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

TL;DR

This work formalizes tabletop time-reversibility (TTR) for quantum channels as the possibility of implementing the Petz recovery map using the same forward hardware. It provides an exact criterion for TTR, showing that exact reversal requires a Kraus-operator-level equivalence between Petz and tabletop reverse maps, and illustrates this with concrete channel examples. It then develops two practical, approximate TTR frameworks via first- and second-order expansions in the interaction time, plus a Lindbladian collision-model approach that supports sequential reversal with controllable error. The results highlight when cost-free, near-optimal reversal is feasible (e.g., under steady states, certain symmetries, or thermal operations) and point to open questions about the necessity of product preservation and platform-specific resource accounting. Collectively, the paper advances both theoretical and operational understanding of reversing open quantum dynamics with minimal extra resources.

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 $\hat{\mathcal{N}}_γ^{(\texttt{P})}$ is a systematic construction that depends only on $\mathcal{N}$ and on a reference state $γ$, which will be recovered exactly. If the real input state was different from $γ$, 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 $γ$ 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.

Figures (3)

• Figure 1: Tabletop Time-reversibility. The shaded in orange is a system of interest, and the shaded in blue is an environmental system which is initially uncorrelated with the system. The shaded in purple after an interaction depicts potential correlations between the system and the environment.
• Figure 2: Approximate Tabletop Time-reversibility.
• Figure 3: Random-time Collision Model.

Theorems & Definitions (16)

• Definition 1: Petz recovery map
• Definition 2: Tabletop reverse map
• Theorem 1
• proof
• Example 1: Factorizable channels
• Example 2: Controlled-$X$ gates
• Example 3: $XX$ Hamiltonian
• Definition 3
• Theorem 2: First-order approximation
• proof