Exact and approximate conditions of tabletop reversibility: when is Petz recovery cost-free?

TL;DR

This paper investigates conditions under which quantum channels can be reversed with the Petz recovery map, focusing on exact and approximate scenarios, and explores the resource requirements for implementing such reversibility.

Contribution

It provides the first comprehensive analysis of exact and approximate tabletop reversibility conditions for Petz recovery in quantum channels.

Findings

Exact TTR requires time-sensitive ancilla control.

Approximate TTR does not need time-sensitive control.

Lindbladian TTR conditions derived for a collision model.

Abstract

Channels $\mathcal{N}$ that describe open quantum dynamics are inherently irreversible: it is impossible to undo their effect completely, but one can study partial recovery of the information. The Petz recovery map $\hat{\mathcal{N}}_{\gamma}^{(\texttt{P})}$ is a systematic construction that depends only on $\mathcal{N}$ and on a reference state $\gamma$, which will be recovered exactly. If the real input state was different from $\gamma$, the recovery is partial, with a guarantee of near-optimality. Generically, an implementation of the Petz recovery map would look very different from the implementation of the channel. It is natural to study under which conditions the two maps require similar or even identical resources. The noisy forward channel $\mathcal{N}$ is called ``tabletop time-reversible'' for a given $\gamma$ when the corresponding Petz recovery map is realizable in such a way. First, we study the exact tabletop reversibility (TTR) conditions. We show in particular that a time-sensitive control of an ancilla system is needed. Second, we present the approximate TTR conditions, which do not require such a time-sensitive control. Third, we derive Lindbladian TTR conditions under a random-time collision model.