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Physical Review A

Enhanced quantumness via non-Markovianity

Haiping Li, Jian Zou, Bin Shao

Published Nov 1, 2021• 7 citations

Abstract

In this paper we address the issue of the interplay between non-Markovianity and nonclassicality of a system. We consider the action of non-Markovian environments on a qubit in terms of two models, a classical noise model and a quantum microscopic collision model. For both models we use the violation of the Leggett-Garg inequality (LGI) as a criterion of quantumness, and use the conditional past-future (CPF) correlation to measure non-Markovianity. Besides for the collision model, we also use violation of the nondiscord generating and detecting (NDGD) dynamics to indicate the nonclassicality. And LGI, NDGD, and CPF are probed sequentially by three projective measurements of some observable. In this paper we do not use the assumption of the quantum regression theorem and especially for the collision model the measurement backaction on the environment is explicitly considered. We find that compared with the Markovian dynamics the LGI can be violated for a longer time interval, i.e., non-Markovianity can preserve the quantumness of the system. For the collision model we find that for a specific measurement process, although the non-Markovianity cannot be detected, the effect of non-Markovianity of the system dynamics on nonclassicality still exists, and for some specific measurement operator the violation of both LGI and NDGD will never occur for the Markovian dynamics while with the enhancement of non-Markovianity the quantumness can appear. We also find that the NDGD can be violated for a much wider parameter regime than the LGI.

DOI: 10.1103/physreva.104.052201

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Nonclassicality of temporal correlations pertaining to noncommutative sequential measurements is defined through the violation of macrorealistic inequalities, known as Leggett-Garg inequalities (LGI). We investigate the energy cost of the process associated with the Leggett-Garg test in the context of noiseless and Markovian noise for arbitrary initial states. We prove that in noiseless and in certain noisy scenarios, the maximal violations of LGI under the energy constraint occurs when the average energy of the process is equal to the negative of the energy of the initial state. Such a dependence of LGI on the choice of the initial state is not seen in the unconstrained case. Moreover, we find that in the presence of a moderate amount of Markovian noise, the amount of violation of LGI remains almost unaltered with a suitable choice of the evolution and dephasing operators in the neighborhood of the maximal violation line, thereby showing the robustness of temporal correlations under environmental effects.

We study the violations of Leggett-Garg (LG) inequality in a qubit subjected to non-Markovian noisy channels such as Random Telegraph Noise (RTN) and Ornstein-Uhlenbeck Noise (OUN). Quite generally, the state-independence of the violation in the noiseless case is preserved under the application of noise. Within a given family of noisy channels (in specific, RTN or OUN), we find an enhancement in the violation in the non-Markovian case as compared to the Markovian case. We thus find that non-Markovianity provides a stronger demonstration of quantumness of the system.

For classical Markovian stochastic systems, past and future events become statistically independent when conditioned to a given state at the present time. Memory non-Markovian effects break this condition, inducing a nonvanishing conditional past-future correlation. Here, this classical memory indicator is extended to a quantum regime, which provides an operational definition of quantum non-Markovianity based on a minimal set of three time-ordered quantum system measurements and postselection. The detection of memory effects through the measurement scheme is univocally related to departures from Born-Markov and white noise approximations in quantum and classical environments, respectively.

We introduce an operational measure of nonclassicality for qubit systems based on the violation of Kolmogorov consistency conditions in sequential measurements. In contrast to previous work by Milz et al.~\cite{Milz-2020}, who characterized classicality via NCGD maps, and Sakuldee et al.~\cite{Sakuldee-2022a, Sakuldee-2022b}, who studied quantum correlations under measurement disturbance, our witness explicitly quantifies nonclassicality in terms of the POVM unsharpness parameter

a_z

. For projective measurements on an initially diagonal state, the witness vanishes identically, showing that such measurements cannot reveal nonclassicality. However, by generalizing to positive operator-valued measures (POVMs), we find that non-projective measurements can reveal nonclassicality even for diagonal initial states. We derive explicit expressions for the witness for general initial states, including coherences, and show that its maximum value is

1/4

, which is a new result achieved for unbiased POVMs, maximal dephasing, and equal initial populations. Our results connect Kolmogorov consistency, Leggett-Garg inequalities, and POVMs, providing an experimentally accessible tool for detecting nonclassicality in qubit systems with potential applications in quantum technology certification.

Memory effects can be studied through a conditional past-future correlation, which measures departure with respect to a conditional past-future independence valid in a memoryless Markovian regime. In a quantum regime this property leads to an operational definition of quantum non-Markovianity based on three consecutive system measurement processes and postselection [Phys. Rev. Lett. 121, 240401 (2018)]. Here, we study the conditional past-future correlation for a qubit system coupled to different dephasing environments. Exact solutions are obtained for a quantum spin bath as well as for classically fluctuating random Hamiltonian models. The developing of memory effects and departures from Born-Markov or white-noise approximations are related to a measurement back action that changes the system dynamics between consecutive measurements. It is shown that this effect may develop even when the former system evolution is given by a time-independent Lindblad equation. This unusual non-Markovian case arises when the characteristic parameters of the dynamics become Lorentzian random distributed variables.

We investigate a two-qubit quantum system in contact with an environment modeled by a microscopic collision model with added ancilla-ancilla collisions in the non-Markovian regime. Two schemes of the two-qubit collision model with carried-forward correlations are introduced. In one scheme, a single stream of ancillae interacts with only one of the qubits of the two-qubit system; in the other, both the qubits interact with two independent sequences of ancillae, which could be at the same or different temperatures. The system's non-Markovian evolution is examined using the trace distance measure, and the non-classicality of the system is studied using the Wigner function, non-classical volume, and concurrence. Also, interesting steady-state behavior is observed when both the independent ancillae are kept at the same temperature.