Rational-Valued, Small-Prime-Based Qubit-Qutrit and Rebit-Retrit Rank-4/Rank-6 Conjectured Hilbert-Schmidt Separability Probability Ratios

Abstract

We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by both K. Życzkowski and the group of A. Khvedelidze, I. Rogojin and V. Abgaryan, for the generation of random (complex or real) N × N density matrices of rank k ≤ N with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix (of normal variates) A of dimensions k × k + 2(N − k), while for a real scenario, one employs a Ginibre matrix B of dimensionsThen, the k × k product AA † or BB T is diagonalized-padded with zeros to size N × N -and rotated by a random unitary or orthogonal matrix to obtain a random density matrix with respect to HS measure. Implementing the indicated procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value of 387 5000 = 3 2 •43 2 3 •5 4 = .0774. A prior conjecture for the HS separability probability of rebit-retrit systems of full rank is 860 6561 = 2 2 •5•43 3 8 ≈ 0.1310775 (while the two-rebit counterpart has been proven to be 29 64 = 29 2 6 , and the two-qubit one, very strongly indicated to be 8 33 = 2 3 3•11 ). Subject to these two conjectures, the ratio of the rank-4 to rank-6 probabilities would be 59049 1000000 = 3 10 2 6 •5 6 ≈ 0.059049, with the common factor 43 cancelling. As to the intermediate rank-5 probability, application of a 2006 theorem of Szarek, Bengtsson and Życskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be 27 1000 = 3 3 2 3 •5 3 , while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a 70 2673 = 2•5•7 3 5 •11 ≈ 0.0261878 rank-4 to rank-6 probability ratio.

Density Matrix Diagonal-Block Lovas-Andai-type singular-value ratios for qubit-qudit separability/PPT probability analyses

“…where f (α) = P sep/P P T (α) − P sep/P P T (α + 1) = q(α)2 −4α−6 Γ(3α + 5 2 )Γ(5α + 2) 3Γ(α + 1)Γ(2α + 3)Γ(5α + 13 2 )…”

“…Here, we begin by investigating whether the lines of analysis recently pursued by Lovas and Andai and Slater in[2,3] can be "lifted" in a productive manner to the 20-dimensional rebit-retrit and 35-dimensional qubit-qutrit settings. (Prior conjectures for the associated HS separability probabilities are 860 6561 = 2 2 •5•43 3 = 0.027, respectively[5][6][7].) Central to the Lovas-Andai analysis-and motivating our present research-was the singularvalue ratio[2, Lemma 5, p. 7] …”