Operational separability criteria and matrix reorderings

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.