Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.23622 · Physical Review Letters

Optimality Condition for the Petz Map

Bikun Li, Zhaoyou Wang, Guo Zheng, Yat Wong, Liang Jiang

Published May 19, 2025• 7 citations
arXivHTMLPDF

Abstract

In quantum error correction, the Petz map serves as a perfect recovery map when the Knill-Laflamme conditions are satisfied. Notably, while perfect recovery is generally infeasible for most quantum channels of finite dimension, the Petz map remains a versatile tool with near-optimal performance in recovering quantum states. This work introduces and proves, for the first time, the necessary and sufficient conditions for the optimality of the Petz map in terms of entanglement fidelity. In some special cases, the violation of this condition can be easily characterized by a simple commutator that can be efficiently computed. We provide multiple examples that substantiate our new findings.

DOI
10.1103/physrevlett.134.200602

Related papers

General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels

Cédric Bény, Ognyan Oreshkov

We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.

arXiv:0907.5391•Physical Review Letters•Mar 23, 2010•126 citations

Quantum Algorithm for Petz Recovery Channels and Pretty Good Measurements

András Gilyén, Seth Lloyd, Iman Marvian +2 more

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This Letter sets out to rectify this lack: Using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.

arXiv:2006.16924•Physical Review Letters•Jun 1, 2022•66 citations

Petz map recovery for long-range entangled quantum many-body states

Yangrui Hu, Yijian Zou

Given a tripartite quantum state on $A,B,C$ and the erasure channel on $C$, the rotated Petz map is a recovery channel that acts on $B$ to recover the erased quantum information. The infidelity of the best recovery is upper bounded by the conditional mutual information (CMI). In this work, we study the infidelity of the rotated Petz map on several physically relevant long-range entangled quantum states. Specifically, we study three classes of quantum phases: (i) steady states of measurement-induced phase transitions, (ii) critical ground state under local measurements, and (iii) chiral states under local measurements. We find that the averaged infidelity of the Petz map recovery sharply distinguishes the three classes: (i) and (ii) are distinguished by the scaling of the infidelity with CMI and (iii) is characterized by an asymmetry of the fidelity with the rotation parameter. We also study Petz map recovery for topological order and find an operational interpretation of the topological entanglement entropy. Our result indicates that the recovery fidelity of the Petz map is a useful diagnostic of quantum phases of matter.

arXiv:2408.00857•Physical review. B./Physical review. B•Nov 4, 2024•6 citations

Universal Optimization and Tighter Fidelity Bounds for Approximate Quantum Error Correction

Jing Wu, Michele Grossi, Doga Kurkcuoglu +1 more

Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.

quant-pharXiv:2607.24968•Jul 29, 2026•0 citations

Experimental Tabletop Petz recovery of a photonic qubit

Hui Li, Jinyan Chen, Yue Pan +4 more

The quantum information lost in open evolutions cannot be fully recovered, but partial recovery is possible. The Petz recovery map guarantees almost optimal recovery, notably if the chosen reference state is close to the real one. This map has been widely used in theoretical studies, but has been the object of only a handful of experimental realisations, typically under a single fixed noise model. In this work, we describe and implement the Petz recovery map for a versatile class of qubit channels with tunable decoherence and dissipation. The setup we realize is also the first experimental example of ``tabletop reversibility'': for a good range of choices of the reference state, the Petz recovery map can be implemented with the same devices as the forward dissipative evolution, whose effect it is partially undoing. Our results demonstrate that the Petz recovery map can be resource-efficiently realized without requiring complex ancillary resources, providing a feasible pathway for mitigating information loss in quantum systems.

quant-pharXiv:2606.12020•Jun 11, 2026•0 citations

Optimal recovery for quantum error correction

Sun Woo P. Kim

The calculation of the error threshold of quantum error correcting codes typically proceeds as follows. First, syndromes are measured. Then, a decoder infers the error chain and the corresponding correction is applied. The threshold is then defined as the largest correctable error rate, with the maximum-likelihood decoder corresponding to the ``optimal'' threshold. However, a broader set of operations could be used to recover quantum information. The true optimal threshold should be optimised over all possible recovery schemes, which can be described by quantum channels. Here, we study such optimal recovery channels and their thresholds $p_\mathrm{th}^\mathrm{opt}$. We introduce an information-theoretic quantity, mutual trace distance, which provides a necessary and sufficient diagnostic for sharply determining $p_\mathrm{th}^\mathrm{opt}$ without explicit optimisation. In contrast, previous works give a lower bound on $p_\mathrm{th}^\mathrm{opt}$ by specifying particular recovery schemes, e.g. Schumacher-Westmoreland (SW) which provides coherent information as a diagnostic to lower bound $p^\mathrm{opt}_\mathrm{th}$. We prove that the Petz and SW recovery schemes are optimal, i.e. their threshold is $p_\mathrm{th}^\mathrm{opt}$. With their optimality established, we explore the structure of optimal and non-optimal recovery schemes and their phase diagrams.

quant-pharXiv:2603.06520•Mar 6, 2026•0 citations
In quantum error correction, the Petz map serves as a perfect recovery map when the Knill-Laflamme conditions are satisfied. Notably, while perfect recovery is generally infeasible for most quantum channels of finite dimension, the Petz map remains a versatile tool with near-optimal performance in recovering quantum states. This work introduces and proves, for the first time, the necessary and sufficient conditions for the optimality of the Petz map in terms of entanglement fidelity. In some special cases, the violation of this condition can be easily characterized by a simple commutator that can be efficiently computed. We provide multiple examples that substantiate our new findings.
We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.
The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This Letter sets out to rectify this lack: Using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.
Given a tripartite quantum state on A,B,CA,B,CA,B,C and the erasure channel on CCC, the rotated Petz map is a recovery channel that acts on BBB to recover the erased quantum information. The infidelity of the best recovery is upper bounded by the conditional mutual information (CMI). In this work, we study the infidelity of the rotated Petz map on several physically relevant long-range entangled quantum states. Specifically, we study three classes of quantum phases: (i) steady states of measurement-induced phase transitions, (ii) critical ground state under local measurements, and (iii) chiral states under local measurements. We find that the averaged infidelity of the Petz map recovery sharply distinguishes the three classes: (i) and (ii) are distinguished by the scaling of the infidelity with CMI and (iii) is characterized by an asymmetry of the fidelity with the rotation parameter. We also study Petz map recovery for topological order and find an operational interpretation of the topological entanglement entropy. Our result indicates that the recovery fidelity of the Petz map is a useful diagnostic of quantum phases of matter.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.
The quantum information lost in open evolutions cannot be fully recovered, but partial recovery is possible. The Petz recovery map guarantees almost optimal recovery, notably if the chosen reference state is close to the real one. This map has been widely used in theoretical studies, but has been the object of only a handful of experimental realisations, typically under a single fixed noise model. In this work, we describe and implement the Petz recovery map for a versatile class of qubit channels with tunable decoherence and dissipation. The setup we realize is also the first experimental example of ``tabletop reversibility'': for a good range of choices of the reference state, the Petz recovery map can be implemented with the same devices as the forward dissipative evolution, whose effect it is partially undoing. Our results demonstrate that the Petz recovery map can be resource-efficiently realized without requiring complex ancillary resources, providing a feasible pathway for mitigating information loss in quantum systems.
The calculation of the error threshold of quantum error correcting codes typically proceeds as follows. First, syndromes are measured. Then, a decoder infers the error chain and the corresponding correction is applied. The threshold is then defined as the largest correctable error rate, with the maximum-likelihood decoder corresponding to the ``optimal'' threshold. However, a broader set of operations could be used to recover quantum information. The true optimal threshold should be optimised over all possible recovery schemes, which can be described by quantum channels. Here, we study such optimal recovery channels and their thresholds pthoptp_\mathrm{th}^\mathrm{opt}pthopt​. We introduce an information-theoretic quantity, mutual trace distance, which provides a necessary and sufficient diagnostic for sharply determining pthoptp_\mathrm{th}^\mathrm{opt}pthopt​ without explicit optimisation. In contrast, previous works give a lower bound on pthoptp_\mathrm{th}^\mathrm{opt}pthopt​ by specifying particular recovery schemes, e.g. Schumacher-Westmoreland (SW) which provides coherent information as a diagnostic to lower bound pthoptp^\mathrm{opt}_\mathrm{th}pthopt​. We prove that the Petz and SW recovery schemes are optimal, i.e. their threshold is pthoptp_\mathrm{th}^\mathrm{opt}pthopt​. With their optimality established, we explore the structure of optimal and non-optimal recovery schemes and their phase diagrams.