We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria -such as the computable cross norm and local uncertainty criteria -as well as their limitations. Entanglement plays a central role in applications of quantum information science as well as in the foundations of quantum theory. Quite naturally, one of the problems that have received a significant amount of attention is the question to decide whether a given state is entangled or separable. In fact, the development of separability criteria [1,2,3,4,5,6] has been one of the key activities in quantum information theory: On top of certifying a given state to be entangled, often in an experimental context, they often provide physically valuable intuition concerning the structure of the entanglement in a given state. Among these attempts, methods using positive maps [1] or convex geometry [4,5,6] turned out to be fruitful.In this work, we propose a unifying approach to finding criteria for separability for finite dimensional systems in terms of covariance matrices (CMs). In the infinitedimensional setting (in particular for Gaussian states) such CMs constitute a well-established and powerful tool, not least due to the experimental accessibility of quadrature measurements using homodyning [7,8,9]. In contrast, for finite-dimensional systems, the theory is yet hardly developed [10,11]. We introduce a framework for CMs for finite-dimensional systems, formulate a general separability criterion and evaluate it for various scenarios. The merits of this approach are two-fold: (i) The resulting criteria are very strong and allow, notably, to detect many bound entangled states. (ii) Our approach provides a framework to link and understand several existing criteria like a recent criterion using the Bloch representation [12], the one based on local uncertainty relations (LURs) [13], or the cross-norm or realignment (CCNR) criterion [2], the latter being an immediate corollary of our theory.The main idea. -Let us start by defining CMs. Let ̺ be a given quantum state and let {M k : k = 1, . . . , N } be some observables. Then the N × N CM γ -dependent on the state ̺ and the choice for {M k } -is given by
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion -the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others -are in fact a corollary of the covariance matrix criterion.
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