Entropic trade-off relations for quantum operations

TL;DR

This paper establishes fundamental entropic trade-off relations for quantum operations, linking the decoherence and informational properties of quantum channels through spectral entropy bounds.

Contribution

It introduces new entropic bounds for quantum maps, connecting the spectral properties of dynamical matrices and superoperators, and explores their implications for quantum information theory.

Findings

Sum of entropies ≥ ln N for any quantum map

Sum of entropies ≥ 2 ln N for bistochastic maps

Provides bounds for Rényi entropies and analyzes associated entanglement

Abstract

Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Phi. We prove that for any map acting on a N--dimensional quantum system the sum of both entropies is not smaller than ln N. For any bistochastic map this lower bound reads 2 ln N. We investigate also the corresponding R\'enyi entropies, providing an upper bound for their sum and analyze entanglement of the bi-partite quantum state associated with the channel.