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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

A criterion for spontaneous state reduction

Donald Bedford, Derek Wang

Published Jan 1, 1977• 11 citations
Publisher
DOI
10.1007/bf02727957

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A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.

arXiv:quant-ph/0312185•Physical Review A•Dec 17, 2003•63 citations
In this paper, we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.
We propose a family of entanglement witnesses. These witnesses are constructed by using local orthogonal observables, and therefore can be easily measured by means of local measurements and classical communications. From the Jamiokowski isomorphism we obtain the corresponding positive maps that are not completely positive. These maps lead to a new separability criterion——the O-reduction criterion, which includes the reduction criterion and the realignment criterion as special cases. In addition, we find that the O-reduction criterion can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables. As applications, we construct an explicit entanglement witness for the first bound entangled state found by Horodecki in 1997,and as well introduce a family of dd bound entangled states, whose entanglement can be detected by permuting local orthogonal observables.
A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.