Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Previously, a formula, incorporating a $5F4$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $\left\langle \left\vert \rho^{PT}\right\vert ^{n}\left\vert \rho\right\vert ^{k}\right\rangle /\left\langle \left\vert \rho\right\vert ^{k}\right\rangle$ of $4 \times 4$ density-matrices ($\rho$), and their partial transposes ($\rho^{PT}|$) was applied with $k=0$ to the generalized two-qubit separability-probability question. The formula can, further, be viewed we note here, as an averaging over"induced measures in the space of mixed quantum states". The associated induced-measure separability probabilities ($k =1, 2,\ldots$) are found--{\it via} a high-precision density approximation procedure--to assume interesting, relatively simple rational values in the two-re[al]bit ($\alpha = \frac{1}{2}$), (standard) two-qubit ($\alpha = 1$) and two-quater[nionic]bit ($\alpha =2$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,\ldots) formulas that successfully reproduce the rational values assumed for {\it general} $k$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

The question of the probability that a generic quantum system is separable/disentangled was raised in a 1998 paper ofŻyczkowski, Sanpera, Lewenstein and Horodecki, entitled "Volume of the set of separable states" [5]. Certainly, any particular answer to this question will crucially depend upon the measure that is attached to the systems in question. A large body of literature has arisen from the 1998 study, and we seek to make a significant contribution to it, addressing heretofore unsolved problems. Let us point out the work of Aubrun, Szarek and Ye [6], which addresses questions of a somewhat similar nature to those examined below, while employing the same class of measures. However, their work is set in an asymptotic framework, while we will be concerned with obtaining exact finite-dimensional results. On the other hand, Singh, Kunjal and Simon [7] did focus on finite-dimensional scenarios, but with a distinct form of measure, the one originally used in [5].
We have investigated the possibility of extending to the class of "induced measures in the space of mixed quantum states" [3,8] the line of analysis reported in [1,2], the principal separability probability findings of which have recently been robustly supported, with the use of extensive Monte-Carlo sampling, by Fei and Joynt [9], as well as by Milz  In [1, p. 30], a central role was played by the (not yet formally proven) determinantal moment formula obtained there ρ P T n |ρ| k / |ρ| k = (k + 1) n (k + 1 + α) n (k + 1 + 2α) n 2 6n k + 3α + 3 2 n 2k + 6α + 5 2 2n on the basis of extensive computations. Here ρ P T denotes the partial transpose [11] of the density matrix ρ, and |ρ|, its determinant. The brackets represent averaging with respect to Hilbert-Schmidt measure [12]. Furthermore, α is a random-matrix Dyson-index-like parameter [13], assuming, in particular, the value 1 for the standard density matrices with complex-valued off-diagonal entries.
It subsequently occurred to us that this formula could be readily adapted to the broader class of induced measures by considering, in the notation of [3,8] that where K is the dimension of the ancilla/environment state.
Pursuing this idea, we employed, as in [1,2], the determinantal moment formula above in the Legendre-polynomial-based (Mathematica-implemented) density approximation (inverse) procedure of Provost [4], which possesses a least-squares rationale. We speed the program as originally presented, by incorporating the well-known recursion formula for Legendre polynomials, so that successive polynomials do not have to be computed ab initio.
It is interesting to observe, additionally, that for k = −1 (that is, K = 3), a value not apparently susceptible to use of the principal 5F 4-hypergeometric determinantal moment formula and the density approximation (inverse) procedure of Provost [4], the three basic formulas yield, the (now smaller than Hilbert-Schmidt) further simple rational values 1 8 , 1 14 and 11 442 , respectively, for the rebit, qubit and quaterbit cases (cf. [6, p. 130]). Further, for k = −2 (K = 2), the rebit formula has a singularity, the qubit formula yields 0, and the quaterbit one gives 1 429 = 1 3×11×13 ≈ 0.002331. We have been able to formally extend this series of three formulas to other values of α, as well, including α = 3 2 , 5 2 , 3, 7 2 , 4, 9 2 , 5, 6, . . . , 13 obtaining similarly structured (increasingly larger) formulas. A major challenge that we are continuing to address is to find a single master formula that encompasses these several results, and can itself yield the formula for any specific half-integer or integer value of α (Appendix I).
In pursuit of such a goal, we have developed an alternative (Appendix II) to the Legendrepolynomial-based density approximation procedure of Provost [4], which we have made abundant use of above and in our earlier work [1,2,14]. Though well-conditioned, it perhaps is relatively slow to converge for our purposes, since it takes as the baseline (starting) distribution, the uniform one, which is far from the sharply-peaked ones, with vanishing endpoints, we have encountered in our separability probability investigations. The approach outlined in Appendix II uses base functions of the form ((x − a)(b − x)) α where α is a small positive integer. (Provost does present a number of codes, other than the Legendrepolynomial one, including one based on Jacobi polynomials [4, pp. 15, 24]. It employs an adaptive strategy of matching the first and second given moments to those of a Beta distribution. But we have found this algorithm to be highly ill-conditioned in our specific applications.)  This appendix is based on the random induced separability probability formulas we have The purpose is to find P ρ P T > 0 with respect to the normalized measure |ρ| k with parameter α. The values α = 1 2 , 1, 2 correspond to the real, complex, quaternion cases respectively. The obtained formulas have the form P ρ P T > 0 = 1 − F (α, k) .
The first observation: when α is integer or half-integer is a rational function of k, that is, a ratio of polynomials.

This coefficient is
The second coefficient of p α is c 2 (α) − c 2 (α); the sequence of values for α = 1 . . . 14 is The third observation: when α is a half-integer then where p α (k) is a polynomial of degree 5α − 3 2 with leading coefficient and to apply it to the approximation of (where a < 0 < b) By orthogonality, for m = 0, 1, 2, . . .
The main problem is to approximate b ξ f (x) dx for some given ξ: so and now we observe that the sum over n is the coefficient of Truncate the infinite series to obtain an approximation.
Suppose the process is terminated at some m, then (approximate values) Since polynomial interpolation tends to be not numerically well-conditioned (a lot of cancellation) it is suggested to compute the quantities {q j } , {B j,i } to high precision, or even better, in exact arithmetic for α = 0, 1, 2, . . ..