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arXiv:1306.2376 · Physical Review A

Universal framework for entanglement detection

Michał Oszmaniec, Marek Kuś

Published Nov 22, 2013• 25 citations
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Abstract

We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons, and fermions. In each case properties of an entanglement measure are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many-particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a nontrivial lower bound of thus obtained generalized concurrence.

DOI
10.1103/physreva.88.052328

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We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons, and fermions. In each case properties of an entanglement measure are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many-particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a nontrivial lower bound of thus obtained generalized concurrence.
We introduce geometric measures of entanglement for indistinguishable particles, which apply to mixed states, multipartite systems, and arbitrary dimensions. They are based on generalized (i.e., not necessarily finite) norms on the set of quantum states and lead to the first necessary and sufficient computational separability criterion in this general setting. The coherent approach developed in the paper allows us to compare, in particular, entanglement for fermionic and distinguishable particles: The entanglement measure for fermionic particles coincides with the corresponding entanglement measure for distinguishable particles up to a factor of k!k!k! where kkk is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.
To determine whether a given multipartite quantum state is separable with respect to some partition, we construct a family of entanglement measures {Rm(ρ)}. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Furthermore, we provide an analytically computable lower bound for {Rm(ρ)} via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.
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.