Terminologically, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators.
[1]) 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.
with the following properties:[2] The adjectives completely positive and trace preserving used to describe a map are sometimes abbreviated CPTP.
between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend
The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.
As stated in the introduction, the input and output of a channel can include classical information as well.
A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators: that is unital and completely positive (CP).
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
Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define
[4] The dual map in the Heisenberg picture is Consider a composite quantum system with state space
For a state the reduced state of ρ on system A, ρA, 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.
Furthermore, viewing states as normalized functionals, and invoking the Riesz representation theorem, we can put 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.
According to the message he receives, B prepares his (quantum) system in a specific state.
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.
The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver.
This measurement results in classical information which must be sent to the receiver to complete the teleportation.
Experimentally, a simple implementation of a quantum channel is fiber optic (or free-space for that matter) transmission of single photons.
Single photons can be transmitted up to 100 km in standard fiber optics before losses dominate.
To make the operator norm even a more undesirable candidate, the quantity may increase without bound as
The quantity of interest is the best case scenario: with the infimum being taken over all possible encoders and decoders.
To transmit words of length n, the ideal channel is to be applied n times, so we consider the tensor power The
The quantity is therefore a measure of the ability of the channel to transmit words of length n faithfully by being invoked m times.
can be viewed as representing a message consisting of possibly infinite number of words.
Thus for a purely n dimensional quantum system, the ideal channel is the identity map on the space of n × n matrices
As a slight abuse of notation, this ideal quantum channel will be also denoted by
is This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel.
The entanglement-assisted teleportation scheme allows one to transmit quantum information using a classical channel.