A priori probabilities of separable quantum states

Zyczkowski, Horodecki; Sanpera and Lewenstein (ZHSL) recently proposed a 'n atural measure' on the N-dimensional quantum systems, but expressed surpris e when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632 +/- 0.002) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a soun d theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. W e arrive at this conclusion in two quite distinct ways; the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts t o adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoret ic analysis-which yields the smallest probabilities of disentanglement-uses the minimum number of parameters possible, that is N-2 - 1, as opposed to N-2 + N - 1 (although this 'over-parametrization', as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we a re not able to obtain precise estimates of these probabilities and the need for additional (possibly supercomputer) analyses is indicated-particularly so for higher-dimensional quantum systems (such as the 2 x 3 ones, which w e also study here).