]

* {#Kraus} [[Karl Kraus]], *States, Effects, and Operations -- Fundamental Notions of Quantum Theory*, Lecture Notes in Physics **190** Springer (1983) [[doi:10.1007/3-540-12732-1](https://doi.org/10.1007/3-540-12732-1)]

  > (emphasis on [[effect algebras]] and [[quantum operations]])

* [[Jun John Sakurai]], Jim Napolitano, *Modern Quantum Mechanics*, Cambridge University Press (1985, 2020) [[doi:10.1017/9781108587280](https://doi.org/10.1017/9781108587280), [Wikipedia](https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics)]

More recent textbook accounts:

* [[Paul Busch]], Marian Grabowski, [[Pekka J. Lahti]], *Operational Quantum Physics*, Lecture Notes in Physics Monographs **31**, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

  > (perspective of [[quantum probability theory]] via [[POVMs]])

* [[Chris Isham]], *Lectures on Quantum Theory -- Mathematical and Structural Foundations*, World Scientific (1995) [[doi:10.1142/p001](https://doi.org/10.1142/p001), [ark:/13960/t4xh7cs99](https://archive.org/details/lecturesonquantu0000isha)]

* [[Klaas Landsman]], *[[Mathematical Topics Between Classical and Quantum Mechanics]]*, Springer (1998) [[doi:10.1007/978-1-4612-1680-3](https://doi.org/10.1007/978-1-4612-1680-3)]

* [[Robert B. Griffiths]], *Consistent Quantum Theory*, Cambridge University Press (2002) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), [webpage](https://quantum.phys.cmu.edu/CQT)]

* [[Mikio Nakahara]], Chapter 1 of: _[[Geometry, Topology and Physics]]_, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf]

* [[Serge Haroche]], [[Jean-Michel Raimond]], *Exploring the Quantum: Atoms, Cavities, and Photons*, Oxford University Press (2006) [[doi:10.1093/acprof:oso/9780198509141.001.0001](https://doi.org/10.1093/acprof:oso/9780198509141.001.0001)]

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

  > (focus on the [[geometry]] of [[quantum state spaces]] and culminating in a chapter on [[quantum entanglement]])

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], *The Theory of Open Quantum Systems*, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

  > (focus on [[open quantum systems]])

* [[Teiko Heinosaari]], [[Mário Ziman]], *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

* [[Nik Weaver]], *Mathematical Quantization*, Routledge (2011) [[ISBN 9781584880011](https://www.routledge.com/Mathematical-Quantization/Weaver/p/book/9781584880011)]

* [[Thomas L. Curtright]], [[David B. Fairlie]], [[Cosmas K. Zachos]], *A Concise Treatise on Quantum Mechanics in Phase Space*, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]

  > (in [[Weyl quantization]])

* [[Paul Busch]], [[Pekka J. Lahti]], Juha-Pekka Pellonpää, Kari Ylinen, *Quantum Measurement*, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

  > (perspective of [[quantum probability]] via [[POVMs]])

* {#Landsman17} [[Klaas Landsman]], *Foundations of quantum theory -- From classical concepts to Operator algebras*, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf)]

  > With a focus on the relationship between quantum mechanics and [[representation theory]]

* [[Peter Woit]], *Quantum Theory, Groups and Representations: An Introduction*, Springer 2017 [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7]

On the [[interpretation of quantum mechanics]]:

* [[Roland Omnès]], *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

Lecture notes:

* [[Robert Geroch]], *Geometrical Quantum Mechanics*, University of Chicago (1974) [[pdf](http://strangebeautiful.com/other-texts/geroch-geom-qm.pdf), [[Geroch-GeometricalQuantumMechanics.pdf:file]]]

* Uni Bonn, _[Lecture scripts and Online courses -- Quantum mechanics](https://web.archive.org/web/20001101031920/http://www-cip.physik.uni-bonn.de/~baehren/scripts/quantum.html)_

* [[Valter Moretti]], *Mathematical Foundations of Quantum Mechanics: An Advanced Short Course*, Int. J. Geom. Methods Mod. Phys. **13** Supp. 1 (2016) 1630011 [[arXiv:1508.06951](http://arxiv.org/abs/1508.06951), [doi:10.1142/S0219887816300117](https://doi.org/10.1142/S0219887816300117)]

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

  > (with an eye towards [[quantum probability]] and [[quantum computation]])

* [[Stéphane Attal]], *Quantum Mechanics*, Lecture 5 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Quantum_Mechanics.pdf), [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

  > (with an eye towards [[quantum probability]] and [[quantum noise]])

Further references:

* {#BatesWeinstein} Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_ AMS (1997) [[pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)]

  > (on [[geometric quantization]])

* [[Pierre Cartier]], [[Cécile DeWitt-Morette]], _Functional integration: action and symmetries_, Cambridge Monographs on Mathematical Physics (2006) [[ISBN:9780521143578](https://www.cambridge.org/ae/academic/subjects/physics/theoretical-physics-and-mathematical-physics/functional-integration-action-and-symmetries?format=PB#contentsTabAnchor)]

  > (on rigorous [[path integrals]])

* Leon Takhtajan, _[[Quantum mechanics for mathematicians]]_, Amer. Math. Soc. (2008)

* Michael Movshev, *Concepts of Quantum Mechanics* (2008) [[web](http://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/syllabusfinal.html)]

* [[Franco Strocchi]], *An introduction to the mathematical structure of quantum mechanics*, Advanced Series in Mathematical Physics **28**, World Scientific (2008) [[doi:10.1142/7038](https://doi.org/10.1142/7038)]

* [[Steven Weinberg]], *Lectures on Quantum Mechanics*, Cambridge University Press (2015) [[doi:10.1017/CBO9781316276105](https://doi.org/10.1017/CBO9781316276105)]

* [[Valter Moretti]], _Spectral Theory and Quantum Mechanics -- Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation_, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]

* [[Valter Moretti]], *Fundamental Mathematical Structures of Quantum Theory* -- *Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation*, Springer (2020) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]

* Jan Perina, Z. Hradil, [[Branislav Jurčo]], _[[Quantum optics and fundamentals of physics]]_, Kluwer 1994

Introduction to mathematical foundations of quantum physics in [[quantum probability]], [[operator algebra]]:

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

* {#FroehlichSchubnel15} [[Jürg Fröhlich]],  B. Schubnel, _Quantum Probability Theory and the Foundations of Quantum Mechanics_. In: Blanchard P., Fröhlich J. (eds.) _The Message of Quantum Science_. Lecture Notes in Physics, vol 899. Springer 2015 ([arXiv:1310.1484](https://arxiv.org/abs/1310.1484), [doi:10.1007/978-3-662-46422-9_7](https://doi.org/10.1007/978-3-662-46422-9_7))

* {#Froehlich} [[Jürg Fröhlich]], _The structure of quantum theory_, Chapter 6 in _The quest for laws and structure_, EMS 2016  ([doi](https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure), [doi:10.4171/164-1/8](https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8)).

* {#Landsman17} [[Klaas Landsman]], _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

see also

* {#Hardy01} [[Lucien Hardy]], _Quantum Theory From Five Reasonable Axioms_ ([arXiv:quant-ph/0101012](https://arxiv.org/abs/quant-ph/0101012))

Generalization of the algebraic perspective to [[quantum field theory]] is discussed in

* [[Rudolf Haag]], _[[Local Quantum Physics -- Fields, Particles, Algebras]]_

for more on this see at _[[AQFT]]_ and at _[[perturbative AQFT]]_

Different incarnations of this [[C*-algebra|C*-algebraic]] locality condition are discussed in section 3 of

* Sander Wolters, _Quantum toposophy_,

relating it to the [[topos theory|topos-theoretic]] formulation in

* [[Joost Nuiten]], _[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]_

Aspects of quantum mechanics in [[category theory]] and [[topos theory]] are discussed in

* [[Hans Halvorson]] (ed.) _Deep Beauty -- Understanding the quantum world through mathematical innovation_ Cambridge (2011) ([pdf](http://www.tacron.com/uploads/Deep_Beauty_-_Understg_the_Quantum_World_Through_Mathem._Innov._-_H._Halvorson__Cambridge__2011__WW.pdf))

This discusses for instance [[higher category theory and physics]] and the [[Bohr topos]] of a quantum system.

[[!include quantum information theory via string diagrams -- references]]

[[!redirects quantum physics]]
[[!redirects quantum theory]]

[[!redirects quantum mechanical system]]
[[!redirects quantum mechanical systems]]


Quantum mechanics for mathematicians

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

This is a book entry for

* [[Leon A. Takhtajan]]:

  \linebreak

  **Quantum mechanics for mathematicians**

  \linebreak

  Graduate Studies in Mathematics __95__

  Amer. Math. Soc. (2008)

  [ISBN:978-0-8218-4630-8](https://bookstore.ams.org/gsm-95)

  chapters 1 & 2: [pdf](http://www.ams.org/bookstore/pspdf/gsm-95-prev.pdf)

  [MAA review](https://www.maa.org/press/maa-reviews/quantum-mechanics-for-mathematicians)

on [[quantum mechanics]] in [[mathematical physics]].

category: reference


quantum noise

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#### Measure and probability theory
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#Contents#
* table of contents
{:toc}

## Idea

Broadly, quantum noise is [[noise]] in the context of [[quantum physics]]. (...)

## Examples

* [[bit flip channel]]

## Related concepts

* [[quantum channel]]

* [[error correction]]

* [[quantum error correction]]

## References

* [[Stéphane Attal]], *Quantum Noises*, in: *Open Quantum Systems II*, Lecture Notes in Mathematics **1881** (2006)  [[doi:10.1007/3-540-33966-3_3](https://doi.org/10.1007/3-540-33966-3_3), [pdf](http://math.univ-lyon1.fr/~attal/Mesarticles/Attal.pdf)]

* [[Stéphane Attal]], *Lectures on Quantum Noises* (under construction) [[webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

* A. Clerk, *Quantum noise and quantum measurement*, Chapter 2 in: *Quantum Machines: Measurement and Control of Engineered Quantum Systems*, Oxford University Press (2014) 61-112 [[doi:10.1093/acprof:oso/9780199681181.003.0002](https://doi.org/10.1093/acprof:oso/9780199681181.003.0002)]

See also:

* Wikipedia, *[Quantum noise](https://en.wikipedia.org/wiki/Quantum_noise)*


quantum observable

> See also at *[[algebra of observables]]*.

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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[observable]] in [[quantum physics]].

## Definition in geometric quantization

We consider the notion of quantum observables in the the context of [[geometric quantization]]. See also [[quantum operator (in geometric quantization)]].

### On a symplectic manifold
 {#OnASymplecticManifold}

Let $(X, \omega)$ be a ([[presymplectic manifold|pre]]-)[[symplectic manifold]], thought of as the [[phase space]] of a [[classical mechanics|physical system]].

Assume that $\omega$ is prequantizable (integral) and let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a
[[prequantum bundle]] $E \to X$ [[connection on a bundle|with connection]] for $\omega$, hence with [[curvature]] $F_\nabla = \omega$. Write $\Gamma_X(E)$ for the space of smooth [[sections]] of the [[associated bundle|associated]] [[complex line bundle]]. This is  the _prequantum space of states_.

+-- {: .num_defn}
###### Definition

For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding **pre-quantum operator** is the linear map on prequantum states

$$
  \hat f : \Gamma_X(E) \to \Gamma_X(E)
$$

given by

$$
  \psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi
  \,,
$$

where

* $v_f$ is the [[Hamiltonian vector field]]
corresponding to $f$;

* $\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)$ is the [[covariant derivative]] of sections along $v_f$ for the given choice of prequantum connection;

* $f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)$ is the operation of degreewise multiplication pf sections.

=--

+-- {: .num_remark}
###### Remark

In terms of [[schreiber:Higher geometric prequantum theory]] we may, as discussed there, identify the [[Poisson bracket]] [[Lie algebra]] $\mathfrak{Poisson}(X,\omega)$ with the Lie algebra of the group of automorphism $\exp(O) \colon \nabla \stackrel{\simeq}{\to} \nabla$ regarded in the [[slice (∞,1)-topos|slice]] over $\mathbf{B}U(1)_{conn}$. Moreover, the space of sections is equivalently the space of maps $\Psi \colon \nabla \to \mathbb{C}//U(1)_{conn}$ in the slice from $\nabla$ into the differential refinement of the smooth universal line bundle $\mathbb{C}//U(1) \to \mathbf{B}U(1)$. In this formulation the action of prequantum operators is just the precomposition action

$$
  \widehat{\exp(O)} \colon (\nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn})
  \mapsto
  (\nabla \stackrel{exp(O)}{\to} \nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn})
  \,.
$$

=--

Now after a choice of [[polarization]] a [[quantum state]] is a prequantum [[wave function]] which is covariantly constant along the [[Lagrangian submanifolds]] of the foliation. Not all prequantum operators will respect the space of such quantum states inside all quantum states. Those that do become genuine quantum operators.

+-- {: .num_defn #GeometricQuantumOperator}
###### Definition

Let $\mathcal{P}$ be a [[polarization]] of the [[symplectic manifold]] $(X,\omega)$. then a [[quantum state]] or [[wavefunction]] is a prequantum state $\psi$ such that $\nabla \Psi$ vanishes along the [[leaves]] of the polarization.

A **quantum operator** is a prequantum operator which preserves quantum states among all prequantum states.

=--

+-- {: .num_prop #CompatibiltyOfFlowWithPolarization}
###### Proposition

A prequantum operator given by a [[Hamiltonian]] function $f$ with [[Hamiltonian vector field]] $v_f$ is a quantum operator, def. \ref{GeometricQuantumOperator}, with respect to a given [[polarization]] $\mathcal{P}$ precisely if its flow preserves $\mathcal{P}$, hence precisely if

$$
  [v_f, \mathcal{P}] \subset \mathcal{P}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

If $\mathcal{P}$ is a [[Kähler polarization]] then its underlying [[almost complex structure]] it induces a [[spin^c structure]], as discussed there. If $ \rho \colon G \to QuantMorph(X,\nabla)$ is a [[Hamiltonian action]] (a homomorphism to the [[quantomorphism group]]) such that each prequantum operator $\rho(g)$ is a quantum operator in that it preserves the polarization, by prop. \ref{CompatibiltyOfFlowWithPolarization}, then the corresponding [[spin^c structure]] is $G$-invariant. Accordingly the [[index]] of the [[spin^c Dirac operator]] which gives the [geometric quantization by cohomological quantization](http://ncatlab.org/nlab/show/geometric%20quantization#AsIndexOfSpinCDiracOperator) exists not just in [[K-theory]], where it yields the [[space of quantum states]], but even in $G$-[[equivariant K-theory]], exhibiting a [[representation]] of $G$ on the Hilbert space. This is the action of the quantum observables given by $\rho$ from the point of view of cohomological quantization.

=--

+-- {: .num_example}
###### Example

Over a phase space which is a [[cotangent bundle]] and with respect to the corresponding canonical vertical polarization, a Hamiltonian function is a quantum operator precisely if it is at most linear in the [[canonical momenta]].

=--

See for instance ([Blau, around p. 35](#Blau))

### On an $n$-plectic smooth $\infty$-groupoid

(...)

### Irreducible representations and superselection sectors

The [[space of quantum states]] forms a linear [[representation]] of a given [[algebra of observables]]. The decomposition of that into [[irreducible representations]] is physically the decomposition into [[superselection sectors]].

## Related concepts

[[!include products in pQFT -- table]]

* [[Hamiltonian operator]]

* [[self-adjoint operator]], [[daseinisation]]

[[!include states and observables -- content]]

## References

* [[Roland Omnès]], §3 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

### In algebraic quantum theory

In traditional [[AQFT]]:

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, *Algebra of Observables and State Space* [[doi:10.1007/978-94-009-0491-0_6](https://doi.org/10.1007/978-94-009-0491-0_6)],  Chapter in: *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

Lecture notes in [[perturbative quantum field theory]]:

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory]]_ -- _[Observables](geometry+of+physics+--+perturbative+quantum+field+theory#Observables)_

### In geometric quantization

See also the references at [[geometric quantization]].

Standard facts are recalled for instance around p. 35 of

* {#Blau} [[Matthias Blau]], _Symplectic Geometry and Geometric Quantization_ ([[BlauGeometricQuantization.pdf:file]])

Computation of quantum observables by [[index]] maps in [[equivariant K-theory]] is in (see specifically around p. 8 and 9)

* [[Michèle Vergne]], _Geometric quantization and equivariant cohomology_ ([pdf](http://www.math.jussieu.fr/~vergne/pageperso/articles/CONGEURO.pdf))

[[!redirects quantum observables]]

[[!redirects quantum operator]]
[[!redirects quantum operators]]


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#Contents#
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{:toc}

## Idea

In the contect of [[geometric quantization]] a _prequantum operator_ is a [[linear operator]] is a canonical [[action]] of a [[Hamiltonian]] [[observable]] on the [[space of sections]] of the [[prequantum line bundle]]. After a choice of [[polarization]], if this restricts to the subspace of polarized sections, hence to the actual [[space of states (in geometric quantization)|space of quantum states]] ("[[wavefunctions]]") then this is the operator that [[representation|represents]] the given [[observable]].

More in detail, the [[quantomorphism group]] $\mathbf{Aut}(\mathbf{c}_{conn})$ naturally acts on the [[space of sections]] $\mathbf{\Gamma}_X(E)$ of the [[prequantum line bundle]].

$$
  \widehat {(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E)
  \,.
$$

For $O \in \mathbf{Aut}(\mathbf{c}_{conn})$ a given [[Hamiltonian symplectomorphism]] with [[Hamiltonian]], the corresponding map

$$
  \widehat{O} : \mathbf{\Gamma}_X(E) \to \mathbf{\Gamma}_X(E)
$$

is the _prequantum operator that quantizes $O$_.

Given a choice of [[polarization]] the actual _quantum operator_ corresponding to this is the restriction, if it exists, to the sub-space of polarized sections.

Typically this is considered for [[infinitesimal]] elements of the [[quantomorphism group]], hence for elements of the [[Poisson bracket Lie algebra]] (which then typically end up as [[unbounded operators]]), see def. \ref{PrequantumOperator} below.

## Definition

### For ordinary phase spaces

Let $(X,\omega)$ be a [[presymplectic manifold]].

Let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a
[[prequantum line bundle]] $E \to X$ [[connection on a bundle|with connection]] for $\omega$. Write $\Gamma_X(E)$ for its space of smooth [[sections]], the _[[prequantum space of states]]_.

+-- {: .num_defn #PrequantumOperator}
###### Definition
**([[prequantum operators]])**

For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding **[[quantum operator (in geometric quantization)|quantum operator]]** is the linear map

$$
  \hat f \colon \Gamma_X(E) \to \Gamma_X(E)
$$

given by

$$
  \label{FormulaForPrequantumOperator}
  \psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi
  \,,
$$

where

* $v_f$ is the [[Hamiltonian vector field]]
corresponding to $f$;

* $\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)$ is the [[covariant derivative]] of sections along $v_f$ for the given choice of prequantum connection;

* $f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)$ is the operation of degreewise multiplication pf sections.

=--

+-- {: .num_remark #OriginOfTheFormulaForPrequantumOperators}
###### Remark
**(origin of the formulas for [[prequantum operators]])*+

The formula (eq:FormulaForPrequantumOperator) may look a bit mysterious on first sight. The correction term to the [[covariant derivative]] appearing in this formula is ultimately due to the fact that with $v$ the [[Hamiltonian vector field]] corresponding to a [[Hamiltonian]] $H_v$ via

$$
  \iota_v \omega = d H_v
$$

then the [[Lie derivative]] of $\theta$ (the symplectic potentiation, related by $d \theta = \omega$) is

$$
  \mathcal{L}_v \theta = d \tilde H_v
$$

for

$$
  \tilde H_v = H_v + \iota_v \theta
  \,.
$$

Here the second term on the right is what yields the [[covariant derivative]] in (eq:FormulaForPrequantumOperator), while the first summand is the correction term in (eq:FormulaForPrequantumOperator).

A derivation of these formulas from first principles is given in ([Fiorenza-Rogers-Schreiber 13a](#FiorenzaRogersSchreiber13a), example 3.2.3 and remark 3.3.16).

=--

## Related concepts

[[!include states and observables -- content]]

## References

* {#FiorenzaRogersSchreiber13a} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]],  _[[schreiber:Higher geometric prequantum theory|Higher U(1)-gerbe connections in geometric prequantization]]_, Reviews in Mathematical Physics, Volume 28, Issue 06, July 2016 ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

* {#FiorenzaRogersSchreiber13b} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]]  _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107  142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

[[!redirects prequantum operators]]

[[!redirects prequantum operator]]


quantum optics

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## Idea

__Quantum optics__ is an area of applied [[quantum mechanics]]/[[quantum many-body physics]] concentrating on nonlinear phenomena and coherence of [[light]] and other [[electromagnetic waves]]. Main optical devices exploring nonlinear regime are lasers and their variants, e.g. masers. These phenomena are also important in [[quantum computing]] (and quantum information theory in general), plasma fusion research and study of principal questions in quantum mechanics. Quantum optics also studies microscopic quantum and semiclassical models of interaction of light with matter.

## Related concepts

* [[photon]]

* [[Bose-Einstein condensate]]

## References

Textbook accounts:

* Jan Peřina, Zdeněk Hradil, [[Branislav Jurčo]], _[[Quantum optics and fundamentals of physics]]_, Kluwer, Dodrecht-Boston 1994.

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

See also:

* Wikipedia, *[Quantum optics](https://en.wikipedia.org/wiki/Quantum_optics)*

With an eye towards [[integrable systems]]:

* [[Branislav Jurčo]], *On quantum integrable models related to nonlinear quantum optics. An algebraic Bethe ansatz approach*,  J. Math. Phys. 30, 1739 (1989) ([doi:10.1063/1.528262](http://dx.doi.org/10.1063/1.528262))


This page is about the book

* Jan Peřina, Zdeněk Hradil, Branislav Jurčo, __Quantum optics and fundamentals of physics__, Kluwer, Dodrecht-Boston 1994.

* [Springer page](http://www.springer.com/us/book/9780792330004)

CONTENTS

* Preface

* 1 Introduction 1-4
* 2  Fundamentals of quantum theory 5-53
* 3 Quantum theory of measurement 54-115
* 4 Coherent states 116-155
* 5 Nonclassical optical phenomena and their relations 156-196
* 6 Photon interferences and correlations 197-237
* 7 Quantum optical and Bell's inequalities 238-259
* 8 Quantum optical experiments supporting quantum theory 260-313
* 9 Conclusions 314-315


quantum parallelism

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#### Computation
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[[!include constructivism - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] and specifically in [[quantum information theory]], by *quantum parallelism* one means the property --- to the extent that it holds, see below ---  of quantum processes and hence in particularly of [[quantum computations]] to potentially run "in parallel", analogous to classical [[parallel computing]].

### Traditional accounts

In the literature, the notion of quantum parallelism is traditionally introduced either informally or by way of some explicit example, typically followed by some equally vague warning about not taking the parallelism too literally (e.g. [Haroche & Raimond (2006), p. 96](#HarocheRaimond06)).

But it is easy to be fully precise about this phenomenon:

### Formalization in quantum modal logic

Quantum parallelism is about [[direct sums]] of [[spaces of quantum states]], hence about [[additive disjunction]] in their [[linear logic]] (in direct analogy to how [[quantum entanglement]] is about their [[tensor products]] and hence the [[multiplicative conjunction]] in their [[linear logic]]):

A quantum process $F \;\colon\; \mathscr{H} \multimap \mathscr{H}'$ (a [[linear map]] between [[spaces of quantum states]], such as a [[quantum logic gate]] or [[quantum circuit]]) exhibits quantum parallelism to the extent that it is the [[image]] under the [[direct sum]]-[[functor]] of some $B$-[[indexed set]] of linear maps $F_b \;\colon\;\mathscr{H}_b \multimap \mathscr{H}'_b$:

$$
  F \;=\;
  \underset{b \colon B}{\oplus} F_b
  \;\colon\;
  \underset{b \colon B}{\bigoplus}
  \mathscr{H}_b
  \xrightarrow{\;\;}
  \underset{b \colon B}{\bigoplus}
  \mathscr{H}'_b
  \,.
$$

In [[matrix calculus]] terms this means that we have a block-[[diagonal matrix]]. In [[quantum field theory]] such direct summands are also referred to as *[[superselection sectors]]*.

The caveat which traditional references try to allude to is that few *interesting* processes in [[experiment]] are of this pure direct sum form, while many interesting processes are [[composites]] of one of this form for the case that $b \colon B \;\vdash\; \mathscr{H}'_b = \mathscr{H}'_0$ with the [[codiagonal map]] on $\mathscr{H}'_0$

$$
  \array{
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}'_0
    &
    \xrightarrow{\; codiag \;}
    &
    \mathscr{H}'_0
    \\
    \underset{b \colon B}{\oplus} \vert \psi_b \rangle
    &\mapsto&
    \underset{b \colon B}{\sum} \vert \psi_b \rangle
  }
$$

This latter operation turns the "parallel" quantum states into a single [[quantum superposition]] from which the previous parallel components may not in general be recovered anymore.

(The ingenuity of quantum algorithms which do exhibit [[quantum advantage]] (such as [[Grover's algorithm]] or [[Shor's algorithm]]) is typically all in cleverly arranging the input to the computation such that some partial direct sum-decomposition is being retained after this superposition step.)

Conversely, if one does have a pure direct sum process pre-composed by the [[diagonal map]] on $\mathscr{H}_0$

$$
  \array{
    \mathscr{H}'_0
    &
    \xrightarrow{\; diag \;}
    &
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}_0
    \\
    \vert \psi \rangle
    &\mapsto&
    \underset{b \colon B}{\oplus} \vert \psi \rangle
  }
$$

then this pre-composition may be regarded as preparing a quantum states for parallel [[quantum computation]] on it.

The various fundamental quantum physics phenomena at play here may be identified with four [[modality|modal]]-[[unit of a monad|units]] of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) (as discussed at *[[quantum circuits via dependent linear types]]*):

\begin{imagefromfile}
    "file_name": "QCwthLHT-QuantumModalUnits-221110.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

|  |  |
|--|--|
| [[quantum measurement]] | [[quantum state preparation]] |
| [[quantum superposition]] | quantum parallelism |

## References

Traditional accounts:

* {#HarocheRaimond06} [[Serge Haroche]], [[Jean-Michel Raimond]], *Exploring the Quantum: Atoms, Cavities, and Photons*, Oxford University Press (2006) [[doi:10.1093/acprof:oso/9780198509141.001.0001](https://doi.org/10.1093/acprof:oso/9780198509141.001.0001)]


quantum phase transition

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#Contents#
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{:toc}

## Idea

In [[solid state physics]], a *quantum phase transition* is a [[phase transition]] in a [[quantum material]] that occurs at [[absolute zero temperature]] as some other non-[[temperature]] [[thermodynamics|thermodynamic]] variable is varied (such as [[pressure]] or an external [[magnetic field]]).

Quantum phase transition tend to be associated to (dis-)appearance of [[topological order]], detected by [[topological entanglement entropy]].

## References

Review:

* Matthias Vojta, *Quantum phase transitions*, Rep. Prog. Phys. 66, 2069 (2003) ([arXiv:cond-mat/0309604](https://arxiv.org/abs/cond-mat/0309604))

See also:

* Wikipedia, *[Quantum phase transition](https://en.wikipedia.org/wiki/Quantum_phase_transition)*

[[!redirects quantum phase transitions]]


This is a book entry on:

\linebreak

*  [[James Glimm]], [[Arthur Jaffe]]:

   \linebreak

   **Quantum physics: a functional integral point of view**

   \linebreak

   Springer 1981, 1987

   [doi:10.1007/978-1-4612-4728-9](https://doi.org/10.1007/978-1-4612-4728-9)

   535 pages

on the rigorous [[path integral]] formulation of [[quantum mechanics]] and (mostly [[free quantum field|free]]) [[quantum field theory]] ([[constructive quantum field theory]]).

In order to construct a fully rigorous definition of a [[quantum field theory]], the authors use the Osterwalder-Schrader axioms to define a notion of a [[quantum field]] that is equivalent to the [[Wightman axioms]], see [[Osterwalder-Schrader theorem]].

Note that the ultimate goal of the program, namely the fully rigorous construction of the [[standard model of particle physics]], has not been achived (yet) and has become one of the famous Millennium Problems of the Clay institute.

Online *reviews* of the first edition of the book (the first one along with a book of B. Simon):

* Edward Nelson, Bull. AMS __8__, Number 2 (1983), 378-380.  [link](http://projecteuclid.org/euclid.bams/1183550144)

* [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0461.46051&format=complete)

[[!redirects Glimm-Jaffe]]

[[!redirects Quantum physics -- A functional integral point of view]]

category: reference


quantum probability theory

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#### Measure and probability theory
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=--
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# Contents
* table of contents
{: toc}

## Idea

In [[probability theory]],
the concept of _noncommutative probability space_ or _quantum probability space_ is the generalization of that of _[[probability space]]_ as the concept of "space" is generalized to [[non-commutative geometry]].

The basic idea is to encode a would-be [[probability space]] [[Isbell duality|dually]] in its [[algebra of functions]] $\mathcal{A}$, typically regarded as a [[star algebra]], and encode the [[probability measure]] as a [[state on a star-algebra|state on this star algebra]]

$$
  \langle
    -
  \rangle
  \;\colon\;
  \mathcal{A}
    \longrightarrow
  \mathbb{C}
  \,.
$$

Hence this primarily [[axiom|axiomatizes]] the concept of _[[expectation values]]_ $\langle A\rangle$ ([Segal 65](#Segal65), [Whittle 92](#Whittle92)) while leaving the nature of the underlying [[probability measure]] implicit (in contrast to the classical formalization of [[probability theory]] by [[Andrey Kolmogorov]]).

Often $\mathcal{A}$ is assumed/required to be a [[von Neumann algebra]] (e.g. [Kuperberg 05, section 1.8](#Kuperberg05)). Often $\mathcal{A}$ is taken to be the full algebra of [[bounded operators]] on some [[Hilbert space]] (e.g. [Attal, def. 7.1](#Attal)).

In [[quantum physics]], $\mathcal{A}$ is an [[algebra of observables]] (or a [[local net of observables|local net]] thereof) and $\langle (-)\rangle$ is a particular [[quantum state]], for instance a [[vacuum state]].

The formulation of [[non-perturbative quantum field theory]] from the algebraic perspective of quantum probability is known as _[[algebraic quantum field theory]]_ ([[AQFT]]).

The formulation of [[perturbative quantum field theory]] from the algebraic perspective of quantum probability is known as _[[perturbative algebraic quantum field theory]]_ ([[pAQFT]]).

The sentiment that [[quantum physics]] _is_ quantum probability theory is also referred to as the _[[Bayesian interpretation of quantum mechanics]]_ ("QBism").

## Properties

### As classical probability theory internal to a Bohr topos
 {#AsClassicalProbabilityInBohrTopos}

The idea that

> [[quantum probability]] is "just as" classical [[probability theory]] but generalized to [[non-commutative geometry|non-commutative]] [[probability spaces]], hence, for [[quantum physics]], to quantized [[phase spaces]]

may be made precise and fully manifest by understanding quantum probability theory as being classical [[probability theory]] _[[internalization|internal]]_ to the _[[Bohr topos]]_ of the given [[quantum mechanical system]].

For details see at _[[Bohr topos]]_ the section _[Kinematics in a Bohr topos](Bohr+topos#KinematicsOnBohrTopos)_.

For going deeper, see at _[[order-theoretic structure in quantum mechanics]]_.

### Conditional expectation and Wave function collapse
 {#ConditionalExpectationAndWaveFunctionCollapse}

Quantum probability theory shows that "[[wave function collapse]]" is just part of the formula for [[conditional expectation values]] in [[quantum probability theory]] (e.g. [Kuperberg 05, section 1.2](#Kuperberg05), [Yuan 12](#Yuan12)):

Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.

This means that for $A \in \mathcal{A}$ any [[observable]], its _[[expectation value]]_ in the given [[state on a star-algebra|state]] is

$$
  \mathbb{E}(A) \;\coloneqq\;  \langle A \rangle \in \mathbb{C}
  \,.
$$

More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]]

$$
  \label{RealIdempotent}
  P^\ast = P
  \,,
  \phantom{AAA}
  P P = P
$$

thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is (e.g. [Redei-Summers 06, section 7.3](#RedeiSummers06), see also [Fröhlich-Schubnel 15, (5.49)](#FroehlichSchubnel15), [Fröhlich 19 (45)](#Froehlich16))

$$
  \label{ConditionalExpectation}
  \mathbb{E}(A \vert P)
  \;\coloneqq\;
  \frac{
    \left \langle P A P \right\rangle
  }{
    \left\langle P \right\rangle
  }
  \,.
$$

Now assume a [[star-representation]] $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ of the [[algebra of observables]] by [[linear operators]] on a [[Hilbert space]] $\mathcal{H}$ is given, and that the state $\langle -\rangle$ is a [[pure state]], hence given by an [[vector]] $\psi \in \mathcal{H}$ ("[[wave function]]") via the [[Hilbert space]] [[inner product]] $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as

$$
  \begin{aligned}
    \langle A \rangle
    & \coloneqq
    \left\langle\psi
      \vert A \vert \psi
    \right\rangle
    \\
    & \coloneqq
    \left\langle\psi,
      A \psi
    \right\rangle
  \end{aligned}
  \,.
$$

In this case the expression for the [[conditional expectation value]] (eq:ConditionalExpectation) of an observable $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the [[representation]] $\rho$)

$$
  \begin{aligned}
    \mathbb{E}(A\vert P)
    & =
    \frac{
      \left\langle
        \psi \vert P A P\vert \psi
      \right\rangle
    }{
      \left\langle
        \psi \vert P \vert \psi
      \right\rangle
    }
    \\
    & =
    \frac{
      \left\langle
        P \psi \vert A \vert P \psi
      \right\rangle
    }{
     \left\langle
        P \psi \vert P \psi
     \right\rangle
    }
    \,,
  \end{aligned}
$$

where in the last step we used (eq:RealIdempotent).

This says that _assuming_ that $P$ has been observed in the [[pure state]] $\vert \psi\rangle$, then the corresponding [[conditional expectation values]] are the same as actual [[expectation values]] but for the new pure state $\vert P \psi \rangle$.

This is the statement of "[[wave function collapse]]":

$$
  \vert \psi \rangle \mapsto P \vert \psi \rangle
  \,.
$$

The original [[wave function]] is $\psi \in \mathcal{H}$, and after observing $P$ it "collapses" to $P \psi \in \mathcal{H}$ (up to normalization).

## Related concepts

* [[noncommutative measure theory]]

* [[probability amplitude]]

* [[quantum statistical mechanics]]

* [[quantum information theory]], [[quantum mechanics]]

* [[Bell's inequalities]]

* [[Bayesian interpretation of quantum mechanics]]

* [[quantum logic]]

* [[Bohr topos]]

[[!include states and observables -- content]]

## References

The understanding of [[quantum physics]] as a [[probability theory|probabilistic]] [[theory (physics)|theory]] originates with the formulation of the [[Born rule]] and was made fully explicit in:

* [[Pascual Jordan]], *Über eine neue Begründung der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

* {#vonNeumann32} [[John von Neumann]], *Die quantenmechanische Statistik*, Part III in:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

see also:

* [[Jagdish Mehra]], Helmut Rechenberg, *The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932*, Part 1 in:  *The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941*, Springer (2001)   [[ISBN:978-0-387-98971-6](https://link.springer.com/book/9780387989716)]

with emphasis on [[POVMs]]:

* [[Paul Busch]], Marian Grabowski, [[Pekka J. Lahti]], *Operational Quantum Physics*, Lecture Notes in Physics Monographs **31**, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

* [[Paul Busch]], [[Pekka J. Lahti]], Juha-Pekka Pellonpää, Kari Ylinen, *Quantum Measurement*, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

The axiomatization of [[probability theory]] in terms of the concept of [[expectation values]] (instead of [[probability measures]]) is amplified in:

* [vonNeumann 1932, §IV.1](#vonNeumann32)

*  {#Segal65} [[Irving Segal]], _Algebraic integration theory_, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 ([eculid:1183526903](https://projecteuclid.org/euclid.bams/1183526903))

* {#Whittle92} [[Peter Whittle]], _Probability via expectation_, Springer 1992 ([doi:10.1007/978-1-4612-0509-8](https://link.springer.com/book/10.1007/978-1-4612-0509-8))

* [[Rolando Rebolledo]], *An algebraic view on Probability*, Section 1.2 in: *Complete Positivity and the Markov structure of Open Quantum Systems*, chapter in: [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*,  Lecture Notes in Mathematics **1881**, Springer (2006) 149-182 [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]

Dedicated discussion of quantum probability theory:

* [[Itamar Pitowsky]], *Quantum Probability -- Quantum Logic*, Lecture Notes in Physics **321**, Springer (1989) [[doi:10.1007/BFb0021186](https://doi.org/10.1007/BFb0021186)]

  > (emphasis on the [[Bell's inequalities]] and [[quantum logic]])

* {#Meyer95} [[Paul-André Meyer]], *Quantum Probability for Probabilists*,  Lecture Notes in Mathematics **1538**, Springer 1995 ([doi:10.1007/BFb0084701](https://link.springer.com/book/10.1007/BFb0084701))

* [[Hans Maassen]], _Quantum Probability Theory_, Lecture notes 1998 ([pdf](https://www.math.ru.nl/~maassen/lectures/qp.pdf), [[MaassenQuantumProbability.pdf:file]])

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

* {#RedeiSummers06} [[Miklos Redei]], [[Stephen Summers]], _Quantum Probability Theory_, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics Volume 38, Issue 2, June 2007, Pages 390-417 ([arXiv:quant-ph/0601158](https://arxiv.org/abs/quant-ph/0601158), [doi:10.1016/j.shpsb.2006.05.006](https://doi.org/10.1016/j.shpsb.2006.05.006))

* {#Attal} [[Stéphane Attal]], *Quantum Probability*, Lecture 7 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Quantum_Probability.pdf), [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

* [[Franco Strocchi]], Section 2.4 in: *An introduction to the mathematical structure of quantum mechanics*, Advanced Series in Mathematical Physics **28**, World Scientific (2008) [[doi:10.1142/7038](https://doi.org/10.1142/7038)]

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

* [[Qiaochu Yuan]], _[Noncommutative probability](https://qchu.wordpress.com/2012/08/18/noncommutative-probability/)_, 2012

* {#Yuan12} [[Qiaochu Yuan]], _[Finite noncommutative probability, the Born rule, and wave function collapse](https://qchu.wordpress.com/2012/09/09/finite-noncommutative-probability-the-born-rule-and-wave-function-collapse/)_, 2012

* A. Ibort, V.I. Manko, G. Marmo, A. Simoni, F. Ventriglia, _A pedagogical presentation of a $C^\ast$-algebraic approach to quantum tomography_, Phys. Scr., 84 (2011) 065006 ([arXiv:1204.5231](https://arxiv.org/abs/1204.5231))

* {#FroehlichSchubnel15} [[Jürg Fröhlich]],  B. Schubnel, _Quantum Probability Theory and the Foundations of Quantum Mechanics_. In: Blanchard P., Fröhlich J. (eds.) _The Message of Quantum Science_. Lecture Notes in Physics, vol 899. Springer 2015 ([arXiv:1310.1484](https://arxiv.org/abs/1310.1484), [doi:10.1007/978-3-662-46422-9_7](https://doi.org/10.1007/978-3-662-46422-9_7))

* {#Froehlich16} [[Jürg Fröhlich]], _The structure of quantum theory_, Chapter 6 in _The quest for laws and structure_, EMS 2016  ([doi](https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure), [doi:10.4171/164-1/8](https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8))

* [[Jan Swart]], _Introduction to Quantum Probability_, 2017 ([pdf](http://staff.utia.cas.cz/swart/dict.pdf), [[SwartQuantumProbability.pdf:file]])

* {#Landsman17} [[Klaas Landsman]], _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

Further discussion of quantum probability:

* [[Dror Bar-Natan]], *Two examples in noncommutative probability*, Foundations of Physics, **19** (1989) 97–104  ([doi:10.1007/BF00737769](https://doi.org/10.1007/BF00737769))

* Nicolò Drago, [[Valter Moretti]], _The notion of observable and the moment problem for $\ast$-algebras and their GNS representations_ ([doi:1903.07496](https://arxiv.org/abs/1903.07496), [spire:1725528](http://inspirehep.net/record/1725528))

Discussion of [[density matrices]] and [[entropy]] in quantum probability, via the [[GNS construction]]:

* [[A. P. Balachandran]], T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, _Algebraic approach to entanglement and entropy_, Phys. Rev. A 88, 022301 (2013) ([arXiv:1301.1300](http://arxiv.org/abs/1301.1300))

* [[A. P. Balachandran]], A. R. de Queiroz, S. Vaidya, *Entropy of Quantum States: Ambiguities*, Eur. Phys. J. Plus 128, 112 (2013) ([arXiv:1212.1239](https://arxiv.org/abs/1212.1239), [doi:10.1140/epjp/i2013-13112-3](https://doi.org/10.1140/epjp/i2013-13112-3))

* Paolo Facchi, Giovanni Gramegna, Arturo Konderak, *Entropy of quantum states* ([arXiv:2104.12611](https://arxiv.org/abs/2104.12611))

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quantum programming language

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _quantum programming language_ is a [[programming language]] for [[quantum computation]].

In as far as [[data types]] in [[classical physics|classical]] [[programming languages]] may and often are understood in terms of ([[dependent type theory|dependent]]) [[type theory]], quantum programming languages concern quantum data types such as "[[qbits]]" which may be understood via [[linear logic]] ([Pratt (1992)](#Pratt92)) in ([[dependent linear type theory|dependent]]) [[linear type theory]] (e.g. [Abramsky & Duncan (2005)](#AbramskyDuncan05); [Duncan (2006)](#Duncan06); [Dal Lago & Faggian (2012)](linear+logic#DalLagoFaggianQuote); [Green et al (2013)](#GLRSV13); [Staton (2015)](#Staton15)):

> "A [[quantum programming language]] captures the ideas of [[quantum computation]] in a [[linear type theory]]." [[Staton (2015), p. 1](#Staton15)]

Many existing quantum programming languages are in fact [[domain specific programming languages]] for the description of [[quantum circuits]] compiled from [[quantum logic gates]], and as such often [[domain specific embedded programming language|embedded]] into ambient [[type theories]] (eg. *[[Quipper]]* and *[[QWIRE]]*).

Other QPLs are more algorithmic (such as *[[Q Sharp]]*).

## Related concepts

* [[programming language]]

* [[quantum logic]]

* [[quantum logic gate]], [[quantum circuit]]

* [[quantum computing]], [[quantum simulation]]

* [[linear logic]], [[linear type theory]], [[dependent linear type theory]]

## References

[[!include quantum programming languages -- references]]

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