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AIP conference proceedings

Measures of Quantum Entanglement for Three-Level Bipartite Systems

César Herreño F.

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Quantum entanglement is of great current interest because of its key role in the modern theory of quantum information. Entanglement in two level bipartite systems is well understood. There exist several useful measures for these systems which include concurrence and reduced entropy. Recently entanglement has been studied for bipartite three‐level atomic systems. These authors have developed interesting measures using extensions of Wootters concurrence and the generators of group SU(3) for the spin operators. In this work we generalize concurrence using the extension of Pauli spin operators to three‐level atomic systems. We explore the above concurrence for three level atomic systems and their extension to n‐level atomic systems. We compare our results with other measures of entanglement.

DOI
10.1063/1.1834426

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Quantum entanglement is of great current interest because of its key role in the modern theory of quantum information. Entanglement in two level bipartite systems is well understood. There exist several useful measures for these systems which include concurrence and reduced entropy. Recently entanglement has been studied for bipartite three‐level atomic systems. These authors have developed interesting measures using extensions of Wootters concurrence and the generators of group SU(3) for the spin operators. In this work we generalize concurrence using the extension of Pauli spin operators to three‐level atomic systems. We explore the above concurrence for three level atomic systems and their extension to n‐level atomic systems. We compare our results with other measures of entanglement.
We define the concurrence hierarchy as d − 1 independent invariants under local unitary transformations in d -level quantum system. The first one is the original concurrence defined by Wootters (1998 Phys. Rev. Lett. 80 2245) and Hill and Wootters (1997 Phys. Rev. Lett. 78 5022) in a two-level quantum system and generalized to the d -level pure quantum state case. We propose to use this concurrence hierarchy as a measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.
We propose an entanglement measure for pure M ⊗ N bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a 2 ⊗ 2 system, via a 2 ⊗ 3 system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure and for experimentally measuring the entanglement. The form of the measure is similar to the generalized concurrence. In the case of 2 ⊗ 3 systems, we prove that our measure, which is directly measurable, equals the concurrence. It is also shown that, in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of M ( M − 1) N ( N − 1)/2 Bell states.
ABSTRACT Characterizing the entanglement in multipartite quantum systems remains a central challenge in the theory of quantum entanglement. We establish a comprehensive framework for constructing ‐entanglement measures by leveraging bipartite entanglement measures and general ‐entanglement measures () with arithmetic and geometric averaging schemes. Numerical examples demonstrate that these measures outperform the existing ones, such as ‐ME concurrence, geometric mean (k‐GM) concurrence, and the generalized ‐entanglement measure by inducing distinct entanglement orderings and offering superior resolution. Furthermore, we reveal the operational relationship between the ‐entanglement and the local quantum coherence, verifying that the persistence of entanglement is intrinsically constrained by the system's capacity to maintain the local coherence. Finally, to overcome the inherent limitation of the conventional three‐tangle in characterizing the genuine multipartite entanglement of the GHZ‐W mixed states, we design a novel GME measure. Numerical results confirm that this measure effectively eliminates the detection blind spots where vanishes, providing a complete and practical mathematical tool for the quantitative analysis of genuine multipartite quantum correlations.
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.
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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.