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Noiseless Circuits for the Peres Criterion

Hilary A. Carteret

Published Sep 29, 2003• 2 citations
OpenAlex

Abstract

In this paper we give a method for constructing circuits that can determine the spectrum of a partially transposed density matrix, without requiring either a tomographically complete POVM or the addition of noise to make the spectrum non-negative. These circuits depend only on the dimension of the Hilbert space and are otherwise independent of the state.

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Entanglement in SU(2)-invariant quantum systems: The positive partial transpose criterion and others

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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.

arXiv:quant-ph/0503123•Physical Review A•Jul 11, 2005•36 citations
In this paper we give a method for constructing circuits that can determine the spectrum of a partially transposed density matrix, without requiring either a tomographically complete POVM or the addition of noise to make the spectrum non-negative. These circuits depend only on the dimension of the Hilbert space and are otherwise independent of the state.
We present a method for checking the Peres separability criterion in an arbitrary bipartite quantum state ${\ensuremath{\rho}}_{\mathrm{AB}}$ within local operations and classical communication scenario. The method does not require noise operation which is needed in making the partial transposition map physically implementable. The main task for the two observers, Alice and Bob, is to measure some specific functions of the partial transposed matrix. With these functions, they can determine the eigenvalues of ${\ensuremath{\rho}}_{\mathrm{AB}}^{{T}_{\mathrm{B}}}$, among which the minimum serves as an entanglement witness.
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.