quantum programming languages -- references

### Quantum programming languages

The first proposal towards formalizing quantum programming languages was:

* {#Knill96} [[Emanuel Knill]], *Conventions for quantum pseudocode*, Los Alamos Technical Report (1996) $[$[LA-UR-96-2724](https://www.osti.gov/biblio/366453-conventions-quantum-pseudocode), [doi:10.2172/366453](https://doi.org/10.2172/366453), [pdf](https://www.osti.gov/servlets/purl/366453), [[Knill-QuantumPseudocode.pdf:file]]$]$

and early formalization via [[quantum lambda-calculus]] invoking [[linear logic]]/[[linear type theory|linear types]] (cf. [Pratt 1992](#Pratt92) etc.):

* [[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)$]$

> [[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*.

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 $[$[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)$]$

On [[quantum programming languages]] ([[programming languages]] for [[quantum computation]]):

General:

* [[Jarosław Adam Miszczak]], *Models of quantum computation and quantum programming languages*, Bull. Pol. Acad. Sci.-Tech. Sci., Vol. 59, No. 3 (2011), pp. 305-324 ([arXiv:1012.6035](https://arxiv.org/abs/1012.6035))

* {#Miszczak12} [[Jarosław Adam Miszczak]], *High-level Structures for Quantum Computing*, Morgan&Claypool 2012 ([doi:10.2200/S00422ED1V01Y201205QMC006](https://doi.org/10.2200/S00422ED1V01Y201205QMC006), [pdf](https://www.morganclaypool.com/doi/pdf/10.2200/S00422ED1V01Y201205QMC006))

See also:

* Wikipedia, _[Quantum programming](https://en.wikipedia.org/wiki/Quantum_programming)_

Surveys of existing languages:

* Simon Gay, _Quantum  programming languages: Survey and bibliography_, Mathematical Structures in Computer Science16(2006) ([doi:10.1017/S0960129506005378](https://doi.org/10.1017/S0960129506005378),  [pdf](http://www.dcs.gla.ac.uk/~simon/publications/QPLsurvey.pdf))

* Sunita Garhwal, Maryam Ghorani , Amir Ahmad, *Quantum Programming Language: A Systematic Review of Research Topic and Top Cited Languages*,  Arch Computat Methods Eng 28, 289–310 (2021) ([doi:10.1007/s11831-019-09372-6](https://doi.org/10.1007/s11831-019-09372-6))

* B. Heim et al., *Quantum programming languages*, Nat Rev Phys **2** (2020) 709–722 $[$[doi:10.1038/s42254-020-00245-7](https://doi.org/10.1038/s42254-020-00245-7)$]$

Quantum programming via [[quantum logic]] understood as [[linear type theory]] [[categorical semantics|interpreted in]] [[symmetric monoidal categories]]:

* {#Pratt92} [[Vaughan Pratt]], *Linear logic for generalized quantum mechanics*, in Proc. of *[Workshop on Physics and Computation (PhysComp'92)](https://www.computer.org/csdl/proceedings/phycmp/1992/12OmNx19jVS)*, Dallas (1992) $[$[pdf](http://boole.stanford.edu/pub/ql.pdf), [[Pratt-LinearLogic.pdf:file]], [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.7817)$]$

* {#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))

* {#AbramskyDuncan05} [[Samson Abramsky]], [[Ross Duncan]], _A Categorical Quantum Logic_, Mathematical Structures in Computer Science, **16** 3 (2006) 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 Computing*, (2006) $[$[pdf](http://personal.strath.ac.uk/ross.duncan/papers/rduncan-thesis.pdf)$]$

* {#LagoFaggian12} [[Ugo Dal Lago]], [[Claudia Faggian]], _On Multiplicative Linear Logic, Modality and Quantum Circuits_, EPTCS 95 (2012) 55-66 $[$[arXiv:1210.0613](http://arxiv.org/abs/1210.0613), [doi:10.4204/EPTCS.95.6](https://doi.org/10.4204/EPTCS.95.6)$]$

* {#Staton15} [[Sam Staton]], *Algebraic Effects, Linearity, and Quantum Programming Languages*, POPL '15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) $[$[doi:10.1145/2676726.2676999](https://doi.org/10.1145/2676726.2676999), [pdf](http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl2015.pdf)$]$

  > "A [[quantum programming]] language captures the ideas of [[quantum computation]] in a [[linear type theory]]."

The corresponding [[string diagrams]] are known in quantum computation as *[[quantum circuit diagrams]]*:

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], Chapter 3 of: *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

* [Miszczak 12, Sec. 4.3](#Miszczak12)

[[type theory|Typed]]$\;$[[functional programming languages]] for [[quantum computation]]:

**[[QPL]]:**

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

**[[QML]]:**

* {#AltenkirchGrattage05} [[Thorsten Altenkirch]], [[Jonathan Grattage]], _A functional quantum programming language_, Logic in Computer Science, 2005. Proceedings. 20th Annual IEEE Symposium on, 26-29 June 2005 Page(s):249 - 258 ([arXiv:quant-ph/0409065](https://arxiv.org/abs/quant-ph/0409065), [doi:10.1109/LICS.2005.1](https://doi.org/10.1109/LICS.2005.1), [pdf](http://www.cs.nott.ac.uk/~txa/publ/qml.pdf))

* [[Jonathan Grattage]], _An overview of QML with a concrete implementation in Haskell_, talk at *Quantum Physics and Logic 2008*, ENTCS: Proceedings of QPL V - DCV IV, 157-165, Reykjavik, Iceland, 2008 ([arXiv:0806.2735](https://arxiv.org/abs/0806.2735))

**[[quantum IO monad]]:**

* {#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)$]$

**[[Quipper]]:**

* [[Peter Selinger]], _The Quipper Language_ ([web](http://www.mathstat.dal.ca/~selinger/quipper/))

* {#GLRSV13} [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices **48** 6  (2013) 333-342 $[$[arXiv:1304.3390](https://arxiv.org/abs/1304.3390)$]$

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

* Francisco Rios, [[Peter Selinger]], _A Categorical Model for a Quantum Circuit Description Language_, EPTCS 266, 2018, pp. 164-178 ([arXiv:1706.02630](https://arxiv.org/abs/1706.02630))

**[[QWIRE]]:**

* {#PaykinRandZdancewic17} [[Jennifer Paykin]], [[Robert Rand]], [[Steve Zdancewic]], *QWIRE: a core language for quantum circuits*, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages (2017) 846–858 ([doi:10.1145/3009837.3009894](https://doi.org/10.1145/3009837.3009894))

  (using the [[exponential modality]] between [[linear type theory]] and [[intuitionistic type theory]])

* [[Jennifer Paykin]], *Linear/non-Linear Types For Embedded Domain-Specific Languages*, 2018 ([upenn:2752](https://repository.upenn.edu/edissertations/2752))

* {#PaykinZdancewic19} [[Jennifer Paykin]], [[Steve Zdancewic]], *A HoTT Quantum Equational Theory*, [talk at QPL2019](http://qpl2019.org/a-hott-quantum-equational-theory/) $[$[arXiv:1904.04371](https://arxiv.org/abs/1904.04371), talk slides: [pdf](https://jpaykin.github.io/talks/HoTT-3-8-2019.pdf), [[PaykinZdancewic-HoTTQuantum.pdf:file]]$]$

  > (using ambient [[homotopy type theory]])

**[[CoqQ]]:**

* {#YingEtAl22} [[Li Zhou]], [[Gilles Barthe]], Pierre-Yves Strub, Junyi Liu, [[Mingsheng Ying]], *CoqQ: Foundational Verification of Quantum Programs* $[$[arXiv:2207.11350](https://arxiv.org/abs/2207.11350)$]$

\linebreak

On [[Q Sharp]]:

*  Kartik Singhal, Kesha Hietala, Sarah Marshall, Robert Rand, *Q# as a Quantum Algorithmic Language*, Proceedings of *Quantum Physics and Logic* (2022) $[$[arXiv:2206.03532](https://arxiv.org/abs/2206.03532)$]$

On [[classically controlled quantum computation]]:

* [[Bernhard Ömer]], *Structured Quantum Programming*, 2003/2009 ([pdf](http://www.itp.tuwien.ac.at/~oemer/doc/structquprog.pdf))

Quantum programming via [[dependent linear type theory]]/[[indexed monoidal (∞,1)-categories]]:

* {#Schreiber14} [[Urs Schreiber]], _Quantization via Linear Homotopy Types_ ([arXiv:1402.7041](http://arxiv.org/abs/1402.7041))

* {#FKS20} [[Peng Fu]], [[Kohei Kishida]], [[Peter Selinger]], _Linear Dependent Type Theory for Quantum Programming Languages_, LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 ([arXiv:2004.13472](https://arxiv.org/abs/2004.13472), [doi:10.1145/3373718.3394765](https://dl.acm.org/doi/10.1145/3373718.3394765), [pdf](https://depend.cs.uni-saarland.de/lics-icalp/papers/B5.F), [video](https://www.youtube.com/watch?v=GUT8j4V6Nzg))

specifically with [[Quipper]]:

* {#FKRS20} [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 ([arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9))

See also [QS](https://ncatlab.org/schreiber/show/QS).

On quantum **software engineering**:

* Manuel A. Serrano, Ricardo Pérez-Castillo, Mario Piattini, *Quantum Software Engineering*, Springer (2022) $[$[doi:10.1007/978-3-031-05324-5](https://doi.org/10.1007/978-3-031-05324-5)$]$

On **[[software verification]]**:

with [[QWIRE]]:

* [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS 266, 2018, pp. 119-132 ([arXiv:1803.00699](https://arxiv.org/abs/1803.00699))

* [[Robert Rand]], [[Jennifer Paykin]], Dong-Ho Lee, [[Steve Zdancewic]], *ReQWIRE: Reasoning about Reversible Quantum Circuits*, EPTCS **287** (2019) 299-312 $[$[arXiv:1901.10118](https://arxiv.org/abs/1901.10118), [doi:10.4204/EPTCS.287.17](https://doi.org/10.4204/EPTCS.287.17)$]$

for [[Quipper]] with [[QPMC]]:

* Linda Anticoli, Carla Piazza, Leonardo Taglialegne, Paolo Zuliani, _Towards Quantum Programs Verification: From Quipper Circuits to QPMC_, In: Devitt S., Lanese I. (eds.) *Reversible Computation. RC 2016*, Lecture Notes in Computer Science **9720** Springer (2016) $[$[arXiv:1708.06312](https://arxiv.org/abs/1708.06312), [doi:10.1007/978-3-319-40578-0_16](https://doi.org/10.1007/978-3-319-40578-0_16)$]$

with [[CoqQ]]: [Ying et al. (2022)](#YingEtAl22)


quantum propagator

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] and in the [[Schrödinger picture]], the term _propagator_ refers to the [[linear map]] between [[spaces of quantum states]] ([[wave functions]]) which describes how these [[states]] change in [[time]] ([[spacetime]]), hence how the corresponding [[quantum system]] _propagates_.

In [[quantum mechanics]] the time [[propagator]] is given by the [[parallel transport]] along the [[worldline]] given by the [[Hamiltonian]] (energy) [[quantum operator]].

In [[perturbative quantum field theory]] of [[free fields]] propagation is typically encoded in the [[Green functions]] of the [[wave equation]]/[[Klein-Gordon operator]] and here it is these Green functions that are referred to as _propagators_.

[[!include propagators - table]]

## Further examples

* [[Chern-Simons propagator]]

## Related concepts

* [[path integral]]

* [[Hamiltonian]]

## References

An overview of the [[Green functions]] of the [[Klein-Gordon operator]], hence the [[Feynman propagator]], [[advanced propagator]], [[retarded propagator]], [[causal propagator]] etc. is given in

* {#Kocic16} [[Mikica Kocic]], _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

[[!redirects quantum propagators]]

[[!redirects propagator]]
[[!redirects propagators]]


quantum reader monad

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#### Categorical algebra
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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

By the *linear reader monad* or *quantum reader monad* we shall mean the [[reader monad]] but on the [[category]] of [[vector spaces]] (in applications specifically of [[complex vector spaces]], though essentially all discussion here applies over any [[ground field]]):

$$
  B \,\colon\, Set
  \;\;\;\;\;
  \vdash
  \;\;\;\;\;
  \bigcirc_B
  \;\coloneqq\;
  Map(B,-)
  \;=\;
  \underset{B}{\prod}(-)
  \;\colon\;
  Vect
  \to
  Vect
  \,.
$$

The reason for the [[quantum physics]]-terminology is that this monad is closely related to --- in fact arguably is the fundamental formalization in [[categorical logic]] of --- the concept of *[[quantum measurement]]*, including its subtle peculiarities such as the dependence on a choice of *measurement basis* and the phenomenon of [[wavefunction collapse]] entailed by the measurement process. It would in fact make good sense to speak of the **quantum measurement monad** --- but notice that the conceptual difference between "reading" (say a global parameter, as is common interpretation for the [[reader monad]] when used as a [[monad in computer science]]) and "measuring" is small already informally and seems to all but vanish in the [[modal logic]] of the quantum reader (its "[quantum modal logic](necessity+and+possibility#ModalQuantumLogic)", if you wish).

Often one is interested in the case that the parameter set $B$ here is a [[finite set]] (explicitly so in applications to [[quantum information theory]] and [[quantum computing]] where eg. $B =$ [[Bool]] for [[qbits]], but it could probably be argued that any *real* measurement in [[experiment]] can only ever resolve a [[finite set]] of potential outcomes), and this is the case we shall focus on now unless otherwise specified.

In this case of [[finite set|finite]] parameter set $B$, the existence of [[biproducts]] in [[VectorSpaces]] --- namely of *[[direct sums]]* $\oplus_B$ --- implies that the [[underlying]] [[functor]] of the quantum $B$-reader monad coincides with that of the $B$-[[coreader comonad]] $\star_B \;\coloneqq\; \coprod_B (-) \;\colon\; Vect \to Vect$:

\[
  \label{IsomorphismsOfUnderlyingFunctors}
  \array{
  \star_B \mathscr{V}
  &\simeq&
  \oplus_B \mathscr{V}
  &\simeq&
  \bigcirc_B \mathscr{V}
  \\
  &&
  \simeq
  \\
  &&
  \mathbb{1}^B \otimes \mathscr{V}
  }
  \,.
\]

In fact, the induced joint [[monad]]- and [[comonad]]-[[structure]] on this functor are compatible to jointly make the quantum reader be a *[[Frobenius monad]]*:

An organized way to see this [[Frobenius monad]]-[[structure]] on the quantum reader monad over a finite parameter set is to observe (which is a straightforward inspection, but we spell it out [below](#Details)) that the quantum $B$-reader monad is also [[isomorphism|isomorphic]] to the *$\mathbb{1}^B$-[[writer monad]]* corresponding to the [[associative algebra]]
\[
  \label{AlgebraOfProjectorsInIntroduction}
  \mathbb{1}^B
    \;=\;
  \oplus_B k
  \;\in\;
  CMon\big(Vect_k\big)
  \,=\,
  CAlg_k
  \,,
\]
which is [[generators and relations|generated]] by a $B$-[[indexed set]] of mutually orthogonal [[projection operators]] $\{P_b\}_{b \colon B}$ (over the [[ground field]] $k$).

It is immediate to check that (eq:AlgebraOfProjectorsInIntroduction) canonically carries the further [[structure]] of a [[Frobenius algebra]] (see the example [there](Frobenius+algebra#VectorSpaceWithLinearBasis))
with [[coproduct]] $\delta \;\colon\; \mathbb{1}^B \to \mathbb{1}^B \otimes \mathbb{1}^B$ given by duplication over the $B$-[[linear basis|basis]]: $\delta \;\colon\; P_b \,\mapsto\, P_B \otimes P_b$. But this makes manifest that the [[underlying]] [[functor]] of [[tensor product of vector spaces|tensoring]] $\mathbb{1}^B \otimes (-)$ also carries the [[structure]] of a [[comonad]] in a compatible way to make a [[Frobenius monad]].

\[
  \label{EquivalencesOfTheQuantumReaderMonad}
\]
\begin{imagefromfile}
    "file_name": "QuantumReaderFrobeniusMonad-221112.jpg",
    "width": "600",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In this its incarnation as the $\mathbb{1}^B$-[[writer monad|writer]] [[Frobenius monad]], the quantum $B$-reader monad is famous in the literature on [[quantum information theory via dagger-compact categories]] as encoding [[quantum measurement]] with respect to the $B$-basis -- known as **classical structures** or **Frobenius structures**  in these references [[Coecke & Pavlović (2008), §1.5.1](#CoeckePavlović08), [Coecke & Paquette (2008), §2.3](#CoeckePaquette08)], as well as [[classically controlled quantum gates]] [[Heunen & Vicary (2019), around Lem.  5.61](#HeunenVicary19), albeit not making the monadic algebra explicit] (See also at *[[no-cloning theorem]]* [this Remark](no-cloning+theorem#ExampleOfNonCloningOfCoalgebraMap).)

In this guise, the quantum reader monad is reflected by the "green spiders" in the [[ZX-calculus]].

## Properties
 {#Properties}

### Relation to dependent linear types

We discuss the quantum reader monad as the reader monad induced by $B$-[[dependent linear types]] which have [[biproducts]], such as $B$-[[indexed sets]]  of [[vector spaces]]. Notation and graphics follows [CQTS (2022)](#CQTS22).

\begin{proposition}
\label{QuantumBReaderAlgebrasAreBDependentLinearTypes}
**(quantum $\bigcirc_B$-algebras are $B$-[[dependent linear types]])**
\linebreak
For $B \;\in\;$ [[FiniteSets]], the [[category]] of [[algebra over a monad|algebras]] for the $B$-[[reader monad]]

$$
  \array{
    \bigcirc_B
    \colon
    &
    Vect
    &\xrightarrow{\phantom{---}}&
    Vect
    \\
    &
    \mathscr{H}
    &\mapsto&
    Maps\big(B, \mathscr{H} \big)
    \mathrlap{
      \;
      \simeq
      \;
      \underset{b \colon B}{\bigoplus}
      \mathscr{H}
    }
  }
$$

on the category [[Vect]] of [[vector spaces]] (over any given [[ground field]]) is [[equivalence of categories|equivalent]] to that of [[vector bundles]] over $B$ (hence to $B$-[[indexed sets]] of [[vector spaces]]):
$$
  Vect^{\bigcirc_B}
  \;\;\simeq\;\;
  Vect_B
  \,.
$$

\end{proposition}

\begin{imagefromfile}
    "file_name": "LinearTypesAsModalModules-221113.jpg",
    "width": "840",
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}
(Something close to this statement, in its [[adjoint comonad|adjoint]] form for the quantum [[coreader comonad]] $\bigstar$, is the content of [Coecke & Pavlović 2008, Thm. 1.5](#CoeckePavlović08).)
\begin{proof}
 We state first an abstract proof and then a concrete proof.

Abstractly, observe that [[Vect]] has (finite) [[biproducts]] (given by the [[direct sum]] of vector spaces) which together with the assumption that $B$ is [[finite set|finite]] implies that $\underset{B}{\prod} \;\colon\; Vect_B \xrightarrow{\;} Vect$:

1. is not only a [[right adjoint]] but also a [[left adjoint]] (hence an [[ambidextrous adjunction|ambidextrous adjoint]] to [[pullback of vector bundles]] along $B \to \ast$), [[left adjoints preserve colimits|hence]] in particular it [[preserved colimit|preserves]] all [[colimits]];

1. is a [[conservative functor]] (since the $B$-components of any morphism of vector bundles over $V$ can still be extracted via ([[coprojection|co-]])[[projections]] from their image under forming the [[biproduct]] over $B$).

Therefore the conditions for the [[monadicity theorem]] are met (see [there](monadicity+theorem#CrudeMonadicityTheorem)), implying that the functor is [[monadic functor]], which in turn implies the claim.

\linebreak

Alternatively, we now check the claim more concretely by unwinding what it means for a vector space to carry $\bigcirc_B$-algebra structure and for a linear map between vector spaces to be a [[homomorphism]] for this structure:

First observe that the $\bigcirc_B$-[[monad]] product $join^{\bigcirc_B} \;\colon\; \bigcirc_B \bigcirc_B \xrightarrow{join^{\bigcirc}} \bigcirc_B$ on a [[vector space]] $\mathscr{H} \,\in\, Vect$ is explicitly given by:

$$
  \array{
  \underset{b \colon B}{\bigoplus}
  \;
  \underset{b' \colon B}{\bigoplus}
  \;
  \mathscr{H}
  &
  \xrightarrow{\;\; join^{\bigcirc}_{\mathscr{H}} \;\;}
  &
  \underset{b'' \colon B}{\bigoplus}
  \mathscr{H}
  \\
  \big(
     \psi_{b,b'}
  \big)_{b,b' \colon B}
  &\mapsto&
  \big(
    \psi_{b'' ,b''}
  \big)_{b'' \colon B}
  }
$$

and that a $\bigcirc_B$-algebra structure on $\mathscr{H}$ is a [[linear map]] of this form:

\[
  \label{MapsUnderlyingAlgebraStructureOverLinearReaderMonad}
  \rho^\bigcirc_{\mathscr{H}}
  \;\colon\;
  \underset{b \colon B}{\oplus}
  \mathscr{H}
  \xrightarrow{\phantom{-}(P_b)_{b \colon B}\phantom{-}}
  \mathscr{H}
  \,.
\]

Of such maps, the $\bigcirc_B$-action property ([here](algebra+over+a+monad#eq:ActionProperty)) demands that the following two linear maps are equal:

$$
  \array{
    &
    \underset{b \colon B}{\bigoplus}
    \;
    \underset{b' \colon B}{\bigoplus}
    \;
    \mathscr{H}
      &\xrightarrow{\; join^\bigcirc_{\mathscr{H}}  \;}&
    \underset{b'' \colon B}{\bigoplus}
    \mathscr{H}
      &\xrightarrow{\; \rho^\bigcirc_{\mathscr{H}} \;}&
    \mathscr{H}
    \\
    &
    \big(
       \psi_{b,b'}
    \big)_{b,b' \colon B}
    &\mapsto&
    \big(
       \psi_{b'',b''}
    \big)_{b'',b'' \colon B}
    &\mapsto&
    \underset{b'' \colon B}{\sum}
    P_{b''}(\psi_{b'',b''})
    \\
    \text{and}
    &
    \\
    &
    \underset{b \colon B}{\bigoplus}
    \;
    \underset{b' \colon B}{\bigoplus}
    \;
    \mathscr{H}
      &\xrightarrow{\; \bigcirc \rho^\bigcirc_{\mathscr{H}}  \;}&
    \underset{b'' \colon B}{\bigoplus}
    \mathscr{H}
      &\xrightarrow{\; \rho^\bigcirc_{\mathscr{H}} \;}&
    \mathscr{H}
    \\
    &
    \big(
       \psi_{b,b'}
    \big)_{b,b' \colon B}
    &\mapsto&
    \big(
       \underset{b \colon B}{\sum}
       P_{b}(\psi_{b,b'})
    \big)_{b' \colon B}
    &\mapsto&
    \underset{b, b' \colon B}{\sum}
    P_{b'}\big(P_{b}(\psi_{b,b'})\big)
    \mathrlap{\,.}
  }
$$

By considering the value of these maps on [[tuples]] of [[vectors]] $\big(\psi_{b,b'}\big)_{b,b' \colon B}$ which are non-[[zero]] only for single elements $(b,b') \in B \times B$ one finds that the above condition is equivalent to the following condition:

$$
  \underset{
    b, b' \colon B
  }{
    \forall
  }
  \;\;\;\;
    P_b \circ P_{b'}
  \;=\;
  \left\{
    \array{
      P_b &\text{if} \; b = b'
      \\
      0  & \text{otherwise}
      \,.
    }
  \right.
$$

Moreover, the $\bigcirc_B$-[[unit of a monad|unit]] $\id \xrightarrow{\;\; ret^{\bigcirc_B} \;\;} \bigcirc_B$ is given by the [[diagonal map]] into the [[biproduct]]

$$
  \array{
    \mathscr{H}
    &\xrightarrow{\;\;
      ret^{\bigcirc_B}_{\mathscr{H}}
    \;\;}
    &
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}
    \\
    \psi
    &\mapsto&
    (\psi_b \coloneqq \psi)_{b \colon B}
  }
$$

so that the unitality property ([here](algebra+over+a+monad#eq:UnitProperty)) of the $\bigcirc_B$-algebra structure $(P_b)_{b \colon B}$ demands equivalently that the [[composition|composite]]

$$
  \array{
    \mathscr{H}
    &\xrightarrow{\; ret^{\bigcirc_B}_{\mathscr{H}} \;}&
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}
    &\xrightarrow{\; \rho^{\bigcirc_B}_{\mathscr{H}} \;}&
    \mathscr{H}
    \\
    \psi
    &\mapsto&
    \big(
      \psi
    \big)_{b \colon B}
    &\mapsto&
    \underset{b \colon B}{\sum}
    \;
    P_b(\psi)
  }
$$

is the [[identity function]] on $\mathscr{H}$, hence that this system of [[projection operators]] is "complete" in that

$$
  \underset{b \colon B}{\bigoplus} P_b
  \;=\;
  id_{\mathscr{H}}
  \,.
$$

In summary, this means that for $B$-[[tuples]] of [[linear maps]] $(P_b \colon \mathscr{H} \to \mathscr{H})_{b \colon B}$ as in (eq:MapsUnderlyingAlgebraStructureOverLinearReaderMonad) to constitute a $\bigcirc_B$-module structure is equivalent to them being
systems of orthogonal linear [[projection operators]], whose [[images]]

$$
  \mathscr{H}_b \;\coloneqq\; im(P_b) \;\subset\; \mathscr{H}
$$

[[linear span|span]]$\;$ $\mathscr{H}$ under [[direct sum]]:

$$
  \mathscr{H}
  \;\simeq\;
  \underset{b \colon B}{\bigoplus}
  \mathscr{H}_b
  \,.
$$

Such structures, of course, are exactly the images under the right [[base change]] $\underset{B}{\prod} \;\colon\; Vect_B \longrightarrow{\;} Vect$ of $B$-[[indexed sets]] of vector spaces.

Finally, the [[homomorphism]]-property on a [[linear map]] $\mathscr{H} \xrightarrow{\phi} \mathscr{H}'$ between the underlying vector spaces of two such $\bigcirc_B$-modules demands that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}
    &\xrightarrow{\;\;\;
      \underset{b \colon B}{\oplus}
    \;\;\;}&
    \underset{b \colon B}{\bigoplus}
    \mathscr{H}
    \\
    \mathllap{
      {}^{
      \big(
        P_b
      \big)_{b \colon B}
      }
    }
    \big\downarrow
    &&
    \big\downarrow
    \mathrlap{
      {}^{
      \big(
        P'_b
      \big)_{b \colon B}
      }
    }
    \\
    \mathscr{H}
    &
    \underset{\phantom{---} \phi \phantom{---} }
      {\longrightarrow}
    &
    \mathscr{H}'
    \,.
  }
$$

The $B$-indexed components of this condition require that $\phi$ [[commutator|commutes]] with all these [[projection operators]], in that

$$
  b \colon B
  \;\;\;\;
  \vdash
  \;\;\;\;
  P'_b \circ \phi
  \;=\;
  \phi \circ P_b
  \,.
$$

But this evidently means that $\phi$ itself is the image under the [[direct sum]]-[[functor]]
$$
  \phi
  \;=\;
  \underset{b \colon}{\oplus}
  \phi_b
$$
of a $B$-[[tuple]] of [[linear maps]] between the $B$-indexed component spaces:
$$
  b \colon B
  \;\;\;
  \colon
  \;\;\;
  \mathscr{H}_b \longrightarrow \mathscr{H}'
  \,.
$$
This is the defining property of morphisms in $Vect_B$ and hence shows that $\bigcirc_B$-algebra homomorphisms are equivalently morphisms of $B$-[[indexed sets]] of vector spaces, hence that the two [[categories]] are [[equivalence of categories|agree]].
\end{proof}

\begin{example}\label{FreeQuantumReaderAlgebras}
**(free quantum $\bigcirc_B$-algebras and quantum measurement bases)**
\linebreak
  The *[free](algebra+over+a+monad#FreeAlgebras)* $\bigcirc_B$-algebras in [[Vect]] are those whose underlying vector space is of the form $\bigcirc_B \mathscr{B} = \underset{b \colon B}{\bigoplus} \mathscr{B}$ for any $\mathscr{B} \,\in \, Vect$, hence, under the equivalence of Prop. \ref{QuantumBReaderAlgebrasAreBDependentLinearTypes}, are those $B$-[[dependent linear types]] which happen to assign the same vector space $\mathscr{B}$ for all $b \colon B$.

But [[homomorphisms]] of such free $\bigcirc_B$-algebras are still allowed to be non-trivially $B$-dependent families of linear maps. This is directly a special case of Prop. \ref{QuantumBReaderAlgebrasAreBDependentLinearTypes} but is also immediate under the [Kleisli equivalence](Kleisli+category#KleisliEquivalence) from observing that $\bigcirc_B$-[[Kleisli morphisms]] are of the following form:
$$
  \array{
    \mathscr{B}
    &\xrightarrow{
      \big(
        \phi_b
      \big)_{b \colon B}
    }&
    \underset{b \colon B}{\bigoplus}
    \mathscr{B}'
    \,.
  }
$$

### Relation to quantum measurement
 {#RelationToQuantumMeasurement}

In [[quantum information theory]] the $\bigcirc_B$-algebras in [[complex vector spaces]] which are free on the [[tensor unit]] $\mathbb{C} \,\in\, \mathbb{C} Vect$ (the [[dimension of a vector space|1-dimensional]] [[complex vector space]]) play the role of  ([[finite dimensional vector space|finite dimensional]] [[complex vector spaces]] [[underlying]]) [[finite-dimensional Hilbert spaces]] [[space of quantum states|of quantum states]] which are [[linear space|spanned]] by the set $B$ regarded as a "[[quantum measurement]] [[linear basis|basis]]".

For example, the [[dependent linear type]] of [[qbits]] is the free $\bigcirc_B$-algebra $\bigcirc_{Bool} \mathbb{C}$ spanned by the [[intuitionistic type theory|classical type]] [[Bool]] $= \{0,1\}$ of [[bits]], reflecting the two canonical [[quantum measurement]]-outcomes ($\vert 0\rangle$, $\vert 1 \rangle$) of [[qbits]].

{#QuantumMeasurementAsIndefinitenessHandling} Curiously, from this one finds that *[[quantum measurement]] is exactly indefiniteness handling*

where

* "indefiniteness" is understood as the the effect/[[modality]] expressed by the reader monad $\bigcirc_B$ (see [here](necessity+and+possibility#RelationToPotentiality)),

* "handling" is understood as *effect handling* in the sense of effect-[[monads in computer science]] (see [here](monad+in+computer+science#MonadModulesInIdeaSection));

namely:

\begin{imagefromfile}
    "file_name": "QMeasurementViaIndefinitenessHandling-221109c.jpg",
    "width": "730",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Notice that this process may be understood as the "[[dynamic lifting]]" of [[quantum measurement]]-results into the [[context]] (here: $B = $ [[Bool]]) of the ambient [[dependent linear type theory]].

For more on this see at *[[quantum circuits via dependent linear types]]*.
\end{example}

\linebreak

This naturally relates to the discussion of [[quantum measurement]] via [[Frobenius algebra]]-structures which is popular in the context of [[quantum information theory via dagger-compact categories]] -- as follows:

\begin{remark}\label{QuantumrReaderMonadIsFrobenius}
**(quantum reader monad is Frobenius)**
\linebreak
Since the [[direct sum]] of [[vector spaces]] is a [[biproduct]] and using our running assumption that $B$ is a [[finite set]], it follows that the [[underlying]] [[functor]] of the quantum [[reader monad]] from Prop. \ref{QuantumBReaderAlgebrasAreBDependentLinearTypes} coincides with that of the [[coreader comonad]], and hence that the quantum reader monad is a *[[Frobenius monad]]* (see there).
\end{remark}

\begin{remark}\label{QuantumReaderMonadIsSpecialFrobeniusWriterMonad}
**(quantum reader monad is special Frobenius writer monad)**
\linebreak
  Consider the $B$-[[indexed set|indexed]] the [[direct sum]] $k^{\otimes^B}$ of the [[ground field]] with itself as a $k$-[[associative algebra|algebra]] and write

$$
  {k^{\otimes^B}}\text{-}Writer
  \;\colon\;
  Vect \to Vect
$$

for the [[writer monad]] corresponding to this [[monoid object]] in [[Vect]]: The monad which acts by forming the [[tensor product]] $Writer_{k^{\otimes^B}}(V) \;\coloneqq\; V \otimes \big( k^{\otimes^B} \big) $ and whose monad multiplication and [[unit of a monad]] are induced from the [[multiplication]] and [[unit]] in this monoid.

The above proof of Prop. \ref{QuantumBReaderAlgebrasAreBDependentLinearTypes} shows at once that this [[writer monad]] is [[isomorphism|isomorphic]] to the $B$-reader monad:
$$
  \bigcirc_B
  \;\simeq\;
  Writer_{k^{\otimes^B}}
  \,.
$$
Now the [[module over a monad|monad modules]] over a [[writer monad]] are just the ordinary [[modules]] over the corresponding [[monoid]], so that
$$
  \bigcirc_B Mod(Vect)
  \;\simeq\;
  \big(
    k^{\otimes^B}
  \big)Mod(Vect)
  \,.
$$
This provides a rather transparent re-derivation of and alternative perspective on Example \ref{FreeQuantumReaderAlgebras}.

[[formal duality|Dually]], as in Prop. \ref{QuantumReaderMonadIsFrobenius}, the quantum [[coreader comonad]] is isomorphic to the [[coreader comonad]] corresponding to the [[coalgebra]]-[[structure]] on $k^{\otimes^B}$.
Hence, as a [[Frobenius monad]] (Prop. \ref{QuantumReaderMonadIsFrobenius}), the quantum reader corresponds to $k^{\otimes^B}$ regarded as a [[Frobenius algebra]], in fact as a commutative and special Frobenius algebra. In this form the quantum  reader monad is prominent in the literature on [[quantum information theory via dagger-compact categories]] [[Coecke & Pavlović (2008), §1.5.1](#CoeckePavlović08), [Coecke & Paquette (2008), §2.3](#CoeckePaquette08)].
\end{remark}

### Monoidal monad structure and Decoherence
 {#MonoidalMonadStructure}

\begin{proposition}
\label{SymmetricMonoidalMonadStructure}
  The above quantum reader monad carries the [[structure]] of a [[symmetric monoidal functor|symmetric]] [[monoidal monad]] with respect to the [[tensor product of vector spaces]] (or whatever the [[tensor product]] of the given linear types) as follows:

\begin{imagefromfile}
    "file_name": "LaxMonoidalStructureOnQuantumReader-230930.jpg",
    "width": 710,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

{#ComomoidalStructure} [[formal duality|Dually]], the quantum coreaders is comonoidal comonadic:

\begin{imagefromfile}
    "file_name": "LaxCoMonoidalStructureOnQuantumCoreader-230930.jpg",
    "width": 710,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

\end{proposition}
(Here we have assumed [[strict monoidal category|strict monoidal structure]] cf. [[Schauenburg 2001](strinct+monoidal+category#Schauenburg01)], to notationally suppress its [[associators]] and [[unitors]].)

\begin{remark}
\label{MonoidalMonadStructureAndDecoherence}
  The monoidal monad structure from Prop. \ref{SymmetricMonoidalMonadStructure} extends the [above](#RelationToQuantumMeasurement) quantum measurement typing from [[pure states]] to [[mixed states]], by implementing the [[decoherence]] process which eliminates the off-diagonal terms in the density matrix, as indicated in the following:

<center>
<img src="/nlab/files/QuantumReaderMonoidalnessAsDecoherence-230903.jpg" width="780">
</center>

This operation hence sends quantum states (pure or not) to their corresponding [[probability distributions]] (under [[quantum measurement]]) just as demanded by the *[[Born rule]]* postulate.

\end{remark}

\linebreak

## Related entries

* [[quantum costate comonad]]

* [[quantum circuits via dependent linear types]]

* [[doubly closed monoidal category]]

## References

The [[quantum reader monad]] -- implicitly, in its incarnation as the $\mathbb{1}^B$-[[writer monad|writer]] [[Frobenius monad]] -- is highlighted in the literature on [[quantum information theory via dagger-compact categories]] as formalizing "classical structures" (namely [[linear bases]] for [[quantum measurement]]):

* {#CoeckePavlović08} [[Bob Coecke]], [[Duško Pavlović]], §1.5.1 of: *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]], §2.3 in: *POVMs and Naimark's theorem without sums*, Electronic Notes in Theoretical Computer Science **210** (2008) 15-31 [[arXiv:quant-ph/0608072](https://arxiv.org/abs/quant-ph/0608072), [doi:10.1016/j.entcs.2008.04.015](https://doi.org/10.1016/j.entcs.2008.04.015)]

* [[Bob Coecke]], [[Eric Paquette]], [[Duško Pavlović]], Def. 2.8 in: *Classical and quantum structures* (2008) [[pdf](http://www.comlab.ox.ac.uk/files/627/RR-08-02.pdf), [[Coecke-Paquette-Pavlovic-CQS.pdf:file]]]

* [[Bob Coecke]], [[Eric Oliver Paquette]], [[Duško Pavlović]], *Classical and quantum structuralism* [[arXiv:0904.1997](https://arxiv.org/abs/0904.1997)]

and the evolution of the "classical structures"-monad into the "spider"-ingredient 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)]

Also, albeit without explicit mentioning of [[monad|monadic]] structures:

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

See also:

* [[Stefano Gogioso]], *Finite-dimensional Quantum Observables are the Special Symmetric Dagger-Frobenius Algebras of CP Maps*, EPTCS **394** (2023) 432-441 [[arXiv:2110.07074](https://arxiv.org/abs/2110.07074)]

The above discussion follows:

* {#CQTS22} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:The Quantum Monadology|The Quantum Monadology]]* [[arXiv:2310.15735](https://arxiv.org/abs/2310.15735)]

[[!redirects quantum reader monads]]

[[!redirects quantum measurement monad]]
[[!redirects quantum measurement monads]]

[[!redirects classical structure]]
[[!redirects classical structures]]


quantum relation

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of *quantum relations* of [Weaver 2010](#Weaver10), [Weaver & Kuperberg 2011](#WeaverKuperberg11) is a "[[quantum physics|quantum]]" or [[noncommutative geometry]]/[[noncommutative measure theory]]-version of the basic notion of [[relations]] on sets.

Concretely, they define a quantum relation on a [[von Neumann algebra]] $\mathcal{A}$ equipped with an embedding $\mathcal{A} \subset \mathcal{B}\mathscr{H}$ (into the [[bounded operators]] on some [[Hilbert space]]) as an operator [[bimodule]] over the [[commutant]] of $\mathcal{A}$ which is $\ast$-closed in the [[weak operator topology]], and explain why this (non-obvious) definition is sensible; for instance it has good application to [[quantum error correction]].

According to [Kornell 2020](#Kornell20), and in mild paraphrase (following the discussion at *[[dependent linear type]]* and *[[quantum circuits via dependent linear types]]*):

* **[[quantum sets]]** are  [[indexed sets]] of [[finite-dimensional Hilbert spaces]], hence finite-[[rank of a vector bundle|rank]] [[Hilbert bundle|Hilbert]]-[[vector bundles]] over a [[discrete topological spaces]] regarded as a [[sets]], and regarded as equipped with the [[external tensor product]] $\boxtimes$ [of vector bundles](VectBund#TensorMonoidalStructure);

* a **quantum relation** between quantum sets $(x \colon X) \times \mathscr{H}_x$ and $(x' \colon X') \times \mathscr{H}'_x$ is a [[monomorphism]] from a quantum set $\big((x,x') \colon X \times X'\big) \times \mathscr{R}_{(x,x')}$ to $\mathscr{H}^\ast_\bullet \boxtimes \mathscr{H}'_\bullet$:

  $$
    (x,x') \colon X \times X'
    \;\;\;
    \vdash
    \;\;\;
    \mathscr{R}_{(x,y)}
    \hookrightarrow
    \mathscr{H}^\ast_{(x,y)}
      \otimes
    \mathscr{H}^'_{(x,y)}
  $$

With [[composition]] the evident [[matrix multiplication]] ([Kornell 2020 (5)](#Kornell20)), quantum relations between quantum sets form a [[category]] $qRel$, which is a [[dagger-compact category]].

As such, this serves as [[categorical semantics]] for [[quantum programming languages]] like [[Quipper]] equipped with term recursion, via [[quantum CPOs]] ([Kornell, Lindenhovius & Mislove 2021](#KornellLindenhoviusMislove21)).

## Related concepts

* [[quantum set]]

* [[quantum CPO]]

## References

* {#Weaver10} [[Nik Weaver]], *Quantum relations* [[arXiv:1005.0354](https://arxiv.org/abs/1005.0354)]

* {#WeaverKuperberg11} [[Nik Weaver]], [[Greg Kuperberg]], *A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations*, Memoirs of the AMS **215** (2011) 1010 [[ams:memo-215-1010](https://bookstore.ams.org/memo-215-1010)]

* {#Kornell20} [[Andre Kornell]], *Quantum Sets*, J. Math. Phys. **61** 102202 (2020) [[doi:10.1063/1.5054128](https://doi.org/10.1063/1.5054128)]

  > (cf. *[[quantum set]]*)

* [[Andre Kornell]], *Discrete quantum structures* [[arXiv:2004.04377](https://arxiv.org/abs/2004.04377)]

* [[Andre Kornell]], *Discrete quantum structures I: Quantum predicate logic*, J. Noncommut. Geom. (2023) [[doi:10.4171/jncg/531](https://doi.org/10.4171/jncg/531)]

* {#KornellLindenhoviusMislove21} [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], §2 in: *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

  > (in the context of [[quantum CPOs]])

[[!redirects quantum relations]]


quantum Schubert cell

Quantum Schubert cells are a quantum group analogue of [[Schubert cell]]s.

* C. De Concini, C. Procesi, _Quantum Schubert cells and representations at roots of 1_, [pdf](https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=2DE1620582664C79507AE3BDB2E4455F?doi=10.1.1.30.4615&rep=rep1&type=pdf)

* S. Levendorskii, [[Yan Soibelman]], _Algebras of functions on compact quantum groups, Schubert cells and quantum tori_ Commun.Math. Phys. __139__, 141–170 (1991) [doi](https://doi.org/10.1007/BF02102732)

* A. Berenstein, J. Greenstein, _Canonical bases of quantum Schubert cells and their symmetries_, Sel. Math. New Ser. 23, 2755–2799 (2017) [doi](https://doi.org/10.1007/s00029-017-0316-8)

* Milen Yakimov, _Spectra and catenarity of quantum Schubert cells_, Glasgow Mathematical Journal, 55(A), 169-194 (2013) [doi](https://doi.org/10.1017/S0017089513000578)


quantum secret sharing

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[quantum error correction]]

## References

* M. Hillery, V. Buzek, A. Berthiaume, *Quantum secret sharing*, Phys. Rev. **A59** (1999) 1829 ([doi:10.1103/PhysRevA.59.1829](https://doi.org/10.1103/PhysRevA.59.1829))

* Richard Cleve, Daniel Gottesman, Hoi-Kwong Lo, *How to share a quantum secret*, Phys. Rev. Lett. 83 (1999) 648-651 ([arXiv:quant-ph/9901025](https://arxiv.org/abs/quant-ph/9901025))


quantum sensing

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

#Contents#
* table of contents
{:toc}

## Idea

By *quantum sensing* (or *quantum metrology*) one means the theory technology of making high-resolution [[measurements]] by making use of [[quantum physics]] effects.

## Related concepts

* [[nuclear magnetic resonance]]

* [[nitrogen-vacancy center in diamond]]

* [[quantum measurement]]

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

* [[quantum communication]]

* [[quantum information theory]]

## References

* Vittorio Giovannetti, [[Seth Lloyd]], Lorenzo Maccone, *Quantum metrology*,  	Phys. Rev. Lett. **96** (2006) 010401 [[arXiv:quant-ph/0509179](https://arxiv.org/abs/quant-ph/0509179), [doi:10.1103/PhysRevLett.96.010401](https://doi.org/10.1103/PhysRevLett.96.010401)]

* C. L. Degen, F. Reinhard, P. Cappellaro, *Quantum sensing*, Rev. Mod. Phys. **89** (2017) 035002 [[arXiv:1611.02427](https://arxiv.org/abs/1611.02427), [doi:10.1103/RevModPhys.89.035002](https://doi.org/10.1103/RevModPhys.89.035002)]

See also:

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

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

Via [[spin resonance]] on a [[nitrogen-vacancy center in diamond]]:

* Jixing Zhang, Lixia Xu, Guodong Bian, Pengcheng Fan, Mingxin Li, Wuming Liu, Heng Yuan, *Diamond Nitrogen-Vacancy Center Magnetometry: Advances and Challenges* [[arXiv:2010.10231](https://arxiv.org/abs/2010.10231)]

* [[Mouzhe Xie]] et al., *Biocompatible surface functionalization architecture for a diamond quantum sensor*, PNAS **119** 8 (2022) e2114186119 [[arXiv:2108.04843](https://arxiv.org/abs/2108.04843), [doi:10.1073/pnas.2114186119](https://doi.org/10.1073/pnas.2114186119)]

Survey:

* [[Mouzhe Xie]], *Diamond-based quantum sensor for molecular analytics*, [talk at](Center+for+Quantum+and+Topological+Systems#XieApr2023) [[CQTS]] (Apr 2023) [slides: [[Xie-CQTS-Apr2023.pdf:file]]]

* [[Dominik Bucher]], *Quantum Sensing with Spin Defects in Diamond*, [talk at](CQTS#BucherOct2023) [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/ArN3Ed78sSc)]

[[!redirects quantum sensor]]
[[!redirects quantum sensors]]

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quantum set

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## Idea

According to [Kornell 2020](#Kornell20), and in mild paraphrase (following the discussion at *[[dependent linear type]]* and *[[quantum circuits via dependent linear types]]*):

* **quantum sets** are  [[indexed sets]] of [[finite-dimensional Hilbert spaces]] --- hence finite-[[rank of a vector bundle|rank]] [[Hilbert bundle|Hilbert]]-[[vector bundles]] over [[discrete topological spaces]] regarded as a [[sets]] --- and regarded as equipped with the [[external tensor product]] $\boxtimes$ [of vector bundles](VectBund#TensorMonoidalStructure);

* a **[[quantum relation]]** between quantum sets $(x \colon X) \times \mathscr{H}_x$ and $(x' \colon X') \times \mathscr{H}'_x$ is a [[monomorphism]] from a quantum set $\big((x,x') \colon X \times X'\big) \times \mathscr{R}_{(x,x')}$ to $\mathscr{H}^\ast_\bullet \boxtimes \mathscr{H}'_\bullet$:

  $$
    (x,x') \colon X \times X'
    \;\;\;
    \vdash
    \;\;\;
    \mathscr{R}_{(x,y)}
    \hookrightarrow
    \mathscr{H}^\ast_{(x,y)}
      \otimes
    \mathscr{H}^'_{(x,y)}
  $$

With [[composition]] the evident [[matrix multiplication]] ([Kornell 2020 (5)](#Kornell20)), quantum relations between quantum sets form a [[category]] $qRel$, which is a [[dagger-compact category]].

As such, this serves as [[categorical semantics]] for [[quantum programming languages]] like [[Quipper]] equipped with term recursion, via [[quantum CPOs]] ([Kornell, Lindenhovius & Mislove 2021](#KornellLindenhoviusMislove21)).

## Related concepts

* [[quantum CPO]]

## References

* {#Kornell20} [[Andre Kornell]], *Quantum Sets*, J. Math. Phys. **61** 102202 (2020) [[doi:10.1063/1.5054128](https://doi.org/10.1063/1.5054128)]

* {#KornellLindenhoviusMislove21} [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], §2 in: *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

  > (in the context of [[quantum CPOs]])

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quantum simulation

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## Idea

By "quantum simulation" one broadly means the simulation *of* [[quantum systems]]. The term is used in two rather different ways:

1. Without further qualification, "quantum simulation" typically refers to the simulation of (complicated) quantum systems appearing in nature, such as (large) [[molecules]], *by* other (more controllable) quantum systems, notably by [[quantum computers]] (cf. *[[quantum chemistry]]*).

1. *Classical* quantum simulation refers to the simulation of [[quantum computers]] themselves by classical computers, such as for testing ([[software verification|verifying]]) [[quantum circuit]]-designs and more generally for testing [[quantum programming language|quantum programs]] and [[quantum computing]]-architectures.

## Related concepts

* [[quantum computing]]

* [[quantum sensing]]

* [[quantum communication]]

* [[quantum information]]

## References

### Quantum simulation by quantum systems

General discussion:

* Tomi H Johnson, Stephen R Clark, Dieter Jaksch, *What is a quantum simulator?*, EPJ Quantum Technol. **1** 10 (2014) [[doi:10.1140/epjqt10](https://doi.org/10.1140/epjqt10)]

* I. M. Georgescu, S. Ashhab, Franco Nori, *Quantum Simulation*, Rev. Mod. Phys. **86** 154 (2014) [[arXiv:1308.6253](https://arxiv.org/abs/1308.6253), [doi:10.1103/RevModPhys.86.153](https://doi.org/10.1103/RevModPhys.86.153)]

See also:

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

On quantum simulation of ([[lattice QFT|lattice]]) [[quantum field theory]]:

* [[John Preskill]], *Simulating quantum field theory with a quantum computer*, 36th Annual International Symposium on Lattice Field Theory, PoS **334** (2019) [[doi:10.22323/1.334.0024](https://doi.org/10.22323/1.334.0024),
[arXiv:1811.10085](https://arxiv.org/abs/1811.10085)]

specifically of  [[scattering amplitudes]] of [[bound states]]:

* Matteo Turco, Gonçalo M. Quinta, João Seixas, Yasser Omar, *Towards Quantum Simulation of Bound States Scattering* [[arXiv:2305.07692](https://arxiv.org/abs/2305.07692)]

### Quantum simulation by classical systems

* Yongshan Ding, Frederic T. Chong, *Classical Simulation of Quantum Computation*, in: *Quantum Computer Systems*, Synthesis Lectures on Computer Architecture. Springer (2020) [[doi:10.1007/978-3-031-01765-0_9](https://doi.org/10.1007/978-3-031-01765-0_9)]

* Ya-Qian Zhao, Ren-Gang Li, Jin-Zhe Jiang, Chen Li, Hong-Zhen Li, En-Dong Wang, Wei-Feng Gong, Xin Zhang, Zhi-Qiang Wei, *Simulation of Quantum Computing on Classical Supercomputers*, Phys. Rev. A **104** 032603 (2021) [[arXiv:2010.14962](https://arxiv.org/abs/2010.14962), [doi:10.1103/PhysRevA.104.032603](https://doi.org/10.1103/PhysRevA.104.032603)]

* Robert Willie, *Classical simulation of quantum circuits* (2022) [[pdf](https://www.cda.cit.tum.de/files/eda/2022_oag_classical_simulation_of_quantum_circuits.pdf)]

* Xiaosi Xu, Simon Benjamin, Jinzhao Sun, Xiao Yuan, Pan Zhang, *A Herculean task: Classical simulation of quantum computers* [[arXiv:2302.08880](https://arxiv.org/abs/2302.08880)]

* Kieran Young, Marcus Scese, Ali Ebnenasir, *Simulating Quantum Computations on Classical Machines: A Survey* [[arXiv:2311.16505](https://arxiv.org/abs/2311.16505)]

See also:

* IONQ: *[The Value of Classical Quantum Simulators](https://ionq.com/resources/the-value-of-classical-quantum-simulators)*

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quantum spin Hall effect

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## Idea

A _quantum spin Hall effect_, is

1. a variant of the [[spin Hall effect]] where [[quantum physics|quantum effetcs]] play a role;

1. a variant of the _[[quantum Hall effect]]_ where [[spin-orbit coupling]] in the underlying material (typically a [[topological insulator]]) takes the role of the external [[magnetic field]].

The fact that no external field is needed in a quantum spin Hall effect make the materials that exhibit this effect (e.g. certain [[topological insulators]]) be attractive substrates for operating on the corresponding [[anyon]]-excitations. This makes the quantum spin Hall materials potential hardware for [[topological quantum computation]].

## Related concepts

* [[Hall effect]], [[quantum Hall effect]]

* [[spin Hall effect]]

## References

The original proposal/prediction of a quantum spin Hall effect:

in [[graphene]]:

* [[Charles Kane]], [[Eugene Mele]], _Quantum Spin Hall Effect in Graphene_, Phys. Rev. Lett. 95, 226801 (2005) ([arXiv:cond-mat/0411737](https://arxiv.org/abs/cond-mat/0411737), [doi:10.1103/PhysRevLett.95.226801](https://doi.org/10.1103/PhysRevLett.95.226801))

in [[topological insulators]] based on mercury telluride ("[HgTe quantum wells](https://en.wikipedia.org/wiki/Quantum_spin_Hall_effect#In_HgTe_quantum_wells)"):

* [[Andrei Bernevig]], [[Shou-Cheng Zhang]], _Quantum Spin Hall Effect_, Phys. Rev. Lett. 96, 106802 (2006) ([arXiv:cond-mat/0504147](https://arxiv.org/abs/cond-mat/0504147), [doi:10.1103/PhysRevLett.96.106802](https://doi.org/10.1103/PhysRevLett.96.106802))

* [[Andrei Bernevig]], [[Taylor Hughes]], [[Shou-Cheng Zhang]], _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science 15 December 2006: __314__, n. 5806, pp. 1757-1761 ([doi:10.1126/science.1133734](http://dx.doi.org/10.1126/science.1133734), [arXiv:cond-mat/0611399](https://arxiv.org/abs/cond-mat/0611399))

  > [[Xiao-Gang Wen]] commented in Oct. 2013 ([rev 2](https://ncatlab.org/nlab/revision/diff/quantum+spin+Hall+effect/2), [rev 3](https://ncatlab.org/nlab/revision/diff/quantum+spin+Hall+effect/3)):  In fact, the above material have no quantum spin Hall effect since the spin is not conserved due to the spin-orbital interaction that makes the material non trivial, or the _Quantum spin Hall effect_ defined in the paper is different from that defined in the Wiki: http://en.wikipedia.org/wiki/Quantum_spin_Hall_effect

Experimental detection:

* {#KWBRB07} Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, _Quantum Spin Hall Insulator State in HgTe Quantum Wells_, Science  02 Nov 2007, Vol. 318, Issue 5851, pp. 766-770 ([doi:10.1126/science.1148047 ](https://science.sciencemag.org/content/318/5851/766))

Review:

* Wikipedia, _[Quantum spin Hall effect](https://en.wikipedia.org/wiki/Quantum_spin_Hall_effect)_

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quantum spin liquid

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## Idea

A [[topological state of matter]] governed by a kind of $\mathbb{Z}/2$-[[gauge theory]] and thought to exhibit [[topological order]].

## Related concepts

* [[topological phase of matter]]

* [[topological order]]

## References

The state was originally conceived under the name *resonating valence bond states* in:

* [[Philip W. Anderson]], *Resonating valence bonds: A new kind of insulator?*, Materials Research Bulletin
**8** 2 (1973) 153-160 [doi:10.1016/0025-5408(73)90167-0]

Further discussion:

* [[Nicholas Read]], B. Chakraborty, *Statistics of the excitations of the resonating-valence-bond state*, Phys. Rev. B **40** (1989) 7133 [[doi:10.1103/PhysRevB.40.7133](https://doi.org/10.1103/PhysRevB.40.7133)]

Understanding as [[topological order]] with [[anyon]]-excitations:

* V. Kalmeyer, [[Robert B. Laughlin]], *Equivalence of the resonating-valence-bond and fractional quantum Hall states*, Phys. Rev. Lett. **59** (1987) 2095 [[doi:10.1103/PhysRevLett.59.2095](https://doi.org/10.1103/PhysRevLett.59.2095)]

* [[Xiao-Gang Wen]], [[Frank Wilczek]], [[Anthony Zee]], *Chiral spin states and superconductivity*, Phys. Rev. B **39** (1989) 11413 [[doi:10.1103/PhysRevB.39.11413](https://doi.org/10.1103/PhysRevB.39.11413)]

* Andrew M. Essin, Michael Hermele, *Classifying fractionalization: symmetry classification of gapped $\mathbb{Z}_2$ spin liquids in two dimensions*, Phys. Rev. B **87** 104406 (2013) [[arXiv:1212.0593](https://arxiv.org/abs/1212.0593), [doi:10.1103/PhysRevB.87.104406](https://doi.org/10.1103/PhysRevB.87.104406)]

Review:

* Steven A. Kivelson, Shivaji Sondhi, *50 years of quantum spin liquids*, Nature Reviews Physics, May 26, 2023 [[arXiv:2305.18103](https://arxiv.org/abs/2305.18103), [doi:10.1038/s42254-023-00596-x](https://doi.org/10.1038/s42254-023-00596-x)]

* Wikipedia, *[Quantum spin liquid](https://en.wikipedia.org/wiki/Quantum_spin_liquid)*

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quantum state

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## Idea

A _quantum state_ is a [[state]] of a [[physical system|system]] of [[quantum mechanics]].

## Definitions

The precise mathematical notion of state depends on what mathematical formalization of quantum mechanics is used.

### Hilbert spaces

In the simple formulation over a [[Hilbert space]] $H$, a [[pure state]] is a ray in $H$.  Thus, the [[pure states]] form the space $(H \setminus \{0\})/\mathbb{C}$, where we mod out by the [[action]] of $\mathbb{C}$ on $H \setminus \{0\}$ by scalar multiplication; equivalently, we can use $S(H)/\mathrm{U}(1)$, the [[unit sphere]] in $H$ modulo the action of the [[unitary group]] $\mathrm{U}(1)$.  Often by abuse of language, one calls $H$ the 'space of states'.

The [[mixed states]] are [[density matrix|density matrices]] on $H$.  Every pure state may be interpreted as a mixed state; taking a representative normalised vector ${|\psi\rangle}$ from a ray in Hilbert space, the operator ${|\psi\rangle}{\langle\psi|}$ is a density matrix.

In principle, any quantum mechanical system can be treated using Hilbert spaces, by imposing [[superselection rule]]s that identify only some [[self-adjoint operators]] on $H$ as [[observables]].  Alternatively, one may take a more abstract approach, as follows.

### In AQFT

In [[AQFT]], a [[quantum mechanical system]] is given by a $C^*$-[[C-star-algebra|algebra]] $A$, giving the [[algebra of observables]].  Then a **state** on $A$ is

* a $\mathbb{C}$-[[linear function]] $\rho\colon A \to \mathbb{C}$

* such that

  * it is **positive**: for every $a \in A$ we have $\rho(a^\ast a) \geq 0 \in \mathbb{R}$;

  * it is **normalized**: $\rho(1) = 1$.

See also [[state in AQFT and operator algebra]].

If $H$ is a Hilbert space, then the [[bounded operators]] on $H$ form a $C^*$-algebra $\mathcal{B}H$, and states on the Hilbert space correspond directly to states on $\mathcal{B}H$.  Classical mechanics can also be formulated in AQFT; the classical space of states $X$ gives rise to a commutative [[von Neumann algebra]] $L^\infty(X)$ as the algebra of observables.

Arguably, the correct notion of state to use is that of [[quasi-state]]; every state gives rise to a unique quasi-state, but not conversely.  However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state *is* a state (at least if the Hilbert space is not of very low dimension, by [[Gleason's theorem]]).  See also the Idea-section at [[Bohr topos]] for a discussion of this point.

### In FQFT

In the [[FQFT]] formulation of [[quantum field theory]], a physical system is given by a [[cobordism]] [[representation]]

$$
  Bord_n^S \to \mathcal{C}
  \,.
$$

In this formulation the [[k-morphism|(n-1)-morphism]] in $\mathcal{C}$ assigned to an $(n-1)$-dimensional [[manifold]] $\Sigma_{n-1}$ is the _space of states_ over that manifold. A state is accordingly a [[generalized element]] of this object.

## Related concepts

* [[potentiality]]

* [[quantum state tomography]]

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## References

Discussion in [[algebraic quantum field theory]]:

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

See also:

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

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