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arXiv:2401.05499 · Annalen der Physik

Facets of Correlated Non‐Markovian Channels

Vivek Balasaheb Sabale, Nihar Ranjan Dash, Atul Kumar, Subhashish Banerjee

Published Jul 29, 2024• 6 citations
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Abstract

Abstract The domain of correlated non‐Markovian channels are investigated, exploring the potential memory arising from the correlated action of channels and the inherent memory due to non‐Markovian dynamics. The impact of channel correlations is studied using different non‐Markovianity indicators and measures. In addition, the dynamical aspects of correlated non‐Markovian channels, including entanglement dynamics as well as changes in the volume of accessible states, are explored. The analysis is presented for both unital and non‐unital correlated channels. A new correlated channel constructed with modified Ornstein–Uhlenbeck noise (OUN) is also presented and explored. Further, the geometrical effects of the non‐Markovianity of the correlated non‐Markovian channels are discussed with a study of change in the volume of the accessible states. The link between the correlation factor and error correction success probability is highlighted.

DOI
10.1002/andp.202400151

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Abstract The domain of correlated non‐Markovian channels are investigated, exploring the potential memory arising from the correlated action of channels and the inherent memory due to non‐Markovian dynamics. The impact of channel correlations is studied using different non‐Markovianity indicators and measures. In addition, the dynamical aspects of correlated non‐Markovian channels, including entanglement dynamics as well as changes in the volume of accessible states, are explored. The analysis is presented for both unital and non‐unital correlated channels. A new correlated channel constructed with modified Ornstein–Uhlenbeck noise (OUN) is also presented and explored. Further, the geometrical effects of the non‐Markovianity of the correlated non‐Markovian channels are discussed with a study of change in the volume of the accessible states. The link between the correlation factor and error correction success probability is highlighted.
We evaluate the dynamics of quantum and classical correlations in the presence of non-Markovian noises. By considering an entangled pair of spins in the noise environment described by Ornstein—Uhlenbeck processes, we show that the quantum discord of some states is completely unaffected by independent Ornstein—Uhlenbeck noises for long intervals of time, and that the inevitable onset of the sudden decrease of the quantum discord can be substantially delayed by the decrease of the noise bandwidth γ, where γ −1 = τ c defines the environment's finite correlation time of the noise.
Memory effects play a fundamental role in the study of the dynamics of open quantum systems. There exist two conceptually distinct notions of memory discussed for quantum channels in the literature. In quantum information theory quantum channels with memory are characterized by the existence of correlations between successive applications of the channel on a sequence of quantum systems. In open quantum systems theory memory effects arise dynamically during the time evolution of quantum systems and define non-Markovian dynamics. Here we relate and combine these two different concepts of memory. In particular, we study the interplay between correlations between multiple uses of quantum channels and non-Markovianity as nondivisibility of the ttt-parametrized family of channels defining the dynamical map.
Quantum non-Markovianity of a quantum noisy channel is typically identified with information backflow or, more generally, with departure of the intermediate map from complete positivity. But here, we also indicate certain non-Markovian channels that cannot be witnessed by the CP-divisibility criterion. In complex systems, non-Markovianity becomes more involved on account of subsystem dynamics. Here we study various facets of non-Markovian evolution, in the context of coined quantum walks, with particular stress on disambiguating the internal vs. environmental contributions to non-Markovian backflow. For the above problem of disambiguation, we present a general power-spectral technique based on a distinguishability measure such as trace-distance or correlation measure such as mutual information. We also study various facets of quantum correlations in the transition from quantum to classical random walks, under the considered non-Markovian noise models. The potential for the application of this analysis to the quantum statistical dynamics of complex systems is indicated.
The problem of defining quantum non-Markovianity has proven elusive, with various inequivalent criteria put forth to address it. We consider the question: what is the weakest notion of quantum non-Markovianity of system dynamics that would account for any aspect of memory? That is, a process indicated to be non-Markovian according to any existing criterion is also non-Markovian according to this criterion, but the converse may not be true. Correspondingly, we identify true memorylessness with the idea of the quantum dynamics being temporally self-similar, whereby the form of the intermediate map that propagates the state between two arbitrary time intervals is independent of the initial time t0t_0t0​. This is identified with the quantum dynamical semigroup (QDS), renewing advocacy for a familiar candidate. Of practical importance is that our weaker criterion is useful for indicating the non-Markovianity of CP-divisible channels that manifest an effect of counteracting decoherence. This is illustrated with the examples of power-law noise (PLN) and Ornstein-Uhlenbeck Noise (OUN). For physically well motivated channels, we propose a geometric quantification of non-Markovianity according to our definition, which is shown to be applicable to stronger manifestations of quantum non-Markovianity, demonstrated here by application to various channels, including those that are CP-indivisible, P-indivisible and eternally non-Markovian. Finally, we indicate a few open questions that come up as natural consequences of this work.
In correlated noisy channels, the global memory effects on the dynamics of a quantum system depend on both non-Markovianity of the single noisy channel (intrinsic memory) and classical correlations between multiple uses of the channel itself (correlation-based memory). We show that the Hilbert–Schmidt speed (HSS), a measure of non-Markovianity, serves as a reliable figure of merit for evaluating the role of this correlation-based memory on the global memory effects, for both unital and non-unital channels irrespective of initial basis. The intensity of the correlation-based memory is ruled by a classical correlation strength between consecutive applications of the channel. We establish that classical correlation between multiple uses of a channel does not alter the duration of revivals in unital channels, whereas in non-unital channels, it does. Therefore, HSS can serve as an effective tool for distinguishing between unital and non-unital correlated channels. We demonstrate that, for unital noisy channels, increasing the number of qubits of the system significantly weakens the sensitivity of the HSS to this classical correlation strength. Such a pattern indicates that the state evolution of large quantum systems may be less prone to be affected by classical correlations between noisy channels. Moreover, assuming that the qubits are affected by independent or classically correlated local non-Markovian unital channels, we observe that, as the number of qubits increases, the collective behavior of the multiqubit system inhibits the non-Markovian features of the overall system dynamics.
In correlated noisy channels, the global memory effects on the dynamics of a quantum system depend on both non-Markovianity of the single noisy channel (intrinsic memory) and classical correlations between multiple uses of the channel itself (correlation-based memory). We show that the Hilbert-Schmidt speed (HSS), a measure of non-Markovianity, serves as a reliable figure of merit for evaluating the role of this correlation-based memory on the global memory effects, for both unital and non-unital channels. The intensity of the correlation-based memory is ruled by a classical correlation strength between consecutive applications of the channel. We demonstrate that, for unital noisy channels, increasing the number of qubits of the system significantly weakens the sensitivity of the HSS to this classical correlation strength. Such a pattern indicates that the state evolution of large quantum systems may be less prone to be affected by classical correlations between noisy channels. Moreover, assuming the qubits are affected by independent or classically correlated local non-Markovian unital channels, we observe that, as the number of qubits increases, the collective behavior of the multiqubit system inhibits the non-Markovian features of the overall system dynamics.