Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fruitfully used in our n=4 three-parameter entropy-maximization-based study quant-ph/0507203 -- we obtain a wide-ranging variety of exact HS-probabilities. For n>6, the probabilities are those of having a partial positive transpose (PPT). For the n=6 case, we also obtain biseparability probabilities; in the n=8,9 instances, bi-PPT probabilities; and for n=8, tri-PPT probabilities. By far, the most frequently recorded probability for n>4 is π/4 = 0.785398. We also conduct a number of related analyses, pertaining to the (one-dimensional) boundaries (both exterior and interior) of the separability and PPT domains, and attempt (with limited success) some exact calculations pertaining to the 9-dimensional (real) and 15-dimensional (complex) convex sets of two-qubit density matrices -- for which HS-separability probabilities have been conjectured, but not verified.
We seek to gain insight into the nature of the determinantal moments 〈|ρ PT | n |ρ| k 〉 Bures of generic (nine-dimensional) two-rebit and (fifteen-dimensional) two-qubit systems (ρ), PT denoting partial transpose. Such information—as it has proved to be in the Hilbert–Schmidt counterpart—should be useful, employing probability-distribution reconstruction (inverse) procedures, in obtaining Bures 2 × 2 separability probabilities. The (regularizing) strategy we first adopt is to plot the ratio of numerically-generated (Ginibre ensemble) estimates of the Bures moments to the corresponding (apparently) exactly-known Hilbert–Schmidt moments (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). Then, through a combination of symbolic and numerical computations, we obtain strong evidence as to the exact values (and underlying patterns) of certain Bures determinantal moments. In particular, the first moment (average) of |ρ PT | (where ) for the two-qubit systems is, remarkably, . The analogous value for the two-rebit systems is . While in the Hilbert–Schmidt case is the ratio of 3 n -degree polynomials in k , it appears to be the ratio of 5 n -degree polynomials in k in the Bures case.
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of
338=3⋅1123≈0.242424 and (the 9-dimensional) two-rebits of
6429=2629≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of
100027=(103)3=23⋅5333=0.027 and a rebit-retrit one of
6561860=3822⋅5⋅43≈0.131078. These four values for
2×m systems (
m=2,3) suggest certain numerator/denominator sequences involving powers of
m, which we further investigate for
m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit
X-states all equal
3π216≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of
X-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is
31ε2(4−ε2), yielding
338, its
k=1 induced measure counterpart is
41ε2(3−ε2)2, yielding
14361=11⋅1361≈0.426573, where
ε is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".