Hilbert-Schmidt Orthogonality of det(rho) and det(rho^{PT}) over the Two-Rebit Systems rho and Further Determinantal Moment Analyses

Abstract

A complete description of the multitudinous ways in which quantum particles can be entangled requires the use of high-dimensional abstract mathematical spaces. We report here a particularly interesting feature of the nine-dimensional convex set-endowed with Hilbert-Schmidt (Euclidean/flat) measure-composed of two-rebit (4 × 4) density matrices (ρ). To each ρ is assigned the product of its (nonnegative) determinant |ρ| and the determinant of its partial transpose |ρ P T |negative values of which, by the results of Peres and Horodecki, signify the entanglement of ρ.Integrating this product, |ρ||ρ P T | = |ρρ P T |, over the nine-dimensional space, using the indicated (HS) measure, we obtain the result zero. The two determinants, thus, form a pair of multivariate orthogonal polynomials with respect to HS measure. However, orthogonality does not hold, we find, if the symmetry of the nine-dimensional two-rebit scenario is broken slightly, nor with the use of non-flat measures, such as the prominent Bures (minimal monotone) measure-nor in the full HS extension to the 15-dimensional convex set of two-qubit density matrices. We discuss relationsinvolving the HS moments of |ρ P T |-to the long-standing problem of determining the probability that a generic pair of rebits/qubits is separable.

Joint Hilbert-Schmidt Determinantal Moments of Product Form for the Two-Rebit and Two-Qubit and Higher-Dimensional Quantum Systems

“…, 13, and similarly for |ρ| k |ρ P T | 4 2−rebit/HS , we were not able to determine, in the same manner, encompassing expressions for them.) As a special case (k = 1) of (3), we obtain the rather remarkable moment result, zero, already reported in [8]. The immediate interpretation of this finding is that for the generic two-rebit systems, the two determinants |ρ| and |ρ P T | comprise a pair of nine-dimensional orthogonal polynomials [9][10][11] with respect to Hilbert-Schmidt measure.…”

“…The real part of one of the roots of A 4 is 2.999905, suggesting to us some possible interesting asymptotic behavior of the roots of these numerators, κ → ∞. In our previous related study [8][sec. II.B.2], we were also able to discern the general structure that the denominators of certain "intermediate [rational] functions" used in computing the (univariate) moments of ρ P T | κ 2−rebit/HS , κ = 1, .…”

“…Since these ratios (6) are so comparatively simple, it suggested to us that we might be more able to progress in a series of analyses [12][13][14][15][16][17][18][19] (mainly devoted to the determination of separability probabilities), by making our initial goal the computation of these ratios for still higher-order moments-rather than the direct computation of the very small values, having lengthy multi-digit denominators, of the moments (|ρ||ρ P T |) k 2−rebit/HS themselves. (In [8][eqs, (33)-(41)], we were able to report and analyze the first nine moments |ρ P T | k -the first two of which, − 1 858 and 27 2489344 , can be obtained by directly setting k = 0 in (3) and (4), respectively. However, to this point, we have not found any associated similarly compact sequences of moment ratios, as above.)…”

“…From our four new two-rebit moment results (3), ( 4), ( 7) and ( 12), we see that the constant terms in the 3κ-degree numerator A κ are −16, 4860, −3612816 and 6610161600 for κ = 1, 2, 3, 4. Since we had previously computed [8][eqs, (33)-( 41)] the moments of |ρ P T | κ 2−rebit/HS , κ = 1, . .…”

Radial and Azimuthal Profiles of Two-Qubit/Rebit Hilbert-Schmidt Separability Probabilities and Related 3-D Visualization Analyses

“…3)-of one-eighth of the unit ball (centered at the origin, of course) in 15-dimensional Euclidean space. Further, for the two-rebit states-that is those assigned 4 × 4 density matrices, the entries of which are restricted to real values [18][19][20]--the oneto-one correspondence is with the points of an analogous one-eighth part (a "hemidemisemihypersphere") of the unit ball in 9-dimensional space.…”