Quantum channel

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 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.

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 and be the state spaces of the sending and receiving ends, respectively, of a channel. will denote the family of operators on In the Schrödinger picture, a purely quantum channel is a map between density matrices acting on and with the following properties:

As required by postulates of quantum mechanics, needs to be linear.
Since density matrices are positive, must preserve the cone of positive elements. In other words, is a positive map.
If an ancilla of arbitrary finite dimension n is coupled to the system, then the induced map where I_n is the identity map on the ancilla, must also be positive. Therefore, it is required that is positive for all n. Such maps are called completely positive.
Density matrices are specified to have trace 1, so 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 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 between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend uniquely to the full space of operators. This leads to the adjoint map, which describes the action of in the Heisenberg picture:
The spaces of operators L and L are Hilbert spaces with the Hilbert–Schmidt inner product. Therefore, viewing as a map between Hilbert spaces, we obtain its adjoint ^* given by
While takes states on A to those on B, 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 is assumed to be trace preserving, is unital, that is,. 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:
that is unital and completely positive. 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:
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 on some set. We assume is finite so can be identified with the n-dimensional Euclidean space with entry-wise multiplication.
Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define to include the relevant classical observables. An example of this would be a channel
Notice is still a C*-algebra. An element of a C*-algebra is called positive if for some. 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
where 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

Restriction

Consider a composite quantum system with state space For a state
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:
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
where A is an observable of system A.

Observable

An observable associates a numerical value to a quantum mechanical effect. 's are assumed to be positive operators acting on appropriate state space and. In the Heisenberg picture, the corresponding observable map maps a classical observable
to the quantum mechanical one
In other words, one integrates f against the POVM to obtain the quantum mechanical observable. It can be easily checked that is CP and unital.
The corresponding Schrödinger map takes density matrices to classical states:
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

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 be the effects associated to an observable. In the Schrödinger picture, an instrument is a map with pure quantum input and with output space :
Let
The dual map in the Heisenberg picture is
where is defined in the following way: Factor then.
We see that is CP and unital.
Notice that gives precisely the observable map. The map
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 system in a specific state. In the Schrödinger picture, the first part of the channel ₁ simply consists of A making a measurement, i.e. it is the observable map:
If, in the event of the i-th measurement outcome, B prepares his system in state R_i, the second part of the channel ₂ takes the above classical state to the density matrix
The total operation is the composition
Channels of this form are called measure-and-prepare or entanglement-breaking.
In the Heisenberg picture, the dual map is defined by
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 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 in the Heisenberg picture. With the assumption that everything is finite-dimensional, is a unital CP map between spaces of matrices
By Choi's theorem on completely positive maps, must take the form
where N ≤ nm. The matrices K_i are called Kraus operators of . The minimum number of Kraus operators is called the Kraus rank of. A channel with Kraus rank 1 is called pure. 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.

Teleportation

In quantum 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.