View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762901.png" />, quantum information is represented by a density operator (state) on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762902.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762903.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762904.png" /> of probability distributions on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762905.png" /> of input signals into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762906.png" /> of all density operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762907.png" />. Properly speaking, a quantum communication channel is an affine mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762908.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q0762909.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629012.png" /> are the Hilbert spaces describing, respectively, the input and output of the channel. A quantum decoding is an affine mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629014.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629016.png" /> is the space of output signals. Transmission of information is described, as in classical information theory, by the scheme

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

An important problem is the determination of an optimal method for transmitting information over a given quantum channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629018.png" />. For a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629019.png" />, the conditional distribution of the output signal with respect to the input signal is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629020.png" /> of the encoding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629021.png" /> and the decoding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629022.png" />. Given some functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629023.png" />, it is required to find its extremum with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629025.png" />. The case most often studied is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629026.png" /> is fixed as well, and it is required to determine the optimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629027.png" />. The scheme (1) then reduces to the simpler version:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

To determine the encoding it suffices to give the images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629029.png" /> of the distributions concentrated at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629030.png" />. The decoding is conveniently described by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629032.png" />-measurement, defined as a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629034.png" /> with values in the set of non-negative Hermitian operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629035.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629036.png" /> is the identity operator. A one-to-one correspondence between the decoding and the measurements is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629037.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629038.png" /></td> </tr></table>

In the case of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629040.png" />, in order that the measurement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629041.png" /> be optimal it is necessary for the operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629042.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629043.png" /></td> </tr></table>

to be Hermitian and to satisfy the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629045.png" /> is an affine function (as in the case of Bayesian risk), then for optimality (in the sense of the minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629046.png" />) it is necessary and sufficient that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629047.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629049.png" />, as well as (3). Similar conditions hold for fairly arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629051.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076290/q07629052.png" /> 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|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|Gaussian channel]]), for which an explicit description of optimal measurements was obtained in a number of cases.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.W. Helstrom,   "Quantum detection and estimation theory" , New York  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.S. Kholevo,   "Problems in the mathematical theory of quantum communication channels"  ''Rep. Math. Phys.'' , '''12'''  (1977)  pp. 273–278</TD></TR></table>