quantum adiabatic theorem

[[!redirects adiabatic theorem]]

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## Idea

In [[quantum physics]], the *quantum adiabatic theorem* for a [[parameterized quantum system]] says that under a sufficiently slow motion of the external parameters along a [[path]] $\gamma \,\colon\, [0,1] \to P$ in parameter space $P$, [[eigenstates]] for [[commutator|mutually commuting]] [[quantum observables]] of the [[quantum system]] at $\gamma(0)$ will approximately evolve to eigenstates at $\gamma(1)$, even if the corresponding [[eigenvalues]] change significantly.

For adiabatic transport along a [[loop]], $\gamma(0) = \gamma(1)$, this implies that the [[eigenspaces]] of the system are acted on by [[unitary operators]], which (at least for multiplicity/eigen-dimension 1) are  known as *[[Berry phases]]*.

{#AsAModelForQuantumComputation} The adiabatic parameter-action on [[ground states]] of ([[topological order|topologically ordered]]) [[quantum materials]] is one model for [[quantum computation]], see at *[[adiabatic quantum computation]]*. If these adiabatic [[quantum gates]] depend only on the [[isotopy]] [[equivalence class|class]] of the parameter path, then one speaks of *[[topological quantum computation]]* (made explicit in [Arovas, Schrieffer, Wilczek & Zee 1985, p. 1](#ArovasSchriefferWilczekZee85), [Freedman, Kitaev, Larsen & Wang 2003, pp. 6](#FreedmanKitaevLarsenWang03) and [Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6)](#NayakSimonSternFreedman08), see also at *[braiding of anyonic defects](braid+group+statistics#AsBraidingOfDefects)*).

The following graphics is meant to illustrate this general idea for the case of adiabatic transformation along the [[braid group|braiding]] of nodal points in the [[Brillouin torus]] of [[semi-metal]] [[quantum materials]] (see the discussion at *[[braid group statistics]] -- [anyonic band nodes](braid+group+statistics#AnyonicBandNodes)*):

\begin{imagefromfile}
    "file_name": "AdiabaticBraidingOfBandNodes-220604.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

> (graphics from [[schreiber:Topological Quantum Computation in TED-K|SS22]])

## Related concepts

* [[Berry connection]], [[Berry phase]], [[Zak phase]]

* [[adiabatic]]

* [[adiabatic quantum computation]]

  * [[anyon]]

  * [[braid group statistics]]

* [[adiabatic switching]]

## References

### General

The original formulation:

* [[Max Born]], [[Vladimir A. Fock]],  *Beweis des Adiabatensatzes*, Zeitschrift für Physik **51**  (1928) 165–180 ([doi:10.1007/BF01343193](https://doi.org/10.1007/BF01343193))

* [[Tosio Kato]], *On the Adiabatic Theorem of Quantum Mechanics*, J. Phys. Soc. Jpn. **5** (1950) 435-439 ([doi:10.1143/JPSJ.5.435](https://journals.jps.jp/doi/10.1143/JPSJ.5.435))

* [[Gheorghe Nenciu]], *On the adiabatic theorem of quantum mechanics*,  J. Phys. A: Math. Gen. **13** (1980) L15 [[doi:10.1088/0305-4470/13/2/002](https://iopscience.iop.org/article/10.1088/0305-4470/13/2/002)]

Making explicit the unitary action on degenerate eigenspaces by non-abelian [[Berry phases]]:

* [[Gustavo Rigolin]], [[Gerardo Ortiz]], *The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy*, Phys. Rev. A **85** 062111 (2012) [[arXiv:1111.5333](https://arxiv.org/abs/1111.5333), [doi:10.1103/PhysRevA.85.062111](https://doi.org/10.1103/PhysRevA.85.062111)]

See also:

* Wikipedia, *[Adiabatic theorem](https://en.wikipedia.org/wiki/Adiabatic_theorem)*

On adiabatic transport subject to repeated [[quantum measurement]]:

* Ludmila Viotti, Ana Laura Gramajo, Paula I. Villar, Fernando C. Lombardo, Rosario Fazio, *Geometric phases along quantum trajectories* [[arXiv:2301.04222](https://arxiv.org/abs/2301.04222)]

### In condensed matter theory

Discussion in [[solid state physics]] and in view of gapped [[topological phases of matter]] ([[quantum Hall effect]], [[topological insulators]]):

* [[Joseph E. Avron]], R. Seiler & L. G. Yaffe, *Adiabatic theorems and applications to the quantum Hall effect*, Communications in Mathematical Physics **110** (1987) 33–49 [[doi:10.1007/BF01209015](https://doi.org/10.1007/BF01209015)]

* [[Joseph E. Avron]], A. Raveh, B. Zur, *Adiabatic quantum transport in multiply connected systems*, Rev. Mod. Phys. **60** (1988) 873-915 [[doi:10.1103/RevModPhys.60.873](https://doi.org/10.1103/RevModPhys.60.873)]

* [[Mathew B. Hastings]], [[Xiao-Gang Wen]], *Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance*, Phys. Rev. B **72** (2005) 045141  [[arXiv:cond-mat/0503554](https://arxiv.org/abs/cond-mat/0503554), [doi:10.1103/PhysRevB.72.045141](https://doi.org/10.1103/PhysRevB.72.045141)]

* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 [[arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0)]

* Jan Carl Budich, Björn Trauzettel, *From the adiabatic theorem of quantum mechanics to topological states of matter*, physica status solidi (RRL) **7** 1-2 (2013) 109-129  [[arXiv:1210.6672](https://arxiv.org/abs/1210.6672), [doi:10.1002/pssr.201206416](https://doi.org/10.1002/pssr.201206416)]

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *Adiabatic Theorem for Quantum Spin Systems*, Phys. Rev. Lett. **119** (2017) 060201 [[doi:10.1103/PhysRevLett.119.060201](https://doi.org/10.1103/PhysRevLett.119.060201)]

Review:

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *The adiabatic theorem in a quantum many-body setting*, in: *Analytic Trends in Mathematical Physics*, Contemporary Mathematics **741** (2020) 43-58 ([arXiv:1808.09985](https://arxiv.org/abs/1808.09985), [ISBN:978-1-4704-5388-6](https://bookstore.ams.org/conm-741))

See also:

* David Aasen, [[Zhenghan Wang]], [[Matthew B. Hastings]], *Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes* [[arXiv:2203.11137](https://arxiv.org/abs/2203.11137)]

### For topological quantum computation

References which make explicit that the standard model of [[topological quantum computation]] by [[braid group statistics|braiding]] of [[anyons]] is a form of [[adiabatic quantum computation]]:

* {#ArovasSchriefferWilczekZee85} [[Daniel P. Arovas]], [[Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](anyon#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

* {#FreedmanKitaevLarsenWang03} [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], pp. 6 of *Topological quantum computation*,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

* {#NayakSimonSternFreedman08} [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], §II.A.2 (p. 6) of: _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. **80** 1083 (2008) $[$[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)$]$

* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 $[$[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)$]$

* {#CLBFN15} Chris Cesare, Andrew J. Landahl, Dave Bacon, Steven T. Flammia, Alice Neels, *Adiabatic topological quantum computing*, Phys. Rev. A **92** (2015) 012336  $[$[arXiv:1406.2690](https://arxiv.org/abs/1406.2690), [doi:10.1103/PhysRevA.92.012336](https://doi.org/10.1103/PhysRevA.92.012336)$]$

[[!redirects adiabatic theorems]]


quantum affine algebra

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## Related concepts

* [[affine Lie algebra]]

* [[quantum group]]

* [[Yangian]]

## References

### Ordinary

* [[Igor Frenkel]], Naihuan Jing, _Vertex representations of quantum affine algebras_, Proc. Nat'l. Acad. Sci. USA 85 (1988) , 9373–9377.

* Naihuan Jing, _Twisted vertex representations of quantum affine algebras_, Invent. Math. 102 (1990), 663–690.

* Naihuan Jing, _Quantum Kac-Moody Algebras and Vertex Representations_ ([arXiv:math/9802036](http://arxiv.org/abs/math/9802036))

### Categorified
 {#ReferencesCategorified}

Discussion of [[categorification]] of quantum Kac-Moody algebras is in:

* Raphael Rouquier, _2-Kac-Moody algebras_ ([arXiv:0812.5023](http://arxiv.org/abs/0812.5023))

* [[Aaron Lauda]], _Diagrammatic categorification of quantum
groups III: categorifying quantum Kac-Moody algebras_ (2010) ([pdf](http://www.math.columbia.edu/~lauda/talks/vienna/2010-vienna3.pdf))

* Seok-Jin Kang, Se-jin Oh, Euiyong Park, _Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases_ ([arXiv:1102.5165](http://arxiv.org/abs/1102.5165))

* David Hill, Weiqiang Wang, _Categorification of quantum Kac-Moody superalgebras_ ([arXiv:1202.2769](http://arxiv.org/abs/1202.2769))

[[!redirects quantum affine algebras]]


quantum anomaly

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## Idea

There are at least two things that are called _quantum anomalies_ in the context of [[quantum field theory]]

* **anomalous action functional**: the [[action functional]] (in [[path integral|path integral quantization]]) is not a globally well defined [[function]], but instead a [[section]] of a [[line bundle]] on [[configuration space]];

* **anomalous symmetry** (gauge anomaly): a symmetry of the [[action functional]] does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the [[action]] of a group on classical [[phase space]] is not preserved by [[deformation quantization]].

## Definition

### Anomalous action functional
 {#AnomalousActionFunctional}

There are two major kinds of [[action functionals]] that may be anomalous in that they are not actually [[function]]s/[[nonlinear functional|functional]]s on the configuration space of fields, but just [[section]]s of some [[line bundle]]:

* theories with fermions (see e.g. [[spinors in Yang-Mills theory]]) whose [[action functional]] is given by a [[Dirac operator]], or else other fields whose action functional is given by a [[Fredholm operator]].

* [[gauge theory|gauge theories]] with higher degree gauge fields ([[differential cohomology|differential cocycles]] of higher degree.)

#### Fermionic anomalies

The [[path integral]] for a [[quantum field theory]] with [[fermions]] can be decomposed into a [[fermionic path integral]] (see there for more details) over the [[fermionic fields]] followed by that over the [[bosonic fields]]. The former, a [[Berezin integral]], is typically well defined for a fixed configuration of the bosonic fields, but does not produce a well defined function on the space of all bosonic fields: but a _twisted function_, a [[section]] of some [[line bundle]] called a [[determinant line bundle]] or, in $8k+2$ dimensions, its [[square root]], the [[Pfaffian line bundle]].

So to even start making sense of the remaining path integral over the bosonic degree of freedom, this [[determinant line bundle]] or the corresponding [[Pfaffian line bundle]] has to be trivializable. Its non-trivializability is the _fermionic anomaly_ .

More in detail ([Freed 86](#Freed86)), the [[path integral]] over an [[Lagrangian]] of the form $(\overline \phi, D \phi)$ for

$$
  D \;\colon\; V \longrightarrow W
$$

a [[Fredholm operator]] computes the [[determinant]] of that operator. Formally this is a [[section]] of the [[determinant line bundle]] over the remaining [[field (physics)|fields]]

$$
  (det V)^\ast \otimes (det W)
  \simeq
  (det ker D)^\ast \otimes (det coker D)
  \,,
$$

where the left hand side makes sense and the equivalence holds for $V$ and $W$ finite dimensional, and where the right hand side is the definition of the expression for general [[Fredholm operators]]. (([Freed 86, 1.](#Freed86)))

In more detail this [[determinant line bundle]] also carries a [[connection]] on a bundle. To make the formal [[path integral]], which is a [[section]] of this bundle, into an actual function, one this bundle with connection needs to be trivializable and trivialized. The [[obstruction]] to this is the anomaly.

#### Higher gauge-theoretic anomalies

For the moment see [[Green-Schwarz mechanism]] for more.

### Anomalous symmetry
 {#AnomalousSymmetry}

> under construction

Let

$$
  S : C \to \mathbb{R}
$$

be a (well defined) [[action functional]]. Write $P$ for its resolved [[covariant phase space]] in [[dg-geometry]] and

$$
  S^{BV} : P \to \mathbb{R}
$$

for the BV-action functional, both as given by [[BRST-BV formalism]].

If the action functional is local (comes from a [[Lagrangian]] on a [[jet bundle]]) the [[covariant phase space]] $P$ a priori only carries a [[presymplectic structure]]. But by BV-theory there exists an equivalent (homotopical) derived action functional $S_\Psi^{BV} : P \to \mathbb{R}$ such that $S_\Psi^{BV}$ does induce a genuine [[symplectic structure]] on the [[derived geometry|derived]] space $P$.

For ordinary [[Poisson manifold]]s and hence [[symplectic manifold]]s [[Maxim Kontsevich]]'s theorem says that their [[deformation quantization]] always exist. But if $S$ is the [[action functional]] of a [[gauge theory]] then $P$ is in general a nontrivial derived [[infinity-Lie algebroid]] (its function algebra has "ghosts" and "ghosts of ghost": the [[Chevalley-Eilenberg algebra]] generators) and the theorem does not apply. Instead, the quantization of the derived symplectic space $P$ exists only if the first and second [[infinity-Lie algebroid cohomology]] of $P$ vanishes:

These two cohomology groups

$$
 Anom_{gauge} = H^1(CE(P)) \oplus H^2(CE(P))
$$

are called the **gauge anomaly** of the system. Only if they vanish does the [[quantization]] of the [[gauge theory]] encoded by $S$ exist.

More concretely, the function algebra on $P$ is a graded-commutative [[dg-algebra]] equipped with a graded Poisson bracket $\{-,-\}_{BV}$ and an element $Q \in C^\infty(P)$ (the BV-BRST charge) whose [[Hamiltonian vector field]] is the [[derivation]] that is the [[differential]] of the dg-algebra $C^\infty(X)$. If the gauge anomaly does not vanish, then, while the deformation quantization of the graded algebra $C^\infty(P)$ to a non commutative graded algebra with commutator $[-,-]$ will exist, it may happen that the image $S$ of $Q$ under the quantization no longer satisfies the [[quantum master equation]] $[S,S] = \hbar \Delta S$.

Therefore the [[derivation]] $[S,-]$ will not define a quantized [[differential]] and therefore the quantization of the graded-commutative [[dg-algebra]] $C^\infty(P)$ will only be a noncommutative algebra, not a non-commutative dg-algebra, hence will not be functions on a non-commutative space in [[derived geometry]].

## Examples

### Anomalous action functional

#### Spinning particles and super-branes
  {#SpinningParticlesAndSuperBranes}

The [[sigma-model]] for a [[supersymmetry|supersymmetric]] fundamental [[brane]] on a target space $X$ has an anomaly coming from the nontriviality of [[Pfaffian line bundle]]s associated with the [[fermion]]ic fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the [[tangent bundle]] of $X$ has a sufficiently high lift through the [[Whitehead tower]] of $O(n)$.

* **Spin structure** the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has [[Spin structure]]

  This is a classical result. A concrete derivation is in

  * [[Edward Witten]], _Global anomalies in String theory_ in _Symposium on anomalies, geometry, topology_ , World Scientific Publishing, Singapore (1985)

* **String structure** the worldsheet anomaly for the spinning string/superstring in [[heterotic string theory]] vanishes (essentially) when $X$ has [[String structure]]

  This is originally due to Killingback and Witten. A commented list of literature is [here](http://golem.ph.utexas.edu/string/archives/000572.html). Recently [[Ulrich Bunke]] gave the rigorous proof

  * [[Ulrich Bunke]], _String structures and trivialisations of a Pfaffian line bundle_ ([arXiv](http://arxiv.org/abs/0909.0846))

in terms of [[differential cohomology]] in general and [[differential string structure]]s in particular.

* **Fivebrane structure** the worldvolume anomaly for the super-5-brane in [[dual heterotic string theory]] vanishes (essentially) when $X$ has [[Fivebrane structure]]. See there.

#### Gravitational anomaly

* [[gravitational anomaly]]

#### Axial anomaly

* [[axial anomaly]]

#### Conformal anomaly of the string

The [[2d CFT]] on the [[worldsheet]] of the [[bosonic string]] (in flat space, without further background fields) has an
anomaly unless the [[dimension|dimensional]] [[target space]] is $d = 26$.

This is discussed as a condition of trivialization of a bundle in
([Freed 86, section 2](#Freed86)). A brief summary is stated [this comment on MO](http://mathoverflow.net/a/99667/381).

For more see at [[conformal anomaly]] for more.

#### Freed-Witten anomaly

see at _[[Freed-Witten anomaly]]_.

#### Diaconescu-Moore-Witten anomaly

see at _[[Diaconescu-Moore-Witten anomaly]]_

#### M5-Brane anomaly

see at _[M5-brane anomaly](M5-brane#AnomalyCancellation)_

### Anomalous symmetry

#### Conformal anomaly

For the moment see [[Liouville cocycle]].

### Other

* [[RR-field tadpole cancellation]]

* [[C-field tadpole cancellation]]

## Related concepts

* [[classical anomaly]]

* [['t Hooft anomaly]]

* [[fiber bundles in physics]]

## References

* [References on anomalous action functionals](#AnomalousActionFunction)

* [References on gauge anomalies](#ReferencesGaugeAnomaly)

Anomalies originate in the 1949 article by Steinberger:

* J. Steinberger, _On the Use of Subtraction Fields and the Lifetimes of Some Types of Meson Decay_, Phys. Rev. 76, 1180 – Published 15 October 1949.  [doi](https://doi.org/10.1103/PhysRev.76.1180).

Another early reference is

* Stephen L. Adler, _Axial-Vector Vertex in Spinor Electrodynamics_, Phys. Rev. 177, 2426 – Published 25 January 1969.  [doi](https://doi.org/10.1103/PhysRev.177.2426).

### Anomalous action functional
 {#AnomalousActionFunction}

The original articles on anomalous action functionals are

* [[Luis Alvarez-Gaumé]], [[Edward Witten]], _Gravitational Anomalies_, Nucl. Phys. B234 (1984) 269 (doi:10.1016/0550-3213(84)90066-X, [spire:192309](http://inspirehep.net/record/192309))

* [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], _The structure of gauge and gravitational anomalies_, Ann. Phys. 161 (1985) 423. (doi:10.1016/0003-4916(85)90087-9, [spire:202565](http://inspirehep.net/record/202565))

* [[Edward Witten]], _Global gravitational anomalies_, Commun. Math. Phys. 100 (1985) 197. ([doi:10.1007/BF01212448](https://doi.org/10.1007/BF01212448), [euclid:1103943444](http://projecteuclid.org/euclid.cmp/1103943444))

Review

* [[Paolo Di Vecchia]], _Green-Schwarz anomaly cancellation_ (2010) (slides [[DiVecchiaAnomaly.pdf:file]])

* [[Jeffrey Harvey]], _TASI 2003 Lectures on Anomalies_ ([arXiv:hep-th/0509097](https://arxiv.org/abs/hep-th/0509097), [spire:692082](http://inspirehep.net/record/692082))

The mathematical formulation of this in terms of [[index theory]] is due to

* [[Michael Atiyah]], [[Isadore Singer]], _Dirac operators coupled to vector potentials_, Proc. Nat. Acad. Sci. USA __81__, 2597-2600 (1984) ([doi:10.1073/pnas.81.8.2597]( https://doi.org/10.1073/pnas.81.8.2597), [jstor:23378](https://www.jstor.org/stable/23378))

* [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. I. Metrics and connections on determinant bundles_ , Comm. Math. Phys. 106 (1986), no. 1, 159–176 ([doi:10.1007/BF01210930](https://doi.org/10.1007/BF01210930), [euclid:1104115586](https://projecteuclid.org/euclid.cmp/1104115586))

* [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem_ , Comm. Math. Phys. 107 (1986), no. 1, 103–163 ([doi:10.1007/BF01206955](https://doi.org/10.1007/BF01206955), [euclid:1104115934](https://projecteuclid.org/euclid.cmp/1104115934))

and a clear comprehensive account of the situation (topological anomaly, geometric anomaly) is in

* [[Raphael Flauger]], _Anomalies and the Atiyah-Singer Index Theorem_ ([[FlaugerAnomalies.pdf:file]])

* {#Freed86} [[Daniel Freed]], _Determinants, torsion, and strings_,  Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. ([doi:10.1007/BF01221001](https://doi.org/10.1007/BF01221001), [euclid:1104116145](http://projecteuclid.org/euclid.cmp/1104116145))

Slick formulation of these anomalies as [[invertible topological field theories]] is discussed in

* [[Daniel Freed]], _Anomalies and Invertible Field Theories_, talk at [StringMath2013](http://scgp.stonybrook.edu/events/event-pages/string-math-2013) ([arXiv.1404.7224](https://arxiv.org/abs/1404.7224))

A physicists' monograph is

* Reinhold A. Bertlmann, _Anomalies in quantum field theory_, Oxford Science Publ., 1996, 2000

Review:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Anomalies and the Green-Schwarz Mechanism*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.06467](https://arxiv.org/abs/2211.06467)]

A clear description of the quantum anomalies for higher gauge theories is in

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_ ([arXiv](http://arxiv.org/abs/hep-th/0011220))

As an application of this, a detailed discussion of the cancellation of the anomaly of the [[supergravity C-field]] in 11-dimensional [[supergravity]] is in

* [[Dan Freed]], [[Greg Moore]], _Setting the quantum integrand of M-theory_ ([arXiv:hep-th/0409135](http://arxiv.org/abs/hep-th/0409135))

The role of [[spin structure]]s as the anomaly cancellation condition for the spinning particle is discussed in

* [[Edward Witten]], _Global anomalies in String theory_ in _Symposium on anomalies, geometry, topology_ , World Scientific Publishing, Singapore (1985)

### Gauge anomaly
 {#ReferencesGaugeAnomaly}

The original work on the [[chiral anomaly]] is due to

* [[Stephen Adler]]. _Axial-Vector Vertex in Spinor Electrodynamics_ Physical Review 177 (5): 2426. (1969)

* [[John Bell]], [[Roman Jackiw]], _A PCAC puzzle: π0→γγ in the σ-model". Il Nuovo Cimento A 60: 47. (1969)

See also

* L. Faddeev and S. Shatashvili, "Algebraic and Hamiltonian Methods in the theory of Nonabelian Anomalies," Theor. Math. Fiz., 60 (1984) 206; english transl. Theor.
Math. Phys. 60 (1984) 770.

*  B. Zumino, "Chiral anomalies and differential geometry," in Relativity, Groups and Topology II, proceedings of the Les Houches summer school, B.S. DeWitt and R. Stora, eds. North-Holland, 1984.

* [[Mikio Nakahara]], Chapter 13 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

#### In BV-BRST formulation

General discussion in the context of [[BRST-BV formalism]] (breaking of the [[quantum master equation]] by quantum corrections) is discussed in

* W. Troost, P. van Nieuwenhuizen, A. van Proeyen, _Anomalies and the Batalin-Vilkovisky lagrangian formalism_ ([web](http://adsabs.harvard.edu/abs/1990NuPhB.333..727T))

* [[Paul Howe]], [[Ulf Lindström]], P. White, _Anomalies And Renormalization In The BRST-BV Framework_ , Phys. Lett. B246 (1990) 430.

* J. Paris, W. Troost, _Higher loop anomalies and their consistency conditions in nonlocal regularization_ , Nucl. Phys. B482 (1996) 373 ([arXiv:hep-th/9607215](http://arxiv.org/abs/hep-th/9607215))

* [[Glenn Barnich]], _Classical and quantum aspects of the
extended antifield formalism_ ([arXiv:hep-th/0011120](http://arxiv.org/abs/hep-th/0011120))

The fact that the anomaly sits in degree-1 BRST cohomology corresponds to the consistency condition discussed in

* [[Julius Wess]] and B. Zumino, _Consequences of anomalous Ward identities_ , Phys. Lett. B37 (1971) 95 doi
* R. Stora, _The Wess Zumino consistency condition: a paradigm in renormalized perturbation theory_, Fortsch. Phys. 54:175-182 (2006) [doi](http://dx.doi.org/10.1002/prop.200510266)

Discussion of  special applications in

* F. De Jonghe, J. Paris and W. Troost, _The BPHZ renormalised BV master equation and Two-loop Anomalies in Chiral Gravities_ ,  Nucl. Phys. B476
(1996) 559 [arXiv:hep-th/9603012](http://arxiv.org/abs/hep-th/9502140)

* J. Paris, _Nonlocally regularized antibracket - antifield formalism and anomalies in chiral $W(3)$ gravity_ , Nucl. Phys. B450 (1995) 357 ([arXiv:hep-th/9502140](http://arxiv.org/abs/hep-th/9502140))

* R. Amorim, N.R.F.Braga, R. Thibes, _Axial and gauge anomalies in the field antifield quantization of the generalized Schwinger model_ ([arXiv:hep-th/9712014](http://arxiv.org/abs/hep-th/9712014))

Discussion in the context of [[AQFT]] with [[functional analysis]] taken into account is in

* section 5.3.3. in [[Katarzyna Rejzner]], _Batalin-Vilkovisky formalism in locally covariant field theory_ PhD thesis, Hamburg (2011) ([arXiv:1111.5130](http://arxiv.org/abs/1111.5130))

* [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_ ([arXiv:1110.5232](http://arxiv.org/abs/1110.5232))

#### Other

An interpretation of gauge anomalies as failures of [[Hamiltonian]]s to have [[self-adjoint extension]]s is in

* J. G. Esteve, _Origin of the anomalies: the modified Heisenberg equation_ ([arXiv:hep-th/0207164](http://arxiv.org/abs/hep-th/0207164))

### Anomaly inflow

* [[Curtis Callan]], [[Jeffrey Harvey]], _Anomalies and Fermion Zero Modes on Strings and Domain Walls_, Nucl. Phys. B250 (1985) 427-436 (doi:10.1016/0550-3213(85)90489-4, [spire:15691](http://inspirehep.net/record/15691))

* [[Edward Witten]], [[Kazuya Yonekura]], *Anomaly Inflow and the $\eta$-Invariant* [[arXiv:1909.08775](https://arxiv.org/abs/1909.08775), [spire:1755070](http://inspirehep.net/record/1755070)]

### In finite-dimensional quantum mechanics

* A. P. Balachandran, Amilcar R. de Queiroz, _Mixed states from anomalies_, Phys. Rev. __D85__ (2012) 025017
[arxiv/1108.3898](https://arxiv.org/abs/1108.3898) [doi](https://doi.org/10.1103/PhysRevD.85.025017)
* Carlos Alcalde, Daniel Sternheimer, _Analytic vectors, anomalies and star representations_, Lett. Math. Phys. __17__ (1989), no. 2, 117–127. [MR90h:22012](http://www.ams.org/mathscinet-getitem?mr=993017), [doi](http://dx.doi.org/10.1007/BF00402326) (the last section has also the field theory case)

[[!redirects anomaly]]
[[!redirects anomalies]]
[[!redirects quantum anomalies]]

[[!redirects gauge anomaly]]
[[!redirects gauge anomalies]]

[[!redirects anomaly inflow]]

[[!redirects quantum anomaly cancellation]]


quantum biology

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## Idea

The field of _quantum biology_ studies non-trivial effects of [[quantum mechanics]] in biological systems, typically in molecular biology.

To the extent that biology is based on [[chemistry]] it is ultimately based on [[quantum chemistry]]. But since biological systems are typically "warm" and "wet" relative to the cold and evacuated experimental setups in which quantum effects are usually seen and studied, quantum effects beyond those that account for the existence of [[molecules]] etc. in the first place have been thought and argued to be generally negligible in biology. But there have been arguments that some key mechanisms in biology might rely on macroscopic quantum coherence/[[quantum superposition]] in a crucial way.

{#Photosynthesis} This includes notably [[photosynthesis]], which has been argued to depend, if not qualitatively at least quantitatively, on macroscopic [[quantum entanglement]]/[[quantum superposition]] [[Sarovar et al. 2010](#SIFW10)]. However, meanwhile this claim seems to have been conclusively dismissed [cf. [Duan et al. 2017](#DuanEtAl17), [Cao et al. 2020](#Cao20)]

Very speculative but prominently promoted by mathematician [[Roger Penrose]], based on joint work with [[Stuart Hameroff]], is the idea that macroscopic [[quantum superposition]] in [[microtubules]] inside [[neurons]] is the source of the still mostly mysterious characteristic 40[[Hz]] [[oscillations]] seen in brain [[EEG]]s. (In fact, [[Penrose]] goes much further and suggests that [[quantum state collapse|collapse]] of this superposition by [[quantum gravity]]-effects is a source of non-[[computability]] in [[physics]] and also somehow the source of [[consciousness]] in [[biology]], see at _[orch OR](http://en.wikipedia.org/wiki/Orchestrated_objective_reduction)_ for further pointers on this).

## Related concepts

* [[quantum chemistry]]

* [[quantum cryptography]]

## References
 {#References}

The suggestion that quantum biology is relevant for fundamentally understanding life goes back all the way to

* [[Erwin Schrödinger]], *What is life?*, Cambridge (1948) [cover:[web](https://zbphysik.univie.ac.at/ausstellung/schroedinger/dokumenteOG/whatislife1.htm), retyped:[pdf](https://www.arvindguptatoys.com/arvindgupta/whatislife-schrodinger.pdf), [Wikipedia entry](https://en.wikipedia.org/wiki/What_Is_Life%3F)]

Review:

* N. Lambert et al.,  *Quantum biology*, Nature Physics **9** (2013) 10–18 [[doi:10.1038/nphys2474](https://doi.org/10.1038/nphys2474)]

* Y. Kim et al., *Quantum Biology: An Update and Perspective*, Quantum Reports **3** 1 (2021) 80-126 [[doi:10.3390/quantum3010006](https://doi.org/10.3390/quantum3010006)]

Specifically on the existence and relevance -- or not -- of non-negligible coherent quantum entanglement in photosynthesis:

* {#SIFW10} M. Sarovar et al.,  *Quantum entanglement in photosynthetic light-harvesting complexes*, Nature Physics **6** 6  (2010) 462-467 [[arXiv:0905.3787](http://arxiv.org/abs/0905.3787), [doi:10.1038/nphys1652](https://doi.org/10.1038/nphys1652)]

* {#DuanEtAl17} H.-G. Duan et al., *Nature does not rely on long-lived electronic quantum coherence for photosynthetic energy transfer*, PNAS **114** 32 (2017) 8493-8498 [[doi:10.1073/pnas.1702261114](https://doi.org/10.1073/pnas.1702261114)]

* {#Cao20} J. Cao et al., *Quantum biology revisited*, Science Advances **6** 14 (2020) [[doi:10.1126/sciadv.aaz4888](https://doi.org/10.1126/sciadv.aaz4888)]

An argument in favour of quantum biology which claims that cellular [[information theory|information processing]] must employ [[quantum coherence]] as a resource for [[reversible computation|reversibility]] in order to maintain a biologically reasonable [[free energy]] budget:

* [[Chris Fields]], [[Michael Levin (biologist)|Michael Levin]], _Metabolic limits on classical information processing by biological cells_ Biosystems **209** (2021) 104513 [[doi:10.1016/j.biosystems.2021.104513](https://doi.org/10.1016/j.biosystems.2021.104513)]


quantum channel

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## Idea
 {#Idea}

In [[quantum physics]] -- and specifically in [[quantum information theory]] and [[quantum probability theory]] -- by a *quantum operation* or *quantum channel* one means any physically reasonable operation on, or transformation of *[[mixed states]]* (in contrast to *[[quantum gates]]* operating on [[pure states]]), notably such as sending information through a "communication channel" (in the sense of [[information theory]]), whence the terminology *quantum channel*.

More concretely, the physical nature of quantum channels is that they unify "loss-less" [[unitary operator|unitary]] transformations on [[quantum states]] (as known [[Schrödinger equation|Schrödinger evoluation]] and [[quantum gates]]) with stochastic effects such as due to [[quantum noise]] and [[quantum state collapse]] due to [[quantum measurement]].

In short, just as the notion of [[mixed states]] generalizes the notion of [[pure state|pure]] [[quantum states]] with their objective, intrinsic and fundamental stochasticity (expressed the [[Born rule]]) to include also subjective, [[thermodynamics|thermodynamical]] [[probability distribution|classical stochasticity]], so quantum channels generalize [[quantum gates]] from pure to mixed states.

Mathematically, with [[mixed states]] represented by [[density matrices]] and generally by [[positive linear operators]], a quantum channel is just a suitable [[map]] between spaces of such [[matrices]] or [[linear operators]], whence they are sometimes also called *[[superoperators]]* (in the sense of "operators operating on operators").

But in the context of [[quantum information theory]] the relevant [[spaces of quantum states]] are all [[finite-dimensional Hilbert space|finite-dimensional]], in which case quantum channels are traditionally discussed as (special) [[linear maps]] between [[vector spaces]] of [[square matrix|square]] [[matrices]]:

$$
  chan
  \;\colon\;
  Mat_{n_1}(\mathbb{C})
    \longrightarrow
  Mat_{n_2}(\mathbb{C})
  \,.
$$

Slightly more abstractly, such as in the formulation of [[quantum information theory via dagger-compact categories]], these are certain [[morphisms]] in a [[compact closed category]] of the form

$$
  chan
  \;\colon\;
  \mathscr{H}_1 \otimes \mathscr{H}_1^\ast
    \longrightarrow
  \mathscr{H}_2 \otimes \mathscr{H}_2^\ast
$$

(where above $n_i = dim(\mathscr{H}_i)$ is the [[dimension of a vector space|dimension]] of the given [[finite-dimensional Hilbert space]]).

The key point is that such linear maps are to qualify as quantum channels iff they suitably restrict to maps between the *[[convex set|convex]]* [[subsets]] of [[density matrices]] (the [[mixed states]]) inside $\mathscr{H}_i\otimes\mathscr{H}_i^\ast$, which is a non-linear condition.

There is slight variation in the exact list of properties demanded of a quantum channel, but the key demand is that it  be a "*positive map*" in that it takes [[positive operators]] (such as [[density operators]]) to positive operators --- and in fact a *completely positive map*, meaning that it remains positive after [[tensor product of vector spaces|tensoring]] with any [[identity map|identity transformation]].

This is discussed below at:

* *[In terms of positivity conditions](#InTermsOfPositivityConditions)*

Often demanded is also that a quantum channel preserves the [[trace]] of matrices, which in [[quantum probability]] means that it preserves total probability, hence that it is the quantum analog of a *[[stochastic map]]*  --- through what fundamentally matters is that a quantum channel at most lowers the probability (the channel need not describe all possible outcomes, but it must not make new outcomes appear out of nowhere).

Less often demanded (but usually the case anyway) is that a quantum channel also preserves the [[identity matrix]], in which case it is the quantum analog of a [[doubly stochastic map]].

Beyond these abstract characterizations, the [[Stinespring factorization theorem]] characterizes quantum channels more explicitly as those maps on matrices arising as [[sums]] of [[conjugations]]
$$
  chan
  \;\colon\;
  \rho
  \;\mapsto\;
  \underset{w}{\sum}
  E_w \cdot \rho \cdot E_w^\dagger
$$
by certain [[tuples]] $(E_w)_{w \colon W}$ of linear operators ("*Kraus operators*"). Much of the discussion of quantum channels in the literature proceeds by manipulating such *Kraus decompositions* of quantum channels.

This is discussed below at:

* *[In terms of Kraus decompositions](#InTermsOfKrausDecompositions)*

For example, a [[unitary quantum channel]] describing a loss-less [[quantum gate]] is given by a single [[unitary operator|unitary]] Kraus operator as

\[
  \label{UnitaryChannelInIdeaSection}
  chan_U
  \;\colon\;
  \rho
  \;\mapsto\;
  U \cdot \rho \cdot U^\dagger
  \,,
\]

which on [[pure states]] among [[mixed states]], $\rho_{|\psi\rangle} \coloneqq \left\vert \psi \right\rangle \left\langle \psi \right\vert$, restricts to an ordinary [[quantum gate]]

$$
  chan_U
  \;\colon\;
  \rho_{\vert\psi \rangle}
  \;\mapsto\;
  \rho_{ U \vert \psi \rangle }
  \,.
$$

On the other extreme, a [[quantum measurement]] in a measurement [[basis of a vector space|basis]] $W$, $\underset{W}{\oplus} \mathbb{C}  \simeq \mathscr{H}$ is given by the corresponding [[projection operators]] $P_w$ as

\[
  \label{MeasurementChannelInIdeaSection}
  meas_W
  \;\colon\;
  \rho
  \;\mapsto\;
  \underset{w}{\sum}
  P_w \cdot \rho \cdot P_w
  \,.
\]

Remarkably (from the discussion at *[[quantum decoherence]]*) one finds that such a measurement channel (eq:MeasurementChannelInIdeaSection) may equivalently be understood as the result of a unitary evolution (eq:UnitaryChannelInIdeaSection) of the state $\rho$ coupled to an environment state $\omega$ *followed by* the [[partial trace quantum channel|partial trace]] over the environment's Hilbert space. This observation turns out to generally lead to yet another characterization of quantum channels:

Quantum channels equivalently act on a density matrix $\rho$ by

1. tensoring it to another state $\omega$

   (coupling the system to an environment/"bath")

1. sending the tensor state through a unitary channel (eq:UnitaryChannelInIdeaSection)

   ([[Schrödinger equation|Schrödinger evolution]] of the couplesystem)

1. applying the [[partial trace quantum channel|partial trace]] over the Hilbert space of $\omega$

   (averaging the outcome over all states of the environment/bath).

In this perspective, quantum channels are understood as a kind of unitary quantum gates after all, but acting on [[open quantum systems]] including their environment with the stochasticity induced (only) by (deliberate) ignorance of the environment's state.

This is discussed below at:

* *[Quantum channels and Decoherence](#QuantumChannelsAndDecoherence)*

In this last form, the formulation of quantum channels lends itself to formulation in the [[string diagram]]-calculus of [[quantum information theory via dagger-compact categories]].

This is discussed below at :

* [In terms of dagger-compact closed categories](#InTermsOfCompactClosedCategories)

\linebreak

## Definition
 {#Definition}

### In terms of positivity conditions
 {#InTermsOfPositivityConditions}

Let $\mathscr{H}_i$ be a [[complex vector space|complex]] [[finite-dimensional Hilbert space]], with

$$
  Mat(\mathscr{H}_i)
  \,\simeq\,
  \mathscr{H}_i
  \otimes
  \mathscr{H}_i^\ast
$$

the corresponding space of [[matrices]]  --- including as a [[convex set|convex]] [[subset]] the [[density matrices]] representing the [[mixed states]] of the [[quantum system]] described by $\mathscr{H}$.

\begin{definition}\label{PositiveMaps}
  A $\mathbb{C}$-[[linear map]]
$$
  \Phi
  \;\colon\;
  \mathscr{H}_1
  \otimes
  \mathscr{H}_1^\ast
  \longrightarrow
  \mathscr{H}_2
  \otimes
  \mathscr{H}_2^\ast
$$
is called

* *hermitian* iff it preserves [[Hermitian matrices]],

* *positive* iff it preserves [[positive operator|positive matrices]],

* *$n$-positive* if $\Phi \otimes Id_{\mathbb{C}^n}$ is positive for $n \in \mathbb{N}$,

* *completely positive* if $\Phi$ is $n$-positive for all $n \in \mathbb{N}$.

A positive $\Phi$ is furthermore called:

* *[[stochastic map|stochastic]]* if it preserves the [[trace]] of matrices

* *doubly stochastic* if it preserves also the identity matrix.

Finally, a completely positive $\Phi$ is called:

* a *quantum channel* if it is stochastic,

* a *[[unital quantum channel]]* if it is doubly stochastic.

\end{definition}

[[Arveson 1969 §1](#Arveson69), [Landau & Streater 1993 p. 107-108](#LandauStreater93)]

### In terms of operator-sum decompositions
 {#InTermsOfKrausDecompositions}

\begin{theorem}
\label{OperatorSumDecompositionOfQuantumChannels}
**(operator-sum decomposition of quantum channels)**
\linebreak
For $\mathscr{H}_i$ [[finite-dimensional Hilbert spaces]], a  [[linear map]]

$$
  chan
  \;\colon\;
  \mathscr{H}_1 \otimes \mathscr{H}_1^\ast
  \longrightarrow
  \mathscr{H}_2 \otimes \mathscr{H}_2^\ast
$$

is completely positive (Def. \ref{PositiveMaps}) precisely if there exists an [[indexed set]]

\[
  R \,\colon\, FinSet
  ,\;\;
  r \,\colon\, R
  \;\;\;\;\;
  \vdash
  \;\;\;\;\;
  E_r \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2
\]

of [[linear operators]] such that $chan$ is the sum of [[conjugation actions]] with these operators ("Kraus form"):

\[
  \label{OperatorSumDecompositionFormula}
  chan(\rho)
  \;=\;
  \underset{r}{\sum}
  \,
  E_r \cdot \rho \cdot E_r^\dagger.
\]
\end{theorem}
Accordingly, in addition:

* $chan$ preserves the [[trace]] and is hence a quantum channel iff

  $$
  \sum_r \, E_r^\dagger \cdot E_r \,=\, id_{\mathscr{H}_1}
  $$

* $chan$ preserves also the identity matrix and is hence a [[unital quantum channel]] iff (in addition)

  $$
  \sum_r \, E_r \cdot E_r^\dagger \,=\, id_{\mathscr{H}_2}
  \,.
  $$

The idea goes back to [Stinespring 1955](#Stinespring55). The decomposition (eq:OperatorSumDecompositionFormula) is also called *Kraus decomposition* by *Kraus operators*, after [Kraus 1971](#Kraus71). The fully explicit statement of Thm. \ref{OperatorSumDecompositionOfQuantumChannels} is due to [Choi 1975 Thm. 1](#Choi75).

Review includes: [Nielsen & Chuang 2000 Thm. 8.1](#NielsenChuang00), [Kuperberg 2005 Thm. 1.5.1](#Kuperberg05).

A general abstract proof in terms of [[dagger category|†-categories]] is claimed by [Selinger 2005](#Selinger05). A characterization of completely positive maps entirely in terms of $\dagger$-categories is given in [Coecke 2007](#Coecke07).

\begin{proposition}\label{CanonicalKrausDecomposition}
**(canonical operator sum-decomposition)**
\linebreak
The operator-sum decomosition $(E_r)_{r \colon R}$ in Thm. \ref{OperatorSumDecompositionOfQuantumChannels} may always be chosen such as to be [[orthogonality|orthogonal]] under the [[trace]]:
$$
  r,r' \,\colon\, R
  \;\;\;\;
  \vdash
  \;\;\;\;
  trace^{ \mathscr{H} }\big(
    E_{r'}^\dagger \cdot E_{r}
  \big)
  \;=\;
  d_r \delta_{r}^{r'}
$$
([[Kronecker delta]]) for some [[non-negative number|non-negative]] [[real numbers]] $(d_r)_{r \colon R}$.

Moreover, this *canonical Kraus form* is unique up to [[permutation]] of $R$.
\end{proposition}
[Bengtsson & Życzkowski 2006 p. 256](#BengtssonŻyczkowski06)

[[!include quantum channels and decoherence -- section]]

### In terms of $\dagger$-compact closed categories
 {#InTermsOfCompactClosedCategories}

... due to ([Selinger 05](#Selinger05)) ... see for instance ([Coecke-Heunen 11, section 2](#CoeckeHeunen11)) for a quick summary ...

## Properties

### Universal property

The [[category]] whose [[objects]] are indexed by [[natural numbers]] $n,m, \cdots$ and whose [[morphisms]] are quantum operations from $n \times n$ to $m \times m$ matrices is a [[semicartesian monoidal category]] with the [[monoidal category|monoidal structure]] given by multiplication of numbers. Being semicartesian, the monoidal [[tensor unit]] (the number $1$) has a unique morphism to it from any object: this morphism is the trace.

In fact, this category has the [[universal property]] of the semicartesian reflection of the monoidal category of [[isometries]]. This is the category whose objects are natural numbers, considered as [[Hilbert spaces]], and whose [[morphisms]] are isometries between them, where an [[isometry]] $m\to n$ is an $m\times n$ complex matrix $V$ such that $V V* = I$.

In detail, the [[universal property]] says that for any strict semicartesian monoidal category $\mathcal{D}$ and any monoidal functor $\mathbf{Isometries}\to \mathcal{D}$, there is a unique symmetric monoidal functor making the following diagram commute:
$$
\array{
{Isometries} &\rightarrow& {Quantum Channels} \\
&\searrow&\downarrow\\
&& \mathcal{D}
}
$$

This fits a physical intuition as follows. Suppose that the isometries are a model of reality, as in the many worlds interpretation and the Church of the larger Hilbert space. But in practice the observer cannot access the entirety of reality, and so some bits are hidden. The canonical way to model this hiding is to do it freely, which is to form the semicartesian reflection.

## Examples
 {#Examples}

### Unitary quantum channels

A *[[unitary quantum channel]]* is a quantum channel whose restriction to [[pure states]] acts by a [[unitary transformation]] just as a loss-less [[quantum gate]] does.

Concretely, in terms of [operator-sym decomposition](#InTermsOfKrausDecompositions), a quantum channel

$$
  \array{
    \mathscr{H}_1
    \otimes
    \mathscr{H}_1^\ast
    &
    \longrightarrow
    &
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    \\
    \rho
    &\mapsto&
    ch(\rho)
  }
$$

is unitary iff there exists a [[unitary operator]] $U \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2$ such that $ch$ is given by [[conjugation]] with this operator:

$$
  ch(\rho) \;=\; U \cdot \rho \cdot U^\dagger
  \,.
$$

### Mixed unitary quantum channels

More generally, a *[[mixed unitary quantum channel]]* is given by an $S$-[[indexed set]] of [[unitary operators]] $(U_s)_{s \colon S}$ and a [[probability distribution]] $p \,\colon\, S \to [0,1]$ as

$$
  \rho
    \,\mapsto\,
  \underset{s}{\sum}
  \,
  U_s \cdot \rho U_s^\dagger
  \,.
$$

This class includes for instance the [[bit-flip quantum channels]].

### Averaging quantum channels
 {#DecoherenceExample}

For $\mathscr{B}$ a [[finite-dimensional Hilbert space]], the operation of [[partial trace quantum channel|partial trace]] over $\mathscr{B}$ is a quantum channel, the *[[averaging quantum channel]]*.

$$
  \array{
    \mathscr{H}
    \otimes
    \mathscr{B}
    \otimes
    \mathscr{B}^\ast
    \otimes
    \mathscr{H}^\ast
    &
    \xrightarrow{
      trace^{\mathscr{B}}
    }
    &
    \mathscr{H} \otimes \mathscr{H}^\ast
    \\
    \left\vert \psi, \beta \right\rangle
    \left\langle \beta', \psi' \right\vert
    &\mapsto&
    \left\vert \psi \right\rangle
    \left\langle \beta' \vert \beta \right\rangle
    \left\langle \psi' \right\vert
    \mathrlap{\,.}
  }
$$

See also at *[[quantum decoherence]]*.

### Quantum measurement channels
 {#QuantumMeasurementAndPOVM}

Consider

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

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

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

for the corresponding [[projection operator]].

By construction this is such that

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

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

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

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

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

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

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

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

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

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

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

### Noise channels

Examples of [[quantum noise]] channels:

* [[bit flip channel]]

## Related concepts

* [[extremal quantum channel]]

* [[quantum information theory via dagger-compact categories]]

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

## References

The operator-sum decomposition characterization of completely positive maps is due to:

* {#Stinespring55} [[W. Forrest Stinespring]], *Positive functions on $C^\ast$-algebras*, Proc. Amer. Math. Soc. **6** 2 (1955) 211-216 [[doi:2032342](https://www.jstor.org/stable/2032342), [doi:10.2307/2032342](https://doi.org/10.2307/2032342)]

* {#Kraus71} [[Karl Kraus]], *General state changes in quantum theory*, Ann. Physics **64** 2 (1971) 311-335 [doi:10.1016/0003-4916(71)90108-4]

* {#Choi75} [[Man-Duen Choi]], *Completely positive linear maps on complex matrices*, Linear Algebra and its Applications **10** 3 (1975) 285-290 [doi:10.1016/0024-3795(75)90075-0]

with early review in:

* [[David E. Evans]], [[John T. Lewis]],  *Dilations of irreversible evolutions in algebraic quantum theory*, Communications of the Dublin Institute for Advanced Studies, Series A: Theoretical Physics **24** (1977) [[eprint:34031](https://orca.cardiff.ac.uk/id/eprint/34031), [pdf](https://orca.cardiff.ac.uk/id/eprint/34031/1/Evans_Lewis_DIAS.pdf)]

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

See also

* {#Arveson69} [[William B. Arveson]], *On subalgebras of $C^\ast$-algebras*, Bull. Amer. Math. Soc. **75** (1969) 790-794 [[doi:1969-75-04/S0002-9904-1969-12293-7](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7), [pdf](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7/S0002-9904-1969-12293-7.pdf)]

The [environmental representation](#QuantumChannelsAndDecoherence) of completely positive maps originates with

* {#Lindblad75} [[Göran Lindblad]], *Completely positive maps and entropy inequalities*,  Commun. Math. Phys. **40** (1975) 147–151 [[doi:10.1007/BF01609396](https://doi.org/10.1007/BF01609396)]

and was then eventually rediscovered for the special case of [[quantum measurement channels]] following the 1980s discussion of *[[decoherence]]*.

The terminology "quantum operation" for linear maps on the linear dual of a [[C-star algebra|$C^\ast$-algebra]] which preserve the subset of [[states on a star-algebra]]:

* {#HaagKastler64} [[Rudolf Haag]], [[Daniel Kastler]], pp. 850 in: *An algebraic approach to quantum field theory*, Journal of Mathematical Physics, **5** (1964) 848-861 [[doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124)]

The terminology "quantum channel":

* {#ŻyczkowskiBengtsson04} [[Karol Życzkowski]], [[Ingemar Bengtsson]], Section 3 of: *On Duality between Quantum Maps and Quantum States*, Open Systems & Information Dynamics **11** 01 (2004) 3-42 [[doi:10.1023/B:OPSY.0000024753.05661.c2](https://doi.org/10.1023/B:OPSY.0000024753.05661.c2)]

* [[Teiko Heinosaari]], [[Mário Ziman]], Section 4 of: *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

Analysis of [[extremal quantum channels]]:

* [Choi 1975. Thm. 2.11](#Choi75)

* {#LandauStreater93} [[L. J. Landau]], [[Raymond F. Streater]],  *On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras*, Linear Algebra and its Applications **193** (1993) 107-127 [doi:10.1016/0024-3795(93)90274-R]

* [[Christian B. Mendl]], [[Michael M. Wolf]], *Unital Quantum Channels -- Convex Structure and Revivals of Birkhoff's Theorem*, Commun. Math. Phys. **289**  (2009) 1057-1096 [[arXiv:0806.2820](http://arxiv.org/abs/0806.2820)]

* James Miller, S. T. da Silva, *On the Extremality of the Tensor Product of Quantum Channels* [[arXiv;2305.05795](https://arxiv.org/abs/2305.05795)]

Review and survey:

In the context of [[quantum computation]]:

* {#NielsenChuang00} [[Michael A. Nielsen]], [[Isaac L. Chuang]], §8.2 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* [[John Preskill]], §3.2 in: *Measurement and Evolution*, chapter 3 of: *Quantum Information*, lecture notes, since 2004 [[pdf](http://www.theory.caltech.edu/~preskill/ph219/chap3_15.pdf), [web](http://theory.caltech.edu/~preskill/ph229/)]

* {#Selinger04} [[Peter Selinger]], §6.3 in: _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)]

* {#Kuperberg05} [[Greg Kuperberg]], §1.5 of _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

in the context of [[quantum probability]]:

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

and in [[quantum information theory]]:

* [[Mark M. Wilde]], *Quantum Information Theory*, Cambridge University Press (2013) [[doi:10.1017/CBO9781139525343](https://doi.org/10.1017/CBO9781139525343), [arXiv:1106.1445](https://arxiv.org/abs/1106.1445)]

* [[Joseph M. Renes]], 4.32 in: *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)] De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]

* [[Sumeet Khatri]], [[Mark M. Wilde]], §3.2 in: _Principles of Quantum Communication Theory: A Modern Approach_ [[arXiv:2011.04672](https://arxiv.org/abs/2011.04672)]

Further:

* {#BengtssonŻyczkowski06} [[Ingemar Bengtsson]], [[Karol Życzkowski]], Chapter 10 of: *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048), [ResearchGate](https://www.researchgate.net/publication/266435541_Geometry_of_Quantum_States_An_Introduction_to_Quantum_Entanglement)]

* [[Robert B. Griffiths]], *Quantum Channels, Kraus Operators, POVMs* (2012) [[pdf](https://quantum.phys.cmu.edu/QCQI/qitd412.pdf), [[Griffiths-QuantumChannels.pdf:file]]]

* {#AttalLectureNotes} [[Stéphane Attal]], *Quantum Channels*, Lecture 6 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Quantum_Channels.pdf), [[Attal-QuantumChannels.pdf:file]], [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

See also

* Caleb J. O'Loan, *Topics in Estimation of Quantum Channels*, PhD thesis, University of St. Andrews (2009) [[arXiv:1001.397](http://arxiv.org/abs/1001.3971)]

* John A. Smolin, Frank Verstraete, Andreas Winter, *Entanglement of assistance and multipartite state distillation*, Phys. Rev. A **72** (2005) 052317 ([arXiv:quant-ph/0505038](http://arxiv.org/abs/quant-ph/0505038))

* John Watrous, _Mixing doubly stochastic quantum channels with the completely depolarizing channel_ (2008) ([arXiv](http://arxiv.org/abs/0807.2668))

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

The description of completely positive maps in terms of [[dagger-categories]] (see at _[[quantum information theory via dagger-compact categories]]_) goes back to

* {#Selinger05} [[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), [[SelingerPositiveMaps.pdf:file]]]

* {#Coecke07} [[Bob Coecke]], _Complete positivity without compactness_, 2007 ([pdf](http://www.comlab.ox.ac.uk/files/666/RR-07-05.pdf))

This is further explored in:

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

* {#CoeckeHeunen11} [[Bob Coecke]], [[Chris Heunen]], _Pictures of complete positivity in arbitrary dimension_, EPTCS 95, 2012, pp. 27-35 ([arXiv:1110.3055](http://arxiv.org/abs/1110.3055))

* [[Bob Coecke]], [[Chris Heunen]], [[Aleks Kissinger]], _Categories of Quantum and Classical Channels_ ([arXiv:1305.3821](http://arxiv.org/abs/1305.3821))

For the universal property, see

* Mathieu Huot, Sam Staton, _Universal properties in quantum theory_ (QPL 2018) ([pdf](https://www.mathstat.dal.ca/qpl2018/papers/QPL_2018_paper_68.pdf)).

On quantum channel capacity:

* Alexander S. Holevo, *Quantum Systems, Channels, Information -- A Mathematical Introduction*, Studies in Mathematical Physics **16**, De Gruyter (2013) [[doi:10.1515/9783110273403](https://doi.org/10.1515/9783110273403)]

* Alexander S. Holevo, *Quantum channel capacities*, Quantum Electron. **50** 440 (2020) [[doi:10.1070/QEL17285/meta](https://iopscience.iop.org/article/10.1070/QEL17285/meta)]

[[!redirects quantum channel]]
[[!redirects quantum channels]]
[[!redirects quantum operation]]
[[!redirects quantum operations]]
[[!redirects quantum operations and channels]]

[[!redirects completely positive map]]
[[!redirects completely positive maps]]

[[!redirects Kraus decomposition]]
[[!redirects Kraus decompositions]]


quantum channels and decoherence -- section

### Environmental representation of quantum channels
 {#QuantumChannelsAndDecoherence}

The crux of dynamical [[quantum decoherence]] is that *fundamentally* the (time-)evolution of any [[quantum system]] $\mathscr{H}$ may be assumed [[unitary quantum channel|unitary]] (say via a [[Schrödinger equation]]) *when taking the whole evolution of its environment* $\mathscr{B}$ (the "bath", ultimately the whole [[observable universe]]) into account, too, in that the evolution of the total system $\mathscr{H} \otimes \mathscr{B}$  is given by a [[unitary operator]]

$$
  \array{
    \mathllap{
      evolve
      \;\colon\;
    }
    \mathscr{H} \otimes \mathscr{B}
    &\longrightarrow&
    \mathscr{H} \otimes \mathscr{B}
    \\
    \left\vert \psi, \beta \right\rangle
    &\mapsto&
    U_{tot}
    \left\vert \psi, \beta \right\rangle
    \mathrlap{\,,}
  }
$$

after understanding the [[mixed states]] $\rho \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast$ ([[density matrices]]) of the given [[quantum system]] as coupled to any given mixed state $env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast$ of the bath (via [[tensor product]])

$$
  \array{
    \mathllap{
      couple
      \;\colon\;
    }
    \mathscr{H} \otimes \mathscr{H}^\ast
    &
    \longrightarrow
    &
    (\mathscr{H} \otimes \mathscr{B})
    \otimes
    (\mathscr{H} \otimes \mathscr{B})^\ast
    \\
    \rho &\mapsto&
    \rho \otimes env
    \mathrlap{\,;}
  }
$$

...the only catch being that one cannot --- and in any case does not (want or need to) ---  keep track of the precise [[quantum state]] of the environment/bath, instead only of its *average* effect on the given [[quantum system]], which by the rule of [[quantum probability]] is the [[mixed state]] that remains after the [[partial trace quantum channel|partial trace]] over the environment:
\[
  \label{PartialTraceOverBathTowardsQuantumChannels}
  \array{
    \mathllap{
    average
    \;\colon\;
    }
    (\mathscr{H} \otimes \mathscr{B})
    \otimes
    (\mathscr{H} \otimes \mathscr{B})^\ast
    &\longrightarrow&
    \mathscr{H} \otimes \mathscr{H}^\ast
    \\
    \widehat{\rho}
      &\mapsto&
    Tr_{\mathscr{B}}\big(\widehat{\rho}\big)
    \mathrlap{\,.}
  }
\]

In summary this means *for practical purposes* that the probabilistic evolution of quantum systems $\mathscr{H}$ is always of the composite form

$$
  \array{
    \mathscr{H}
    \otimes
    \mathscr{H}^\ast
    &
    \xrightarrow{
      \array{
        \text{couple to}
        \\
        \text{environment}
      }
    }
    &
    \left(
    \array{
      \mathscr{H}
      \\
      \otimes
      \\
      \mathscr{B}
    }
    \right)
    \otimes
    \left(
    \array{
      \mathscr{H}
      \\
      \otimes
      \\
      \mathscr{B}
    }
    \right)^\ast
    &
    \xrightarrow{
      \array{
        \text{total unitary}
        \\
        \text{evolution}
      }
    }
    &
    \left(
    \array{
      \mathscr{H}
      \\
      \otimes
      \\
      \mathscr{B}
    }
    \right)
    \otimes
    \left(
    \array{
      \mathscr{H}
      \\
      \otimes
      \\
      \mathscr{B}
    }
    \right)^\ast
    &
    \xrightarrow{
      \array{
        \text{average over}
        \\
        \text{environment}
      }
    }
    &
    \;\;\;\;
    \mathscr{H} \otimes \mathscr{H}^\ast
    \\
    \rho
    &\mapsto&
    \rho \otimes env
    &\mapsto&
    \mathclap{
    U_{tot}
      \cdot
    (\rho \otimes env)
      \cdot
    U_{tot}^\dagger
    }
    &\mapsto&
    \;\;\;\;\;\;\;\;\;\;\;\;
    \mathclap{
    Tr_{\mathscr{B}}
    \big(
      U_{tot}
        \cdot
      (\rho \otimes env)
        \cdot
      U_{tot}^\dagger
   \big)
   }
  }
$$

This composite turns out to be a "*[[quantum channel]]*"

The realization of a quantum channel in the form (eq:QuantumChannelVieDecoherence) is also called an *environmental representation* (eg. [Życzkowski & Bengtsson 2004 (3.5)](quantum+channel#ŻyczkowskiBengtsson04)).

In fact all quantum channels on a fixed Hilbert space have such an evironmental representation:

\begin{proposition}
\label{QuantumChannelsAsPartialTracesOfUnitariesOnTensors}
**(environmental representation of quantum channels)**

Every [[quantum channel]]

$$
  chan
  \;\;\colon\;\;
  \mathscr{H} \otimes \mathscr{H}^\ast
  \longrightarrow
  \mathscr{H} \otimes \mathscr{H}^\ast
$$

may be written as

1. a [[unitary quantum channel]], induced by a [[unitary operator]] $U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B} $

1. on a [[compound system]] with some $\mathscr{B}$ (the "bath"), yielding a *total system* [[Hilbert space]] $\mathscr{H} \otimes \mathscr{B}$ ([[tensor product of Hilbert spaces|tensor product]]),

1. and acting on the given [[mixed state]] $\rho$ coupled (tensored) with any [[pure state]] of the bath system,

1. followed by [[partial trace quantum channel|partial trace]] (averaging) over $\mathscr{B}$ (leading to [[decoherence]] in the remaining state)

in that

\[
  \label{QuantumChannelVieDecoherence}
  chan(\rho)
  \;\;=\;\;
  Tr_{\mathscr{B}}
  \big(
    U_{tot}
    \cdot
    (\rho \otimes env)
    \cdot
    U_{tot}^\dagger
  \big)
  \,.
\]

Conversely, every operation of the form (eq:QuantumChannelVieDecoherence) is a quantum channel.
\end{proposition}

This is originally due to [Lindblad 1975 ](quantum+channel#Lindblad75) (see top of p. 149 and inside the proof of Lem. 5). For exposition and review see: [Nielsen & Chuang 2000 §8.2.2-8.2.3](quantum+channel#NielsenChuang00).
An account of the infinite-dimensional case is in [Attal, Thm. 6.5 & 6.7](quantum+channel#AttalLectureNotes). These authors focus on the case that the environment is in a [[pure state]], the (parital) generalization to [[mixed state|mixed]] environment states is discussed in [Bengtsson & Życzkowski  2006 pp. 258](quantum+channel#BengtssonŻyczkowski06).

\begin{proof}
\label{ProofOfEnvironmentalRepresentationOfQuantumChannels}
We spell out the proof assuming [[finite-dimensional Hilbert spaces]]. (The general case follows the same idea, supplemented by arguments that the following [[sums]] [[convergence|converge]].)

Now given a [[completely positive map]]:

$$
  chan
    \,\colon\,
  \mathscr{H} \otimes \mathscr{H}^\ast
  \longrightarrow
  \mathscr{H} \otimes \mathscr{H}^\ast
  \,,
$$

then by  [operator-sum decomposition](quantum+channel#OperatorSumDecompositionOfQuantumChannels) there exists a [[set]] ([[finite set|finite]], under our assumptions) [[inhabited set|inhabited]] by at least one element
$$
  s_{ini} \,\colon\, S
  \,,
$$
and an $S$-[[indexed set]] of [[linear operators]]
\[
  \label{KrausDecompositionInConstructionOfEnvRep}
  s \,\colon\, S
  \;\;\;
    \vdash
  \;\;\;
  E_s
  \;\colon\;
  \mathscr{H}
  \longrightarrow
  \mathscr{H}
  \,,\;\;\;\;
  \text{with}
  \;\;\;\;
  \underset{s}{\sum}
  E_s^\dagger \cdot E_s \,=\, Id
  \mathrlap{\,,}
\]

such that

$$
  chan(\rho)
  \;=\;
  \underset{s}{\sum}
  \,
  E_s \cdot \rho \cdot E_s^\dagger
  \,.
$$

Now take

$$
  \mathscr{B}
  \,\equiv\,
  \underset{S}{\oplus} \mathbb{C}
$$

with its canonical [[Hermitian form|Hermitian]] [[inner product]]-structure with [[orthonormal linear basis]] $\big(\left\vert s \right\rangle\big)_{s \colon S}$ and consider the linear map

$$
  \array{
    \mathllap{
      V
      \;\colon\;\;
    }
    \mathscr{H}
    &\longrightarrow&
    \mathscr{H}
    \otimes
    \mathscr{B}
    \\
    \left\vert \psi \right\rangle
    &\mapsto&
    \underset{s}{\sum}
    \,
    E_s
    \left\vert
      \psi
    \right\rangle
    \otimes
    \left\vert s \right\rangle
    \mathrlap{\,.}
  }
$$

Observe that this is a [[linear isometry]]

$$
  \begin{array}{ll}
    \left\langle \psi \right\vert
      V^\dagger
      V
    \left\vert \psi \right\rangle
    \\
    \;=\;
    \underset{s,s'}{\sum}
    \left\langle \psi \right\vert
      E_{s'}^\dagger
      E_s
    \left\vert \psi \right\rangle
    \underset{
      \delta_s^{s'}
    }{
      \underbrace{
        \left\langle s' \vert s \right\rangle
      }
    }
    \\
    \;=\;
    \left\langle \psi \right\vert
    \underset{
      Id
    }{
      \underbrace{
        \underset{s}{\sum}
        E_{s}^\dagger
        E_s
      }
    }
    \left\vert \psi \right\rangle
    \\
    \;=\;
    \left\langle \psi \vert \psi \right\rangle
    \mathrlap{\,.}
  \end{array}
$$

This [implies](linear+isometryLinearIsometriesAreInjective) that $V$ is [[injective map|injective]] so that we have a [[direct sum]]-decomposition of its [[codomain]] into its [[image]] and its [[cokernel]] [[orthogonal complement]], which is unitarily isomorphic to $dim(\mathscr{B})-1$ summands of $\mathscr{H}$ that we may identify as follows:

$$
  \mathscr{H} \otimes \mathscr{B}
  \;\simeq\;
  V\big( \mathscr{H} \big)
  \oplus
  \Big(
    \mathscr{H}
    \otimes
    \big(
      \mathscr{B}
        \ominus
      \mathbb{C}\left\vert s_0 \right\rangle
    \big)
  \Big)
  \,.
$$

In total this yields a unitary operator

$$
  U
  \;\colon\;
  \mathscr{H} \otimes \mathscr{B}
  \,\simeq\,
  \mathscr{H}
  \oplus
  \Big(
    \mathscr{H}
      \otimes
    \big(
      \mathscr{B}
        \ominus
      \mathbb{C}\left\vert s_{ini} \right\rangle
    \big)
  \Big)
  \underoverset{}{}{\longrightarrow}
  V\big(
    \mathscr{H}
  \big)
  \oplus
  \Big(
    \mathscr{H}
      \otimes
    \big(
      \mathscr{B}
        \ominus
      \mathbb{C}\left\vert s_{ini} \right\rangle
    \big)
  \Big)
  \;\simeq\;
  \mathscr{H} \otimes \mathscr{B}
$$

and we claim that this has the desired action on couplings of the $\mathscr{H}$-system to the pure bath state $\left\vert s_{ini} \right\rangle$:

$$
  \begin{array}{l}
    trace^{\mathscr{B}}
    \Big(
      U
      \big(
        \left\vert s_{ini} \right\rangle
          \rho
        \left\langle s_{ini} \right\vert
      \big)
      U^\dagger
    \Big)
    \\
    \;=\;
    \underset{s,s'}{\sum}
    trace^{\mathscr{B}}
    \big(
        \left\vert s \right\rangle
          E_s \cdot \rho \cdot E_{s'}^\dagger
        \left\langle s' \right\vert
    \big)
    \\
    \;=\;
    \underset{s,s'}{\sum}
    \underset{
      \delta_{s}^{s'}
    }{
      \underbrace{
        \left\langle s' \vert s \right\rangle
      }
    }
    E_s \cdot \rho \cdot E_{s'}^\dagger
    \\
    \;=\;
    \underset{s}{\sum}
    E_s \cdot \rho \cdot E_{s}^\dagger
    \\
    \;=\;
    chan(\rho)
    \,.
  \end{array}
$$
This concludes the construction of an environmental representation where the environment is in a pure state.
\end{proof}

\begin{remark}
  The above theorem is often phrased as "... and the environment can be assumed to be in a pure state". But in fact the proof crucially uses the assumption that the environment is in a pure state. It is not clear that there is a proof that works more generally.

In fact, if the environment is taken to be in the maximally mixed state, then the resulting quantum channels are called *noisy operations* or *[[unistochastic quantum channel|unistochastic quantum channels]]* and are not expected to exhaust all quantum channels.
\end{remark}


quantum chemistry

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=--
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* table of contents
{:toc}

## Idea

Since the nature of [[atoms]] is described by [[quantum mechanics]] and since [[chemistry]] deals with [[bound states]] of atoms -- [[molecules]] -- it is ultimately based on [[quantum physics]], but typically the detailed quantum mechanical processes are heavily coarse-grained to effective laws of chemistry. In contrast to this _quantum chemistry_ studies chemical effects explicitly with more of the detailed quantum mechanics underlying them taken into account. For example, quantum mechanics may be used to explain the [[periodic table of chemical elements]] through the occupation of their electron shells ([Feynman63](#Feynman63)).

## Related concepts

* [[quantum simulation]]

* [[quantum biology]]

* [[quantum cryptography]]

## References

### General

Textbook accounts:

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

* [[Attila Szabo]], [[Neil S. Ostlund]], *Modern Quantum Chemistry -- Introduction to Advanced Electronic Structure Theory*, Macmillan (1982), McGraw-Hill (1989), Dover (1996) $[$[pdf](https://chemistlibrary.files.wordpress.com/2015/02/modern-quantum-chemistry.pdf)$]$

See also:

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

* {#Feynman63} [[Richard Feynman]], _The Feynman Lectures on Physics, vol III_, Chapter 19, [Lectures](http://www.feynmanlectures.caltech.edu/III_toc.html)

Potential application of [[quantum computation]] ([[quantum supremacy]]) to quantum chemistry:

* {#Lloyd96} [[Seth Lloyd]], *Universal Quantum Simulators*, Science New Series **273** 5278 (1996) 1073-1078 [[jstor:2899535](https://www.jstor.org/stable/2899535), [doi:10.1126/science.273.5278.1073](https://doi.org/10.1126/science.273.5278.1073)]

* {#QuantumChemistryInAgeOfQuantumComputing} *Quantum Chemistry in the Age of Quantum Computing*, Chem. Rev. (2019) **119** 19 (2019) 10856–10915 [[doi:10.1021/acs.chemrev.8b00803](https://doi.org/10.1021/acs.chemrev.8b00803)]

Argument that quantum advantage in quantum chemistry may be elusive:

* [[Garnet Kin-Lic Chan]] et al., *Is there evidence for exponential quantum advantage in quantum chemistry?* [[arXiv:2208.02199](https://arxiv.org/abs/2208.02199), [talk video](https://youtu.be/DZPH7ENcRLU)]

### Topological quantum chemistry

In relation to nontrivial [[topology]] ([[knot]]-structure) of [[molecules]]:

* [[Nils Baas]], [[Nadrian Seeman]], _On the chemical synthesis of new topological structures_, J Math Chem (2012) 50: 220 ([doi:10.1007/s10910-011-9907-3](https://doi.org/10.1007/s10910-011-9907-3))

* [[Nils Baas]], [[Nadrian Seeman]], [[Andrew Stacey]], _Synthesising Topological Links_, J Math Chem. 2015 Jan; 53(1): 183–199 (doi:10.1007\%2Fs10910-014-0420-3)

In relation to [[topological phases of matter]]:

* Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo & B. Andrei Bernevig, _Topological quantum chemistry_, Nature volume 547, pages 298–305 (2017) ([doi:10.1038/nature23268](https://doi.org/10.1038/nature23268))

### Application of tensor network states

Application of [[tensor networks]] (specifically [[tree tensor networks]]) in quantum chemistry:

* Naoki Nakatani, Garnet Kin-Lic Chan, _Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm_, J. Chem. Phys. 138, 134113 (2013) ([arXiv:1302.2298](https://arxiv.org/abs/1302.2298))

* Klaas Gunst, [[Frank Verstraete]], Sebastian Wouters, Örs Legeza, Dimitri Van Neck, _T3NS: three-legged tree tensor network states_, Chem. Theory Comput. 2018, 14, 4, 2026-2033 ([arXiv:1801.09998](https://arxiv.org/abs/1801.09998))

* Henrik R. Larsson, _Computing vibrational eigenstates with tree tensor network states (TTNS)_, J. Chem. Phys. 151, 204102 (2019) ([arXiv:1909.13831](https://arxiv.org/abs/1909.13831))


quantum circuit diagram

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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Broadly a *quantum logic circuit* is a [[logic circuit]] but in the context of [[quantum information theory]]/[[quantum computation]].
Concretely, due to the [[linear logic]]/[[linear type theory]] underlying [[quantum physics]]:

In the context of [[quantum computation]], a *quantum circuit diagram* (originally: "quantum computational network", [Deutsch 1989](#Deutsch89)) is a kind of [[string diagram]] ([Abramsky & Coecke 2004](#AbramskyCoecke04)) in [[finite-dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]], typically used to express a sequence of low-level [[quantum logic gate|quantum gates]] mapping between [[space of quantum states|spaces of]] [[pure quantum state|pure]] [[quantum states]].

A generic *pure* quantum circuit might look as follows:

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

> [figure from [[schreiber:TQP|Myers et al. 2023]]]

Here, for instance, the "[[quantum gate]]" $U_{34}$ is a [[linear operator]] on the [[tensor product]] [[Hilbert space]] $\mathscr{H} \oplus \mathscr{H}$.

Notice that the quantum circuit is understood to be the actual [[string diagram]], up to the usual admissible topological deformations, not just the [[composition|composite]] [[linear transformation]] which it encodes: The compositeness of the diagram encodes how (e.g. in which order) available [[quantum gate]]-operations would be operated on an actual [[quantum computer]], and the composite linear map only encodes the inpute-to-output-specification of the resulting [[quantum computation]]. In this sense, quantum circuits constitute a [[quantum programming language]] and one also speaks of the "quantum circuit model of quantum computation" (e.g. [Nielsen & Chuang 2000, §II.4.6](#NielsenChuang00); [Miszczak 2011, §3](#Miszczak11), [2012, §4.3](#Miszczak12); [Beneti & Casati 2018, §3.2](#BenetiCasati18)).

Accordingly, many existing [[quantum programming languages]] are actually [[domain specific programming languages]] for quantum circuits, and as such they are often [[domain specific embedded programming language|embedded]] into ambient [[type theories]] (for instance *[[Quipper]]* and *[[QWIRE]]*).

Often the basic Hilbert space building block here is taken to be [[complex vector space|complex]] 2-[[dimension of a vector space|dimensional]], $\mathscr{H} \,\coloneqq\, \mathbb{C}^2$, in which case one speaks of a quantum circuit on *[[q-bits]]*.

Sometimes all [[quantum gates]] involved, and hence also the resulting [[composition|composite]] [[linear maps]], are assumed to be [[invertible]], as befits [[reversible computation]] by [[unitary operator|unitary]] quantum state evolution in [[quantum mechanics]].

But more generally and certainly in the broader context of [[quantum information theory]] (such as in formulating the [[quantum teleportation protocol]] and similar), quantum circuits are admitted to also contain crucial [[classical mechanics|classical]]/[[quantum mechanics]]-interaction, in the form of:

1. *[[quantum measurement]]-gates*,

1. *[[quantum state preparation]]-gates*,

1. *classically [[controlled quantum gates]]*.

{#IBMResearchBlogGraphics}
\begin{imagefromfile}
    "file_name": "IBMOnMidCircuitMeasurement.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 10,
        "right": 0,
        "left": 10
    },
    "caption": "(example from [IBM Research Blog 2021/02](https://www.ibm.com/blogs/research/2021/02/quantum-mid-circuit-measurement/))"
\end{imagefromfile}

Here [[quantum measurement]], in particular, is not only non-[[reversible computation|reversible]] (due to the [[quantum state collapse]] involved) but also non-deterministic, meaning that it is not immediately represented by a fixed linear map at all -- at least not between the original Hilbert spaces.

One may model the $B$-[[tuple]] of [[projection operators]] corresponding to the possible set $B$ of measurement outcomes as a [[linear map]] between the original [[Hilbert space]] $\mathscr{H}$ and its $B$-[[indexed set|indexed]] [[direct sum]] (one rare place where this is made explicit, albeit without comment, is [Selinger 2004](#Selinger04), [p. 39](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf#page=39)):

$$
  \array{
    meas
    &
      \colon
    &
    \mathcal{H}
    &
      \xrightarrow{\phantom{--}}
    &
    \underset{
      b \colon B
    }{\oplus}
    \mathcal{H}
    \\
    &&
    \vert \psi \rangle
    &\mapsto&
    \underset{b \colon B}{\oplus}
    \,
    P_{b} \vert \psi \rangle
  }
  \,.
$$

Therefore, as soon as these three operations crossing the [[classical physics|classical]]/[[quantum physics]]-boundary are involved (and made explicit), quantum circuits are no longer just [[string diagrams]] in [[finite-dimensional vector space|finite-dimensional]] [[HilbertSpaces]], but something richer, involving besides [[tensor products]] also some kind of reflection of [[direct sums]] of [[spaces of quantum states]].

For a critical assessment of the (lack of) literature on this issue see [Gurevich & Blass 2021](#GurevichBlass21).

One approach for handling these more general quantum circuits is ([Aharonov, Kitaev & Nisan 1998](#AharonovKitaevNisan98), [Selinger 2004](#Selinger04)) to understand them, in the tradition on [[quantum probability theory]], as diagrams of "[[quantum operations]]" on [[space of quantum states|spaces of]] [[mixed quantum states]], namely as a kind of [[string diagrams]] of [[completely positive maps]] between [[convex spaces]] of [[density matrices]].

Other approaches for giving [[categorical semantics]] for quantum circuits in this general sense have been proposed (and implemented) in the discussion of the [[quantum programming language]] *[[Quipper]]* (see the references [there](Quipper#ReferencesDependentLinearTypesAndCategoricalSemantics)). A natural formulation in terms of [Quantum Modal Logic](necessity+and+possibility#ModalQuantumLogic) in terms of [[dependent linear type theory]] is discussed at:

* *[[quantum circuits via dependent linear types]]*.

These turn out to be related: From a suitably rich formulation of quantum circuits of pure states with measurement and classical control, the [[density matrix]]-formulation may be recovered (...).

At the time of writing (2021) most of the actual programming of [[experiment|experimental]] [[quantum computers]] is conceived through quantum circuit diagrams, while more high-level [[quantum programming languages]] are are awaiting the rise of more powerful quantum hardware.

## Examples
 {#Examples}

### Bell state preparation

In [[qbit]]-based [[quantum computation]], the elementary [[Bell state]] is usually prepared via the following small quantum circuit consisting of a [[Hadamard gate]] followed by a [[quantum CNOT gate]]:

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

### Quantum teleportation

The standard (cf. [GLRSV13, p. 5](Quipper#GLRSV13)) [[qbit]]-form of the [[quantum teleportation]]-protocol is the following quantum ciruit on [[qbits]] (where "$H$" denotes the [[Hadamard gate]], the circles denote the [[quantum CNOT gates]] and the boxed pointers denote [[quantum measurement]] in the [[qbit]]-basis):

\begin{imagefromfile}
    "file_name": "QBitQuantumTeleportationProtocol-221026.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Properties

### Deferred measurement principle

See at *[[deferred measurement principle]]*.

## Related concepts

* [[logic circuit]]

* [[QRAM]]

* [[quantum computation]], [[quantum error correction]]

* [[string diagram]], [[proof net]]

* [[DisCoPy]]

## References

### Quantum circuit diagrams

The notion of [[quantum gates]] and quantum circuits acting on [[pure quantum states]] originates with:

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

* {#Deutsch89} [[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)]

On quantum circuits for [[mixed quantum states]] (with [[density matrices]]) and rigorous propsals for the classical/quantum interaction:

* {#AharonovKitaevNisan98} [[Dorit Aharonov]], [[Alexei Kitaev]], [[Noam Nisan]], *Quantum Circuits with Mixed States*, *Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC)* (1998) 20-30 [[arXiv:quant-ph/9806029](https://arxiv.org/abs/quant-ph/9806029), [doi:10.1145/276698.276708](https://doi.org/10.1145/276698.276708)]

* {#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)]

* {#GurevichBlass21} [[Yuri Gurevich]], [[Andreas Blass]], *Quantum circuits with classical channels and the principle of deferred measurements*, Theoretical Computer Science **920** (2022) 21–32 [[arXiv:2107.08324](https://arxiv.org/abs/2107.08324), [doi:10.1016/j.tcs.2022.02.002](https://doi.org/10.1016/j.tcs.2022.02.002)]

Standard textbook accounts:

* {#NielsenChuang00} [[Michael A. Nielsen]], [[Isaac L. Chuang]], Section II.4 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

*  {#Miszczak11} [[Jarosław Adam Miszczak]], Sec. 3 of: *Models of quantum computation and quantum programming languages*, Bull. Pol. Acad. Sci.-Tech. Sci., **59** 3 (2011)305-324 [[arXiv:1012.6035](https://arxiv.org/abs/1012.6035), [doi:10.2478/v10175-011-0039-5](https://doi.org/10.2478/v10175-011-0039-5)]

* {#Miszczak12} [[Jarosław Adam Miszczak]], Section 4.3 in: *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)]

* *Quantum Circuit Model of Computation*, 2015 ([pdf](https://ipgold.epfl.ch/_media/en/courses/2015-2016/calculquant/chapter3corrected.pdf))

* {#BenetiCasati18} [[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)]

Standard lecture notes:

* [[John Preskill]], *Classical and quantum circuits* ([pdf](http://theory.caltech.edu/~preskill/ph219/chap5_13.pdf)), Chapter 5 in: *Quantum Computation*, lecture notes ([web](http://theory.caltech.edu/~preskill/ph229/))

* Ryan O’Donnell, *Introduction to the Quantum Circuit Model* (2015) ([pdf](https://www.cs.cmu.edu/~odonnell/quantum15/lecture01.pdf))

* [[Scott Aaronson]], §4.2 in: *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

Survey of simulating and designing quantum circuits:

* Alwin Zulehner and Robert Wille: *Simulation and Design of Quantum Circuits*, in: *Reversible Computation: Extending Horizons of Computing. RC 2020* Lecture Notes in Computer Science **12070**, Springer (2020) [[doi:10.1007/978-3-030-47361-7_3](https://doi.org/10.1007/978-3-030-47361-7_3)]

The ([[dagger-compact category|dagger-compact]] [[monoidal category|monoidal]])  [[category|category theoretic]] formulation is due to:

* {#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](https://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

with exposition and further development in:

* {#Coecke05} [[Bob Coecke]], *Kindergarten Quantum Mechanics*, in *Quantum Theory: Reconsideration of Foundations* (QTRF 3) Vaxjo, Sweden, June 6-11, 2005, AIP Conf. Proc. **810** (2006) [[arXiv:quant-ph/0510032](https://arxiv.org/abs/quant-ph/0510032), [doi:10.1063/1.2158713](https://doi.org/10.1063/1.2158713)]

* {#Coecke10} [[Bob Coecke]], *Quantum Picturalism*, Contemporary Physics **51** (2010) 59-83 [[arXiv:0908.1787](https://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]

For more on this see at *[[finite quantum mechanics in terms of dagger-compact categories]]*.

See also:

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

* Stephen P. Jordan, Sec. 1.2 in: *Quantum Computation Beyond the Circuit Model* ([arXiv:0809.2307](https://arxiv.org/abs/0809.2307))

On quantum circuits as ([[geodesic]], if optimal) [computational paths](computation#ReferencesConceptualizationViaPathLifting):

* {#NielsenDowlingGuDoherty06} [[Michael A. Nielsen]], [[Mark R. Dowling]], [[Mile Gu]], [[Andrew C. Doherty]], *Quantum Computation as Geometry*, Science, **311** 5764 (2006) 1133-1135 [[doi:10.1126/science.1121541](https://doi.org/10.1126/science.1121541), [arXiv:quant-ph/0603161](https://arxiv.org/abs/quant-ph/0603161)]

* [[Mark R. Dowling]], [[Michael A. Nielsen]], *The geometry of quantum computation*, Quantum Information & Computation **8** 10 (2008) 861–899 [[doi:10.5555/2016985.2016986](https://dl.acm.org/doi/abs/10.5555/2016985.2016986), [arXiv:quant-ph/0701004](https://arxiv.org/abs/quant-ph/0701004)]

Survey, examples, and implementations:

* Michal Charemza, *Examples of Quantum Circuit Diagrams*, 2006 ([pdf](https://warwick.ac.uk/fac/sci/physics/research/cfsa/people/pastmembers/charemzam/pastprojects/mcharemza_quant_circ.pdf))

* Qiskit, *[Defining Quantum Circuits](https://qiskit.org/textbook/ch-algorithms/defining-quantum-circuits.html)*

* {#SKQHackathon20} [South Korea Qiskit Hackathon'20 - Qiskit Metal exercise](https://qiskit.org/documentation/metal/circuit-examples/full-design-flow-examples/Exercise-for-the-South-Korea-Hackathon'20.html)

  > (tutorial for a 2-[[qbit]] quantum chip using Qiskit)

* Microsof Build, *[Quantum circuit diagrams](https://docs.microsoft.com/en-us/azure/quantum/concepts-circuits)*

* Vladimir P. Gerdt, Alexander N. Prokopenya, *The Circuit Model of Quantum Computation and Its Simulation with Mathematica*, In: Adam G., Buša J., Hnatič M. (eds.) *Mathematical Modeling and Computational Science* MMCP 2011. Lecture Notes in Computer Science, vol 7125. Springer 2011 ([doi:10.1007/978-3-642-28212-6_4](https://doi.org/10.1007/978-3-642-28212-6_4))

With an eye towards [[quantum complexity theory]]:

* Richard Cleve, Section 1.2 in: *An Introduction to Quantum Complexity Theory* ([pdf](https://cds.cern.ch/record/392006/files/9906111.pdf))

Formal [[quantum programming language]]-perspective on quantum circuits:

* [[Yuri Gurevich]], [[Andreas Blass]], *Software science view on quantum circuit algorithms* [[arXiv:2209.13731](https://arxiv.org/abs/2209.13731)]

See also:

* {#ACGMMR23} M. Amy, M. Crawford, A. N. Glaudell, M. L. Macasieb, S. S. Mendelson, [[Neil J. Ross]], *Catalytic Embeddings of Quantum Circuits* [[arXiv:2305.07720](https://browse.arxiv.org/abs/2305.07720)]

### Parameterized quantum circuits
 {#ReferencesParameterizedQuantumCircuits}

Discussion of classically-[[parameterized quantum system|parameterized]] quantum circuits:

(...)

Discussion of quantumly-[[parameterized quantum systems|parameterized]] quantum circuits:

* Evan Peters, Prasanth Shyamsundar, *Qarameterized circuits: Quantum parameters for QML* [[web](https://peterse.github.io/groveropt/)]

following

* Guillaume Verdon, Jason Pye, Michael Broughton, *A Universal Training Algorithm for Quantum Deep Learning* [[arXiv:1806.09729](https://arxiv.org/abs/1806.09729)]

* Prasanth Shyamsundar, *Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation* [[arXiv:2102.04975](https://arxiv.org/abs/2102.04975)]

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