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Ratios of maximal concurrence-parameterized separability functions, and generalized Peres–Horodecki conditions

Paul B Slater

2009
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Journal of Physics A: Mathematical and Theoretical
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The probability that a generic real, complex or quaternionic two-qubit state is separable can be considered to be the sum of three contributions. One is from those states that are absolutely separable, that is those (which can not be entangled by unitary transformations) for which the maximal concurrence over spectral orbits (C_max) is zero. The other two contributions are from the states for which C_max in [0,1/2], and for which C_max in [1/2,1]. We have previously (arXiv:0805.0267) found
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... formulas for the absolutely separable contributions in terms of the Hilbert-Schmidt metric over the quantum states, and here advance hypotheses as to the exact contributions for C_max in [1/2,1]. A crucial element in understanding the other two contributions is the nature of the ratio (R) of the C_max-parameterized separability function for the complex states to the square of the comparable function for the real states--both such functions having clearly displayed jump discontinuities at C_max=1/2. For C_max in [0,1/2], R appears to be of the form 1+ k C_max, except near C_max=1/2, while for C_max in [1/2,1], it seems to be a constant (thus according to the Dyson-index pattern of random matrix theory). Related phenomena also occur for the minimally-degenerate two-qubit states and the qubit-qutrit states. Our results have immediate application to the computation of separability probabilities in terms of other metrics, such as the Bures (minimal monotone) metric. The paper begins with continuous embeddings of the separability probability question in terms of four metrics of interest, using "generalized Peres-Horodecki conditions".

doi:10.1088/1751-8113/42/46/465305
fatcat:fat3i4wnprdspb5mobcrahbou4