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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

Nonclassicality and Entanglement with Continuous Variables

W. Vogel

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The problem of characterizing nonclassicality and entanglement in terms of continuous variables is considered. Necessary and sufficient conditions for the nonclassicality, that is the P-functions fails to be a probability, are formulated in terms of measurable characteristic functions or moments. Applications for highly sensitive measurements are discussed. Conditions for entanglement are formulated and the nonclassical correlations of multimode radiation fields are studied. The needed correlation functions can be observed by balanced homodyne correlation measurements.

DOI
10.1364/cqo.2007.cwb3

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The problem of characterizing nonclassicality and entanglement in terms of continuous variables is considered. Necessary and sufficient conditions for the nonclassicality, that is the P-functions fails to be a probability, are formulated in terms of measurable characteristic functions or moments. Applications for highly sensitive measurements are discussed. Conditions for entanglement are formulated and the nonclassical correlations of multimode radiation fields are studied. The needed correlation functions can be observed by balanced homodyne correlation measurements.
A unified characterization of continuous-variable quantum states is presented that gives general insight in their nonclassical features and, in particular, in their entanglement properties. Nonclassicality is characterized in terms of observable characteristic functions. Equivalently, the nonclassicality conditions are reformulated in terms of moments. The latter approach turns out to be a powerful tool for deriving a complete set of entanglement conditions for bipartite quantum systems, which are fully equivalent to the nonpositivity of the partially transposed density operator. The possibilities of observing the needed moments by balanced homodyne correlation measurements are analyzed.
In Quantum Optics, the widely accepted definition of nonclassicality is based on the P function of Glauber and Sudarshan. When it fails to be interpreted as a classical probability density, the corresponding quantum state is said to be a nonclassical one. Here we present the first reconstruction of a nonclassical P function of a single-photon added thermal state. We also consider the nonclassical properties of general spacegtime dependent correlation functions of radiation fields. For the detection of these correlation functions, a balanced homodyne correlation technique was proposed. It is shown that the measurable correlation functions also allow one to completely characterize bipartite entangled quantum states with a negative partial transposition. Finally, we present a method for identifying general bipartite entanglement for continuous variables.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.
Entanglement is a nonclassical correlation of compound quantum systems. We derive Separability-Eigenvalue-Equations. The solution of these equations delivers all observables for the determination of entanglement. On the other hand, it delivers entanglement quasi-probabilities. We conclude from negativities of such distributions, that no classical statistics can describe the state under study. For the quantification of entanglement we compare different entanglement measures. We also consider different kinds of entanglement and the relation to nonclassicality.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
We present how nonclassicality and entanglement can be characterized and detected efficiently for continuous variable systems. Of particular interest is the use of homodyne detections to measure quadrature amplitudes at minimum level to confirm nonclassicality and entanglement beyond Gaussian states. We introduce a systematic method for a functional form of uncertainty relations, which can be efficiently employed to experimentally detect non-Gaussian states comprehensively. Our approach for quantum correlations unifies a framework for quantum entanglement and quantum steering, which include the known results for Gaussian states and provides a better tool for non-Gaussian states than existing methods, e.g. entropic uncertainty relations.