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Physical Review A

Bipartite entanglement measure based on covariance

Isabel Sainz, Gunnar G. Bjork

Published Jun 14, 2007• 16 citations
Publisher

Abstract

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

DOI
10.1103/physreva.75.062317

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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.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
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.