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arXiv:1501.07026 · International Journal of Quantum Information

Heuristic for estimation of multiqubit genuine multipartite entanglement

Paulo E. M. F. Mendonça, Marcelo A. Marchiolli, Gerard J Milburn

Published Apr 1, 2015• 3 citations
arXivHTMLPDF

Abstract

For every N-qubit density matrix written in the computational basis, an associated "X-density matrix" can be obtained by vanishing all entries out of the main- and anti-diagonals. It is very simple to compute the genuine multipartite (GM) concurrence of this associated N-qubit X-state, which, moreover, lower bounds the GM-concurrence of the original (non-X) state. In this paper, we rely on these facts to introduce and benchmark a heuristic for estimating the GM-concurrence of an arbitrary multiqubit mixed state. By explicitly considering two classes of mixed states, we illustrate that our estimates are usually very close to the standard lower bound on the GM-concurrence, being significantly easier to compute. In addition, while evaluating the performance of our proposed heuristic, we provide the first characterization of GM-entanglement in the steady states of the driven Dicke model at zero temperature.

DOI
10.1142/s0219749915500239

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For every N-qubit density matrix written in the computational basis, an associated "X-density matrix" can be obtained by vanishing all entries out of the main- and anti-diagonals. It is very simple to compute the genuine multipartite (GM) concurrence of this associated N-qubit X-state, which, moreover, lower bounds the GM-concurrence of the original (non-X) state. In this paper, we rely on these facts to introduce and benchmark a heuristic for estimating the GM-concurrence of an arbitrary multiqubit mixed state. By explicitly considering two classes of mixed states, we illustrate that our estimates are usually very close to the standard lower bound on the GM-concurrence, being significantly easier to compute. In addition, while evaluating the performance of our proposed heuristic, we provide the first characterization of GM-entanglement in the steady states of the driven Dicke model at zero temperature.
For every possible spectrum of -dimensional density operators, we construct an N -qubit X-state of the same spectrum and maximal genuine multipartite (GM-) concurrence, hence characterizing a global unitary transformation that—constrained to output X-states—maximizes the GM-concurrence of an arbitrary input mixed state of N qubits. We also apply semidefinite programming methods to obtain N -qubit X-states with maximal GM-concurrence for a given purity and to provide an alternative proof of optimality of a recently proposed set of density matrices for the purpose, the so-called X-MEMS. Furthermore, we introduce a numerical strategy to tailor a quantum operation that converts between any two given density matrices using a relatively small number of Kraus operators. We apply our strategy to design short operator-sum representations for the transformation between any given N -qubit mixed state and a corresponding X-MEMS of the same purity.
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Genuine multipartite entanglement (GME) is a valuable resource for quantum technologies, but quantifying it in many-body systems can be demanding since measures defined through optimizations over bipartitions are often assumed to require large joint measurements and many measurement settings. We introduce the concepts of GME evaluation length and GME evaluation number, which capture, respectively, the largest subsystem that must be jointly measured and the number of distinct subsystems that need to be measured to evaluate multipartite entanglement measures defined via bipartition optimizations. Taking the genuine multipartite concurrence (GMC) as a canonical example, we show that, for NNN-qubit Haar-random pure states, the GMC can be evaluated using only NNN single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the NNN single-site bipartitions, rather than the exponentially many distinct bipartitions. We further corroborate this behavior numerically for physically accessible states generated by local random quantum circuits. Our results show that the practical cost of quantifying GME can be dramatically lower than suggested by its formal definition, establishing measurement complexity as a complementary operational lens for multipartite entanglement.
Genuine Multipartite Entanglement Measure Based on ααα-concurrence
Quantifying genuine entanglement is a crucial task in quantum information theory. Based on the geometric mean of bipartite ααα-concurrences among all bipartitions, we present a class of well-defined genuine multipartite entanglement (GME) measures GαααC with one parameter ααα for arbitrary multipartite states. We show that the GαααC is of continuity for any multipartite pure states. By utilizing the related symmetry, analytical results of GαααC are derived for any nnn-qubit GHZ states and W states, which show that the GHZ states are more genuinely entangled than the W states. With explicit examples, we demonstrate that the GαααC can distinguish different GME states that other GME measures fail to. These results justify the potential applications of GαααC in characterizing genuine multipartite entanglements.
The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglements. We present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and classify genuine multipartite entanglement. As applications, genuine multipartite entanglement of tripartite systems is analyzed in detail.