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arXiv:1908.08085 · Physical Review A

Experimental investigation of Markovian and non-Markovian channel addition

Solomon Akpore Uriri, Filip Wudarski, Ilya Sinayskiy, Francesco Petruccione, Mark Tame

Published May 11, 2020• 25 citations
arXivHTMLPDF

Abstract

The study of memory effects in quantum channels helps in developing characterization methods for open quantum systems and strategies for quantum error correction. Two main sets of channels exist, corresponding to system dynamics with no memory (Markovian) and with memory (non-Markovian). Interestingly, these sets have a nonconvex geometry, allowing one to form a channel with memory from the addition of memoryless channels and vice versa. Here, we experimentally investigate this nonconvexity in a photonic setup by subjecting a single qubit to a convex combination of Markovian and non-Markovian channels. We use both divisibility and distinguishability as criteria for the classification of memory effects, with associated measures. Our results highlight some practical considerations that may need to be taken into account when using memory criteria to study system dynamics given by the addition of Markovian and non-Markovian channels in experiments.

DOI
10.1103/physreva.101.052107

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The study of memory effects in quantum channels helps in developing characterization methods for open quantum systems and strategies for quantum error correction. Two main sets of channels exist, corresponding to system dynamics with no memory (Markovian) and with memory (non-Markovian). Interestingly, these sets have a nonconvex geometry, allowing one to form a channel with memory from the addition of memoryless channels and vice versa. Here, we experimentally investigate this nonconvexity in a photonic setup by subjecting a single qubit to a convex combination of Markovian and non-Markovian channels. We use both divisibility and distinguishability as criteria for the classification of memory effects, with associated measures. Our results highlight some practical considerations that may need to be taken into account when using memory criteria to study system dynamics given by the addition of Markovian and non-Markovian channels in experiments.
A profound understanding of decoherence and dissipation in quantum dynamics is crucial for the realistic modeling of the evolution of quantum systems. In open quantum dynamics one distinguishes between a memoryless, so-called Markovian evolution and dynamics incorporating memory effects, termed non-Markovian. In this work we study how prevalent memory effects are in the set of all such dynamics. We thus investigate how often a Markovian description is applicable. This question is approached by investigating randomly generated two-step qubit dynamics with respect to different concepts and witnesses of non-Markovianity. We observe that almost all dynamics are non-Markovian, and only a small (yet finite) fraction is Markovian. Furthermore, we study how this proportion changes when considering certain subclasses such as lower rank or mixed-unitary dynamics. Importantly, our results shed light on the relative ratios of -- and interrelations between -- the sets of dynamics that are non-Markovian with respect to different criteria. Finally, we investigate the fraction of dynamics in which the memory effects are necessarily of quantum nature and establish a connection between two recently developed concepts that characterize the quantumness of memory in non-Markovian dynamics.
Noisy Intermediate Scale Quantum (NISQ) devices have been proposed as a versatile tool for simulating open quantum systems. Recently, the use of NISQ devices as simulators for non-Markovian open quantum systems has helped verify the current descriptions of non-Markovianity in quantum physics. In this work, convex mixtures of channels are simulated using NISQ devices and classified as either Markovian or non-Markovian using the CP-divisibility criteria. Two cases are considered: two Markovian channels being convexly mixed to form a non-Markovian channel and vice versa. This work replicates the experiments performed in a linear optical setup, using NISQ devices, with the addition of a convex mixture of non-Markovian channels that was designed to address some of the problems faced in the experiments performed in the linear optical setup. The NISQ devices used were provided by the IBM Quantum Experience (IBM QE). The results obtained show that, using NISQ devices and within some error, convex mixtures of Markovian channels lead to a non-Markovian channel and vice versa.
Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T({t}_{2},{t}_{0})\ensuremath{\ne}T({t}_{2},{t}_{1})T({t}_{1},{t}_{0})$ in terms of memoryless dynamical map TTT introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.
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.
We experimentally investigate the non-Markovianity problems in a quantum channel implemented by an all-optical setup. The quantum channel simulates the dynamics of a two-level (qubit) system surrounded by bosonic modes, with the system-environment interaction treated without applying the rotating-wave approximation (RWA). Therefore, it allows us to demonstrate the important role of counterrotating-wave terms in influencing the memory effects of the environment. The experimental and theoretical results are in good agreement and show quite different behaviors of the non-Markovianity measure between RWA and non-RWA channels. One can find the nonmonotonic dependence of the non-Markovianity measure on the coupling and detuning parameters in the non-RWA case, which is attributed to the competing mechanisms caused by the counterrotating-wave terms. We emphasize that the non-RWA channel provides a more clear description of the open-system dynamics and the counterrotating-wave terms are indispensable in the study of non-Markovianity.
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.