关键词: 量子相干性度量/
量子计算加速/
离散Wigner函数
English Abstract
Investigating quantum coherence from discrete Wigner function
Lin Yin,Huang Ming-Da,
Yu Ya-Fei,
Zhang Zhi-Ming
1.Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices (SIPSE), Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
Fund Project:Project supported by the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91121023), the National Natural Science Foundation of China (Grant Nos. 11574092, 61378012, 60978009), the National Basic Research Program of China (Grant No. 2013CB921804), and the Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT1243).Received Date:29 November 2016
Accepted Date:01 March 2017
Published Online:05 June 2017
Abstract: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.
Keywords: quantum coherence measure/
quantum computation speed-up/
discrete Wigner function
