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Inquantum information theory, aquantum channel is a communication channel that can transmitquantum information, as well as classical information. An example of quantum information is the general dynamics of aqubit. An example of classical information is a text document transmitted over theInternet.

Terminologically, quantum channels arecompletely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just aquantum operation viewed not merely as thereduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.^[1])

We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional.

Thememoryless in the section title carries the same meaning as in classicalinformation theory: the output of a channel at a given time depends only upon the corresponding input and not any previous ones.

Consider quantum channels that transmit only quantum information. This is precisely aquantum operation, whose properties we now summarize.

Let${\displaystyle H_A}$ and${\displaystyle H_B}$ be the state spaces (finite-dimensionalHilbert spaces) of the sending and receiving ends, respectively, of a channel.${\displaystyle L(H_A)}$ will denote the family of operators on${\displaystyle H_A.}$ In theSchrödinger picture, a purely quantum channel is a map${\displaystyle \mathrm{\Phi }}$ betweendensity matrices acting on${\displaystyle H_A}$ and${\displaystyle H_B}$ with the following properties:^[2]

1. As required by postulates of quantum mechanics,${\displaystyle \mathrm{\Phi }}$ needs to be linear.
2. Since density matrices are positive,${\displaystyle \mathrm{\Phi }}$ must preserve thecone of positive elements. In other words,${\displaystyle \mathrm{\Phi }}$ is apositive map.
3. If anancilla of arbitrary finite dimensionn is coupled to the system, then the induced map${\displaystyle I_n\otimes \mathrm{\Phi },}$ whereI_n is the identity map on the ancilla, must also be positive. Therefore, it is required that${\displaystyle I_n\otimes \mathrm{\Phi }}$ is positive for alln. Such maps are calledcompletely positive.
4. Density matrices are specified to have trace 1, so${\displaystyle \mathrm{\Phi }}$ has to preserve the trace.

The adjectivescompletely positive and trace preserving used to describe a map are sometimes abbreviatedCPTP. In the literature, sometimes the fourth property is weakened so that${\displaystyle \mathrm{\Phi }}$ is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP.

Density matrices acting onH_A only constitute a proper subset of the operators onH_A and same can be said for systemB. However, once a linear map${\displaystyle \mathrm{\Phi }}$ between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend${\displaystyle \mathrm{\Phi }}$ uniquely to the full space of operators. This leads to the adjoint map${\displaystyle {\mathrm{\Phi }}^{\ast }}$, which describes the action of${\displaystyle \mathrm{\Phi }}$ in theHeisenberg picture:^[3]

The spaces of operatorsL(H_A) andL(H_B) are Hilbert spaces with theHilbert–Schmidt inner product. Therefore, viewing${\displaystyle \mathrm{\Phi }:L(H_A)\to L(H_B)}$ as a map between Hilbert spaces, we obtain its adjoint${\displaystyle \mathrm{\Phi }}$^* given by

${\displaystyle ⟨A,\mathrm{\Phi }(\rho )⟩=⟨{\mathrm{\Phi }}^{\ast }(A),\rho ⟩.}$

While${\displaystyle \mathrm{\Phi }}$ takes states onA to those onB,${\displaystyle {\mathrm{\Phi }}^{\ast }}$ maps observables on systemB to observables onA. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.

It can be directly checked that if${\displaystyle \mathrm{\Phi }}$ is assumed to be trace preserving,${\displaystyle {\mathrm{\Phi }}^{\ast }}$ isunital, that is,${\displaystyle {\mathrm{\Phi }}^{\ast }(I)=I}$. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.

So far we have only defined a quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators:

${\displaystyle \mathrm{\Psi }:L(H_B)\to L(H_A)}$

that is unital and completely positive (CP). The operator spaces can be viewed as finite-dimensionalC*-algebras. Therefore, we can say a channel is a unital CP map between C*-algebras:

${\displaystyle \mathrm{\Psi }:\mathcal{B}\to \mathcal{A}.}$

Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions${\displaystyle C(X)}$ on some set${\displaystyle X}$. We assume${\displaystyle X}$ is finite so${\displaystyle C(X)}$ can be identified with then-dimensional Euclidean space${\displaystyle {\mathbb{R}}^n}$ with entry-wise multiplication.

Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define${\displaystyle \mathcal{B}}$ to include the relevant classical observables. An example of this would be a channel

${\displaystyle \mathrm{\Psi }:L(H_B)\otimes C(X)\to L(H_A).}$

Notice${\displaystyle L(H_B)\otimes C(X)}$ is still a C*-algebra. An element${\displaystyle a}$ of a C*-algebra${\displaystyle \mathcal{A}}$ is called positive if${\displaystyle a=x^{\ast }x}$ for some${\displaystyle x}$. Positivity of a map is defined accordingly. This characterization is not universally accepted; thequantum instrument is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in aFrobenius algebra orFrobenius category.

For a purely quantum system, the time evolution, at certain timet, is given by

${\displaystyle \rho \to U\rho U^{\ast },}$

where${\displaystyle U=e^{-iHt/\hslash }}$ andH is theHamiltonian andt is the time. This gives a CPTP map in the Schrödinger picture and is therefore a channel.^[4] The dual map in the Heisenberg picture is

${\displaystyle A\to U^{\ast }AU.}$

Consider a composite quantum system with state space${\displaystyle H_A\otimes H_B.}$ For a state

${\displaystyle \rho \in H_A\otimes H_B,}$

the reduced state ofρ on systemA,ρ^A, is obtained by taking thepartial trace ofρ with respect to theB system:

${\displaystyle {\rho }^A={\mathrm{Tr}}_B\rho .}$

The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture.^[5] In the Heisenberg picture, the dual map of this channel is

${\displaystyle A\to A\otimes I_B,}$

whereA is an observable of systemA.

An observable associates a numerical value${\displaystyle f_i\in \mathbb{C}}$ to a quantum mechanicaleffect${\displaystyle F_i}$.${\displaystyle F_i}$'s are assumed to be positive operators acting on appropriate state space and$\sum _i{F}_i=I$. (Such a collection is called aPOVM.^[6][7]) In the Heisenberg picture, the correspondingobservable map${\displaystyle \mathrm{\Psi }}$ maps a classical observable

${\displaystyle f=\left[\begin{array}{c}{f}_1\\ ⋮\\ f_n\end{array}\right]\in C(X)}$

to the quantum mechanical one

${\displaystyle \mathrm{\Psi }(f)=\sum _i{f}_i{F}_i.}$

In other words, oneintegratesf against the POVM to obtain the quantum mechanical observable. It can be easily checked that${\displaystyle \mathrm{\Psi }}$ is CP and unital.

The corresponding Schrödinger map${\displaystyle {\mathrm{\Psi }}^{\ast }}$ takes density matrices to classical states:^[8]

${\displaystyle \mathrm{\Psi }(\rho )=\left[\begin{array}{c}⟨F_1,\rho ⟩\\ ⋮\\ ⟨F_n,\rho ⟩\end{array}\right],}$

where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalizedfunctionals, and invoking theRiesz representation theorem, we can put

${\displaystyle \mathrm{\Psi }(\rho )=\left[\begin{array}{c}\rho (F_1)\\ ⋮\\ \rho (F_n)\end{array}\right].}$

The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called aquantum instrument. Let${\displaystyle \{F_1,\dots ,F_n\}}$ be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map${\displaystyle \mathrm{\Phi }}$ with pure quantum input${\displaystyle \rho \in L(H)}$ and with output space${\displaystyle C(X)\otimes L(H)}$:

${\displaystyle \mathrm{\Phi }(\rho )=\left[\begin{array}{c}\rho (F_1)\cdot F_1\\ ⋮\\ \rho (F_n)\cdot F_n\end{array}\right].}$

Let

${\displaystyle f=\left[\begin{array}{c}{f}_1\\ ⋮\\ f_n\end{array}\right]\in C(X).}$

The dual map in the Heisenberg picture is

${\displaystyle \mathrm{\Psi }(f\otimes A)=\left[\begin{array}{c}{f}_1{\mathrm{\Psi }}_1(A)\\ ⋮\\ f_n{\mathrm{\Psi }}_n(A)\end{array}\right]}$

where${\displaystyle {\mathrm{\Psi }}_i}$ is defined in the following way: Factor${\displaystyle F_i=M_i^2}$ (this can always be done since elements of a POVM are positive) then${\displaystyle {\mathrm{\Psi }}_i(A)=M_{i}A{M}_i}$.We see that${\displaystyle \mathrm{\Psi }}$ is CP and unital.

Notice that${\displaystyle \mathrm{\Psi }(f\otimes I)}$ gives precisely the observable map. The map

${\displaystyle \tilde{\mathrm{\Psi }}(A)=\sum _i{\mathrm{\Psi }}_i(A)=\sum _i{M}_{i}A{M}_i}$

describes the overall state change.

Suppose two partiesA andB wish to communicate in the following manner:A performs the measurement of an observable and communicates the measurement outcome toB classically. According to the message he receives,B prepares his (quantum) system in a specific state. In the Schrödinger picture, the first part of the channel${\displaystyle \mathrm{\Phi }}$₁ simply consists ofA making a measurement, i.e. it is the observable map:

${\displaystyle {\mathrm{\Phi }}_1(\rho )=\left[\begin{array}{c}\rho (F_1)\\ ⋮\\ \rho (F_n)\end{array}\right].}$

If, in the event of thei-th measurement outcome,B prepares his system in stateR_i, the second part of the channel${\displaystyle \mathrm{\Phi }}$₂ takes the above classical state to the density matrix

${\displaystyle {\mathrm{\Phi }}_2\left(\left[\begin{array}{c}\rho (F_1)\\ ⋮\\ \rho (F_n)\end{array}\right]\right)=\sum _i\rho (F_i)R_i.}$

The total operation is the composition

${\displaystyle \mathrm{\Phi }(\rho )={\mathrm{\Phi }}_2\circ {\mathrm{\Phi }}_1(\rho )=\sum _i\rho (F_i)R_i.}$

Channels of this form are calledmeasure-and-prepare orentanglement-breaking.^{[9][10][11][12]}

In the Heisenberg picture, the dual map${\displaystyle {\mathrm{\Phi }}^{\ast }={\mathrm{\Phi }}_1^{\ast }\circ {\mathrm{\Phi }}_2^{\ast }}$ is defined by

${\displaystyle {\mathrm{\Phi }}^{\ast }(A)=\sum _i{R}_i(A)F_i.}$

A measure-and-prepare channel can not be the identity map. This is precisely the statement of theno teleportation theorem, which says classical teleportation (not to be confused withentanglement-assisted teleportation) is impossible. In other words, a quantum state can not be measured reliably.

In thechannel-state duality, a channel is measure-and-prepare if and only if the corresponding state isseparable. Actually, all the states that result from the partial action of a measure-and-prepare channel are separable, which is why measure-and-prepare channels are also known as entanglement-breaking channels.

Consider the case of a purely quantum channel${\displaystyle \mathrm{\Psi }}$ in the Heisenberg picture. With the assumption that everything is finite-dimensional,${\displaystyle \mathrm{\Psi }}$ is a unital CP map between spaces of matrices

${\displaystyle \mathrm{\Psi }:{\mathbb{C}}^{n\times n}\to {\mathbb{C}}^{m\times m}.}$

ByChoi's theorem on completely positive maps,${\displaystyle \mathrm{\Psi }}$ must take the form

${\displaystyle \mathrm{\Psi }(A)=\sum _{i=1}^N{K}_{i}A{K}_i^{\ast }}$

whereN ≤nm. The matricesK_i are calledKraus operators of${\displaystyle \mathrm{\Psi }}$ (after the German physicistKarl Kraus, who introduced them).^[13][14][15] The minimum number of Kraus operators is called the Kraus rank of${\displaystyle \mathrm{\Psi }}$. A channel with Kraus rank 1 is calledpure. The time evolution is one example of a pure channel. This terminology again comes from the channel-state duality. A channel is pure if and only if its dual state is a pure state.

Inquantum teleportation, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel. The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver. Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle. This measurement results in classical information that must be sent to the receiver to complete the teleportation. Importantly, the classical information can be sent after the quantum channel has ceased to exist.

Experimentally, a simple implementation of a quantum channel isfiber optic (or free-space for that matter) transmission of singlephotons. Single photons can be transmitted up to 100 km in standard fiber optics before losses dominate.^{[citation needed]} The photon's time-of-arrival (time-bin entanglement) orpolarization are used as a basis to encode quantum information for purposes such asquantum cryptography. The channel is capable of transmitting not only basis states (e.g.${\displaystyle |0⟩}$,${\displaystyle |1⟩}$) but also superpositions of them (e.g.${\displaystyle |0⟩+|1⟩}$). Thecoherence of the state is maintained during transmission through the channel. Contrast this with the transmission of electrical pulses through wires (a classical channel), where only classical information (e.g. 0s and 1s) can be sent.

Before giving the definition of channel capacity, the preliminary notion of thenorm of complete boundedness, orcb-norm of a channel needs to be discussed. When considering the capacity of a channel${\displaystyle \mathrm{\Phi }}$, we need to compare it with an "ideal channel"${\displaystyle \mathrm{\Lambda }}$ . For instance, when the input and output algebras are identical, we can choose${\displaystyle \mathrm{\Lambda }}$ to be the identity map. Such a comparison requires ametric between channels.Since a channel can be viewed as a linear operator, it is tempting to use the naturaloperator norm. In other words, the closeness of${\displaystyle \mathrm{\Phi }}$ to the ideal channel${\displaystyle \mathrm{\Lambda }}$ can be defined by

${\displaystyle \Vert \mathrm{\Phi }-\mathrm{\Lambda }\Vert =sup\{\Vert (\mathrm{\Phi }-\mathrm{\Lambda })(A)\Vert |\Vert A\Vert \le 1\}.}$

However, the operator norm may increase when we tensor${\displaystyle \mathrm{\Phi }}$ with the identity map on some ancilla.

To make the operator norm even a more undesirable candidate, the quantity

${\displaystyle \Vert \mathrm{\Phi }\otimes I_n\Vert }$

may increase without bound as${\displaystyle n\to \mathrm{\infty }.}$ The solution is to introduce, for any linear map${\displaystyle \mathrm{\Phi }}$ between C*-algebras, the cb-norm

${\displaystyle \Vert \mathrm{\Phi }{\Vert }_{cb}=\underset{n}{sup}\Vert \mathrm{\Phi }\otimes I_n\Vert .}$

The mathematical model of a channel used here is same as theclassical one.

Let${\displaystyle \mathrm{\Psi }:{\mathcal{B}}_1\to {\mathcal{A}}_1}$ be a channel in the Heisenberg picture and${\displaystyle {\mathrm{\Psi }}_{id}:{\mathcal{B}}_2\to {\mathcal{A}}_2}$ be a chosen ideal channel. To make the comparison possible, one needs to encode and decode Φ via appropriate devices, i.e. we consider the composition

${\displaystyle \hat{\mathrm{\Psi }}=D\circ \mathrm{\Phi }\circ E:{\mathcal{B}}_2\to {\mathcal{A}}_2}$

whereE is an encoder andD is a decoder. In this context,E andD are unital CP maps with appropriate domains. The quantity of interest is thebest case scenario:

${\displaystyle \mathrm{\Delta }(\hat{\mathrm{\Psi }},{\mathrm{\Psi }}_{id})=\underset{E,D}{inf}\Vert \hat{\mathrm{\Psi }}-{\mathrm{\Psi }}_{id}{\Vert }_{cb}}$

with the infimum being taken over all possible encoders and decoders.

To transmit words of lengthn, the ideal channel is to be appliedn times, so we consider the tensor power

${\displaystyle {\mathrm{\Psi }}_{id}^{\otimes n}={\mathrm{\Psi }}_{id}\otimes \cdots \otimes {\mathrm{\Psi }}_{id}.}$

The${\displaystyle \otimes }$ operation describesn inputs undergoing the operation${\displaystyle {\mathrm{\Psi }}_{id}}$ independently and is the quantum mechanical counterpart ofconcatenation. Similarly,m invocations of the channel corresponds to${\displaystyle {\hat{\mathrm{\Psi }}}^{\otimes m}}$.

The quantity

${\displaystyle \mathrm{\Delta }({\hat{\mathrm{\Psi }}}^{\otimes m},{\mathrm{\Psi }}_{id}^{\otimes n})}$

is therefore a measure of the ability of the channel to transmit words of lengthn faithfully by being invokedm times.

This leads to the following definition:

A non-negative real numberr is anachievable rate of${\displaystyle \mathrm{\Psi }}$ with respect to${\displaystyle {\mathrm{\Psi }}_{id}}$ if

For all sequences${\displaystyle \{n_{\alpha }\},\{m_{\alpha }\}\subset \mathbb{N}}$ where${\displaystyle m_{\alpha }\to \mathrm{\infty }}$ and, we have

${\displaystyle \underset{\alpha }{lim}\mathrm{\Delta }({\hat{\mathrm{\Psi }}}^{\otimes m_{\alpha }},{\mathrm{\Psi }}_{id}^{\otimes n_{\alpha }})=0.}$

A sequence${\displaystyle \{n_{\alpha }\}}$ can be viewed as representing a message consisting of possibly infinite number of words. The limit supremum condition in the definition says that, in the limit, faithful transmission can be achieved by invoking the channel no more thanr times the length of a word. One can also say thatr is the number of letters per invocation of the channel that can be sent without error.

Thechannel capacity of${\displaystyle \mathrm{\Psi }}$ with respect to${\displaystyle {\mathrm{\Psi }}_{id}}$, denoted by${\displaystyle C(\mathrm{\Psi },{\mathrm{\Psi }}_{id})}$ is the supremum of all achievable rates.

From the definition, it is vacuously true that 0 is an achievable rate for any channel.

As stated before, for a system with observable algebra${\displaystyle \mathcal{B}}$, the ideal channel${\displaystyle {\mathrm{\Psi }}_{id}}$ is by definition the identity map${\displaystyle I_{\mathcal{B}}}$. Thus for a purelyn dimensional quantum system, the ideal channel is the identity map on the space ofn × n matrices${\displaystyle {\mathbb{C}}^{n\times n}}$. As a slight abuse of notation, this ideal quantum channel will be also denoted by${\displaystyle {\mathbb{C}}^{n\times n}}$. Similarly, a classical system with output algebra${\displaystyle {\mathbb{C}}^m}$ will have an ideal channel denoted by the same symbol. We can now state some fundamental channel capacities.

The channel capacity of the classical ideal channel${\displaystyle {\mathbb{C}}^m}$ with respect to a quantum ideal channel${\displaystyle {\mathbb{C}}^{n\times n}}$ is

${\displaystyle C({\mathbb{C}}^m,{\mathbb{C}}^{n\times n})=0.}$

This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel.

Moreover, the following equalities hold:

${\displaystyle C({\mathbb{C}}^m,{\mathbb{C}}^n)=C({\mathbb{C}}^{m\times m},{\mathbb{C}}^{n\times n})=C({\mathbb{C}}^{m\times m},{\mathbb{C}}^n)=\frac{\log n}{\log m}.}$

The above says, for instance, an ideal quantum channel is no more efficient at transmitting classical information than an ideal classical channel. Whenn =m, the best one can achieve isone bit per qubit.

It is relevant to note here that both of the above bounds on capacities can be broken, with the aid ofentanglement. Theentanglement-assisted teleportation scheme allows one to transmit quantum information using a classical channel.Superdense coding achieves two bits per qubit. These results indicate the significant role played by entanglement in quantum communication.

Using the same notation as the previous subsection, theclassical capacity of a channel Ψ is

${\displaystyle C(\mathrm{\Psi },{\mathbb{C}}^2),}$

that is, it is the capacity of Ψ with respect to the ideal channel on the classical one-bit system${\displaystyle {\mathbb{C}}^2}$.

Similarly thequantum capacity of Ψ is

${\displaystyle C(\mathrm{\Psi },{\mathbb{C}}^{2\times 2}),}$

where the reference system is now the one qubit system${\displaystyle {\mathbb{C}}^{2\times 2}}$.

Another measure of how well a quantum channel preserves information is calledchannel fidelity, and it arises fromfidelity of quantum states. Given two pure states${\displaystyle |\psi ⟩}$ and${\displaystyle |\varphi ⟩}$, their fidelity is the probability that one of them passes a test designed to identify the other:$${\displaystyle F(|\psi ⟩,|\varphi ⟩)=|⟨\psi |\varphi ⟩{|}^2.}$$ This can be generalized to the case where the two states being compared are given by density matrices:^[16][17]$${\displaystyle F(\rho ,\sigma )={\left(\mathrm{t}\mathrm{r}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2.}$$

The channel fidelity for a given channel is found by sending one half of a maximally entangled pair of systems through that channel, and calculating the fidelity between the resulting state and the original input.^[18]

A bistochastic quantum channel is a quantum channel${\displaystyle \mathrm{\Phi }(\rho )}$ that isunital,^[19] i.e.${\displaystyle \mathrm{\Phi }(I)=I}$. These channels include unitary evolutions, convex combinations of unitaries, and (in dimensions larger than 2) other possibilities as well.^[20]

• No-communication theorem
• Amplitude damping channel
• Quantum depolarizing channel

• Bengtsson, Ingemar;Życzkowski, Karol (2017).Geometry of Quantum States: An Introduction to Quantum Entanglement (2nd ed.). Cambridge University Press.ISBN 978-1-107-02625-4.
• Holevo, Alexander S. (2001).Statistical Structure of Quantum Theory.Lecture Notes in Physics. Monographs. Springer.ISBN 3-540-42082-7.
• Keyl, M.;Werner, R. F. (2002). "How to Correct Small Quantum Errors". In Buchleitner, Andreas; Hornberger, Klaus (eds.).Coherent Evolution in Noisy Environments. Lecture Notes in Physics. Vol. 611. Springer. pp. 263–286.arXiv:quant-ph/0206086.doi:10.1007/3-540-45855-7_7.ISBN 978-3-540-44354-4.
• Wilde, Mark M. (2017).Quantum Information Theory (2nd ed.). Cambridge University Press.arXiv:1106.1445.doi:10.1017/9781316809976.ISBN 978-1-316-80997-6.

• Quantum information theory

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