Separability Criteria for Quantum Mixed States in Terms of Trace Norm

Otfried Gühne

Hui Zhao

A new necessary conditions for separability of mixed states in bipartite and multipartite quantum systems of arbitrary dimensions are presented. This method requires only a few local measurements. If a quantum state in a bipartite (or multipartite) system of arbitrary dimension is separable, we show that the trace norm of a constructive matrix is less than or equal to 1.

Wei Cheng

So far the positive partial transpose (PPT) criterion and the computable cross norm (CCN) criterion are two strongest operational separability criteria of bipartite mixed quantum states. In this paper, we exactly connect the PPT criterion and the CCN criterion with the matrix reorderings found by Poluikis and Hill [J. A. Poluikis and R. D. Hill, Linear Algebra and Its Applications 35: 1-10 (1981)] and Oxenrider and Hill [C. J. Oxenrider and R. D. Hill, Linear Algebra and Its Applications 69: 205-212 (1985)]. Motivated by work by Chen etc. [K. Chen, S. Albeverio, and S.-M. Fei, Physical Review Letters 95, 040504 (2005), Physical Review Letters 95, 210501 (2005)], we compute the trace norm of several bound entangled states and illuminate the equivalence among the elements of group generated by the eight matrix reorderings.

Zhao, Hui Hui, Yu +1 more

Paweł Wocjan, Michał Horodecki

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are combinatorially equivalent or not.

Chi‐Shan Yu, H. S. Song