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Title:
Investigations into the a priori probability that pairs of quantum bits (qubits) are separable/disentangled
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Abstract:
In a much-cited 1998 Physical Review A paper, "Volume of the set of separable states", the four authors, K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein (ZHSL) posed "The question of how many entangled or, respectively, separable states there are in the set of all quantum states"? In particular, they suggested considering "the problem of quantum separability or inseparability from a measurement theoretical point of view, and ask about relative volumes of both sets." They gave "philosophical", "practical" and "physical" reasons for so doing. In its most basic mathematical form, this problem concerns the relative volumes of two complementary subsets of the 15-dimensional convex set (Q) of 4 x 4 Hermitian positive semidefinite (density) matrices of unit trace. The high-dimensionality apparently precludes direct analytical computations. I (and, in part recently with Charles Dunkl) have taken a considerable variety of indirect-type approaches to the question. Most notably, a diversity of evidence--though yet no formal proof--has emerged strongly indicating that the probability that a state in Q is separable/disentangled is 8/33. These analyses pertain to the use of the Hilbert-Schmidt (flat/Euclidean) measure on Q. I will survey the development of this body of (in large part, Mathematica-based) evidence. Important tools employed, among others, have been moment-inversion procedures and generalized hypergeometric functions. The clear relevance of "Dyson-indices" has emerged. Recent work of Milz and Strunz--which we have been attempting to extend to a bivariate framework--has shown an interesting (univariate) “Bloch radius”-invariance in the general problem.
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