Operational Quantification of Continuous-Variable Correlations

Abstract

We quantify correlations (quantum and/or classical) between two continuous variable modes in terms of how many correlated bits can be extracted by measuring the sign of two local quadratures. On Gaussian states, such 'bit quadrature correlations' majorize entanglement, reducing to an entanglement monotone for pure states. For non-Gaussian states, such as photonic Bell states, ideal and real de-Gaussified photon-subtracted states, and mixtures of pure Gaussian states, the bit correlations are shown to be a monotonic function of the negativity. This yields a feasible, operational way to quantitatively measure non-Gaussian entanglement in current experiments by means of direct homodyne detection, without a full tomographical reconstruction of the Wigner function.PACS numbers: 03.67. Mn, 42.50.Dv Quantum information with continuous variables (CVs), relying on quadrature entanglement as a resource, has witnessed rapid and exciting progresses recently, also thanks to the high degree of experimental control achievable in the context of quantum optics [1]. While Gaussian states (coherent, squeezed, and thermal states) have been originally the preferred resources for both theoretical and practical implementations, a new frontier emerges with non-Gaussian states (Fock states, Schrödinger's cats, ...). The latter can be highly nonclassical, possess in general more entanglement than Gaussian states [2], and are useful to overcome some limitations of the Gaussian framework such as entanglement distillation [3] and universal quantum computation [4]. Therefore, it is of central relevance to provide proper ways to quantify non-Gaussian entanglement in a way which is experimentally accessible.At a fundamental level, the difficulty in the investigation of entanglement -quantum correlations -can be traced back to the subtle task of distinguishing it from classical correlations [5]. Correlations can be regarded as classical if they can be induced onto the subsystems solely by local operations and classical communication, necessarily resulting in a mixed state. On the other hand, if a pure quantum state displays correlations between the subsystems, they are of genuinely quantum nature (entanglement). We adopt here a pragmatic approach: if two systems are in toto correlated, then this correlation has to be retrieved between the outcomes of some local measurements performed on them. We, therefore, investigate quadrature correlations in CV states. We are also motivated by the experimental adequacy: field quadratures can be efficiently measured by homodyne detection, without the need for complete state tomography. Specifically, in this paper, we study optimal correlations in bit strings obtained by digitalizing the outcomes of joint quadrature measurements on a two-mode CV system. First, we apply our procedure to Gaussian states (GS), finding that bit quadrature correlations provide a clearcut quantification of the total correlations between the two modes. They are monotonic with the entanglement on pure states, and can be arbitraril...

Entropic Entanglement Criteria for Continuous Variables

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Strong subadditivity for log-determinant of covariance matrices and its applications

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Spatial correlations in parametric down-conversion

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