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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 transformations on quantum states (as known 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 quantum states with their objective, intrinsic and fundamental stochasticity (expressed the Born rule) to include also subjective, thermodynamical 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, in which case quantum channels are traditionally discussed as (special) linear maps between vector spaces of square matrices:

chan:Mat n 1()Mat n 2(). 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: 1 1 * 2 2 * chan \;\colon\; \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \longrightarrow \mathscr{H}_2 \otimes \mathscr{H}_2^\ast

(where above n i=dim( i)n_i = dim(\mathscr{H}_i) is the 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 subsets of density matrices (the mixed states) inside i i *\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 tensoring with any identity transformation.

This is discussed below at:

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:ρwE wρE w chan \;\colon\; \rho \;\mapsto\; \underset{w}{\sum} E_w \cdot \rho \cdot E_w^\dagger

by certain tuples (E w) w:W(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:

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

(1)chan U:ρUρU , chan_U \;\colon\; \rho \;\mapsto\; U \cdot \rho \cdot U^\dagger \,,

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

chan U:ρ |ψρ U|ψ. chan_U \;\colon\; \rho_{\vert\psi \rangle} \;\mapsto\; \rho_{ U \vert \psi \rangle } \,.

On the other extreme, a quantum measurement in a measurement basis WW, W\underset{W}{\oplus} \mathbb{C} \simeq \mathscr{H} is given by the corresponding projection operators P wP_w as

(2)meas W:ρwP wρP w. 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 (2) may equivalently be understood as the result of a unitary evolution (1) of the state ρ\rho coupled to an environment state ω\omega followed by the 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 ρ\rho by

  1. tensoring it to another state ω\omega

    (coupling the system to an environment/“bath”)

  2. sending the tensor state through a unitary channel (1)

    (Schrödinger evolution of the couplesystem)

  3. applying the partial trace over the Hilbert space of ω\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:

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 positivity conditions

Let i\mathscr{H}_i be a complex finite-dimensional Hilbert space, with

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

the corresponding space of matrices — including as a convex subset the density matrices representing the mixed states of the quantum system described by \mathscr{H}.


A \mathbb{C}-linear map

Φ: 1 1 * 2 2 * \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 matrices,

  • nn-positive if ΦId n\Phi \otimes Id_{\mathbb{C}^n} is positive for nn \in \mathbb{N},

  • completely positive if Φ\Phi is nn-positive for all nn \in \mathbb{N}.

A positive Φ\Phi is furthermore called:

  • stochastic if it preserves the trace of matrices

  • doubly stochastic if it preserves also the identity matrix.

Finally, a completely positive Φ\Phi is called:

(e.g. Landau & Streater 1993 p. 107-108)

In terms of Kraus decompositions


(operator-sum deomposition of quantum channels)
For i\mathscr{H}_i finite-dimensional Hilbert spaces, a linear map

chan: 1 1 * 2 2 * chan \;\colon\; \mathscr{H}_1 \otimes \mathscr{H}_1^\ast \longrightarrow \mathscr{H}_2 \otimes \mathscr{H}_2^\ast

is completely positive (Def. ) precisely if there exists an indexed set

R:FinSet,r:RE r: 1 2 R \,\colon\, FinSet ,\;\; r \,\colon\, R \;\;\;\;\; \vdash \;\;\;\;\; E_r \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2

of linear operators such that

(3)chan(ρ)=rE rρE r . chan(\rho) \;=\; \underset{r}{\sum} \, E_r \cdot \rho \cdot E_r^\dagger.

Accordingly, in addition:

  • chanchan preserves the trace and is hence a quantum channel iff

    rE r E r=id 1 \sum_r \, E_r^\dagger \cdot E_r \,=\, id_{\mathscr{H}_1}
  • chanchan preserves also the identity matrix and is hence a unital quantum channel iff (in addition)

    rE rE r =id 2. \sum_r \, E_r \cdot E_r^\dagger \,=\, id_{\mathscr{H}_2} \,.

The idea goes back to Stinespring 1955. The decomposition (3) is also called Kraus decomposition by Kraus operators, after Kraus 1971. The fully explicit statement of Thm. is due to Choi 1975 Thm. 1.

Review includes: Nielsen & Chuang 2000 Thm. 8.1, Kuperberg 2005 Thm. 1.5.1.

A general abstract proof in terms of †-categories is given by Selinger 2005. A characterization of completely positive maps entirely in terms of \dagger-categories is given in Coecke 2007.

Quantum channels and decoherence

The crux of dynamical quantum decoherence is that fundamentally the (time-)evolution of any quantum system \mathscr{H} may be assumed 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

evolve: |ψ,β U tot|ψ,β, \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 ρ: *\rho \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast (density matrices) of the given quantum system as coupled to any given mixed state env: *env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast of the bath (via tensor product)

couple: * ()() * ρ ρenv; \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 over the environment:

(4)average:()() * * ρ^ Tr (ρ^). \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

* couple to environment ( )( ) * total unitary evolution ( )( ) * average over environment * ρ ρenv U tot(ρenv)U tot Tr (U tot(ρenv)U tot ) \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” and in fact all quantum channels arise this way:


(quantum channels and decoherence)

Every quantum channel

ch: * * ch \;\;\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:U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B}

  2. on a compound system with some \mathscr{B} (the “bath”), yielding a total system Hilbert space \mathscr{H} \otimes \mathscr{B} (tensor product),

  3. and acting on the given mixed state ρ\rho coupled (tensored) with a fixed mixed state env: *env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast of the bath system,

  4. followed by partial trace (averaging) over \mathscr{B} (leading to decoherence in the remaining state)

in that

(5)ch(ρ)=Tr (U tot(ρenv)U tot ). ch(\rho) \;\;=\;\; Tr_{\mathscr{B}} \big( U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger \big) \,.

Conversely, every operation of the form (5) is a quantum channel.

For exposition: Nielsen & Chuang 2000 §8.2.2-8.2.3

Detailed proof, including the infinite-dimensional case: Attal, Thm. 6.5 & 6.7.

In terms of \dagger-compact closed categories

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


Universal property

The category whose objects are indexed by natural numbers n,m,n,m, \cdots and whose morphisms are quantum operations from n×nn \times n to m×mm \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 11) 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 mnm\to n is an m×nm\times n complex matrix VV such that VV*=IV V* = I.

In detail, the universal property says that for any strict semicartesian monoidal category 𝒟\mathcal{D} and any monoidal functor Isometries𝒟\mathbf{Isometries}\to \mathcal{D}, there is a unique symmetric monoidal functor making the following diagram commute:

Isometries QuantumChannels 𝒟 \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 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.


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.

Noise channels

Examples of quantum noise channels:

quantum probability theoryobservables and states


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

with early review in:

The terminology “quantum operation” for linear maps on the linear dual of a C * C^\ast -algebra which preserve the subset of states on a star-algebra:

Analysis of extremal quantum channels:

Review and survey:

In the context of quantum computation:

in the context of quantum probability:

and in quantum information theory:


See also

  • Caleb J. O’Loan, Topics in Estimation of Quantum Channels, PhD thesis, University of St. Andrews (2009) [arXiv:1001.397]

  • 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

This is further explored in:

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]

Last revised on September 20, 2023 at 19:36:31. See the history of this page for a list of all contributions to it.