Whole and Half Generalized Two-Qubit Hilbert-Schmidt Separability Probabilities

Paul B. Slater, Charles F. Dunkl

Paul B. Slater

Paul B. Slater

The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.

Paul B. Slater

We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of ($2 \times 3$) qubit-qutrit and ($2 \times 4$) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is $(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027$, with the counterpart two-qubit separability probablity being $\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424$. In the ($2 \times 4$) qubit-qudit case, we advance $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ as the PPT-probability. The numerators of the probabilities for $2 \times m$ scenarios are, thus, possibly simply powers of $m$. We also search for the rebit-retrit counterpart to the established two-rebit result of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 \pi^2} \approx 0.54038$, 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 Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.

Paul B. Slater

Paul B. Slater

We seek to develop a Bures (minimal monotone/statistical distinguishability) metric-based series of formulas for the moments of probability distributions over the determinants $|ρ|$ and $|ρ^{PT}|$ of $4 \times 4$ density matrices, $ρ$, for generalized (rebit, quater[nionic]bit,\ldots) two-qubit systems, analogous to a series that has been obtained for the Hilbert-Schmidt (HS) metric. In particular, we desire--using moment-inversion procedures--to be able to closely test the previously-developed conjecture (J. Geom. Phys., 53, 74 [2005]) that the Bures separability probability over the (standard, fifteen-dimensional convex set of) two-qubit states is $\frac{1680 \left(\sqrt{2}-1\right)}{π^8} \approx 0.0733389$--while, in the HS context, strong evidence has been adduced, along the indicated analytical lines, that the counterpart of this value is $\frac{8}{33}$ (J. Phys. A}, 45, 095305 [2012]). Working within the "utility function" framework of Dunkl employed in that latter study, we obtain an interesting $_{10}F_{9}$ balanced hypergeometric function based on a "hybridization" of known Bures and HS terms. This exercise appears to provide an upper bound on the Bures two-qubit separability probability of 0.0798218. We also examine the yet unresolved HS qubit-qutrit scenario. Mathematica calculations indicate that if the same form of hypergeometric paradigm as has been established for the generalized two-qubit HS moments is followed in either the HS qubit-qutrit or Bures two-qubit cases, then a balanced hypergeometric function $_{p}F_{p-1}$ with $p>9$ would be required.