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arXiv:1510.03656 · Physical Review A

Simple permutation-based measure of quantum correlations and maximally-3-tangled states

Udaysinh T. Bhosale, Arul Lakshminarayan

Published Aug 31, 2016• 5 citations
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Abstract

Quantities invariant under local unitary transformations are of natural interest in the study of entanglement. This paper deduces and studies a particularly simple quantity that is constructed from a combination of two standard permutations of the density matrix, namely, realignment and partial transpose. This bipartite quantity, denoted here as ${R}_{12}$, vanishes on large classes of separable states including classical-quantum correlated states, while being maximum for only maximally entangled states. It is shown to be naturally related to the 3-tangle in three-qubit states via their two-qubit reduced density matrices. Upper and lower bounds on concurrence and negativity of two-qubit density matrices for all ranks are given in terms of ${R}_{12}$. Ansatz states satisfying these bounds are given and verified using various numerical methods. In the rank-2 case, it is shown that the states satisfying the lower bound on ${R}_{12}$ versus concurrence define a class of three-qubit states that maximize the tripartite entanglement (the 3-tangle) given an amount of entanglement between a pair of them. The measure ${R}_{12}$ is conjectured, via numerical sampling, to be always larger than the concurrence and negativity. In particular, this is shown to be true for the physically interesting case of $X$ states.

DOI
10.1103/physreva.94.022344

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We derive an inequality for three fermions with six single-particle states which reduces to the sum of the famous Coffman-Kundu-Wootters inequalities when an embedded three-qubit system is considered. We identify the quantities which are playing the role of the concurrence, the three-tangle and the invariant $det{\ensuremath{\rho}}_{A}+det{\ensuremath{\rho}}_{B}+det{\ensuremath{\rho}}_{C}$ for this tripartite system. We show that this latter one is almost interchangeable with the von Neumann entropy and conjecture that it measures the entanglement of one fermion with the rest of the system. We prove that the vanishing of the fermionic ``concurrence'' implies that the two-particle reduced-density matrix is a mixture of separable states. Also, the vanishing of this quantity is only possible in the Greenberger-Horne-Zeilinger class, where some genuine tripartite entanglement is present and in the separable class. Based on this, we conjecture that this ``concurrence'' measures the amount of entanglement between pairs of fermions. We identify the well-known ``spin-flipped'' density matrix in the fermionic context as the reduced-density matrix of a special particle-hole dual state. We show that, in general, this dual state is always canonically defined by the Hermitian inner product of the fermionic Fock space and that it can be used to calculate covariants under stochastic local operations and classical communication (SLOCC). We show that Fierz identities known from the theory of spinors relate SLOCC covariants with reduced-density-matrix elements of the state and its spin-flipped dual.
Abstract We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.
The detection and estimation of quantum entanglement are the essential issues in the theory of quantum entanglement. We construct matrices based on the realignment of density matrices and the vectorization of the reduced density matrices, from which a family of separability criteria are presented for both bipartite and multipartite systems. Moreover, tighter lower bounds of concurrence and convex-roof extended negativity are derived. Criteria are also given to detect the genuine tripartite entanglement. Lower bounds of the concurrence of genuine tripartite entanglement are presented. By detailed examples we show that our results are better than the corresponding ones in identifying and estimating quantum entanglement as well as genuine multipartite entanglement.
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The detection and quantification of quantum entanglement are central problems in quantum information science. Based on the realignment of density matrices and the vectorization of reduced density matrices, we construct a family of enhanced separability criteria for bipartite and multipartite quantum systems. Our criteria recover the previously proposed realignment-based criteria as special cases and detect more entanglement than the earlier ones. Moreover, we derive tighter lower bounds for the concurrence and the convex-roof extended negativity, criteria for detecting the genuine tripartite entanglement, and lower bounds on genuine-multipartite-entanglement concurrence. The advantages of our results are demonstrated with detailed examples.
Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.