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International Conference on Quantum Information

Detecting entanglement with non-hermitian operators

Mark S. Hillery

Published Jan 1, 2011• 0 citations
Publisher

Abstract

There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.

DOI
10.1364/icqi.2011.qmc2

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There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.
We derive several entanglement conditions employing non-Hermitian operators. We start with two conditions that were derived previously for field mode operators and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations such as Gaussian operations. We then apply these conditions to several examples such as the Dicke model. We conclude with a short discussion of how local uncertainty relations with non-Hermitian operators can be used to derive entanglement conditions.
Necessary and sufficient conditions for bipartite entanglement are derived, which apply to arbitrary Hilbert spaces. Motivated by the concept of witnesses, optimized entanglement inequalities are formulated solely in terms of arbitrary Hermitian operators, which makes them useful for applications in experiments. The needed optimization procedure is based on a separability eigenvalue problem, whose analytical solutions are derived for a special class of projection operators. For general Hermitian operators, a numerical implementation of entanglement tests is proposed. It is also shown how to identify bound entangled states with positive partial transposition.
Entanglement finds wide application in the development of quantum information theory. After reviewing states in classical mechanics, this chapter discusses the fundamental aspects of characterization of quantum entanglement like the definition, the detection, the classification and quantification of entanglement. The density operator is used to describe the state of a quantum system in a more general way than provided by the state vector of the Hilbert space. Multipartite entanglement theory is much complex compared to the bipartite case. One cannot generalize the results of bipartite entanglement to multipartite entanglement. The Peres criterion known as positive partial transpose criterion is an important criterion to find the separability of mixed states at least for low-dimensional systems. B.W. Terhal introduced the term entanglement witnesses for operators detecting quantum entanglement states. It is found that any entangled state can be detected by some entanglement witnesses.
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 study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
Quantum correlations in composite and separable quantum systems are characterized by non-vanishing quantum discord. We demonstrate the necessary and sufficient conditions for existence of hermitian witness operators for quantum discord, bypassing the problem of non-convexity of the set of all classical states. We propose an example of a linear witness which can detect quantum correlations in large classes of separable bipartite systems. Decomposition of the witness in terms of a small number of locally measurable operators as shown here, paves the way for experimental detection of quantum correlations beyond entanglement.