Irreversibility and time-translation symmetry are both fundamental in open quantum dynamics, and it is of great importance to study the interplay between them. In this paper, we study tabletop reversibility (TTR) for phase-covariant quantum operations and obtain sharply different conclusions for finite-dimensional and Gaussian systems. For finite-dimensional systems, we prove that any phase-covariant operation that admits a Petz recovery map can be implemented by a dilation which is both time-translational symmetric and tabletop time reversible. In contrast, for phase-covariant Gaussian operations we exhibit a concrete obstruction: any Gaussian dilation of a single-mode Gaussian amplification channel is not tabletop time reversible. As a byproduct, we find that almost every completely positive and trace preserving (CPTP) map admits a dilation that realizes TTR. These results clarify whether and how TTR can be realized under phase-covariant symmetry constraints and reveal a qualitative difference between finite-dimensional and Gaussian systems.
Irreversibility and time-translation symmetry are both fundamental in open quantum dynamics, and it is of great importance to study the interplay between them. In this paper, we study tabletop reversibility (TTR) for phase-covariant quantum operations and obtain sharply different conclusions for finite-dimensional and Gaussian systems. For finite-dimensional systems, we prove that any phase-covariant operation that admits a Petz recovery map can be implemented by a dilation which is both time-translational symmetric and tabletop time reversible. In contrast, for phase-covariant Gaussian operations we exhibit a concrete obstruction: any Gaussian dilation of a single-mode Gaussian amplification channel is not tabletop time reversible. As a byproduct, we find that almost every completely positive and trace preserving (CPTP) map admits a dilation that realizes TTR. These results clarify whether and how TTR can be realized under phase-covariant symmetry constraints and reveal a qualitative difference between finite-dimensional and Gaussian systems.
Abstract Channels 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 N ^ γ ( P ) is a systematic construction that depends only on 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 N is called tabletop time-reversible (TTR) for a given γ when the corresponding Petz recovery map is realizable in such a way. First, we study the exact TTR conditions. We show in particular that a time-sensitive control of an ancilla system is needed for the exact TTR. 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, and demonstrate it with an excitation-exchange Hamiltonian.
Time-translation symmetry strongly constrains physical dynamics, yet systematic characterization for continuous-variable systems lags behind its discrete-variable counterpart. We close this gap by providing a rigorous classification of Gaussian quantum operations that are covariant under time translations, termed Gaussian covariant operations. We show that several key results known for discrete-variable covariant operations break down in the Gaussian optical setting: discrepancies arise in physical and thermodynamic implementation, in the extensivity of asymmetry, and in catalytic advantages. Our results provide comprehensive mathematical and operational toolkits for Gaussian covariant operations, including a peculiar pair of asymmetry measures that are completely non-extensive. Our findings also reveal surprising consequences of the interplay among symmetry, Gaussianity, and thermodynamic constraints, suggesting that real-world scenarios with multiple constraints have a rich structure not accessible from examining individual constraints separately.
Irreversibility, crucial in both thermodynamics and information theory, is naturally studied by comparing the evolution—the (forward) channel—with an associated reverse—the reverse channel. There are two natural ways to define this reverse channel. Using logical inference, the reverse channel is the Bayesian retrodiction (the Petz recovery map in the quantum formalism) of the original one. Alternatively, we know from physics that every irreversible process can be modeled as an open system: one can then define the corresponding closed system by adding a bath (“dilation”), trivially reverse the global reversible process, and finally remove the bath again. We prove that the two recipes are strictly identical, both in the classical and in the quantum formalism, once one accounts for correlations formed between system and the bath. Having established this, we define and study special classes of maps: (including ), for which no such system-bath correlations are formed for some states; and , when the reverse channel can be implemented with the same devices as the original one. We establish several general results connecting these classes, and a very detailed characterization when both the system and the bath are one qubit. In particular, we show that, when reverse channels are well defined, product preservation is a sufficient but not necessary condition for tabletop reversibility; and that the preservation of local energy spectra is a necessary and sufficient condition to generalized thermal operations. Published by the American Physical Society 2024
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 Bayesian inverse of a quantum channel E is a channel F in the reverse direction of E that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse F is defined with respect to a fiducial reference state ρ for the channel E. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states ρ for which a Bayesian inverse of E exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.
Reversibility lies at the heart of physics and information theory, linking entropy, efficiency and information recovery. Yet, definitions of a “reverse process” have been context-specific, relying on physical assumptions rather than a general framework. This thesis develops a universal recipe for reverse processes using Bayesian inference, formalized through Bayes’ rule and the Petz recovery map. We first show how these tools recover conventional notions of reversal while yielding new insights, and justify the Petz transpose as the quantum analog of Bayes’ rule via quasiprobability representations. We then compare Bayesian reversibility with dilation-based approaches, proving their equivalence through the role of correlations. This motivates the notion of tabletop time-reversible processes, special cases where correlations can be neglected with respect to reversal. Beyond the core results, we apply these methods to fluctuation relations, quantum thermal operations, and the Loschmidt paradox, and reflect on implications for the philosophy of time. In sum, Bayesian inference provides a regime-independent foundation for defining and analyzing (ir)reversibility across physics and information theory.