Quantum communication channel

A system of information transmission (transformation) using a quantum-mechanical object as the carrier of the information.

In contrast to classical information, described by means of a probability distribution on the space of signals $ X $, quantum information is represented by a density operator (state) on a Hilbert space $ H $ corresponding to a given quantum-mechanical object. Each communication channel can be regarded as an affine mapping (that is, a mapping preserving convex combinations) from the (convex) set of input signals to the set of output signals. In particular, a quantum encoding is an affine mapping $ C $ from the set $ S ( X) $ of probability distributions on the space $ X $ of input signals into the set $ \Sigma ( H) $ of all density operators on $ H $. Properly speaking, a quantum communication channel is an affine mapping $ L $ from $ \Sigma ( H) $ into $ \Sigma ( H ^ \prime ) $, where $ H $ and $ H ^ \prime $ are the Hilbert spaces describing, respectively, the input and output of the channel. A quantum decoding is an affine mapping $ D $ from $ \Sigma ( H ^ \prime ) $ into $ S ( Y) $, where $ Y $ is the space of output signals. Transmission of information is described, as in classical information theory, by the scheme

$$ \tag{1 } S ( X) \rightarrow ^ { C } \Sigma ( H) \rightarrow ^ { L } \Sigma ( H ^ \prime ) \rightarrow ^ { D } S ( Y) . $$

An important problem is the determination of an optimal method for transmitting information over a given quantum channel $ L $. For a fixed $ L $, the conditional distribution of the output signal with respect to the input signal is a function $ P _ {C,D} ( d y , x ) $ of the encoding $ C $ and the decoding $ D $. Given some functional $ Q \{ P _ {C,D} ( d y \mid x ) \} $, it is required to find its extremum with respect to $ C $ and $ D $. The case most often studied is when $ C $ is fixed as well, and it is required to determine the optimal $ D $. The scheme (1) then reduces to the simpler version:

$$ \tag{2 } S ( X) \rightarrow ^ { C } \Sigma ( H) \rightarrow ^ { D } S ( Y) . $$

To determine the encoding it suffices to give the images $ \rho _ {x} $ of the distributions concentrated at the points $ x \in X $. The decoding is conveniently described by the $ Y $- measurement, defined as a measure $ M ( d y ) $ on $ Y $ with values in the set of non-negative Hermitian operators on $ H $; here $ M ( Y) $ is the identity operator. A one-to-one correspondence between the decoding and the measurements is given by the formula

$$ D _ \rho ( d y ) = \mathop{\rm Tr} \rho M ( d y ) , $$

such that the conditional distribution of the output signal of the scheme (2) with respect to the input signal is

$$ P ( d y \mid x ) = \mathop{\rm Tr} \rho _ {x} M ( d y ) . $$

In the case of finite $ X $ and $ Y $, in order that the measurement $ \{ M ( y) \} $ be optimal it is necessary for the operator

$$ \Lambda = \sum _ { y } F ( y) M ( y) , $$

where

$$ F ( y) = \sum _ { x } \rho _ {x} \frac{\partial Q }{\partial P ( y \mid x ) } , $$

to be Hermitian and to satisfy the condition

$$ \tag{3 } ( F ( y) - \Lambda ) M ( y) = 0 ,\ \ y \in Y . $$

If $ Q $ is an affine function (as in the case of Bayesian risk), then for optimality (in the sense of the minimum of $ Q $) it is necessary and sufficient that the operator $ \Lambda $ satisfies the condition $ \Lambda \leq F ( y) $, $ y \in Y $, as well as (3). Similar conditions hold for fairly arbitrary $ X $ and $ Y $.

There is a parallel between quantum measurements and the decision procedures of classical statistical decision theory, where the simple measurements defined by the projection-valued measures $ M ( d y ) $ correspond to deterministic procedures. However, in contrast to classical statistics, where the optimal procedure reduces, as a rule, to a deterministic one, in the quantum case even for the Bayesian problem with a finite number of decisions the optimal measurements cannot, in general, be chosen to be simple. This is explained geometrically by the fact that the optimum is attained at extreme points of the convex set of all measurements, and in the quantum case the class of simple measurements is properly contained in the set of extreme points.

As in classical statistical decision theory, it is possible to restrict the class of measurements by requiring invariance or unbiasedness. Quantum analogues of the Rao–Cramér inequality are known, giving a lower bound for the mean square error of the measurement. In applications of the theory, much attention has been given to boson Gaussian communication channels (cf. Gaussian channel), for which an explicit description of optimal measurements was obtained in a number of cases.

References

[1]	C.W. Helstrom, "Quantum detection and estimation theory" , New York (1976)
[2]	A.S. [A.S. Kholevo] Holevo, "Investigations in the general theory of statistical decisions" Proc. Steklov Inst. Math. , 3 (1978) Tr. Mat. Inst. Steklov. , 3 (1978)
[3]	A.S. Kholevo, "Problems in the mathematical theory of quantum communication channels" Rep. Math. Phys. , 12 (1977) pp. 273–278