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Chinese Physics Letters

A Measure of Non-Classicality of Even and Odd Coherent States

YE Yong-hua, Li Zhi-Jian, Gao-Jian Zeng

Published Apr 1, 2008• 4 citations
Publisher

Abstract

A measure of non-classicality of even and odd coherent states is studied. We first calculate the Wigner functions of the even and odd coherent states, which consists of two terms: the positive-definite Gaussian term and the wave term with negativity, and then calculate the integrated value ∊ max of the wave term of the Wigner functions of the even and odd coherent states in their area with negativity, and use ∊ max to measure non-classicality of the even and odd coherent states. For the even and odd coherent states with certain photon count, it is very convenient for us to use ∊ max to measure their non-classicality. The methods of our definition and calculation for ∊ max have theoretical reference value.

DOI
10.1088/0256-307x/25/4/004

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A measure of non-classicality of even and odd coherent states is studied. We first calculate the Wigner functions of the even and odd coherent states, which consists of two terms: the positive-definite Gaussian term and the wave term with negativity, and then calculate the integrated value ∊ max of the wave term of the Wigner functions of the even and odd coherent states in their area with negativity, and use ∊ max to measure non-classicality of the even and odd coherent states. For the even and odd coherent states with certain photon count, it is very convenient for us to use ∊ max to measure their non-classicality. The methods of our definition and calculation for ∊ max have theoretical reference value.
Using the coherent state representation of the Wigner operator, the Wigner function for the photon-added even and odd coherent state (PAEOCS) is obtained. In terms of the variations of the Wigner function with the complex parameter α in the phase space, the non-classical properties of the PAEOCS are discussed. It is found that the PAEOCS always exhibits the non-classical properties, and the photon-added even (or odd) coherent state exhibits the non-classical properties more easily when m is odd (or even). By virtue of the marginal distributions of the Wigner function for the PAEOCS, we illuminate the physical meaning of the Wigner function. Finally, based on the intermidate coordinate-momentum presentation the quantum tomogram function for the PAEOCS is obtained.
In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms: the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius |β| = | α | with a separate angle of 2π/N , and the second term is composed of ½ N ( N– 1) Gaussian-cosine surfaces evenly centred in a circular region of radius |β| < | α | . Here, α is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.
We present a detailed discussion of even and odd coherent states defined as eigenstates of the single mode two-photon annihilation operator a 2. We study the non-classical properties such as squeezing, higher-order squeezing, and photon antibunching. Also discussed are the various quantum quasiprobability distributions, namely the P function, which is shown to be highly singular, the Q function, and the Wigner, which can take on negative values for these states. Finally, we present a discussion of a possible mechanism for the generation of such states based on the competition between parametric amplification and the incoherent losses from two-photon resonant absorption.
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.