Quantum channel

Short description: Foundational object in quantum communication theory

In quantum information theory, a quantum channel is a communication channel that can transmit quantum information, as well as classical information. An example of quantum information is the general dynamics of a qubit. An example of classical information is a text document transmitted over the Internet.

Terminologically, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced 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.)

Memoryless quantum channel

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

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

Schrödinger picture

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

Let $H_A$ and $H_B$ be the state spaces (finite-dimensional Hilbert spaces) of the sending and receiving ends, respectively, of a channel. $L(H_A)$ will denote the family of operators on $H_A.$ In the Schrödinger picture, a purely quantum channel is a map $\mathrm{\Phi }$ between density matrices acting on $H_A$ and $H_B$ with the following properties:

1. As required by postulates of quantum mechanics, $\mathrm{\Phi }$ needs to be linear.
2. Since density matrices are positive, $\mathrm{\Phi }$ must preserve the cone of positive elements. In other words, $\mathrm{\Phi }$ is a positive map.
3. If an ancilla of arbitrary finite dimension n is coupled to the system, then the induced map $I_n\otimes \mathrm{\Phi },$ where I_n is the identity map on the ancilla, must also be positive. Therefore, it is required that $I_n\otimes \mathrm{\Phi }$ is positive for all n. Such maps are called completely positive.
4. Density matrices are specified to have trace 1, so $\mathrm{\Phi }$ has to preserve the trace.

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

Heisenberg picture

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

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

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

While $\mathrm{\Phi }$ takes states on A to those on B, ${\mathrm{\Phi }}^{*}$ maps observables on system B to observables on A. 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 $\mathrm{\Phi }$ is assumed to be trace preserving, ${\mathrm{\Phi }}^{*}$ is unital, that is,${\mathrm{\Phi }}^{*}(I)=I$. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.

Classical information

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:

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

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

$\mathrm{\Psi }:{ℬ}\to {𝒜}.$

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 $C(X)$ on some set $X$. We assume $X$ is finite so $C(X)$ can be identified with the n-dimensional Euclidean space ${\mathbb{ℝ}}^n$ with entry-wise multiplication.

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

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

Notice $L(H_B)\otimes C(X)$ is still a C*-algebra. An element $a$ of a C*-algebra ${𝒜}$ is called positive if $a=x^{*}x$ for some $x$. Positivity of a map is defined accordingly. This characterization is not universally accepted; the quantum 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 a Frobenius algebra or Frobenius category.

Examples

Time evolution

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

$\rho \to U\rho U^{*},$

where $U=e^{-iHt/\hslash }$ and H is the Hamiltonian and t is the time. This gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is

$A\to U^{*}AU.$

Restriction

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

$\rho \in H_A\otimes H_B,$

the reduced state of ρ on system A, ρ^A, is obtained by taking the partial trace of ρ with respect to the B system:

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

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

$A\to A\otimes I_B,$

where A is an observable of system A.

Observable

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

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

to the quantum mechanical one

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

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

The corresponding Schrödinger map ${\mathrm{\Psi }}^{*}$ takes density matrices to classical states:

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

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

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

Instrument

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 a quantum instrument. Let $\{F_1,\dots ,F_n\}$ be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map $\mathrm{\Phi }$ with pure quantum input $\rho \in L(H)$ and with output space $C(X)\otimes L(H)$:

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

Let

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

The dual map in the Heisenberg picture is

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

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

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

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

describes the overall state change.

Measure-and-prepare channel

Suppose two parties A and B wish to communicate in the following manner: A performs the measurement of an observable and communicates the measurement outcome to B 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 $\mathrm{\Phi }$₁ simply consists of A making a measurement, i.e. it is the observable map:

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

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

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

The total operation is the composition

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

Channels of this form are called measure-and-prepare or entanglement-breaking.

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

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

A measure-and-prepare channel can not be the identity map. This is precisely the statement of the no teleportation theorem, which says classical teleportation (not to be confused with entanglement-assisted teleportation) is impossible. In other words, a quantum state can not be measured reliably.

In the channel-state duality, a channel is measure-and-prepare if and only if the corresponding state is separable. 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.

Pure channel

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

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

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

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

where N ≤ nm. The matrices K_i are called Kraus operators of $\mathrm{\Psi }$ (after the German physicist Karl Kraus, who introduced them). $$F(\rho ,\sigma )={\left(\mathrm{t}\mathrm{r}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2.$$

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.

Bistochastic quantum channel

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

See also

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

References

• 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". Coherent Evolution in Noisy Environments. Lecture Notes in Physics. 611. Springer. pp. 263–286. 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. doi:10.1017/9781316809976. ISBN 978-1-316-80997-6.