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arXiv:0808.2581 · Physical Review A

Separability bounds on multiqubit moments due to positivity under partial transpose

A. R. Usha Devi, A. K. Rajagopal

Published Dec 19, 2008• 1 citation
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Abstract

Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for noncommuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit moment matrix (based on PPT constraints on bipartite divisions of the density matrix) and show that three-qubit pure states with nonzero tangle violate these PPT moment constraints. Further, we recover a necessary and sufficient condition of separability in a multiqubit Werner state through PPT bounds on moments.

DOI
10.1103/physreva.78.062334

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Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for noncommuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit moment matrix (based on PPT constraints on bipartite divisions of the density matrix) and show that three-qubit pure states with nonzero tangle violate these PPT moment constraints. Further, we recover a necessary and sufficient condition of separability in a multiqubit Werner state through PPT bounds on moments.
We express the positive-partial-transpose (PPT) separability criterion for symmetric states of multiqubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-jjj state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a two-qubit system and the Bell bases used for the teleportation of a one- or two-qubit state.
We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-\ifmmode \dot{Z}\else \.{Z}\fi{}ukowski criterion, and the D\"ur-Cirac criterion) and also by showing their great noise robustness for a variety of multiqubit states, including NNN-qubit Greenberger-Horne-Zeilinger states and Dicke states. Furthermore, for $N\ensuremath{\geqslant}3$ they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only N+1N+1N+1. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the antidiagonal matrix elements of a state in terms of its diagonal matrix elements.
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.
We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N.3N.3N. If it exceeds this number, we show that one can subtract product vectors until it decreases to 3N,3N,3N, while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.
In this Letter, we show that the fulfillment of uncertainty relations is a sufficient criterion for a quantum-mechanically permissible state. We specifically construct two pseudospin observables for an arbitrary nonpositive Hermitian matrix whose uncertainty relation is violated. This method enables us to systematically derive separability conditions for all negative partial-transpose states in experimentally accessible forms. In particular, generalized entanglement criteria are derived from the Schrödinger-Robertson inequalities for bipartite continuous-variable states.
We present a set of inequalities for detecting quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states. For $2\ensuremath{\bigotimes}2$ and $2\ensuremath{\bigotimes}3$ systems, the inequalities give rise to sufficient and necessary separability conditions for both pure and mixed states. For the case of d>3d>3d>3, these inequalities are necessary conditions for separability, which detect all entangled states that are not positive under partial transposition and even some entangled states with positive partial transposition. These inequalities are given by mean values of local observables and present an experimental way of detecting the quantum entanglement of $2\ensuremath{\bigotimes}d$ quantum states and even multiqubit pure states.