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arXiv:quant-ph/0001076 · Journal of Physics A Mathematical and General

Covariance, correlation and entanglement

Richard I. A. Davis, R Delbourgo, P D Jarvis

Published Feb 28, 2000• 17 citations
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Abstract

Some new identities for quantum variance and covariance involving commutators are presented, in which the density matrix and the operators are treated symmetrically. A measure of entanglement is proposed for bipartite systems, based upon covariance. This works for two- and three-component systems but produces ambiguities for multicomponent systems of composite dimension. Its relationship to angular momentum dispersion for symmetric spin states is described.

DOI
10.1088/0305-4470/33/9/312

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Some new identities for quantum variance and covariance involving commutators are presented, in which the density matrix and the operators are treated symmetrically. A measure of entanglement is proposed for bipartite systems, based upon covariance. This works for two- and three-component systems but produces ambiguities for multicomponent systems of composite dimension. Its relationship to angular momentum dispersion for symmetric spin states is described.
Variance is a ubiquitous quantity in quantum information theory. Given a basis, we consider the averaged variances of a fixed diagonal observable in a pure state under all possible permutations on the components of the pure state and call it symmetrized variance. Moreover, we work out the analytical expression of the symmetrized variance and find that such an expression is in the factorized form where two factors separately depend on the diagonal observable and quantum state. By shifting the factor corresponding to the diagonal observable, we introduce the notion referred to as standard symmetrized variance for the pure state which is independent of the diagonal observable. We then extend the standard symmetrized variance to mixed states in three different ways, which characterize the uncertainty, the coherence, and the coherence of assistance, respectively. These quantities are evaluated analytically and the relations among them are established. In addition, we show that standard symmetrized variance is also an entanglement measure for bipartite systems. In this way, these different quantumnesses of quantum states are unified by the variance.
Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.
We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
Abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum sub-systems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states—sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.