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Physica Scripta

Wigner function analysis of the decoherence of adiscrete-time quantum walk in the bit-phase noisechannel

Keng Hong Kua, Sithi Vinayakam Muniandy, Nur Izzati Ishak, Wu Yi Chong

Published Sep 26, 2025• 0 citations
Publisher

Abstract

Abstract The one-dimensional discrete-time Hadamard quantum walk in the bit-phase noise channel is studied using the Wigner function. Decoherence is characterized using the l1-norm of coherence, Wigner function, and complex Wigner entropy. The l1-norm of coherence is used to quantify the persistence of coin coherence, and the non-classicality is verified through negativity of the Wigner function. Analysis of the complex Wigner entropy revealed a non-monotonic behavior in the non-classicality of the quantum walk, with an optimal noise level where the non-classicality is maximized. The Wigner function analysis is able to provide information related to non-classicality that is not possible using the standard Hilbert-space operator formalism.

DOI
10.1088/1402-4896/ae0c6b

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Abstract The one-dimensional discrete-time Hadamard quantum walk in the bit-phase noise channel is studied using the Wigner function. Decoherence is characterized using the l1-norm of coherence, Wigner function, and complex Wigner entropy. The l1-norm of coherence is used to quantify the persistence of coin coherence, and the non-classicality is verified through negativity of the Wigner function. Analysis of the complex Wigner entropy revealed a non-monotonic behavior in the non-classicality of the quantum walk, with an optimal noise level where the non-classicality is maximized. The Wigner function analysis is able to provide information related to non-classicality that is not possible using the standard Hilbert-space operator formalism.
We perform a systematic study of the discrete time Quantum Walk on one dimension using Wigner functions, which are generalized to include the chirality (or coin) degree of freedom. In particular, we analyze the evolution of the negative volume in phase space, as a function of time, for different initial states. This negativity can be used to quantify the degree of departure of the system from a classical state. We also relate this quantity to the entanglement between the coin and walker subspaces.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
Quantum Walk (QW) has very different transport properties to its classical counterpart due to interference effects. Here we study the discrete-time quantum walk (DTQW) with on-site static/dynamic phase disorder following either binary or uniform distribution in both one and two dimensions. For one dimension, we consider the Hadamard coin; for two dimensions, we consider either a 2-level Hadamard coin (Hadamard walk) or a 4-level Grover coin (Grover walk) for the rotation in coin-space. We study the transport properties e.g. inverse participation ratio (IPR) and the standard deviation of the density function (σ) as well as the coin-position entanglement entropy (EE), due to the two types of phase disorders and the two types of coins. Our numerical simulations show that the dimensionality, the type of coins, and whether the disorder is static or dynamic play a pivotal role and lead to interesting behaviors of the DTQW. The distribution of the phase disorder has very minor effects on the quantum walk.
We analyze the quantum walk on a cycle using discrete Wigner functions as a way to represent the states and the evolution of the walker. The method provides some insight on the nature of the interference effects that make quantum and classical walks different. We also study the behavior of the system when the quantum coin carried by the walker interacts with an environment. We show that for this system quantum coherence is robust for initially delocalized states of the walker. The use of phase-space representation enables us to develop an intuitive description of the nature of the decoherence process in this system.
Abstract In this article we investigate the effects of shifting position decoherence, arising from the incoherent tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discrete time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not vanish, in contrast to the coin decoherence for which the walker undergoes a quantum-to-classical transition even for weak noise. In particular, we show that the quadratic dependence of the variance on the time and also the coin–position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of the decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.
Abstract We investigate the behaviour of one-dimensional dissipative Hadamard discrete-time quantum walk (DTQW) in the generalized amplitude damping channel. By manipulating the noise intensity, we uncover intriguing dynamics in the coherence of the coin state such as rebounding and freezing phenomena. The entanglement between the quantum walker’ states diminishes at lower dissipation rates despite the pronounced coherence. The non-classical characters of the dissipative DTQW becomes evident through the manifestation of quantum discord. Our study shows that even under maximum dissipation stemming from both zero and finite-temperature environment, the quantum walker retains rudimentary non-classical behaviours. It is shown that fine-tuning the parameter governing the coin rotation angle optimizes the quantum discord of the walker. These results are useful for optimizing quantum walks in the presence of noisy channels.