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arXiv:2104.03360 · Physical Review Letters

Reversing Lindblad Dynamics via Continuous Petz Recovery Map

Hyukjoon Kwon, Rick Mukherjee, M. S. Kim

Published Jan 11, 2022• 35 citations
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Abstract

An important issue in developing quantum technology is that quantum states are so sensitive to noise. We propose a protocol that introduces reverse dynamics, in order to precisely control quantum systems against noise described by the Lindblad master equation. The reverse dynamics can be obtained by constructing the Petz recovery map in continuous time. By providing the exact form of the Hamiltonian and jump operators for the reverse dynamics, we explore the potential of utilizing the near-optimal recovery of the Petz map in controlling noisy quantum dynamics. While time-dependent dissipation engineering enables us to fully recover a single quantum trajectory, we also design a time-independent recovery protocol to protect encoded quantum information against decoherence. Our protocol can efficiently suppress only the noise part of dynamics thereby providing an effective unitary evolution of the quantum system.

DOI
10.1103/physrevlett.128.020403

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An important issue in developing quantum technology is that quantum states are so sensitive to noise. We propose a protocol that introduces reverse dynamics, in order to precisely control quantum systems against noise described by the Lindblad master equation. The reverse dynamics can be obtained by constructing the Petz recovery map in continuous time. By providing the exact form of the Hamiltonian and jump operators for the reverse dynamics, we explore the potential of utilizing the near-optimal recovery of the Petz map in controlling noisy quantum dynamics. While time-dependent dissipation engineering enables us to fully recover a single quantum trajectory, we also design a time-independent recovery protocol to protect encoded quantum information against decoherence. Our protocol can efficiently suppress only the noise part of dynamics thereby providing an effective unitary evolution of the quantum system.
One of the important issues in developing quantum technology is noise as quantum states are extremely sensitive to it. We propose a physical protocol which introduces reverse dynamics, in order to precisely control an open quantum system against noise. The reverse dynamics is realised by applying a well-controlled Hamiltonian and jump-operators. The control parameters for the Hamiltonian and jump operators are formulated by the Petz recovery map in continuous-time, from which we reverse the noisy quantum dynamics. The time-dependent jump operators can be implemented by considering strongly decaying ancillary qubits which interact with the system in a controlled manner. As a specific application, we demonstrate that our protocol efficiently suppresses only the noise part of dynamics thereby providing an effective unitary evolution of the quantum system.
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.
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.
ABSTRACT Controlling quantum systems in the presence of environmental noise presents significant challenges, primarily because the dissipative dynamics intricately depend on the control fields applied. To address this issue, we introduce a versatile and efficient framework based on dynamical invariants, enabling the analytical design of time‐dependent Hamiltonians tailored for optimal operation in noisy, dissipative environments. By employing a master equation featuring explicitly time‐dependent Lindblad generators, our reverse‐engineering approach allows precise manipulation of state dynamics without expensive iterative state propagation. This method dynamically constructs an effective decoherence‐free subspace, confining the system to a minimally noisy region within the Hilbert space. We illustrate the effectiveness of our technique using two paradigmatic examples: a driven two‐level system and a harmonic oscillator, both coupled to thermal baths. In each case, we achieve substantial fidelity improvements compared to conventional methods, highlighting the robustness and potential of our approach for reliable quantum control in open quantum systems.
The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic Schrödinger equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact reverse stochastic Schrödinger equations admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.
Providing entanglement for the design of quantum technologies in the presence of noise constitutes today's main challenge in quantum information science. A framework is required that assesses the build-up of entanglement in realistic settings. In this work, we put forth a new class of nonlinear quantum master equations in Lindblad form that unambiguously identify dynamical entanglement in open quantum systems via deviations from a separable evolution. This separability Lindblad equation restricts quantum trajectories to classically correlated states only. Unlike many conventional approaches, here the entangling capabilities of a process are not characterized by input-output relations, but separability is imposed at each instant of time. We solve these equations for crucial examples, thereby quantifying the dynamical impact of entanglement in non-equilibrium scenarios. Our results allow to benchmark the engineering of entangled states through dissipation. The separability Lindblad equation provides a unique path to characterizing quantum correlations caused by arbitrary system-bath interactions, specifically tailored for the noisy intermediate-scale quantum era.