Bounding generalized relative entropies: Nonasymptotic quantum speed limits

TL;DR

This paper establishes bounds on generalized relative entropies for quantum channels and introduces a family of quantum speed limits based on these entropies, linking information theory with quantum dynamics.

Contribution

It provides the first bounds on Rènyi and Tsallis relative entropies under quantum channels and derives new quantum speed limits from these bounds.

Findings

Bound on generalized relative entropies for unitary channels

Family of quantum speed limits based on relative entropies

Potential links to thermodynamics and quantum coherence

Abstract

Information theory has become an increasingly important research field to better understand quantum mechanics. Noteworthy, it covers both foundational and applied perspectives, also offering a common technical language to study a variety of research areas. Remarkably, one of the key information-theoretic quantities is given by the relative entropy, which quantifies how difficult is to tell apart two probability distributions, or even two quantum states. Such a quantity rests at the core of fields like metrology, quantum thermodynamics, quantum communication and quantum information. Given this broadness of applications, it is desirable to understand how this quantity changes under a quantum process. By considering a general unitary channel, we establish a bound on the generalized relative entropies (R\'{e}nyi and Tsallis) between the output and the input of the channel. As an application of our bounds, we derive a family of quantum speed limits based on relative entropies. Possible connections between this family with thermodynamics, quantum coherence, asymmetry and single-shot information theory are briefly discussed.