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

Abstract

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 exact 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".