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arXiv:2111.02975 · Physical Review A

Approximating invertible maps by recovery channels: Optimality and an application to non-Markovian dynamics

Lea Lautenbacher, Fernando de Melo, Nadja K. Bernardes

Published Apr 13, 2022• 13 citations
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Abstract

We investigate the problem of reversing quantum dynamics, specifically via optimal Petz recovery maps. We focus on typical decoherence channels, such as dephasing, depolarizing, and amplitude damping. We illustrate how well a physically implementable recovery map simulates an inverse evolution. We extend this idea to explore the use of recovery maps as an approximation of inverse maps, and apply it in the context of non-Markovian dynamics. We show how this strategy attenuates non-Markovian effects, such as the backflow of information.

DOI
10.1103/physreva.105.042421

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We investigate the problem of reversing quantum dynamics, specifically via optimal Petz recovery maps. We focus on typical decoherence channels, such as dephasing, depolarizing, and amplitude damping. We illustrate how well a physically implementable recovery map simulates an inverse evolution. We extend this idea to explore the use of recovery maps as an approximation of inverse maps, and apply it in the context of non-Markovian dynamics. We show how this strategy attenuates non-Markovian effects, such as the backflow of information.
In this dissertation we are interested in the dynamics of open quantum systems in the context of quantum information. First, we investigate the dynamics of one qubit at the regime where an inverse evolution is not well defined. This limit is widely used and explored in this work, called as the limit of non-bijectivity. We demonstrate how to compute a completely positive inverse evolution, based on the theory of recovery maps, here to be the Petz recovery map. The analyzed evolutions are the typical decoherence processes: dephasing, depolarizing and amplitude damping channels in the regime of non-bijectivity. To measure how efficient the Petz can be recovering a random quantum state, we use the fidelity function as the figure of merit. Also, we quantify how non-invertible a dynamics is compared to another. As an application for this formalism, in the second part of this work we show how we can explore recovery maps in the context of non-Markovian evolutions, in order to minimize memory effects. We demonstrate that recovering maps can be useful under certain constraints to simulate a non-Markovian evolution by an “almost” Markovian one. We demonstrate that there exists a strong dependence between the success of this strategy and the initial correlations between system and environment.
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.
This study delves into the efficacy of the Petz recovery map within the context of two paradigmatic quantum channels: dephasing and amplitude-damping. While prior investigations have predominantly focused on qubits, our research extends this inquiry to higher-dimensional systems. We introduce a novel, state-independent framework based on the Choi-Jamiołkowski isomorphism to evaluate the performance of the Petz map. By analyzing different channels and the (non-)unital nature of these processes, we emphasize the pivotal role of the reference state selection in determining the map's effectiveness. Furthermore, our analysis underscores the considerable impact of suboptimal choices on performance, prompting a broader consideration of factors such as system dimensionality.
We consider the problem of quantum error correction (QEC) for non-Markovian noise. Using the well-known Petz recovery map, we first show that conditions for approximate QEC can be easily generalized for the case of non-Markovian noise in the strong-coupling regime where the noise map becomes non-completely-positive at intermediate times. While certain approximate QEC schemes are ineffective against quantum non-Markovian noise, in the sense that the fidelity vanishes in finite time, the Petz map adapted to non-Markovian noise uniquely safeguards the code space even at the maximum noise limit. Focusing on the case of non-Markovian amplitude damping noise, we further show that the non-Markovian Petz map also outperforms the standard, stabilizer-based QEC code. Since implementing such a non-Markovian map poses practical challenges, we also construct a Markovian Petz map that achieves a similar performance, with only a slight compromise on the fidelity. Finally, we also study an example of non-Markovian Pauli noise, namely the non-Markovian dephasing channel. In this case, we show that the non-Markovian Petz recovery preserves the code space even at the high-noise limit for certain noise-adapted codes, whereas the Markovian Petz and stabilizer recoveries are ineffective.
The Petz recovery map provides a near-optimal reversal of quantum noise, yet proposals for its implementation are only recent. We propose a physical realization of the exact state-specific Petz map in an ion trap for qubit decoherence channels. Our circuit constructions require at most 1(2)1 (2)1(2) ancilla qubits and 3(20)3 (20)3(20) CNOT gates for channels with Kraus rank 2(>2)2 (>2)2(>2). We analyze typical ion trap errors and construct corresponding Petz maps, simulating their performance under realistic noise modeled by residual spin-motion coupling. Quantum circuits are provided for depolarizing, dephasing, and amplitude damping channels. Focusing on single-shot recovery, suited for present-day devices, we also quantify the precision of prior knowledge required to achieve a recovery error below 0.01 across varying decoherence levels and state purities.
The Petz recovery map is a central construct in quantum information theory, providing an explicit, channel-aware prescription for reversing the effects of noise. Unlike standard quantum operations, the Petz map is intrinsically dependent on a chosen reference state, which makes its physical implementation and experimental validation particularly challenging. Here, we report an experimental realization of Petz recovery maps on a nuclear magnetic resonance (NMR) quantum processor using the duality quantum computing (DQC) algorithm. We investigate two paradigmatic single-qubit noise models: amplitude damping and phase damping, and construct corresponding families of Petz recovery maps for varying reference states. By systematically tuning the reference state, we experimentally demonstrate the state-adapted nature of Petz recovery, observing both enhanced recovery when the reference state is well matched and fidelity degradation for mismatched choices. Our experimental results show close quantitative agreement with theoretical predictions, providing direct evidence that the Petz recovery map constitutes a physically realizable, reference-state-dependent recovery channel rather than a purely formal inverse of noise. This work bridges the gap between the abstract information-theoretic formulation of Petz recovery and its implementation on a realistic quantum platform, and establishes an experimental benchmark for testing noise-adapted recovery strategies on near-term quantum devices.