2021/04/22 by Slater, Paul B.
#15B52 #60B20 #81P16 #81P40 #81P45 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2104.11071
We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by Życzkowski and Khvedelidze, Rogojin and 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 A of dimensions k × k+ 2 (N-k), while for a real scenario, one employs a Ginibre matrix B of dimensions k × k+1+ 2 (N-k). Then, the k × k product A A† or B BT is diagonalized-padded with zeros to size N × N-and rotated, obtaining a random density matrix. Implementing the 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 (387)/(5000) =(32 ⋅ 43)/(23 ⋅ 54)=.0774. A conjecture for the HS separability probability of rebit-retrit systems of full rank is (860)/(6561) =(22 ⋅ 5 ⋅ 43)/(38) ≈ 0.1310775 (the two-rebit counterpart has been proven to be (29)/(64)=(29)/(26)). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be (59049)/(1000000)=\frac31026 ⋅ 56 ≈ 0.059049, with the common factor 43 cancelling. As to the intermediate rank-5 probability, 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) =(33)/(23 ⋅ 53), 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)/( 35 ⋅ 11) ≈ 0.0261878 rank-4 to rank-6 probability ratio.