A Holevo-Type Bound for a Hilbert Schmidt Distance Measure

Boaz Tamir,Eliahu Cohen

doi:10.4236/jqis.2015.54015

Abstract

We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob using a quantum channel, while Bob is performing some projective measurement. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.

Highlights

Holevo’s theorem [1] is one of the pillars of quantum information theory
We prove a Holevo-type upper bound on the mutual information of X and Y, where the mutual information is written this time in terms of the HS norm instead of the Kullback-Leibler divergence
The very fact that it is smaller than the Tsallis information measure of X means that the quantum channel restricts the rate of classical information transfer, where the mutual information is measured by the HS norm and the source of information X is measured by Tsallis entropy. This is analogous to Holevo’s upper bound in the framework of Tsalis/linear entropy. We find this result similar in spirit to the well-known limitation on the rate of classical information transmission via a quantum channel: one cannot send more than one bit for each use of the channel using a one qubit channel

Summary

Introduction

Holevo’s theorem [1] is one of the pillars of quantum information theory. It can be informally summarized as follows: “It is not possible to communicate more than n classical bits of information by the transmission of n qubits alone”. We prove a Holevo-type upper bound on the mutual information of X and Y, where the mutual information is written this time in terms of the HS norm instead of the Kullback-Leibler divergence. It is recently suggested by Ellerman [8] that employing the HS norm in the formulation of classical mutual information is natural. Note that employing the Kullback-Leibler divergence in the standard form of the Holevo bound gives an expression which can be identified with quantum mutual information, the “coherent information” is considered as a more appropriate expression (see [2] chapter 12). We review some basic properties of quantum logical divergence and use these properties to demonstrate the new Holevo-type bound

The HS Norm and the Holevo-Type Bound

Discussion