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arXiv:1703.08004

Non-Markovian Dynamics of Discrete-Time Quantum Walks

Subhashish Banerjee, N. Pradeep Kumar, R. Srikanth, Vinayak Jagadish, Francesco Petruccione

Published Mar 23, 2017• 4 citations
arXivHTMLPDF

Abstract

In the case of the discrete time coined quantum walk the reduced dynamics of the coin shows non-Markovian recurrence features due to information back-flow from the position degree of freedom. Here we study how this non-Markovian behavior is modified in the presence of open system dynamics. In the process, we obtain useful insights into the nature of non-Markovian physics. In particular, we show that in the case of (non-Markovian) random telegraph noise (RTN), a further discernbile recurrence feature is present in the dynamics. Moreover, this feature is correlated with the localization of the walker. On the other hand, no additional recurruence feature appears for other non-Markovian types of noise (Ornstein-Uhlenbeck and Power Law noise). We propose a power spectral method for comparing the relative strengths of the non-Markovian component due to the external noise and that due to the internal position degree of freedom.

DOI
10.48550/arxiv.1703.08004

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In the case of the discrete time coined quantum walk the reduced dynamics of the coin shows non-Markovian recurrence features due to information back-flow from the position degree of freedom. Here we study how this non-Markovian behavior is modified in the presence of open system dynamics. In the process, we obtain useful insights into the nature of non-Markovian physics. In particular, we show that in the case of (non-Markovian) random telegraph noise (RTN), a further discernbile recurrence feature is present in the dynamics. Moreover, this feature is correlated with the localization of the walker. On the other hand, no additional recurruence feature appears for other non-Markovian types of noise (Ornstein-Uhlenbeck and Power Law noise). We propose a power spectral method for comparing the relative strengths of the non-Markovian component due to the external noise and that due to the internal position degree of freedom.
Non-Markovian quantum effects are typically observed in systems interacting with structured reservoirs. Discrete-time quantum walks are prime example of such systems in which, quantum memory arises due to the controlled interaction between the coin and position degrees of freedom. Here we show that the information backflow that quantifies memory effects can be enhanced when the particle is subjected to uncorrelated static or dynamic disorder. The presence of disorder in the system leads to localization effects in 1-dimensional quantum walks. We shown that it is possible to infer about the nature of localization in position space by monitoring the information backflow in the reduced system. Further, we study other useful properties of quantum walk such as entanglement, interference and its connection to quantum non-Markovianity.
The dynamics of a discrete-time quantum walk subject to time-correlated noise is studied in this paper. The noise, which goes beyond the usual telegraph noise with random variables 1 and $\ensuremath{-}1$, is modeled as a unitary coin-type operator and it is generated by a sample path of the Ornstein-Uhlenbeck process. Up to the first order in the amplitude of noise, a master equation to describe the walk is derived. The dynamics governed by the master equation is in good agreement with those given by numerical simulations within a certain number of steps, depending on the parameters of noise. Two remarkable features of long-time dynamics are observed in the numerical simulations. (1) In the slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. (2) In the fast noise regime, the walker is confined into a few lattice sites, and the width of the wave packet is much narrower compared with that in the slow noise regime.
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.
ABSTRACT In quantum communication, quantum state transfer from one location to another in a quantum network plays a prominent role, where the impact of noise could be crucial. The idea of state transfer can be fruitfully associated with quantum walk on graphs. We investigate the consequences of non‐Markovian quantum noises on periodicity and state transfer induced by a discrete‐time quantum walk on graphs, governed by the Grover coin operator. Different bipartite graphs, such as the path graph, cycle graph, star graph, and complete bipartite graph, present periodicity and state transfer in a discrete‐time quantum walk depending on the topology of the graph. We investigate the effect of quantum non‐Markovian dephasing noises, particularly quantum non‐Markovian Random Telegraph Noise (RTN) and modified non‐Markovian Ornstein‐Uhlenbeck Noise (OUN) on state transfer and periodicity. We demonstrate how the RTN and OUN noises allow state transfer and periodicity for a finite number of steps in a quantum walk. Our investigation brings out the feasibility of state transfer in a noisy environment.
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of an environment such as its frequency spectrum. In this work, the reduced dynamics of a coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a ttt-step quantum walk. Non-Markovianity, in the sense of P indivisibility of the reduced coin dynamics, is inferred from the nonmonotonous behavior of distinguishability of two orthogonal states subjected to it. Furthermore, we study various quantum information-theoretic quantities of a qubit under the action of this channel, putting in perspective the role such channels can play in various quantum information processing tasks.
Abstract We study discrete quantum dynamics where a single evolution step consists of unitary system transformation followed by decoherence via coupling to an environment. Often, non-Markovian memory effects are attributed to structured environments, whereas, here, we take a more general approach within a discrete setting. In addition of controlling the structure of the environment, we are interested in how local unitaries on the open system allow the appearance and control of memory effects. Our first simple qubit model where local unitary is followed by dephasing illustrates how memory effects arise, despite having no structure in the environment the system is coupled with. We, then, elaborate on this observation by constructing a model for an open quantum walk where the unitary coin and transfer operation is augmented with the dephasing of the coin. The results demonstrate tha,t in the limit of strong dephasing within each evolution step, the combined coin-position open system always displays memory effects, and their quantities are independent of the structure of the environment. Our construction makes possible an experimentally realizable open quantum walk with photons exhibiting non-Markovian features.