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Journal of Modern Optics

Phase-sensitive non-classicality criteria for phase-independent quantum states

Th. Richter, W. Vogel

Published Oct 15, 2005• 1 citation
Publisher

Abstract

A recently introduced hierarchy of necessary and sufficient conditions for non-classicality of a quantum state [Th. Richter and W. Vogel, Phys. Rev. Lett. 89 283601 (2002)] is applied to the characterization of phase-insensitive quantum states. Even under such conditions it may be important to analyse the relation between different phase values in the quadrature distributions in order to demonstrate the non-classical nature of the state under study.

DOI
10.1080/09500340500141821

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A recently introduced hierarchy of necessary and sufficient conditions for non-classicality of a quantum state [Th. Richter and W. Vogel, Phys. Rev. Lett. 89 283601 (2002)] is applied to the characterization of phase-insensitive quantum states. Even under such conditions it may be important to analyse the relation between different phase values in the quadrature distributions in order to demonstrate the non-classical nature of the state under study.
The characterization of nonclassicality of quantum states in terms of measurable quadrature distributions is studied, on the basis of recently derived necessary and sufficient conditions for the failure of the Glauber–Sudarshan P -function to be a probability density. When the noise-subtracted quadrature distribution, that is the marginal of the P -function, is not a probability density, the same is true for the Glauber–Sudarshan P -function. Inversely, for a complete characterization of a nonclassical P -function one needs phase-sensitive criteria that connect quadrature distributions for different phases. Even for phase-insensitive quantum states, phase-sensitive nonclassicality conditions are needed, as is illustrated for one-photon added thermal states.
Necessary and sufficient conditions are derived that allow one to completely characterize the nonclassicality of an arbitrary quantum state of a harmonic oscillator. Starting from a theorem by Bochner we derive an infinite hierarchy of conditions for the characteristic functions of the quantum state. Following the hierarchy step by step, the nonclassicality can be classified with respect to increasing orders. The full hierarchy of conditions is completely equivalent to the most commonly accepted criterion for nonclassicality: the failure of the Glauber–Sudarshan P -function to be a probability density. Our reformulated version of the criterion is directly related to measurable quantities that can be observed in phase-sensitive measurements such as homodyne detection. It directly generalizes a criterion for nonclassicality (Vogel 2000 Phys. Rev. Lett. 84 1849) that has already been applied in experiments (Lvovsky and Shapiro 2002 Phys. Rev. A 65 033830). Examples of phase-insensitive quantum states are considered that exhibit no first-order nonclassical effects, but that are nonclassical in second order.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.