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arXiv:0803.0757 · Physical Review A

Unifying several separability conditions using the covariance matrix criterion

Oleg Gittsovich, Otfried Gühne, P. Hyllus, Jens Eisert

Published Nov 12, 2008• 101 citations
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Abstract

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 us 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 that 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.

DOI
10.1103/physreva.78.052319

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Abstract A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.
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 us 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.
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A family of separability criteria based on correlation matrix (tensor) is provided. Interestingly, it unifies several criteria known before like, e.g., computable cross-norm or realignment criterion (CNNR), de Vicente criterion, and derived recently separability criterion based on symmetric informationally complete positive operator valued measures (SIC POVMs). It should be stressed that, unlike the well-known correlation matrix criterion or criterion based on local uncertainty relations, our criteria are linear in the density operator and hence one may find unexplored classes of entanglement witnesses and positive maps. Interestingly, there is a natural generalization to multipartite scenario using multipartite correlation matrix. We illustrate the detection power of the above criteria on several well-known examples of quantum states.
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mnρ\in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†UρU^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρρρ is absolutely separable if and only if UρU†UρU^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.