nLab quantum operation

Idea

Any physical process is supposed to take physical states into physical states (Schrödinger picture). If density matrices are used to describe quantum states in quantum mechanics, then it must be some operation that sends density matrices to density matrices. So for finite-dimensional state spaces a process should be a linear map of vector spaces of matrices

$$U \;\colon\; Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})$$

(so far this is a general “superoperator”) that preserves the subset of density matrices, in that

• it preserves the trace of matrices;

• takes hermitian matrices with non-negative eigenvalues to hermitian matrices with non-negative eigenvalues.

Such a map is then called a quantum operation. The notion of a quantum operation is built from the Stinespring factorization theorem.

Definition

We first give the traditional definition in terms of linear algebra and matrices in

• In terms of matrices

Then we consider the general abstract formulation

• In terms of compact closed categories

In terms of matrices

Let $k,n \in \mathbb{N}$.

A matrix $A \in Mat(n \times n, \mathbb{C})$ is called positive if it is hermitian – if $A^\dagger = A$ – and if all its eigenvalues (which then are necessarily real) are non-negative.

A linear map (morphism of vector spaces of matrices)

$$\Phi \;\colon\; Mat(n \times n, \mathbb{C}) \to Mat(k \times k, \mathbb{C})$$

is called positive if it takes positive matrices to positive matrices.

The map $\Phi$ is called completely positive if for all $p \in \mathbb{N}$ the tensor product

$$\Phi \otimes Id_{Mat(p\times p),\mathbb{C}} \;\colon\; Mat(n \times n , \mathbb{C}) \otimes Mat(p \times p , \mathbb{C}) \longrightarrow Mat(k \times k , \mathbb{C}) \otimes Mat(p \times p , \mathbb{C})$$

is positive.

A quantum operation (or quantum channel) is a map that is both completely positive and trace preserving (often abbreviated to CPTP).

In terms of compact closed categories

… due to (Selinger 05) … see for instance (Coecke-Heunen 11, section 2) for a quick summary …

Properties

Characterization of complete positivity

Theorem

A map $\Phi$ as above is completely positive precisely if there exists a set $I$ and an $I$-family $\{E_i \in Mat(k \times n, \mathbb{C}| i \in I)\}$ of matrices, such that for all $A \in Mat(n \times n, \mathbb{C})$ we have

$$\Phi(A) = \sum_{i \in I} E_i A E_i^\dagger \,.$$

Moreover, such $\Phi$ preserves the trace of matrices precisely if

$$\sum_{i \in I} E_i^\dagger E_i = Id_{Mat(n \times n, \mathbb{C})} \,.$$

This is originally due to (Stinespring 55). The decomposition in the theorem is called Kraus decomposition after (Kraus 71). See also (Choi 76, theorem 1). A brief review is for instance in (Kuperberg 05, theorem 1.5.1). A general abstract proof in terms of †-categories is given in (Selinger 05). A characterization of completely positive maps entirely in terms of $\dagger$-categories is given in (Coecke 07).

The matrices $\{E_i\}$ that are associated to a completely positive and trace-preserving map by the above theorem are called Kraus operators.

In the physics literature the above theorem is then phrased as: Every quantum channel can be represented using Kraus operators .

Notice that the identity map is clearly completely positive and trace preserving, and that the composite of two maps that preserve positivity and trace clearly still preserves positivity and trace. Therefore we obtain a category $QChan \subset Vect$ – a subcategory of Vect${}_{\mathbb{C}}$ – whose

• objects are the vector spaces $Mat(n \times n, \mathbb{C})$ for all $n \in \mathbb{N}$;

• morphism are completely positive and trace-preserving linear maps $\Phi : Mat(n\times n , \mathbb{C}) \to Mat(m \times m, \mathbb{C})$;

• composition of morphisms is, of course, the composition in Vect, i.e. the ordinary composition of linear maps.

See also extremal quantum channels and graphical quantum channels.

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 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 $VV*=I$.

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:

$$\array{ \mathbf{Isometries} &\rightarrow& \mathbf{Quantum Channels} \\ &\searrow&\downarrow\\ && \mathcal{D} }$$

This fits a physical intuition as follows. Suppose that the isometries are a model of reality, as in the 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

Quantum measurement and POVMs

A quantum measurement is formally represented by a quantum operation that is induced by a positive-operator valued probability measure (POVM).

Decoherence and partial traces

For the moment see the references at quantum decoherence.

Noice channels

Examples of quantum noise channels:

• bit flip channel

Related concepts

• quantum information theory via dagger-compact categories

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

The Kraus-decomposition characterization of completely positive maps is due to

• W. Forrest Stinespring, Positive functions on $C^\ast$-algebras, Proc. Amer. Math. Soc. 6 2 (1955) 211-216 [doi:2032342, doi:10.2307/2032342]

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

• M. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications Volume 10, Issue 3, (1975), Pages 285-290

• 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]

Review and survey:

• Michael A. Nielsen, Isaac L. Chuang, §8.2 in : Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

• Greg Kuperberg, section 1.5 of A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)

• Ingemar Bengtsson, Karol Życzkowski, Chapter 10 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

• Robert B. Griffiths, Quantum Channels, Kraus Operators, POVMs (2012) [pdf, pdf]

• Stéphane Attal, Quantum Channels, Lecture 6 in: Lectures on Quantum Noises [pdf, webpage]

See also

• Caleb J. O’Loan (2009), Topics in Estimation of Quantum Channels , PhD thesis, University of St. Andrews, (arXiv)

• Christian B. Mendl, Michael M. Wolf, Unital Quantum Channels - Convex Structure and Revivals of Birkhoff’s Theorem , Commun. Math. Phys. 289, 1057-1096 (2009) (arXiv:0806.2820)

• John A. Smolin, Frank Verstraete, Andreas Winter, Entanglement of assistance and multipartite state distillation, Phys. Rev. A 72 (2005) 052317 (arXiv:quant-ph/0505038)

• John Watrous, Mixing doubly stochastic quantum channels with the completely depolarizing channel (2008) (arXiv)

• Wikipedia, 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

• Peter Selinger, Dagger-compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 (pdf, ps)

• Bob Coecke, Complete positivity without compactness, 2007 (pdf)

This is further explored in

• Bob Coecke, Eric Paquette, Dusko Pavlovic, Classical and quantum structures (2008) [pdf, pdf

  ]

• Bob Coecke, Chris Heunen, Pictures of complete positivity in arbitrary dimension, EPTCS 95, 2012, pp. 27-35 (arXiv:1110.3055)

• Bob Coecke, Chris Heunen, Aleks Kissinger, Categories of Quantum and Classical Channels (arXiv:1305.3821)

For the universal property, see

• Mathieu Huot, Sam Staton, Universal properties in quantum theory (QPL 2018) (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]

• Alexander S. Holevo, Quantum channel capacities, Quantum Electron. 50 440 (2020) [doi:10.1070/QEL17285/meta]