Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 3 / d

The Holevo-Schumacher-Westmoreland theorem gives the classical coding formulaIts ensemble-coding statement makes every rate below a one-use output Holevo quantity achievable with product input encodings and collective output decoding. Under this product-input restriction, the capacity isFor any such input ensemble, rates below its output Holevo quantity are achievable with asymptotically vanishing decoding error. Maximization gives the product-state classical capacity. The converse follows from the Holevo bound, entropy bounds on product outputs, and Fano's inequality.

Define the binary entropy $h_2(x)=-x{\log }_{2}x-(1-x){\log }_2(1-x)$. For an input Bloch radius $r\le 1$, the output eigenvalues are $(1\pm (1-q)r)/2$. Its entropy isOn $[1/2,1]$, $h_2$ is decreasing, so this is minimized by pure inputs $r=1$. Throughout the completely positive interval $0\le q\le 4/3$, the minimum equals $h_2(q/2)$, using symmetry $h_2(x)=h_2(1-x)$. The average output entropy is at most $1$, because it is a qubit. Hence every ensemble satisfiesTwo equally likely orthogonal pure inputs attain this bound: their average output is $I/2$, and each individual output has entropy $h_2(q/2)$. ThereforeThis gives the Holevo capacity of a qubit depolarizing channel in the noise-weight convention used here.

For completeness, correlated classical code labels do not evade this upper bound. Each product input to $n$ uses has product output entropy at least $n{h}_2(q/2)$, whereas the average output entropy is at most $n$. Its Holevo quantity is consequently at most $n[1-h_2(q/2)]$. The Holevo bound and Fano's inequality rule out a reliable rate larger than this value. Combined with HSW achievability, this proves the capacity under the stated product-input restriction. It also remains valid in the extended physical parameter range arising from the original $p$ assumption.