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#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

This page compiles material related to the textbook:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]

  \linebreak

  **Quantum Information Meets Quantum Matter**

  *From Quantum Entanglement to Topological Phases of Many-Body Systems*

  \linebreak

  Quantum Science and Technology (QST)

  Springer (2019)

  [arXiv:1508.02595](https://arxiv.org/abs/1508.02595)

  [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)

on [[topological phases of matter]]/[[topological order]] characterized via concepts from [[quantum information theory]], such as [[short-range entanglement|short]]- or [[long-range entanglement|long]]-range ([[topological entanglement entropy|topological]]) [[entanglement entropy]].

## Part I -- Basic concepts in quantum information theory

* [[quantum information theory]]

### Sec. 1 -- Correlation and Entanglement

* [[entanglement]]

### Sec. 2 --Evolution of Quantum Systems

* [[quantum system]]

* [[quantum mechanics]]

### Sec. 3 -- Quantum Error Correcting Codes

* [[quantum error correcting code]]

## Part II -- Local Hamiltonians, Ground States, and Many-Body Entanglement

* [[Hamiltonian]]

* [[ground state]]

### Sec. 4 -- Local Hamiltonians and Ground States

### Sec. 5 -- Gapped Quantum Systems and Entanglement Area Law

* [[topological insulator]]

* [[holographic entanglement entropy]]

* [[topological entanglement entropy]]

## Part III -- Topological Order and Long-Range Entanglement

### Sec. 6 -- Introduction to Topological Order

* [[topological phases of matter]]

* [[topological order]]

* [[string-net]]

* [[string-net model]]

### Sec. 7 -- Local Transformations and Long-Range Entanglement

* [[long-range entanglement]]

## Part IV -- Gapped Topological Phases and Tensor Networks

* [[tensor network]]

### Sec. 8 -- Matrix Product States and 1D Gapped Phases

* [[matrix product state]]

### Sec. 9 -- Tensor Product States and 2d Gapped Phases

### Sec. 10 -- Symmetry Protected Topological Phases

* [[symmetry protected topological phase]]

## Part V -- Outlook

### Sec. 11 -- A Unification of Information and Matter
 {#Sec11}

* [[string-net theory]]

* [[string-net liquid]]

* [[string-net condensation]]

... an article should be made about the string-net description of [[gauge theory]], as detailed in this chapter.

\linebreak

***

\linebreak

category: reference


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#### Physics
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[[!include physicscontents]]
=--
#### Category theory
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[[!include category theory - contents]]
=--
#### Monoidal categories
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[[!include monoidal categories - contents]]
=--
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=--

> under construction

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Central aspects of [[quantum mechanics]] with [[finite-dimensional Hilbert space|finite-dimensional spaces]] [[spaces of quantum states|of quantum states]] --- such as [[tensor products]] of [[qbit]] states of relevance in [[quantum information theory]] and [[quantum computation]] --- follow from the formal properties of the [[dagger-compact category]] of [[finite-dimensional Hilbert spaces]].

Conversely, much of [[quantum information theory]] and [[quantum computation]] can thus be formulated [[internalization|internal to]] _any_ [[dagger-compact category]], and general reasoning about †-compact categories themselves yields results about quantum mechanics and quantum computation.

A transparent [[string diagram]] calculus in †-compact categories provides an intuitive and powerful tool for reasoning in $\dagger$-compact categories.

## Quantum mechanical concepts in $\dagger$-compact categories

Let $(C,\otimes,I, \dagger)$ be a [[†-compact category]].

We list various concepts in quantum mechanics and their corresponding incarnation in terms of structures in $C$.

### Classical measurement outcomes

An [[observable]] in quantum mechanics formulated on a [[Hilbert space]] is modeled by a [[self-adjoint operator]], and the classical measurement outcomes of this operator provide, at least under some assumptions, an [[orthogonal basis]] on the [[Hilbert space]].

That, more abstractly, the notion of orthogonal basis of an object can be phrased intrinsically inside any suitable $\dagger$-compact category is the point made in ([CoeckePavlovicVicary](#CoeckePavlovicVicary)).

### Complex phases

The underlying "algebra of quantum amplitudes" of the corresponding quantum mechanical system is the [[endomorphism]] [[monoid]] of the tensor unit

$$
  \mathbb{C}_C = End_C(I)
  \,.
$$

In ([Vicary](#Vicary)) it is shown that in $\dagger$-compact categories with all finite limits over certain "tree-like" diagrams compatible with the $\dagger$-structure, this $\mathbb{C}_C$ has the properties that

* it is a [[field]] of characteristic 0 with involution $\dagger$;

* the subfield $\mathbb{R}_C$ fixed under $\dagger$ is [[order]]able.

If furthermore every bounded sequence of measurements in $C$ with values in $\mathbb{R}_C$ has a least upper bound, then it follows that this field coincides with the [[complex number]]s

$$
  \mathbb{C}_C = \mathbb{C}
$$

and moreover

$$
  \mathbb{R}_C = \mathbb{R}
  \,.
$$

### Completely positive maps {#ComplPosMaps}

The behaviour of [[quantum channel]]s and _completely positive maps_ has an elegant categorical description in terms of $\dagger$-compact categories. See ([Selinger](#SelingerPositive) and [Coecke](#CoeckePositive)).

## Quantum logic

[[symmetric monoidal categories|Symmetric monoidal categories]] such as [[†-compact categories]] have as [[internal logic]] a fragment of [[linear logic]] and as [[type theory]] a flavor of [[linear type theory]]. In this fashion everything that can be formally said about quantum mechanics in terms of [[†-compact categories]] has an equivalent expression in [[formal logic]]/[[type theory]]. It has been argued ([Abramsky-Duncan 05](#AbramskyDuncan05), [Duncan 06](#Duncan06)) that this [[linear logic]]/[[linear type theory]] of quantum mechanics is the correct formalization of "[[quantum logic]]". An exposition of this point of view is in ([Baez-Stay 09](#BaezStay09)).

## Related concepts

* [[quantum circuit]]

* [[quantum teleportation]]

* [[order-theoretic structure in quantum mechanics]]

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

* [[JBW-algebraic quantum mechanics]]

* [[tensor network]], [[string diagram]]

## References
 {#References}

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

The role of [[complex numbers]] in general $\dagger$-compact categories:

* {#Vicary} [[Jamie Vicary]], _Completeness of $\dagger$-categories and the complex numbers_ ([arXiv:0807.2927](http://arxiv.org/abs/0807.2927))

[[!redirects quantum information via dagger-compact categories]]

[[!redirects quantum information theory via dagger-compact categories]]

[[!redirects finite quantum mechanics in terms of †-compact categories]]
[[!redirects finite quantum mechanics in terms of dagger-compact categories]]

[[!redirects quantum mechanics in terms of †-compact categories]]
[[!redirects quantum mechanics in terms of dagger-compact categories]]

[[!redirects finite quantum physics in terms of dagger-compact categories]]
[[!redirects quantum information theory in terms of dagger-compact categories]]

[[!redirects categorical quantum mechanics]]


quantum information theory via string diagrams -- references

### Quantum information theory via String diagrams
 {#ReferencesQuantumInformationTheoryViaStringDiagrams}

#### General
 {#GeneralReferencesQuantumInformationTheoryViaStringDiagrams}

The observation that a natural language for [[quantum information theory]] and [[quantum computation]], specifically for [[quantum circuit diagrams]], is that of [[string diagrams]] in [[†-compact categories]] (see *[[quantum information theory via dagger-compact categories]]*):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) $[$[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)$]$

* {#AbramskyCoecke05} [[Samson Abramsky]], [[Bob Coecke]], *Abstract Physical Traces*, Theory and Applications of Categories, **14** 6 (2005) 111-124. \[tac:14-06, [arXiv:0910.3144](https://arxiv.org/abs/0910.3144)\]

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) $[$[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)$]$

* {#Coecke07} [[Bob Coecke]], *De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure*, Proceedings of the [3rd International Workshop on Quantum Programming Languages (2005)](https://www.mathstat.dal.ca/~selinger/qpl2005/), Electronic Notes in Theoretical Computer Science **170** (2007) 49-72 \[doi:10.1016/j.entcs.2006.12.011, arXiv:quant-ph/0506134\]

On the relation to [[quantum logic]]/[[linear logic]]:

* {#AbramskyDuncan05} [[Samson Abramsky]], [[Ross Duncan]], *A Categorical Quantum Logic*, Mathematical Structures in Computer Science **16** 3 (2006) $[$[arXiv:quant-ph/0512114](http://arxiv.org/abs/quant-ph/0512114), [doi:10.1017/S0960129506005275](https://doi.org/10.1017/S0960129506005275)$]$

* {#Duncan06} [[Ross Duncan]], _Types for quantum mechanics_, 2006 $[$[pdf](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf), [slides](http://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf)$]$

Early exposition with introduction to [[monoidal category|monoidal]] [[category theory]]:

* {#CoeckeKindergarten} [[Bob Coecke]], *Kindergarten quantum mechanics* $[$[arXiv:quant-ph/0510032](http://arxiv.org/abs/quant-ph/0510032)$]$

* {#CoeckeIntroducing} [[Bob Coecke]], *Introducing categories to the practicing physicist* $[$[arXiv:0808.1032](http://arxiv.org/abs/0808.1032)$]$

* {#BaezStay09} [[John Baez]], [[Mike Stay]], *Physics, topology, logic and computation: a rosetta stone* in: *New Structures for Physics*, [[Bob Coecke]] (ed.), Lecture Notes in Physics __813__, Springer (2011) 95-174 $[$[arxiv/0903.0340](http://arxiv.org/abs/0903.0340)$]$

* {#CoeckePractising} [[Bob Coecke]], [[Eric Oliver Paquette]], *Categories for the practising physicist*, in: *New Structures for Physics*, Lecture Notes in Physics **813**, Springer (2010) $[$[arXiv:0905.3010](http://arxiv.org/abs/0905.3010), [doi:10.1007/978-3-642-12821-9_3](https://doi.org/10.1007/978-3-642-12821-9_3)$]$

* {#CoeckePicturalism} [[Bob Coecke]], *Quantum Picturalism*, Contemporary Physics **51** 1 (2010) $[$[arXiv:0908.1787](http://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)$]$

Review in contrast to [[quantum logic]]:

* {#AbramskyCoecke2007} [[Samson Abramsky]], [[Bob Coecke]], *Physics from Computer Science: a Position Statement*, [International Journal of Unconventional Computing **3** 3 (2007)](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/) $[$[pdf](https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf), [ijuc-3-3-p-179-197](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/)$]$

and with emphasis on [[quantum computation]]:

* [[Jamie Vicary]], *The Topology of Quantum Algorithms*, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science (2013) 93-102 $[$[arXiv:1209.3917](https://arxiv.org/abs/1209.3917), [doi:10.1109/LICS.2013.14](https://doi.org/10.1109/LICS.2013.14)$]$

Generalization to [[quantum operations]] on [[mixed states]] ([[completely positive maps]] of [[density matrices]]):

* {#Selinger} [[Peter Selinger]], *Dagger compact closed categories and completely positive maps*, Electronic Notes in Theoretical Computer Science **170** (2007) 139-163  $[$[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018), [web](http://www.mscs.dal.ca/~selinger/papers.html#dagger), [pdf](http://www.mscs.dal.ca/~selinger/papers/dagger.pdf)$]$

* [[Bob Coecke]], [[Chris Heunen]], *Pictures of complete positivity in arbitrary dimension*, Information and Computation **250** 50-58 (2016) $[$[arXiv:1110.3055](https://arxiv.org/abs/1110.3055), [doi:10.1016/j.ic.2016.02.007](https://doi.org/10.1016/j.ic.2016.02.007)$]$

* [[Bob Coecke]], [[Chris Heunen]], [[Aleks Kissinger]],
*Categories of Quantum and Classical Channels*, EPTCS **158** (2014) 1-14 $[$[arXiv:1408.0049](https://arxiv.org/abs/1408.0049), [doi:10.4204/EPTCS.158.1](https://doi.org/10.4204/EPTCS.158.1)$]$

Textbook accounts (with background on relevant [[monoidal category|monoidal]] [[category theory]]):

* [[Bob Coecke]], [[Aleks Kissinger]], *Picturing Quantum Processes -- A First Course in Quantum Theory and Diagrammatic Reasoning*, Cambridge University Press (2017) $[$[ISBN:9781107104228](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB&isbn=9781107104228)$]$

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

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], *Lectures on categorical quantum mechanics* (2012) $[$[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]$]$

* [[Bob Coecke]], [[Stefano Gogioso]], *Quantum in Pictures*, [Quantinuum Publications](https://www.quantinuum.com/publications) (2023) $[$[ISBN 978-1739214715](https://www.amazon.co.uk/dp/1739214714), [Quantinuum blog](https://www.quantinuum.com/news/quantum-in-pictures)$]$

  > (focus on [[ZX-calculus]])

#### Measurement & Classical structures
{#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams}

Formalization of [[quantum measurement]] via [[Frobenius algebra]]-[[structures]] ("[[classical structures]]"):

* {#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)$]$

* {#CoeckePaquettePavlović09} [[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)$]$

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)$]$

#### ZX-Calculus
 {#ReferencesZXCalculus}

Evolution of the "classical structures"-Frobenius algebra ([above](#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams)) into the "spider"-ingredient of the [[ZX-calculus]] for specific control of [[quantum circuit]]-diagrams:

* {#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]], *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]], *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]], *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)$]$

Relating the [[ZX-calculus]] to [[braided fusion categories]] for [[anyon]] [[braiding]]:

* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$


quantum integrand

#Contents#
* table of contents
{:toc}

## Idea

The bosonic [[action functional]] of a [[quantum field theory]] with [[fermion]]s is often not naturally given as a [[function]] on the space of field configurations, but a [[section]] of a [[line bundle]]. (See [[fermionic path integral]]). In the absence of a [[quantum anomaly]] this line bundle is trivializable. Any choice of trivialization makes the action functional an actual function. This choice is called sometimes the _setting of the quantum integrand_ .

## References

The quantum integrand of [[11-dimensional supergravity]] is discussed in

* [[Dan Freed]], [[Greg Moore]], _Setting the quantum integrand of M-theory_ , Communications in mathematical physics,  Volume 263, Number 1, 89-132, DOI: 10.1007/s00220-005-1482-7  ([arXiv:hep-th/0409135](http://arxiv.org/abs/hep-th/0409135))


quantum interference

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

## Idea

[[interference]] of [[wave functions]]/[[probability amplitudes]] via [[superposition]] of their [[complex phases]]

## Related concepts

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

[[!redirects quantum interferences]]


quantum IO monad

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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *quantum IO monad* [[Altenkirch & Green (2010)](#AltenkirchGreen10)] is a [[monad in computer science]] (implemented in [[Haskell]]) which means to model the control of a classical computer over a [[quantum computation]]-device that it is connected to: Much like the classical [[IO monad]] (or generally a [[state monad]]) models the effects of writing to and reading from an external device, so the quantum IO monad means to model the effect of instructing a quantum device to prepare a [[qbit]]-state or to read out ([[quantum measurement|measure]]) that qbit state, etc.

## Related concepts

* [[quantum programming languages]]

* [[IO monad]]

## References

* {#AltenkirchGreen10} [[Thorsten Altenkirch]], [[Alexander Green]], *The quantum IO monad*, Ch. 5 of: Simon Gay, Ian Mackie (eds.): *Semantic Techniques in Quantum Computation* (2010) 173-205 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [doi:10.1017/CBO9781139193313.006](https://doi.org/10.1017/CBO9781139193313.006)]

Exposition:

* [[Thorsten Altenkirch]], *The Quantum IO Monad*, Nottingham (2006) [[pdf](http://www.cs.nott.ac.uk/~txa/talks/qnet06.pdf), [[Altenkirch-QIOMonad.pdf:file]]]

* [[Alexander Green]], *The Quantum IO Monad*, Nottingham  (2007) [[pdf](http://drinkupthyzider.co.uk/asg/pdfs/bctcs07.pdf), [[Green-QIOMonad.pdf:file]] ]

* [[Alexander Green]], *Towards a formally verified functional quantum programming language* (2010) [[eprints:11457](http://eprints.nottingham.ac.uk/11457)]

Implementation in [[Haskell]]:

* [[Alexander Green]], *[hackage.haskell.org/package/QIO](https://hackage.haskell.org/package/QIO)*


quantum L-∞ algebra

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[[!include AQFT and operator algebra contents]]
=--
=--
=--

\tableofcontents

## Idea

(...)

## Related concepts

* [[L-∞ algebra]]

* [[BV-BRST formalism]]

* [[quantum master equation]]

* [[homotopy BV-algebra]]

## References

Introduced in:

* [[Martin Doubek]], [[Branislav Jurčo]], [[Ján Pulmann]], *Quantum L∞ Algebras and the Homological Perturbation Lemma*  Comm. Math. Phys. **367** (2019) 215–240 [[arXiv:1712.02696](https://arxiv.org/abs/1712.02696), [doi:10.1007/s00220-019-03375-x](https://doi.org/10.1007/s00220-019-03375-x)]

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [doi:10.1007/JHEP07(2020)003, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

Review:

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

* Sebastian Albrecht: *Formulation of Batalin-Vilkovisky Field Theories as Homotopy Lie Algebras*, Munich (2022) [[pdf](https://www.theorie.physik.uni-muenchen.de/TMP/theses/sebastian_albrecht.pdf)]

[[!redirects quantum L-∞+algebras]]

[[!redirects quantum L-infinity algebra]]
[[!redirects quantum L-infinity algebras]]

[[!redirects homotopy Lie algebra]]
[[!redirects homotopy Lie algebras]]


quantum lambda-calculus

#Contents#
* table of contents
{:toc}

## Idea

A [[quantum computation]]-analog of the classical [[lambda calculus]], using [[linear type theory|linear types]].

(...)

## Related concepts

* [[lambda-calculus]]

* [[linear type theory]]

* [[quantum programming languages]]

* [[QML]], [[Quipper]]

## References

* [[Benoît Valiron]], *A functional programming language for quantum computation with classical control*, MSc thesis, University of Ottawa (2004) [[doi:10.20381/ruor-18372](http://dx.doi.org/10.20381/ruor-18372), [pdf](https://ruor.uottawa.ca/bitstream/10393/26790/1/MR01625.PDF)]

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 [[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)]

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *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)]

> [[Selinger (2016)](Quipper#Selinger16):] When the [QPL workshop series](https://www.mathstat.dal.ca/~selinger/qpl/) was first founded, it was called "*Quantum Programming Languages*". One year I wasn't participating, and while I wasn't looking they changed the name to "*Quantum Physics and Logic*" --- same acronym!

> Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing [[Selinger & Valiron 2004](#SelingerValiron04)], but the sad thing is: In those days nobody really cared. [...]

> Now it's 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. [...]

> So now people are working on [[quantum programming languages]] *again*.

See also:

* Pablo Arrighi, Alejandro Díaz-Caro, [[Benoît Valiron]], *A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus*, EPTCS **88** (2012) 1-15 [[arXiv:1012.4032](https://arxiv.org/abs/1012.4032), [doi:10.4204/EPTCS.88.1](https://doi.org/10.4204/EPTCS.88.1)]

Implementation as an [[embedded programming language]] in [[Haskell]]:

* Richard Eisenberg, [[Benoît Valiron]], Steve Zdancewic, *Typechecking Linear Data: Quantum Computation in Haskell* [[[EisenbergValironZdancewic-LinearData.pdf:file]]]

A [[categorical semantics]] on [[von Neumann algebras]]:

* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]


quantum lattice system

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* table of contents
{:toc}

## Idea

A _quantum lattice system_ is a [[quantum mechanical system]] whose [[states]]/[[observables]] are arise from basic states/observables assigned to each vertex of some [[lattice]], such as $\mathbb{Z}^d$.

Quantum lattice systems are of relevance in [[solid state physics]], where the lattice typically corresponds to an actual physical lattice of [[atoms]], or as approximations to continuous structures, as in [[lattice gauge theory]].

## Description in AQFT

In the context of [[AQFT]] the typical discussion of a quantum lattice system proceeds as follows.

For $\mathbf{L}$ a [[lattice]], let $P_f(\mathbf{L})$ be its [[poset]] of [[finite set|finite subsets]].

Typically there is associated with each lattice site $x \in \mathbf{L}$ a [[Hilbert space]] $\mathcal{H}_x$, typically finite dimensional, $\mathcal{H}_x \simeq \mathbb{C}^d$, representing finitely many physical degrees of freedom at that site (for instance [[spin]] polarizations of a partical). The Hilbert space associated to some $\Lambda \in P_f(\mathbf{L})$ is then typically the [[tensor product]] $\mathcal{H}_\Lambda \coloneqq \otimes_{x \in \Lambda} \mathcal{H}_x$, and hence the corresponding [[algebra of observables]] is that of [[bounded operators]] $\mathcal{A}_\Lambda \coloneqq \mathcal{B}(\mathcal{H}_\Lambda)$.

The complete algebra of observables is then the [[colimit]]/[[inductive limit]]

$$
  \mathcal{A} \coloneqq \underset{\longrightarrow_{\Lambda}}{\lim} \mathcal{A}_\Lambda
  \,.
$$

in the [[category]] of [[C-star algebras]], which is the [[norm]]-closure of the [[union]] of all these algebras. If we call each $\mathcal{A}_{\Lambda}$ a _local algebra_, then this is the algebra of those observables that can be approximated to arbitrary precision in norm by a local algebra. This is called a [[uniformly hyperfinite algebra]].

Next, there is typically for each $\Lambda \in P_f(\mathbf{L})$ a [[Hamiltonian]] $H_\Lambda$ with $\exp(i t H) \in \mathcal{A}_\Lambda$ for all $t \in \mathbb{R}$. Under suitable (subtle) conditions (e.g. [Bratteli-Robinson, part I, chapter 6.2](#BratteliRobinson)) the 1-parameter time evolution $\mathbb{R} \to \mathrm{Aut}(\mathcal{A}_\Lambda)$ induced by this unitary extends to an suitably continuous family of automorphisms of $\mathcal{A}$ (this is no longer given in general by an [[inner automorphism]] even though for each finite $\Lambda$ it is).

## Related concepts

* [[harmonic oscillator]]

* [[hydrogen atom]]

## References

Detailed discussion of quantum lattice systems in [[statistical physics]] formulated with [[AQFT]] tools is in

* O. Bratteli, D. Robinson, _Operator algebras and quantum statistical mechanics_ Texts and monographs in physics, Springer, New York (part I 1987, part II 1997)
 {#BratteliRobinson}

[[!redirects quantum lattice systems]]


quantum linear group

__Quantum linear semigroups__ are [[bialgebra]]s which are deformations of bialgebras of coordinate functions on the groups of $n\times n$-matrices for some $n$. They belong to the class of [[matrix bialgebra]]s. The usual notation for the one-parametric version is $\mathcal{O}(M_q(n))$ or sometimes simply $\mathcal{M}_q(n)$.

Suppose we are given $n\times n$ matrices $P = (p_{ij})$ and
$Q = (q_{ij})$ with invertible entries in the ground field $F$,
for which there exist $q$ such that
$$
  p_{ij} q_{ij} = q^2, q_{ij} = q^{-1}_{ji},\,\,\,\,\,i \lt j,\,\,and \,\,\,\,\,\,
q_{ii} = p_{ii},\,\,\,\,\,\,\, for\,\,\, all\,\,\,\, i.
$$

The multiparametric quantized matrix bialgebra (synonym:
multiparametric quantum linear semigroup) $\mathcal{O}(M_{P,Q}(F,n)):= F \langle T^i_j , i,j = 1,\ldots, n\rangle/I$,
where $I$ is the ideal spanned by the relations
\[\array{
T^k_i T^k_j = q_{ij} T^k_j T^k_i, & i \lt j \\
T^k_i T^l_i = p_{kl} T^l_i T^k_i, & k \lt l \\
q_{ij} T^k_j T^l_i = p_{kl} T^l_i  T^k_j, & i\lt j,\,\,\,\,k\lt l \\
T^k_i T^l_j - q_{ij} q^{-1}_{kl} T^l_j T^k_i
= (q_{ij}-p_{ij}^{-1}) T^k_j T^l_i,& i\lt j,\,\,\,\,k\lt l
}\]
$\mathcal{M} = \mathcal{O}(M_{P,Q}(F,n))$ is a bialgebra
with respect to the "matrix" comultiplication
which is the unique algebra homomorphism $\Delta : \mathcal{M}
\to\mathcal{M} \otimes\mathcal{M}$
extending the formulas which are written in the matrix form as
$\Delta T^i_j = \sum T^i_k \otimes T^k_j$ with
counit $\epsilon T^i_j = \delta^i_j$ (Kronecker delta). This means that it is
a matrix bialgebra with basis $\{T^i_j\}_{i,j=1,\ldots,n}$, in fact a free matrix bialgebra over $F$.

In these conventions, the 1-parametric version $\mathcal{O}(M_q(F))$ is obtained as
a special case when $P = Q$ and
$q_{ij} = q$ for $i \lt j$ and $q_{ij} = q^{-1}$ for $i \gt j$.

__Quantum linear groups__ are Hopf algebras which are quantum deformations of Hopf algebras of coordinate functions on the general linear group or special linear group. There exist one parametric and many parametric versions as well as super analogues. They belong to the class of [[matrix Hopf algebra]]s.

The usual notation for one-parametric versions is $\mathcal{O}(GL_q(n))$, $\mathcal{O}(SL_q(n))$ and variants thereof.

## Bibliography

* related $n$Lab pages: [[quantum group]], [[quantum Gauss decomposition]], [[quantized function algebra]], [[general linear group]]
* Yu. I. Manin, _Quantum groups and non-commutative geometry_, CRM,
Montreal 1988.
* Yu. I. Manin, _Multiparametric quantum deformation of the general
linear supergroup_, Comm. Math. Phys. 123 (1989) 163--175.
* B. Parshall, J.Wang, _Quantum linear groups_, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.
* E. E. Demidov, _Multiparameter quantum deformations
of the group $GL(n)$_, (Russian) Uspehi Mat. Nauk 46 (1991),
no. 4 (280) 147--148; translation in Russian Math. Surveys
46 (1991) no. 4, 169--171.
* M. Hashimoto, T. Hayashi, _Quantum multilinear algebra_, Tohoku Math. J. 44 (1992) 471-521.
* M. Artin, W. Schelter, J. Tate, _Quantum deformations of $GL_n$_,
Commun. Pure Appl. Math. XLIV, 879--895 (1991)
* [[Zoran Škoda]], _Localizations for construction of quantum coset spaces_, [math.QA/0301090](http://front.math.ucdavis.edu/math.QA/0301090), in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.
* Z. Škoda, _Every quantum minor generates an Ore set_,
International Math. Res. Notices 2008, rnn063-8;
[math.QA/0604610](http://arxiv.org/abs/math/0604610)
* Siân Fryer, _From restricted permutations to Grassmann necklaces and back again_, [arxiv/1511.06664](http://arxiv.org/abs/1511.06664)
* Hechun Zhang, R.B.Zhang, _Dual canonical bases for the quantum special linear group and invariant subalgebras_, Letters in Mathematical Physics (2005) __73__:165–181 [doi](https://doi.org/10.1007/s11005-005-0015-9); _Dual canonical bases for the quantum general linear supergroup_, J. Algebra __304__ (2006) 1026–1058 [doi](https://doi.org/10.1016/j.jalgebra.2005.11.023)
* Hans Plesner Jakobsen, Hechun Zhang, _The center of the quantized matrix algebra_, J. Algebra __196__:2 (1997) 458-474 [doi](https://doi.org/10.1006/jabr.1997.7121)
* K.R. Goodearl, E.S. Letzter, _Prime factor algebras of the coordinate ring of quantum matrices_, Proc. Amer. Math. Soc. __121__ (1994) 1017-1025 [doi](https://doi.org/10.1090/S0002-9939-1994-1211579-1)


quantum logic

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### Context
#### Type theory
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=--
#### Quantum systems
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[[!include quantum systems -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Broadly speaking, _quantum logic_ is meant to be a kind of [[formal logic]] that is to traditional formal logic as [[quantum mechanics]] is to [[classical mechanics]]: a formal framework which is supposed to be able to _express_ the statements whose [[semantics]] is the totality of all what is verifiable by [[measurement]] in a [[quantum system]] ([[quantum measurement]]).

In its traditional and default meaning due to ([Birkhoff-vonNeumann 1936](#BirkhoffvonNeumann36)) "quantum logic" refers specifically to the [[orthocomplemented lattice]] of [[closed subspace|closed]] [[linear subspaces]] of a [[Hilbert space]] [[space of quantum states|of quantum states]]. Later it was proposed ([Yetter 90](#Yetter90), [Pratt 92](#Pratt92), [93](#Pratt93), [Abramsky-Duncan 05](#AbramskyDuncan05), [Girard 11](#Girard11)) that a better way to think of the BvN quantum lattices is as the [[propositions]] in [[linear logic]] ([Girard 87](#Girard)), the [[categorical logic]] of [[symmetric monoidal categories]].

There is also the proposal ([Heunen-Landsman-Spitters 07](#HeunenLandsmanSpitters07)) that quantum logic should be understood as being the [[internal logic]] of [[Bohr toposes]].

In [[quantum computing]] the quantum analog of classical [[logic gates]] are called _[[quantum logic gates]]_.

## History and Critique
 {#History}

Typically  in the literature the term "quantum logic" is taken to refer very specifically to the first proposal for such a formalization due to [Birkhoff-von Neumann 1936](#BirkhoffvonNeumann36). In this proposal, given a [[quantum mechanical system]] with a [[Hilbert space]] [[space of states|of states]], the logical [[propositions]] about the system are taken to correspond to (the [[projections]] to) [[closed subspaces]], with [[implication]] given by inclusion of such subspaces.  Hence Birkhoff-von Neumann quantum logic is given by the [[lattice]] of [[closed subspace|closed]] [[linear subspaces]] of [[Hilbert spaces]] (regarded as a [[Hilbert lattice]]).

This formalization is often understood as being the default meaning of "quantum logic". But the proposal has later been much criticised, for its lack of key properties that one would demand of any [[formal logic]].

{#NonLogic} For instance in [Abramsky & Coecke (2007)](#AbramskyCoecke2007) it is called a "non-logic":

> {#NonLogic} The term quantum logic is usually understood in connection with the 1936 Birkhoff-von Neumann proposal to consider the ([[closed subspace|closed]]) [[linear subspaces]] of a [[Hilbert space]] ordered by inclusion as the formal expression of the logical distinction between quantum and classical physics. While in classical logic we have deduction, the linear subspaces of a Hilbert space form a non-distributive lattice and hence there is no obvious notion of implication or deduction. Quantum logic was therefore always seen as logically very weak, or even as a non-logic. In addition, it has never given a satisfactory account of compound systems and entanglement.

Here by "no [[deduction]]" is meant "no [[deduction theorem]]".

And for example in ([Heunen-Landsman-Spitters 07, p. 4](#HeunenLandsmanSpitters07)) it says the following:

> Attractive and revolutionary as this spatial quantum "logic" may appear  it faces severe problems. The main logical drawbacks are:

> 1. Due to its lack of distributivity, quantum 'logic' is difficult to interpret as a logical structure.

> 2. In particular, despite various proposals no satisfactory implication operator has been found (so that there is no deductive system in quantum logic).

> 3. Quantum 'logic' is a [[propositional logic|propositional]] language; no satisfactory generalization to [[predicate logic]] has been found.

> Quantum logic is also problematic from a physical perspective. Since (by various theorems and wide agreement) quantum probabilities do not admit an ignorance interpretation, $[0, 1]$-valued truth values attributed to propositions by pure states via the [[Born rule]] cannot be regarded as sharp (i.e. {0, 1}-valued) truth values muddled by human ignorance. This implies that, if $X = [a \in \Delta]$ represents a quantum-mechanical proposition, it is wrong to say that either $x$ or its [[negation]] holds, but we just do not know which of these alternatives applies. However, in quantum logic one has the law of the excluded middle in the form $x \vee x^\perp = 1$ for all $x$. Thus the formalism of quantum logic does not match the probabilistic structure of quantum theory responsible for its empirical content.

But notice that one may argue that the first three points here are squarely resolved by thinking of BvN-quantum logic as embedded into [[linear logic]], we come back to this in a moment. Concerning the last point one might argue that the propositions in BvN-quantum logic concern the [[quantum measurement]]-outcomes (only), for which, at least in some  [[interpretation of quantum mechanics|interpretations]], it does make sense to speak of a definite result.

In [Girard 2011, page xii](#Girard11) it says:

> {#MagisteralMistake} Among the magisterial mistakes of logic, one will first mention quantum logic, whose ridiculousness can only be ascribed to a feeling of superiority of the language – and ideas, even bad, as soon as they take a written form – over the physical world. Quantum logic is indeed a sort of punishment inflicted on nature, guilty of not yielding to the prejudices of logicians... just like Xerxes had the Hellespont – which had destroyed a boat bridge – whipped.

For more and more objective criticism see [Girard 2011, section 17](#Girard11).

Girard had introduced the class of [[formal logic]] systems called _[[linear logic]]_ and it has been argued that [[linear logic]] and more generally _[[linear type theory]]_ does faithfully capture the essence of [[quantum mechanics]] ([Yetter 90](#Yetter90). [Pratt 92](#Pratt92), [Abramsky-Duncan 05](#AbramskyDuncan05), [Duncan 06](#Duncan06)), see ([Baez-Stay 09](#BaezStay09) for an introductory exposition). This is due to the fact that the [[categorical semantics]] of linear logic is in [[symmetric monoidal categories]] such as those used in the description of [[finite quantum mechanics in terms of dagger-compact categories]]. In particular the [[category]] of (finite dimensional) [[Hilbert spaces]] whose [[subobjects]]/[[propositions]] form the Birkhoff-von Neumann style quantum logic does interpret [[linear logic]].

This is stated explicitly for instance in ([Pratt 92, p.4](#Pratt92)):

> These objections are overcome in the extension of quantum logic to linear logic as a dynamic quantum logic.

Notice that the [[subobjects]] in a category of ([[finite-dimensional vector space|finite dimensional]]) [[Hilbert spaces]], and hence the [[propositions]] in the [[categorical logic]] of [[Hilbert spaces]], are the ([[closed subspace|closed]]) [[linear subspaces]]. Therefore Birkhoff-von Neumann quantum logic is indeed subsumed as a fragment of [[linear logic]]. This (obvious) fact was highlighted in [Crown 1975](#Crown75) and then later with more of [[categorical logic]] in place and emphasizing [[dagger-category|dagger]]-structures in [Heunen 2008](#Heunen08), [Harding 2008](#Harding08) [Heunen-Jacobs 2009](#HeunenJacobs09), [Coecke-Heunen-Kissinger 2013](#CoeckeHeunenKissinger13). Also [CCGP 09, section 9.3](#CCGP09)):

> both orthologic (or weakenings thereof) and linear logic share the failure of lattice distributivity. In particular, the fragment of linear logic that includes just [[negation]] and the additive connectives is nothing but a version of the paraconsistent quantum logic PQL.

That seems to make much of the above-listed criticism appear in a different light. For instance there is also a natural notion of [[dependent linear type theory]] and that does yield a well-behaved kind of [[predicate logic]] with [[quantifiers]] for linear logic.

## Approaches

### Birkhoff-von Neumann quantum logic
 {#BirkhoffVonNeumannQuantumLogic}

The original proposal for quantum logic by [Birkhoff & von Neumann 1936](#BirkhoffvonNeumann36) (BvN quantum logic) is to regard, given a [[quantum system]] with [[Hilbert space]] [[space of quantum states|of states]] $\mathscr{H}$, the [[Hilbert lattice]] of [[closed subspace|closed]] [[linear subspaces]] $V \subset_{cl} \mathscr{H}$ as the set of [[propositions]] about the quantum system.

Then:

* logical [[conjunction]] is given by intersection of  [[linear subspaces]]

  $$
    V_1 \wedge V_2
    \;\coloneqq\;
    V_1 \cap_{\mathscr{H}} V_2
    \,,
  $$

* logical [[disjunction]] is given by forming the [[linear span]]

  $$
    V_1 \vee V_2
    \;\coloneqq\;
    Span_{\mathscr{H}}(V_1 \, V_2)
  $$

  of two [[linear subspaces]].

* logical [[negation]] is given by forming [[orthogonal complement]]:

  $$
    \not V \;\coloneqq\; V^{\perp_{\mathscr{H}}}
    \,.
  $$

Notice that [[quantum states]] in the conjunction $V_1 \vee V_2$ may be [[linear combinations]] of states in $V_1$ and $V_2$, in reflection of the [[superposition principle]] of [[quantum mechanics]].
As a result, the BvN disjunction does not [[distributivity|distribute]] over conjunction, and hence the BvN lattice of quantum propositions is not a [[distributive lattice]].

### Tensor logic of Quantales

The [[tensor product of Hilbert spaces]] -- which in the jargon of [[linear logic]] would be called the *[[multiplicative conjunction]]* $\otimes$ --  makes the [above](#BirkhoffVonNeumannQuantumLogic) [[Hilbert lattice]] a _[[quantale]]_.

### Linear logic and dependent linear type theory
 {#LinearLogicAndDependentLinearTypeTheory}

We point out that --- at least for the [finite dimensional Hilbert spaces](Hilbert+space#FiniteDimensionalInnerProductSpaces) relevant in [[quantum information theory]] --- [BvN quantum logic](#BirkhoffVonNeumannQuantumLogic) is just the [[linear logic]] of a kind of [[linear type|linearly]]-[[dependent type|dependent]] [[linear types]] inside [[dependent linear type theory]] [[categorical semantics|interpreted]] in [[VectBund]] over the [[complex numbers]].

(Notice that it is the linearity of [[linear logic]], which enforces (see [there](linear+logic#AbsenceOfWeakeningAndContraction) for more) both the [[no-cloning theorem]] and the [[no-deleting theorem]] of [[quantum physics]]/[[quantum information theory]].)

Namely, the [[linear types]] in [[VectBund]] form the subcategory [[FinDimVect]]. Any [[finite dimensional vector space]] admits a [[hermitian inner product]] making it a [[Hilbert space]]. For the discussion of logical [[conjunction]] and [[disjunction]] this Hilbert space structure plays no role and we may and will ignore it.

So given $\mathscr{H} \in FinDimVect$, a [[proposition]] about it in the [[internal logic]] sense of [[dependent type theory]] is, in [[categorical semantics]], a [[(-1)-truncation|(-1)-truncated]] [[object]] in the [[slice category]] over $\mathscr{H}$, hence a [[linear subspace]]

$$
  (\mathscr{V} \hookrightarrow \mathscr{H})
  \;\in\;
  \tau_{-1}\big(FinDimVect_{/\mathscr{H}}\big)
  \,.
$$

We need to consider the (co)fiber products of such objects (discussed [here](FinDimVect#FiberCoProducts) at *[[FinDimVect]]*):

Logical [[conjunction]] of such slice objects is given by their [[cartesian product]] in the slice, hence by the [[fiber product]] of their inclusion maps in $FinDimVect$. This is evidently given by the intersection of the subspaces:

\begin{tikzcd}[
  sep=20pt
]
  &
  \mathcal{V}_1 \cap_{_{\mathcal{H}}} \mathcal{V}_2
  \ar[d, phantom, "\simeq"{rotate=-90}]
  \\[-10pt]
  &
  \mathcal{V}_1 \times_{\mathcal{H}} \mathcal{V}_2
  \ar[dl]
  \ar[dr]
  \ar[dd, phantom, "\scalebox{.6}{(pb)}"]
  \\
  \mathcal{V}_1
  \ar[dr, hook]
  &&
  \mathcal{V}_2
  \ar[dl, hook']
  \\
  &
  \mathcal{H}
\end{tikzcd}

Logical [[disjunction]] of such slice objects is given by the [[(-1)-truncation]] of their [[coproduct]] in the slice. Using that coproducts in slices are computed in the ambient category (see [there](over+category#LimitsAndColimits)) this is given by the [[image]] of the abstract [[direct sum]] of the subspaces, which is the [[linear span]]:

\begin{tikzcd}[
  sep=30pt
]
  \mathcal{V}_1
  \ar[r, hook]
  \ar[ddr, hook]
  &
  \mathcal{V}_1 \oplus \mathcal{V}_2
  \ar[d, ->>]
  &
  \mathcal{V}_2
  \ar[l, hook']
  \ar[ddl, hook']
  \\[-10pt]
  &
  \mathrm{Span}_{\mathcal{H}}(\mathcal{V}_1, \mathcal{V}_2)
  \ar[d, hook]
  \\[+10pt]
  &
  \mathcal{H}
\end{tikzcd}

The historical debate [above](#History) about justifying BvN logic as a formal logic is ultimately rooted in the fact that [[Vect]] is not a [[cartesian closed category]] (much less a [[locally cartesian closed category]]) so that its [[internal logic]] is exotic.

{#QLogicInsideVectBund} (...)

> fixing a mistake...

### Bohr toposes

See at _[[Bohr topos]]_ for more.

## Related concepts

* [[linear logic]]

* [[quantum programming language]], [[quantum computation]]

* [[quantum probability]]

* [[interpretation of quantum mechanics]]

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

* [[collapse of the wave function]]

* [[qbit]]

* [[quantum circuit]]

* [[effect algebra]]

* [[quantum mechanics]]

  * [[quantum computing]]

  * [[beable]]

* [[computable physics]]

* [[quantale]],

* [[Bohr topos]]

## Literature
 {#References}

### Original articles

The original proposal on quantum logic:

* {#BirkhoffvonNeumann36} [[Garrett Birkhoff]], [[John von Neumann]]: *The logic of quantum mechanics*, Annals of Mathematics **37** 823-843 (1936) [[doi:10.2307/1968621](https://doi.org/10.2307/1968621), [pdf](http://www.fulviofrisone.com/attachments/article/451/the%20logic%20of%20quantum%20mechanics%201936.pdf)]

Historical account of [[John von Neumann]]'s further reasoning on quantum logic, which led him to develop the theory 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]

Proposal for an enhancement of the Birkhoff-von Neumann axioms by [[probability measures]]:

* [[George Mackey]], Section 2-2 of: *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)]

further discussed in:

* {#Gleason57} [[Andrew Gleason]], _Measures on the closed subspaces of a Hilbert space_, Journal of Mathematics and Mechanics,  Indiana Univ. Math. J. **6** 4 (1957), 885-893 [[IUMJ:56050](http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050), [pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1957/6/56050/pdf)]

  > ([[Gleason's theorem]])

* Samuel S. Holland Jr., _Orthomodularity in infinite dimensions; a theorem of M. Solèr_, Bull. Amer. Math. Soc. (N.S.) __32__ (1995) 205-234 [[arXiv:math.RA/9504224](http://arxiv.org/abs/math/9504224)]

* Pavlos Kazakopoulos, Georgios Regkas, *Infinite Permutation Groups and the Origin of Quantum Mechanics* [[arXiv:2307.13044](https://arxiv.org/abs/2307.13044)]

An account of quantum logic within a comprehensive discussion of [[quantum mechanics]] and its [[interpretation of quantum mechanics]]:

* {#Scheibe73} [[Erhard Scheibe]], pp 71 in: *The logical analysis of quantum mechanics*, Pergamon Press Oxford (1973)

Discussion of [[categorical logic]] in [[symmetric monoidal categories]] (not just of [[vector spaces]] but of [[vector bundles]]) and hence of [[linear logic]] as quantum logic:

* {#Crown75} G. D. Crown, *On some orthomodular posets of vector bundles*, Journ. of Natural Sci. and Math. **15** 1-2 (1975) 11-25

much later followed up in:

* [[John Harding]], Taewon Yang, *The Logic of Bundles*, International Journal of Theoretical Physics **54** 12 (2015) [[doi:10.1007/s10773-015-2760-6](http://dx.doi.org/10.1007/s10773-015-2760-6)]

The introduction of [[linear logic]] with the suggestion of a possible relation to [[quantum physics]]:

* {#Girard87} [[Jean-Yves Girard]], _Linear logic_,   Theoretical Computer Science **50** 1 (1987)  [doi:10.1016/0304-3975(87)90045-4]

but remaining undecided:

> {#WildHopes} [Girard (1987), p. 7](#Girard87): "One of the wild hopes that this suggests is the possibility of a direct connection with quantum mechanics... but let's not dream too much!"

The relation of [[linear logic]] to [[quantales]], which have otherwise been proposed as providing a quantum logic:

* {#Yetter90} [[David Yetter]], *Quantales and (noncommutative) linear logic*, Journal of Symbolic Logic **55** (1990) 41-64 [[doi:10.2307/2274953](https://doi.org/10.2307/2274953)]

Maybe the first to really say that linear logic is a good kind of quantum logic:

* {#Pratt92} [[Vaughan Pratt]], *Linear logic for generalized quantum mechanics*, in Proceedings of *[Workshop on Physics and Computation (PhysComp'92)](https://www.computer.org/csdl/proceedings/phycmp/1992/12OmNx19jVS)* (1992) 166-180 [[doi:10.1109/PHYCMP.1992.615518](https://doi.ieeecomputersociety.org/10.1109/PHYCMP.1992.615518), [pdf](http://boole.stanford.edu/pub/ql.pdf), [[Pratt-LinearLogicForQuantum.pdf:file]]]

* {#Pratt93} [[Vaughan Pratt]], *The second calculus of binary relations*, Mathematical Foundations of Computer Science 1993. MFCS 1993, Lecture Notes in Computer Science **711**, Springer (1993) [[doi:10.1007/3-540-57182-5_9](https://doi.org/10.1007/3-540-57182-5_9)]

  > "Linear logic is seen in its best light as the realization of the [[Curry-Howard isomorphism]] for [[linear algebra]]"

First practical development of this perspective (see *[[quantum information theory in terms of dagger-compact categories]]):

* {#AbramskyCoecke} [[Samson Abramsky]], [[Bob Coecke]], *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

* {#AbramskyDuncan05} [[Samson Abramsky]], [[Ross Duncan]], _A Categorical Quantum Logic_, Mathematical Structures in Computer Science, Volume 16, Issue 3 (2006)  pp. 469 - 489  ([arXiv:quant-ph/0512114](http://arxiv.org/abs/quant-ph/0512114), [doi:10.1017/S0960129506005275](https://doi.org/10.1017/S0960129506005275))

* {#Duncan06} [[Ross Duncan]], _Types for quantum mechanics_, 2006 ([pdf](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf), [slides](http://www.cs.ox.ac.uk/people/ross.duncan/talks/2005/pps-22-05-2005.pdf))

and as a paradigm for [[quantum programming languages]] ([[quantum lambda-calculus]]):

* [[Benoît Valiron]], *A functional programming language for quantum computation with classical control*, MSc thesis, University of Ottawa (2004) [[doi:10.20381/ruor-18372](http://dx.doi.org/10.20381/ruor-18372), [pdf](https://ruor.uottawa.ca/bitstream/10393/26790/1/MR01625.PDF)]

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 [[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)]

* {#SelingerValiron09} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *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)]

That therefore in particular categories of [[phase spaces]] and [[categories of cobordisms]] (the [[domains]] of [[FQFT|functorial quantum field theory]]) interpret quantum logic qua [[linear logic]] has been highlighted in

* {#Baez04} [[John Baez]], *Quantum Quandaries: a Category-Theoretic Perspective*, in D. Rickles et al. (ed.) *The structural foundations of quantum gravity*, Clarendon Press (2006) 240-265 [[arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040), [ISBN:9780199269693](https://global.oup.com/academic/product/the-structural-foundations-of-quantum-gravity-9780199269693)]

* {#Slavnov05} [[Sergey Slavnov]], *From proof-nets to bordisms: the geometric meaning of multiplicative connectives*, Mathematical Structures in Computer Science **15** 06 (2005) 1151-1178 [[doi:10.1017/S0960129505004974](https://doi.org/10.1017/S0960129505004974)]

* [[Sergey Slavnov]], *Geometrical semantics for linear logic (multiplicative fragment)*, Theoretical Computer Science **357** 1-3 (2006) 215-229 [[doi:10.1016/j.tcs.2006.03.020](https://doi.org/10.1016/j.tcs.2006.03.020)]

For more exposition along these lines see [Baez & Stay (2009)](#BaezStay09).

Interpreting the [[linear logic]]-sector of [[bunched logic]] as a quantum logic:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

### Review and Exposition

Early general introductions and survey:

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

  > (including discussion of [[quantum probability theory]])

* [[Roland Omnès]], §5 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

* Pavel Pták , Sylvia Pulmannová , *Orthomodular structures as quantum logics*, Fundamental Theories of Physics **44** Kluwer, Springer (1991) [[ISBN:978-0-7923-1207-9](https://link.springer.com/book/9780792312079)]

* Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) *[[Handbook of Quantum Logic and Quantum Structures]]* Elsevier (2009) [[ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698)]

A bibliography of hundreds of works up to 1992:

* [[Mladen Pavičić]], *Bibliography on quantum logics and related structures*,  Int J Theor Phys **31** (1992) 373–455 [[doi:10.1007/BF00739999](https://doi.org/10.1007/BF00739999), [pdf](http://www.irb.hr/users/mpavicic/papers-ps-pdf/quantum-logic/1992-int-j-theor-phys-1.pdf)]

See also:

* [[Franco Strocchi]], §2.5 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)]

* Wikipedia, _[Quantum logic](http://en.wikipedia.org/wiki/Quantum_logic)_

* Stanford encyclopaedia of philosophy, _[Quantum logic and probability theory](http://plato.stanford.edu/entries/qt-quantlog)_

Critique of Birkhoff-von Neumann logic in contrast to [[quantum information via dagger-compact categories]]:

* {#AbramskyCoecke2007} [[Samson Abramsky]], [[Bob Coecke]], *Physics from Computer Science: a Position Statement*, [International Journal of Unconventional Computing **3** 3 (2007)](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/) [[pdf](https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf), [ijuc-3-3-p-179-197](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/)]

Further exposition of non-cartesian [[monoidal categories]] as [[categorical semantics]] for quantum logic qua [[linear logic]]:

* {#Baez04} [[John Baez]], *Quantum Quandaries: a Category-Theoretic Perspective*, in D. Rickles et al. (ed.) *The structural foundations of quantum gravity*, Clarendon Press (2006) 240-265 [[arXiv:quant-ph/0404040](https://arxiv.org/abs/quant-ph/0404040), [ISBN:9780199269693](https://global.oup.com/academic/product/the-structural-foundations-of-quantum-gravity-9780199269693)]

* {#BaezStay09} [[John Baez]], [[Mike Stay]], *Physics, topology, logic and computation: a rosetta stone*, ([arxiv/0903.0340](http://arxiv.org/abs/0903.0340)); in "New Structures for Physics", ed. Bob Coecke, Lecture Notes in Physics __813__, Springer (2011) 95-174 [[doi:10.1007/978-3-642-12821-9](https://link.springer.com/book/10.1007/978-3-642-12821-9)]

Review of of quantum logic with an eye towards the [[EPR paradox]] and [[Bell's inequalities]]:

* Laura Molenaar, *Quantum logic and the EPR paradox*, Delft (2014) [[uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b](http://resolver.tudelft.nl/uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b)]

### Further development

* {#Heunen08} [[Chris Heunen]], _Quantifiers for quantum logic_ ([arXiv:0811.1457](http://arxiv.org/abs/0811.1457))

* {#Harding08} [[John Harding]], _A link between quantum logic and categorical quantum mechanics_, Int J Theor Phys (2009) 48: 769–802

* {#HeunenJacobs09} [[Chris Heunen]], [[Bart Jacobs]], _Quantum Logic in Dagger Kernel Categories_, Order, July 2010, Volume 27, Issue 2, pp 177-212,  ([arXiv:0902.2355](http://arxiv.org/abs/0902.2355))

* {#CCGP09} Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, section 9 of _Quantum Logic and Nonclassical Logics_, p. 127 in  Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) _Handbook of Quantum Logic and Quantum Structures: Quantum Logic_, 2009 North Holland

* [[Samson Abramsky]], _Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics_, In Goong Chen, [[Louis Kauffman]],
and [[Samuel Lomonaco]] (eds.), _Mathematics of Quantum Computing and Technology_, pages 415–458. Taylor and Francis, 2007. ([arXiv:0910.2737](http://arxiv.org/abs/0910.2737))

* {#Girard11} [[Jean-Yves Girard]], _[[Lectures on Logic]]_, European Mathematical Society 2011

* {#LagoFaffian12} [[Ugo Dal Lago]], Claudia Faggian, _On Multiplicative Linear Logic, Modality and Quantum Circuits_ [[arXiv:1210.0613](http://arxiv.org/abs/1210.0613)]

* {#CoeckeHeunenKissinger13} [[Bob Coecke]], [[Chris Heunen]], [[Aleks Kissinger]], _Compositional Quantum Logic_ ([arXiv:1302.4900](http://arxiv.org/abs/1302.4900))

Discussion of [[Fock space]]-type free [[quantum field theory]] in [[linear logic]] is in

*  [[Marcelo Fiore]], _Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.2328))

The proposal that the [[internal logic]] of [[Bohr toposes]] is a good notion of quantum logic is made in

* {#HeunenLandsmanSpitters07} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], _A topos for algebraic quantum theory_, Comm. Math. Phys. 291:63–110, 2009, free access [doi](http://dx.doi.org/10.1007/s00220-009-0865-6), [arXiv:0709.4364](http://arxiv.org/abs/0709.4364)

* [[Steve Vickers]], slides from Midland Grad. School 2010, quantum topos theory, [web](http://www.cs.bham.ac.uk/~sjv/teaching/MGS2010), most relevant part IV: [pdf](http://www.cs.bham.ac.uk/~sjv/teaching/MGS2010/slides4.pdf)

See also

* [[Yuri Manin]], _A course in mathematical logic_, Springer

*  Д.И. Блохинцев, Принципиальные вопросы квантовой механики, 1966, 162 с.

* A. Sudbery, _Quantum mechanics and the particles of nature_, An outline for mathematicians, Camb. Univ. Press 1986

* Stanford Encyclopedia of Philosophy, [qm: von Neumann vs. Dirac](http://plato.stanford.edu/entries/qt-nvd),

[[!redirects quantum logics]]


quantum logic gate

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Analogously to how a [[classical physics|classical]] [[logic gate]] is a [[function]] between ([[finite set|finite]]) [[sets]] of [[tuples]] of [[bits]] ([[truth values]]), so a *quantum logic gate* is a ([[unitary operator|unitary]]) [[linear operator]] on ([[finite-dimensional vector space|finite-dimensional]]) [[Hilbert spaces]] of [[tensor product of vector spaces|tensor products]] of [[qbits]]:

\begin{imagefromfile}
    "file_name": "GenericLogicGate-221026.jpg",
    "width": "640",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Specifically, one calls such a linear map a *quantum gate* if it is thought of as potentially implemented as a basic operation performed by a [[quantum computing]] machine.

As such, typical quantum logic gates operate on a *small* number of [[qbits]], with more complicated such linear maps obtained by composing a given set of quantum logic gates into *[[quantum logic circuits]]*. Such compilation is hence one model of *[[quantum computation]]*.

## Examples

The first examples are linearizations of classical  [[logic gates]], or rather of their [[reversible computation|reversible]] versions:

\linebreak

**[[NOT]]** or **[[Pauli gate|X]]**:

\begin{imagefromfile}
    "file_name": "NOT_LogicGate-221025.jpg",
    "width": "600",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

\linebreak

**[[XOR]]** and **[[CNOT]]:**

\begin{imagefromfile}
    "file_name": "CNOTGates-221026c.jpg",
    "width": "750",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

**[[AND]]**:

\begin{imagefromfile}
    "file_name": "ANDGates-221026b.jpg",
    "width": "770",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

The following examples have no classical analog:

\linebreak

**[[Hadamard gate]]**:

\begin{imagefromfile}
    "file_name": "HadamardGate-221026a.jpg",
    "width": "560",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

**[[Pauli gate]]**

(...)

\linebreak

(...)

## Related concepts

* [[logic gate]]

* [[controlled quantum gate]]

* [[unitary quantum channel]]

* [[quantum circuit]]

* [[quantum logic]]

* [[qbit]], [[qtrit]], [[qdit]]

## References

The notion of quantum logic gates and [[quantum circuits]] originates with

* [[Richard Feynman]], *Quantum mechanical computers*, Foundations of Physics **16** (1986) 507–531 [[doi:10.1007/BF01886518](https://doi.org/10.1007/BF01886518)]

* [[David E. Deutsch]], *Quantum computational networks*, Proceedings of the Royal Society A, **425** 1868 (1989) 73-90 [[doi:10.1098/rspa.1989.0099](https://doi.org/10.1098/rspa.1989.0099), [jstor:2398494](https://www.jstor.org/stable/2398494)]

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

See also:

* Wikipedia, *[Quantum logic gate](https://en.wikipedia.org/wiki/Quantum_logic_gate)*

Implementation of quantum logic gates on [[qbits]] realized via [[nucleon]]-[[spin]], via pulse protocols in [[nuclear magnetic resonance]]-technology:

* Price, Somaroo, Tseng, Gore, Fahmy,, Havel, Cory: *Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems*, Journal of Magnetic Resonance
**140** 2 (1999) 371-378 [[doi;10.1006/jmre.1999.1851](https://doi.org/10.1006/jmre.1999.1851)]

and analogously on [[electron]]-[[spin]]:

* M. Yu. Volkov and K. M. Salikhov, *Pulse Protocols for Quantum Computing with Electron Spins as Qubits*, Appl Magn Reson **41** (2011) 145–154 [[doi:10.1007/s00723-011-0297-2](https://doi.org/10.1007/s00723-011-0297-2)]

[[!redirects quantum logic gates]]

[[!redirects quantum gate]]
[[!redirects quantum gates]]


quantum Lorentz group

A quantum Lorentz group is a [[quantum group]] deformation of the usual Lorentz group. There are several variants in the literature.

* P. Podleś, [[S. L. Woronowicz]], _Quantum deformation of Lorentz group_, Comm. Math. Phys. __130__ (1990), no. 2, 381–431 [MR91f:46100](http://www.ams.org/mathscinet-getitem?mr=1059324) [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104200517)


quantum many-body physics

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Fock space]]

* [[second quantization]]

* [[quantum entanglement]]

* [[solid-state physics]]

* [[Laughlin wavefuntion]], [[anyon]]

* [[Bose-Einstein condensate]]

* [[tensor network]]

* [[dimer model]]

* [[AdS-CFT in condensed matter physics]]

## References

Textbook accounts:

* Yoshio Kuramoto, *Quantum Many-Body Physics -- A Perspective on Strong Correlations* ([doi:10.1007/978-4-431-55393-9](https://link.springer.com/book/10.1007/978-4-431-55393-9))

In a context of [[quantum optics]]:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], Section 1 of: *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))

In regards to [[tensor networks]]:

* [[Thorsten B. Wahl]], _Tensor network states for the description of quantum many-body systems_ ([arXiv:1509.05984](https://arxiv.org/abs/1509.05984))

[[!redirects quantum many body physics]]


quantum master equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] formulated in terms of [[BV-BRST formalism]], the _classical master equation_ expresses the nilpotency of the [[BV-differential]] before [[quantization]], with the latter regarded as a [[Hamiltonian vector field]] with respect to the _[[antibracket]]_ $\{-,-\}$, for "Hamiltonian" the BV-BRST-extended [[action functional]] "$S + S_{BRST}$":

$$
  \left(
    s_{BV}
  \right)^2 = 0
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \left( \{S + S_{BRST},(-)\}\right)^2 = 0
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \{S + S_{BRST}, S + S_{BRST}\} = 0
  \,.
$$

The _quantum  master equation_ (prop. \ref{QuantumMasterEquation} below) is the version of this equation after [[quantization]], in which case the the [[BV-differential]] picks up a quantum correction of order $\hbar$ ([[Planck's constant]]) by the [[BV-operator]] $\Delta_{BV}$:

$$
  \left(\, \{S + S_{BRST},(-)\} + i \hbar \Delta_{BV} \, \right)^2 = 0
 \,.
$$

## In causal perturbation theory

We discuss the quantum master equation in the rigorous formulation of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] ([Fredenhagen-Rejzner 11b](#FredenhagenRejzner11b), [Rejzner 11](#Rejzner11)).

First we consider all structure just on [[regular polynomial observables]], hence excluding non-linear [[local observables]] such as the usual point-[[interaction]] [[action functionals]].

Then the [[extension]] of all structures from regular to [[local observables]] is the [[renormalization]] step, discussed [furthter below](#RenormalizationAndMasterWardIdentity).

### Background

Throughout, let $(E_{\text{BV-BST}},\mathbf{L}')$ be a [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]] with global [[BV-differential]] ([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal))

$$
  \{-S', -\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

Hence $S'$ denotes the gauge fixed free action functional.

Moreover, let $\Delta_H$ be a compatible choice of [[Wightman propagator]] with associated [[Feynman propagator]] $\Delta_F$.

+-- {: .num_lemma #DerivationBVDifferentialForWickAlgebra}
###### Lemma
**(global [[BV-differential]] is [[derivation]] on [[Wick algebra]])**

The global [[BV-differential]]

$$
  \{-S',-\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
   \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

is a [[derivation]] also with respect to the [[Wick algebra]] [[star product]] $\star_H$:

$$
  \left\{
    -S', A_1 \star_H A_2
  \right\}
  \;=\;
  \left\{
    -S', A_1
  \right\}
    \star_H
  A_2
    \;+\;
  A_1
    \star_H
  \left\{
   -S', A_2
  \right\}
  \,.
$$

=--

([Fredenhagen-Rejzner 11b, below (37)](#FredenhagenRejzner11b), [Rejzner 11, below (5.28)](#Rejzner11)) For **proof** see [this prop](A+first+idea+of+quantum+field+theory#OnMicrocausalObservablesGlobalBVDifferential)

+-- {: .num_defn #OnRegularObservablesPerturbativeSMatrix}
###### Definition
**([[perturbative S-matrix]] on [[regular polynomial observables]])**

The _[[perturbative S-matrix]]_ on [[regular polynomial observables]]
is the [[exponential]] with respect to the [[time-ordered product]]

$$
  \mathcal{S}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))
$$

given by

$$
  \mathcal{S}(S_{int})
  =
  \exp_{\star_F}
  \left(
    \tfrac{1}{i \hbar} S_{int})
  \right)
  \coloneqq
  1
    +
  \tfrac{1}{\i \hbar} S_{int}
    +
  \tfrac{1}{2}
  \tfrac{1}{(i \hbar)^2}
  S_{int} \star_F S_{int}
  +
  \cdots
  \,.
$$

We think of $S_{int}$ here as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].

We write $\mathcal{S}(S_{int})^{-1}$ for the [[inverse]] with respect to the [[Wick algebra|Wick product]] (which exists by [this remark](S-matrix#PerturbativeSMatrixInverse))

$$
  \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int})
  =
  1
  \,.
$$

Notice that this is in general different form the inverse with respect to the [[time-ordered product]] $\star_F$, which is $\mathcal{S}(-S_{int})$:

$$
  \mathcal{S}(-S_{int})
    \star_F
  \mathcal{S}(S_{int})
  =
  1
  \,.
$$

=--

+-- {: .num_defn #MollerOperatorOnRegularPolynomialObservables}
###### Definition
**([[quantum Møller operator]] on [[regular polynomial observables]])**

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] of degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
$$

then the corresponding _[[quantum Møller operator]]_ on [[regular polynomial observables]]

$$
  \mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

is given by the [[derivative]] of [[Bogoliubov's formula]]

$$
  \mathcal{R}^{-1}
    \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1}
    \star_H
  (\mathcal{S}(S_{int}) \star_F (-))
  \,,
$$

where $\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$ is the [[perturbative S-matrix]] from  def. \ref{OnRegularObservablesPerturbativeSMatrix}.

This indeed lands in [[formal power series]] in [[Planck's constant]] $\hbar$ (by [this remark](Bogoliubov's+formula#PowersInPlancksConstant)),  instead of in more general [[Laurent series]] as the [[perturbative S-matrix]] does (def. \ref{OnRegularObservablesPerturbativeSMatrix}).

Hence the inverse map is

$$
  \mathcal{R}
   \;=\;
  \mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) )
  \,.
$$

=--

([Bogoliubov-Shirkov 59](Bogoliubov's+formula#BogoliubovShirkov59); the above terminology follows [Hawkins-Rejzner 16, below def. 5.1](Møller+operator#HawkinsRejzner16))

Notice that compared to Fredenhagen-Rejzner et. al. we have changed notation conventions $\mathcal{R} \leftrightarrow \mathcal{R}^{-1}$ in order to bring out the analogy to (the conventions for the) [[time-ordered product]]  $A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))$ on regular polynomial observables.

notice the implicit dependencies

| [[endomorphism]] of <br/> [[regular polynomial observables]] | meaning | depends on choice of |
|--------|---------|----------------------|
| $\phantom{AA}\mathcal{T}$ | [[time-ordered product|time-ordering]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{S}$ | [[S-matrix]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{R}$ | [[quantum Møller operator]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] and [[interaction]] |

+-- {: .num_defn #FieldAlgebraObservablesInteracting}
###### Definition
**([[interacting field algebra]])**

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
  \,,
$$

then the _[[interacting field algebra]]_ [[structure]] on [[regular polynomial observables]]

$$
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \otimes
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \overset{ \star_{int} }{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

is the [[conjugation]] of the [[Wick algebra]]-[[structure]] by the [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

$$
  A_1 \star_{int} A_2
  \;\coloneqq\;
  \mathcal{R}
  \left(
    \mathcal{R}^{-1}(A_1)
     \star_H
    \mathcal{R}^{-1}(A_2)
  \right)
$$

=--

(e.g. [Fredenhagen-Rejzner 11b, (19)](#FredenhagenRejzner11b))

### Interacting quantum BV-differential

Recall how the global BV-differential

$$
  \{S',-\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

on [[regular polynomial observables]] ([this def.](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal)) is conjugated into the [[time-ordered product]] via the time ordering operator
$\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ ([this prop.](BV-operator#GaugeFixedActionFunctionalTimeOrderedAntibracket)).

In the same way we may use the [[quantum Møller operators]] to conjugate the BV-differential into the regular part of the [[interacting field algebra of observables]]:

+-- {: .num_defn #BVDifferentialInteractingQuantum}
###### Definition
**(interacting quantum [[BV-differential]])**

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] $S_{int}$,
then the _interacting quantum BV-differential_ on the [[interacting field algebra]] (def. \ref{FieldAlgebraObservablesInteracting})
on [[regular polynomial observables]] is the [[conjugation]] of the plain [[BV-differential]] $\{-S',-\}$ by the [[quantum Møller operator]] induced by $S_{int}$ (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

$$
  \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \,.
$$

=--

([Rejzner 11, (5.38)](#Rejzner11))

+-- {: .num_prop #QuantumMasterEquation}
###### Proposition
**([[quantum master equation]] and [[quantum master Ward identity]] on [[regular polynomial observables]])**

Consider an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
  \,,
$$

Then the following are equivalent:

1. The _[[quantum master equation]]_ (QME)

   $$
     \label{OnRegularObservablesQuantumMasterEquation}
      \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
      +
      i \hbar  \Delta_{BV}( S' + S_{int} )
     \;=\;
     0
     \,.
   $$

1. The  [[perturbative S-matrix]] (def. \ref{OnRegularObservablesPerturbativeSMatrix}) is $BV$-closed

   $$
     \{-S', \mathcal{S}(S_{int})\} = 0
     \,.
   $$

1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[retarded products]]_:

   $$
     \label{OnRegularObservablesQuantumMasterWardIdentity}
     \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1}
     \;=\;
     \left\{ -(S' + S_{int}) \,,\, (-) \right\}_{\mathcal{T}}
     -
     i \hbar   \Delta_{BV}
   $$

   ([Dütsch 18, (4.2)](Ward+identity#Duetsch18))

   expressing the interacting quantum [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum}) as the sum of the [[time-ordered product|time-ordered]] [[antibracket]] ([this def.](BV-operator#AntibracketTimeOrdered)) with the _total_ [[action functional]] $S' + S_{int}$ and $i \hbar$ times the [[BV-operator]] ([BV-operator](BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator)).

1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[time-ordered products]]_:

   $$
     \label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
     \mathcal{S}(-S_{int})
       \star_F
     \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
     \;=\;
     -
     \left(
       \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}}
       + i \hbar   \Delta_{BV}
     \right)
   $$

   ([Dütsch 18, (4.8)](Ward+identity#Duetsch18))

=--

([Rejzner 11, (5.35) - (5.38)](#Rejzner11), following [Hollands 07, (342)-(345)](Ward+identity#Hollands07))

+-- {: .proof}
###### Proof

To see that the first two conditions are equivalent, we compute as follows

$$
  \label{QuantumMasterOnRegularObservablesBVDifferentialOfSMatrixInTerms}
  \begin{aligned}
    \left\{
      -S',  \mathcal{S}(S_{int})
    \right\}
    & =
    \left\{
      -S'
      ,
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}
    \\
    & =
    \underset{
      {
        \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}}
      }
      \atop
      {
        \star_F
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar} S_{int}
        \right)
      }
    }{
    \underbrace{
    \left\{
      -S'
      ,
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}_{\mathcal{T}}
    }
    }
    -
    i \hbar
    \underset{
      {
        \left(
          \tfrac{1}{i \hbar}
          \Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2}
          \left\{
            S_{int}, S_{int}
          \right\}_{\mathcal{T}}
        \right)
      }
      \atop
      {
        \star_{F}
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar}
          S_{int}
        \right)
      }
    }{
    \underbrace{
    \Delta_{BV}
    \left(
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right)
    }
    }
    \\
    & =
    \tfrac{-1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \{S',S_{int}\}
      +
      \tfrac{1}{2}\{S_{int}, S_{int}\}
      +
      i \hbar \Delta_{BV}(S_{int})
    \right)
    }
    }
      \star_F
    \exp_{\mathcal{T}}
    \left(
      \tfrac{1}{i \hbar}
      S_{int}
    \right)
  \end{aligned}
$$

Here in the first step we used the definition of the [[BV-operator]] ([this def.](ForGaugeFixedFreeLagrangianFieldTheoryBVOperator)) to rewrite the plain antibracket in terms of the time-ordered antibracket ([this def.](BV-operator#AntibracketTimeOrdered)), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation ([this prop](BV-operator#AntibracketBVOperatorRelation)) and under the first brace the consequence of this statement for application to exponentials ([this example](BV-operator#TimeOrderedExponentialBVOperator)). Finally we collected terms, and to "complete the square" we added the terms on the left of

$$
  \frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}}
  -
  i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0
$$

which vanish because, by definition of [[gauge fixing]] ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)), the free gauge-fixed action functional $S'$ is independent of [[antifields]].

But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the [[inverse]] $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar}  S_{int} \right)$, this implies the claim.

Next we show that the [[quantum master equation]] implies the [[quantum master Ward identities]].

We use that the BV-differential $\{-S',-\}$ is a [[derivation]] of the [[Wick algebra]] product $\star_H$ (lemma \ref{DerivationBVDifferentialForWickAlgebra}).

First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also
$\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.

Thus we compute as follows:

$$
  \begin{aligned}
    \{-S', -\} \circ \mathcal{R}^{-1}(A)
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    \\
    & =
    \left\{
      { \, \atop \, }
      -S',
      \mathcal{S}(S_{int})^{-1}
        \star_H
      \left( \mathcal{S}(S_{int}) \star_F a \right)
      {\, \atop \,}
    \right\}
    \\
    & = \phantom{+}
    \underset{
      = 0
    }{
    \underbrace{
    \left\{
      -S', \mathcal{S}(S_{int})^{-1}
    \right\}
    }
    }
      \star_H
    \left(
      \mathcal{S}(S_{int}) \star_F A
    \right)
    \\
    & \phantom{=}
    +
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left\{
      -S',
      \mathcal{S}(S_{int}) \star_F A
    \right\}
    \\
    & =
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \underset{
        = 1
      }{
      \underbrace{
        \mathcal{S}(+ S_{int})
          \star_F
        \mathcal{S}(- S_{int})
      }
      }
        \star_F
      \left\{
        -S',
        \mathcal{S}(S_{int}) \star_F A
      \right\}
   \right)
   \\
   & =
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \mathcal{S}(+ S_{int})
        \star_F
      \underset{ (\ast) }{
      \underbrace{
        \mathcal{S}(- S_{int})
          \star_F
        \left\{
          -S',
          \mathcal{S}(S_{int}) \star_F A
        \right\}
     }
     }
   \right)
   \\
   & =
   \mathcal{R}^{-1}
   \left(
     \underset{ (\ast) }{
     \underbrace{
       \phantom{\, \atop \,}
       \mathcal{S}(-S_{int})
         \star_F
       \left\{
         -S',
         \mathcal{S}(S_{int}) \star_F A
       \right\}
       }
       }
   \right)
  \end{aligned}
$$

By applying $\mathcal{R}$ to both sides of this equation, this means
first of all that the interacting quantum BV-differential
is equivalently given by

$$
  \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;=\;
  \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
  \,,
$$

hence that if either version (eq:OnRegularObservablesQuantumMasterWardIdentity) or (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered)  of the [[master Ward identity]] holds, it implies the other.

Now expanding out the definition of $\mathcal{S}$ (def. \ref{OnRegularObservablesPerturbativeSMatrix}) and expressing $\{-S',-\}$ via the [[time-ordered product|time-ordered]] [[antibracket]] ([this def.](BV-operator#AntibracketTimeOrdered)) and the [[BV-operator]] $\Delta_{BV}$ ([this prop.](BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator)) as

$$
  \{-S',-\}
  \;=\;
  \{-S',-\}_{\mathcal{T}} - i \hbar \Delta_{BV}
$$

(on [[regular polynomial observables]]), we continue computing as follows:

$$
  \label{QMESecondStep}
  \begin{aligned}
    & \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A )
    \\
    & =
   \exp_{\mathcal{T}}
    \left(
      \tfrac{-1}{i \hbar} S_{int}
    \right)
      \star_F
    \left\{
      -S',
      \exp_{\mathcal{T}}
      \left(
         \tfrac{1}{i \hbar} S_{int}
      \right)
        \star_F
      A
    \right\}
    \\
    & =
    \exp_{\mathcal{T}}
    \left(
      \tfrac{-1}{i \hbar}
        S_{int}
    \right)
      \star_F
    \left(
      \left\{
        -S',
        \exp_{\mathcal{T}}
        \left(
          \tfrac{ 1 }{i \hbar} S_{int}
        \right)
          \star_F
        A
      \right\}_{\mathcal{T}}
      -
      i \hbar
      \Delta_{BV}
      \left(
        \exp_{\mathcal{T}}
        \left(
           \tfrac{ 1 }{i \hbar} S_{int}
        \right)
          \star_F
        A
      \right)
    \right)
    \\
    &
    \phantom{+}
    =
    \tfrac{1}{i \hbar} \{ -S', S_{int}  \}_{\mathcal{T}} \star_F   A
    +
    \{-S', A\}_{\mathcal{T}}
    \\
    &
    \phantom{=}
    -
    i \hbar
    \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right)
    \star_F
    \left(
      \underset{
        {
          \left(
          \tfrac{1}{i \hbar}\Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}
          \right)
        }
        \atop
        {
          \star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right)
        }
      }{
      \underbrace{
        \Delta_{BV}
        \left(
          \exp_{\mathcal{T}}
          \left(
            \tfrac{ 1}{i \hbar} S_{int}
          \right)
        \right)
      }
      }
      \star_F A
      \,+\,
       \exp_{\mathcal{T}}
       \left(
         \tfrac{ 1}{i \hbar} S_{int}
       \right)
       \star_F
       \Delta_{BV}(A)
       \,+\,
       \underset{
         {
           \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)
         }
         \atop
         {
           \star_F
           \tfrac{ 1}{i \hbar}
           \{S_{int}, A\}
         }
       }{
       \underbrace{
         \left\{
           \exp_{\mathcal{T}}
           \left(
             \tfrac{ 1}{i \hbar} S_{int}
           \right)
           \,,\,
           A
         \right\}_{\mathcal{T}}
      }
      }
    \right)
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
        +
      i \hbar \Delta_{BV}(A)
    \right)
    \\
    & \phantom{=}
    -
    \tfrac{1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
      +
      i \hbar  \Delta_{BV}( S' + S_{int} )
    \right)
    }}
      \star_F A
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
        +
      i \hbar \Delta_{BV}(A)
    \right)
  \end{aligned}
$$

Here in the line with the braces we used that the [[BV-operator]] is a [[derivation]] of the [[time-ordered product]] up to correction by the time-ordered [[antibracket]] ([this prop.](BV-operator#AntibracketBVOperatorRelation)), and under the first brace we used the effect of that property on time-ordered exponentials ([this example](BV-operator#TimeOrderedExponentialBVOperator)), while under the second brace we used that $\{(-),A\}_{\mathcal{T}}$ is a derivation of the time-ordered product. Finally we have collected terms, added $0 = \{S',S'\} + i \hbar \Delta_{BV}(S')$ as before, and then used the QME.

This shows that the quantum [[master Ward identities]] follow from the [[quantum master equation]]. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.

To see this, observe that with the BV-differential being nilpotent, also its conjugation by $\mathcal{R}$ is, so that with the above we have:

$$
  \begin{aligned}
  & \left( \{-S',-\}\right)^2 = 0
  \\
  \Leftrightarrow
  \;
  &
  \left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0
  \\
  \Leftrightarrow
  \;
  &
  \underset{
    \left\{
      {\, \atop \,}
      \tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}}
      +
      i \hbar \Delta_{BV}(S' + S_{int})
      \,,\,
      (-)
    \right\}
  }{
  \underbrace{
    \left(
      \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
    \right)^2
  }
  } = 0
  \end{aligned}
$$

Here under the brace we computed as follows:

$$
  \begin{aligned}
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
  \right)^2
  & =
  \phantom{+}
  \underset{
    \tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }{
  \underbrace{
    \{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }}
  \\
  &
  \phantom{=}
  +
  i \hbar
  \underset{
    \{  \Delta_{BV}(S'+ S)\,,\, (-)  \}_{\mathcal{T}}
  }{
  \underbrace{
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV}
    +
    \Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}}
  \right)
  }}
  \\
  &
  \phantom{=}
  +
  (i \hbar)^2
   \underset{= 0}
   {
     \underbrace{
       \Delta_{BV} \circ \Delta_{BV}
      }
    }
  \end{aligned}
  \,.
$$

where, in turn, the term under the first brace follows by the graded [[Jacobi identity]], the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).

=--

+-- {: .num_example #MasterWardIdentityClassical}
###### Example
**([[classical master Ward identity]])**

The [[classical limit]] $\hbar \to 0$ of the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentity) is

$$
  \{-S',(-)\} \circ \mathcal{R}^{-1}
  \;=\;
  -
  \mathcal{R}^{-1}
  \left(
    \left\{ S' + S_{int} \,,\, (-) \right\}
  \right)
  \,.
$$

Applied to an observable which is linear in the [[antifields]]

$$
  A
    \;=\;
  \underset{\Sigma}{\int}
    A^a(x)
    \mathbf{\Phi}^\ddagger_a(x)
  \,
  dvol_\Sigma(x)
$$

this becomes

$$
  \begin{aligned}
    0
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    +
    \mathcal{R}^{-1}
    \left(
      \left\{ S' + S_{int} \,,\, A \right\}_{\mathcal{T}}
    \right)
    \\
    & =
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
    \, dvol_\Sigma(x)
    +
    \mathcal{R}^{-1}
    \left(
      \underset{\Sigma}{\int}
          A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
        \, dvol_\Sigma(x)
    \right)
  \end{aligned}
$$

In this form the _classical Master Ward identity_ was originally identified in ([Dütsch-Fredenhagen 02, (90)](master+Ward+identity#DuetschFredenhagen02), [Brennecke-Dütsch 07, (5.5)](master+Ward+identity#BrennecketDuetsch07), following [Dütsch-Boas 02](master+Ward+identity#DuetschBoas02)).

=--

### Renormalization and Master ward identity
 {#RenormalizationAndMasterWardIdentity}

The quantum master equation in the form of prop. \ref{QuantumMasterEquation} is derived on [[regular polynomial observables]], in particular hence for non-point-[[interaction]] [[action functionals]] $S_{int}$. But the [[interaction]] terms of interest are point-interactions, hence are [[local observables]]. The [[extension]] of the [[time-ordered product]] and hence of the [[perturbative S-matrix]] from regular to local onservables exsists but involves choices, these are the _[[renormalization]]_ choices in the formulation of [[causal perturbation theory]].

Since for [[gauge fixing|gauged fixed]] [[gauge theories]] this physically relevant [[observables]] are not the plain ([[microcausal polynomial observables|mcirocausal]]) [[polynomial observables]], but the [[cochain cohomology]] of the [[BV-BRST differential]] on them, one needs to require for gauge theories that the [[quantum master equation]] still holds after [[renormalization]]. This is closely related to the [[renormalization condition]] called the _[[master Ward identity]]_ ([Rejzner 11 (prop. 5.3.1) and following paragraphs](#Rejzner11)). If the quantum master equation cannot be retained in [[renormalization]] one says that the field theory suffers from a _[[quantum anomaly]]_.

## Related concepts

* [[master Ward identity]]

## References

The concept originates with

* {#BatalinVilkovisky81} [[Igor Batalin]], [[Grigori Vilkovisky]],  _Gauge Algebra and Quantization_, Phys. Lett. B 102 (1): 27–31, 1981 (<a href="https://doi.org/10.1016/0370-2693(81)90205-7">doi:10.1016/0370-2693(81)90205-7</a>)

Traditional review includes

* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.5.3 of _[[Quantization of Gauge Systems]]_, Princeton University Press, 1992

Discussion in the rigorous context of [[relativistic field theory|relativistic]] [[perturbative QFT]] formulated in [[causal perturbation theory]]/[[perturbative AQFT]] is in:

* {#FredenhagenRejzner11a} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in the functional approach to classical field theory_, Commun. Math. Phys. 314(1), 93–127 (2012) ([arXiv:1101.5112](https://arxiv.org/abs/1101.5112))

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. 317(3), 697–725 (2012) ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232))

* {#Rejzner11} [[Katarzyna Rejzner]], _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](https://arxiv.org/abs/1111.5130))

* {#Rejzner13} [[Katarzyna Rejzner]], _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))

and surveyed in

* {#Rejzner16} [[Kasia Rejzner]], section 7 of _Perturbative algebraic quantum field theory_ Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects classical master equation]]


quantum material

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Topological physics
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[[!include topological physics -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], by a *quantum material* one means -- broadly and somewhat vaguely -- a [[phase of matter]] whose properties are deeply controlled by [[quantum physics]] and often by [[topological physics -- contents|topology]], for instance in that their [[quantum state|quantum]] [[ground state]] exhibits [[long-range entanglement]]. Prime examples are [[superconductors]] and, more recently and more prominently, [[topological phases of matter]] possibly controlled by [[topological order]], such as [[topological insulators]]; also [[quasicrystals]]. A concrete example is [[graphene]].

## Related concepts

* [[topological quantum computation]]

* [[topological phase of matter]], [[K-theory classification of topological phases of matter|K-theory classification]]

* [[topological order]]

* [[topological entanglement entropy]]

* [[quantum phase transition]]

* [[Majorana zero mode]]

## References

Textbook accounts:

* [[Tudor D. Stanescu]], *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

Survey:

* Feliciano Giustino et al., *The 2021 quantum materials roadmap*, J. Phys. Mater. 3 042006 (2020) ([doi:10.1088/2515-7639/abb74e](https://iopscience.iop.org/article/10.1088/2515-7639/abb74e))

See also:

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

[[!redirects quantum materials]]