Dyson indices and Hilbert–Schmidt separability functions and probabilities

Abstract

A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15-and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are . We, now, set the in the complexcase-conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory-we take S complex (ν) proportional to S 2 real (ν). We also investigate the real and complex qubit-qutrit cases. Now, there are two variables, ν 1 =

Extended studies of separability functions and probabilities and the relevance of Dyson indices

“…This is exactly the square of the corresponding ratio 24 71 we had conjectured (based on extensive numerical and theoretical evidence) for the full (15-dimensional) complex twoqubit case in [17].…”

“…(Around µ = 1, one must have the evident symmetrical relation S HS (µ) = S HS ( 1 µ ).) Accompanying our estimate in the plot is the (well-fitting) hypothetical true form (according with our Dyson-index ansatz [17]) of the HS two-qubit separability function, that is, the fourth power, allotted to all (separable and nonseparable) density matrices is R 1 = 0.123328. The exact value of m sep is, of course, to begin here, unknown, being a principal desideratum of our investigation.…”

“…II A) to further test the validity of our Dyson-index ansatz, first advanced in [17], as well as possibly develop an enlarged perspective on the still not yet fully resolved problem of the two-qubit HS separability probabilities in all three (real, complex and quaternionic) cases. (In [17], we proposed, combining numerical and theoretical argumentsnot fully rising to the level of a formal demonstration-that in the real two-qubit case, the HS separability probability is 8 17 , and in the complex two-qubit case, 8 33 .) A supplementary treatment of the truncated quaternionic scenario (β = 3) is presented in sec.…”

“…1,2], and observed apparently highly oscillatory behavior in both functions in the vicinity of µ = 1. However, a referee of [17] informed us that this was simply an artifact of using standard machine precision, and that with sufficiently enhanced precision [only recently available for plotting purposes in Mathematica 6.0]-as now employed in Fig. 2-the oscillations could be seen to be, in fact, illusory.)…”

“…Now, we will simply assume-in line with our basic Dyson-index ansatz, substantially supported in [17] and above-that the corresponding separability function is of the form…”