Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum R\'enyi entropy

TL;DR

This paper provides an operational interpretation of quantum Rnyi entropies through a generalized quantum data compression scheme that incorporates exponential penalization, extending the classical understanding of von Neumann entropy.

Contribution

It introduces a quantum encoding framework satisfying a quantum Kraft-McMillan inequality that links quantum Rnyi entropies to optimal lossless compression with exponential penalization.

Findings

Quantum Rnyi entropies characterize optimal codes with exponential length penalties.

Standard quantum data compression results are recovered when using average code length.

The framework generalizes von Neumann entropy as a measure for quantum data compression.

Abstract

Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum R\'enyi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum R\'enyi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.