[Paper Review] Reversibility of stochastic maps via quantum divergences

Fumio Hiai, Milán Mosonyi|arXiv (Cornell University)|Apr 11, 2016

Statistical Mechanics and Entropy62 references3 citations

TL;DR

This paper investigates the reversibility of quantum stochastic maps by analyzing their preservation of various quantum divergences, including f-divergences, sandwiched Rényi divergences, and α-z-Rényi relative entropies. It establishes that reversibility is equivalent to the preservation of these divergences under specific conditions, providing a unified framework for understanding quantum operation reversibility via information-theoretic measures.

ABSTRACT

We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum $f$-divergences, sandwiched R\'enyi divergences, and $\alpha$-$z$-R\'enyi relative entropies.

Motivation & Objective

• To understand the conditions under which quantum stochastic maps are reversible.
• To investigate the role of quantum divergences in characterizing reversibility of quantum operations.
• To unify different divergence measures—f-divergences, sandwiched Rényi divergences, and α-z-Rényi relative entropies—under a common reversibility framework.
• To establish necessary and sufficient conditions for reversibility based on divergence preservation.

Proposed method

• Analyzes the preservation of quantum f-divergences under quantum stochastic maps.
• Applies the concept of sufficiency in quantum statistics to characterize reversibility.
• Uses the monotonicity of quantum divergences under completely positive maps to derive reversibility conditions.
• Considers two versions of f-divergences and their behavior under quantum operations.
• Examines sandwiched Rényi divergences and α-z-Rényi relative entropies as specific instances of divergences in the framework.
• Derives equivalence between reversibility and the invariance of divergences under the action of the map.

Experimental results

Research questions

• RQ1 Under what conditions is a quantum stochastic map reversible if it preserves a given quantum divergence?
• RQ2 How do different quantum divergences—f-divergences, sandwiched Rényi divergences, and α-z-Rényi relative entropies—relate to the reversibility of quantum operations?
• RQ3 Is the preservation of a divergence under a map sufficient to guarantee reversibility of that map?
• RQ4 Can a unified characterization of reversibility be established across multiple divergence measures?
• RQ5 What is the role of the Petz recovery map in the context of divergence preservation and reversibility?

Key findings

• Reversibility of a quantum stochastic map is equivalent to the preservation of quantum f-divergences for all convex functions f.
• The preservation of sandwiched Rényi divergences for all orders α implies the reversibility of the underlying quantum operation.
• For α-z-Rényi relative entropies, reversibility holds if and only if the map preserves the divergence for a specific range of parameters.
• The Petz recovery map is identified as the unique recovery map that preserves these divergences, linking statistical sufficiency to reversibility.
• The results establish a general framework where divergence invariance serves as a necessary and sufficient condition for reversibility across multiple divergence families.