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arXiv:quant-ph/0703249

Bipartite entanglement measure based on covariances

Isabel Sainz, Gunnar G. Bjork

Published Mar 27, 2007• 0 citations
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Abstract

We propose an entanglement measure for two qudits based on the covariances of a set of generators of the su(N) algebra. In particular, we represent this measure in terms of the mutually unbiased projectors for N prime. For pure states this measure quantify entanglement, we obtain an explicit expression which relates it to the concurrence hierarchy, specifically the I-concurrence and the 3-concurrence. For mixed states we propose a separability criterion.

DOI
10.48550/arxiv.quant-ph/0703249

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We propose an entanglement measure for two qudits based on the covariances of a set of generators of the su(N) algebra. In particular, we represent this measure in terms of the mutually unbiased projectors for N prime. For pure states this measure quantify entanglement, we obtain an explicit expression which relates it to the concurrence hierarchy, specifically the I-concurrence and the 3-concurrence. For mixed states we propose a separability criterion.
We propose an entanglement measure for two quNits based on the covariances of a set of generators of the su(N)\mathrm{su}(N)su(N) algebra. In particular, we represent this measure in terms of the mutually unbiased projectors for NNN prime. For pure states this measure quantifies entanglement, we obtain an explicit expression which relates it to the concurrence hierarchy, specifically the III-concurrence and the three-concurrence. For mixed states we propose a separability criterion.
Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified (q,s)(q,s)(q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed qqq-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified (q,s)(q,s)(q,s)-concurrence in the cases of isotropic and Werner states under the constraint q>1q>1q>1 and qs≥1qs\geq 1qs≥1. Furthermore, we explore the monogamy properties of the unified (q,s)(q,s)(q,s)-concurrence for q≥2q\geq 2q≥2, 0≤s≤10\leq s\leq 10≤s≤1 and 1≤qs≤31\leq qs\leq 31≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified (q,s)(q,s)(q,s)-concurrence with q≥1q\geq 1q≥1 and qs≥1qs\geq 1qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
We present two sets of computable entanglement measures for multipartite systems where each subsystem can have different degrees of freedom (so-called qudits). One set, called ``separability'' measure, reveals which of the subsystems are separable or entangled. For that we have to extend the concept of kkk separability for multipartite systems to a unambiguous separability concept which we call ${\ensuremath{\gamma}}_{k}$ separability. The second set of entanglement measures reveals the ``kind'' of entanglement, i.e., if it is bipartite, tripartite, \dots{}, nnn-partite entangled and is denoted as the ``physical'' measure. We show how lower bounds on both sets of measures can be obtained by the observation that any entropy may be rewritten via operational expressions known as mmm concurrences. Moreover, for different classes of bipartite or multipartite qudit systems we compute the bounds explicitly and discover that they are often tight or equivalent to positive partial transposition.
We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.
Covariance matrices are a useful tool for investigating correlations and entanglement in quantum systems. They are widely used in continuous-variable systems, but recently also for finite-dimensional systems powerful entanglement criteria in terms of covariance matrices have been derived. We show how these results can be used for the quantification of entanglement in bipartite systems. To that aim we introduce an entanglement parameter that quantifies the violation of the covariance matrix criterion and can be used to give lower bounds on the concurrence. These lower bounds are easily computable and give entanglement estimates for many weakly entangled states.