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arXiv:quant-ph/0208002 · Physical Review A

Concurrence-based entanglement measures for isotropic states

Pranaw Rungta, Carlton M. Caves

Published Jan 14, 2003• 206 citations
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Abstract

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.

DOI
10.1103/physreva.67.012307

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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.
The entanglement content of superpositions of quantum states is investigated based on a measure called concurrence. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound [N. Linden et al., Phys. Rev. Lett. 97, 100502 (2006)].
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.