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Physical Review A

Criterion for testing multiparticle negative-partial-transpose entanglement

Bei Zeng, Di Zhou, Peng Zhang, Zhen‐Peng Xu, Lixing You

Published Oct 20, 2003• 6 citations
Publisher

Abstract

We revisit the criterion of multiparticle entanglement based on the overlaps of a given quantum state $\ensuremath{\rho}$ with maximally entangled states. For a system of m particles, each with N distinct states, we prove that $\ensuremath{\rho}$ is m-particle negative partial transpose entangled, if there exists a maximally entangled state $|\mathrm{MES}〉,$ such that $〈\mathrm{MES}|\ensuremath{\rho}|\mathrm{MES}〉>1/N.$ While this sufficiency condition is weaker than the Peres-Horodecki criterion in all cases, it applies to multi-particle systems, and becomes especially useful when the number of particles (m) is large. We also consider the converse of this criterion and illustrate its invalidity with counter examples.

DOI
10.1103/physreva.68.042316

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We revisit the criterion of multiparticle entanglement based on the overlaps of a given quantum state \ensuremathρ\ensuremath{\rho}\ensuremathρ with maximally entangled states. For a system of m particles, each with N distinct states, we prove that \ensuremathρ\ensuremath{\rho}\ensuremathρ is m-particle negative partial transpose entangled, if there exists a maximally entangled state ∣MES〉,|\mathrm{MES}〉,∣MES〉, such that 〈MES∣\ensuremathρ∣MES〉>1/N.〈\mathrm{MES}|\ensuremath{\rho}|\mathrm{MES}〉>1/N.〈MES∣\ensuremathρ∣MES〉>1/N. While this sufficiency condition is weaker than the Peres-Horodecki criterion in all cases, it applies to multi-particle systems, and becomes especially useful when the number of particles (m) is large. We also consider the converse of this criterion and illustrate its invalidity with counter examples.
We revisit the criterion of multi-particle entanglement based on the overlaps of a given quantum state ρρρ with maximally entangled states. For a system of mmm particles, each with NNN distinct states, we prove that ρρρ is mmm-particle negative partial transpose (NPT) entangled, if there exists a maximally entangled state ∣MES>|{\rm MES}>∣MES>, such that >1/N>{1}/{N}>1/N. While this sufficiency condition is weaker than the Peres-Horodecki criterion in all cases, it applies to multi-particle systems, and becomes especially useful when the number of particles (mmm) is large. We also consider the converse of this criterion and illustrate its invalidity with counter examples.
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU(2) transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU(2)-invariant density matrix $\ensuremath{\rho}$ is entirely determined by the diagonal elements of $\ensuremath{\rho}$ in a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU(2)-invariant states. A sufficient criterion for $\ensuremath{\rho}$ having a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
A new criterion necessary and sufficient for the separability of pure bipartite systems for arbitrary finite dimensions is demonstrated; and the corresponding finer quantitative measures or characterizations of entanglement (beyond mere separability or non-separability determination) are discussed. Based on this criterion, we proved that the well-known Peres-Horodecki positivity-of-partial-transform criterion is also necessary and sufficient for separability in the case of pure bipartite systems. The maximum value of entanglement, and the corresponding maximally-entangled states are also worked out in detail.
We propose a sufficient entanglement criterion for the multipartite systems based on quantum Fisher information. It complements the well-established positive partial transpose criterion in the sense that it is also able to detect bound entangled states. Moreover, it improves the well-known upper bound on the quantum Fisher information of separable multiqudit states introduced by Toth et al. [G. Toth and T. Vertesi, Phys. Rev. Lett. 120, 020506 (2018)] and hence detects a greater number of metrologically useful entangled states. This criterion is formulated as a simple inequality which must be fulfilled by the quantum Fisher information of separable states of multipartite systems. Violation of this inequality by a state reveals its entanglement and accordingly confirms its usefulness for quantum metrology. The efficiency of this criterion for detection of entanglement is illustrated through some examples.
We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schr\"odinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\ifmmode\times\else\texttimes\fi{}2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.