Advances in Mathematical Physics

Volume 2018, Article ID 9365213, 55 pages

https://doi.org/10.1155/2018/9365213

## Formulas for Generalized Two-Qubit Separability Probabilities

Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA

Correspondence should be addressed to Paul B. Slater

Received 18 April 2017; Revised 31 July 2017; Accepted 10 August 2017; Published 24 May 2018

Academic Editor: Ricardo Weder

Copyright © 2018 Paul B. Slater. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To begin, we find certain formulas
, for
. These yield that part of the* total* separability probability,
, for generalized (real, complex,
, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality
holds. Here denotes a
density matrix, obtained by tracing over the pure states in
-dimensions, and denotes its partial transpose. Further, is a Dyson-index-like parameter with
for the standard (15-dimensional) convex set of (complex) two-qubit states. For
, we obtain the previously reported Hilbert-Schmidt formulas, with
(the real case),
(the standard complex case), and
(the quaternionic case), the three simply equalling
. The factors are sums of polynomial-weighted generalized hypergeometric functions
,
, all with argument
. We find number-theoretic-based formulas for the upper () and lower () parameter sets of these functions and, then, equivalently express in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of
,
, and
, parallel to the one obtained previously (Slater 2013) for
. For nonnegative half-integer and integer values of
,
(as well as
) has descending roots starting at
. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for
itself. The possibility of an analogous “master” formula for
is, then, investigated, and a number of interesting results are found.

#### 1. Introduction

In a previous paper [1], a family of formulas was obtained for the (total) separability probabilities of generalized two-qubit states ( ) endowed with Hilbert-Schmidt ( ) [2] or more generally random induced measure [3, 4]. In this regard, we note that the natural, rotationally invariant measure on the set of all pure states of a composite system ( ) induces a unique measure in the space of mixed states [3, eq. ]. Further, serves as a Dyson-index-like parameter [5, 6], assuming the values for the two-rebit ( ), two-qubit (standard/complex), and two-quaterbit states, respectively.

The concept itself of a “separability probability,” apparently first (implicitly) introduced by Życzkowski et al. in their much cited 1998 paper [7], entails computing the ratio of the volume—in terms of a given measure [8]—of the separable quantum states to all quantum states. Here, we first examine a certain component of . This informs us of that portion—equalling simply in the Hilbert-Schmidt ( ) case [9]—for which the determinantal inequality holds, with denoting a density matrix and denoting its partial transpose. By consequence [10] of the Peres-Horodecki conditions [11, 12], a necessary and sufficient condition for separability in this setting is that . The nonnegativity condition itself certainly holds, independently of any separability considerations. So, the total separability probability can clearly be expressed as the sum of that part for which and that for which . The former quantity will be the one of initial concern here, the ones the formulas will directly yield.

The complementary quantity that for which
can, in the most basic cases of interest, be readily obtained from the* total* separability probability formulas
reported in [1], which took the form
where for integral and half-integral
,
with
Here, for integral
,
is a polynomial of degree
with leading coefficient

In [1], certain -specific formulas ( and , , , and ) had been derived (and we have since continued the integral series to ). Most notably [1, eq. ], Here denotes the total separability probability of the (15-dimensional) standard, complex two-qubit systems endowed with the random induced measure for . Further, in the two-quater[nionic]bit setting [1, eq. ], Also, for the two-re[al]bit scenario [1, eq. ], Tables 1, 2, and 3 in [1] reported for , the, in general, rather simple fractional separability probabilities yielded by these three formulas.

By way of example, we first note that formula (4) yields . Then, since we will find from our analyses below that , we can readily deduce that the corresponding (complementary) separability probability corresponding to the inequalities , for this , scenario, is equal to .

Let us further observe that, for the Hilbert-Schmidt (
) case, strong evidence has been presented [9] that, for the two-rebit, two-qubit, and two-quaterbit cases, the apparent total separability probabilities
of
, and
, respectively, are equally divided between the two forms of determinantal inequalities (cf. [13]). Lovas and Andai have recently formally proven this two-rebit result and presented an integral formula; they hope to similarly yield the two-qubit proportion [14]. (These “half-probabilities,” remarkably, are also the corresponding separability probabilities of the* minimally degenerate* states [13], those for which has a zero eigenvalue.) For
, however, our analyses will indicate that equal splitting is not, in fact, the case. Greater separability probability is associated with the
inequality than
. Thus, in the
,
instance just discussed, we do have
. (On the other hand, if
, then necessarily
, so all the total separability probability
must, it is clear, be assigned to the
component. That is,
. Observations of this nature should help in the further understanding of the intricate geometry of the generalized two-qubit states endowed with random induced measure (cf. [15]).

#### 2. Procedures

##### 2.1. Previous Analyses

To obtain the new formulas to be presented here for the separability probability amounts for which holds, we first employed—as in our prior studies [1, 9, 16, 17]—the Legendre-polynomial-based probability density-approximation (Mathematica-implemented) algorithm of Provost [18] (cf. [19]). In this regard, we utilized the previously obtained determinantal moment formula [1, eq. ] [9, sec. II] (cf. [20]) where the variable has the same sense as indicated above, in equalling , and the bracket notation indicates averaging with respect to the random induced measure. Here, , where the Pochhammer (rising factorial) notation is employed.

On the other hand in [1], a second companion moment formula [16, sec. X.D.6],
had been utilized for density-approximation purposes with the routine of Provost, with the objective of finding the* total* separability probabilities
, associated with the Peres-Horodecki-based inequality
. (These moment formulas had been developed in [16], based on calculations solely for the two-rebit [
] and two-qubit [
] cases. However, they do appear, as well, remarkably, to apply to the two-quater[nionic]bit [
] case, as reported by Fei and Joynt in a highly computationally intensive Monte Carlo study [21]. No explicit formal extension of the Peres-Horodecki positive-partial-transposition separability conditions [11, 12] to two-quaterbit systems seems to have been developed, however [cf. [22–24]]. The value
corresponds, presumably it would seem, to an* octonionic* setting [25, 26].)

##### 2.2. Present Analyses

Here, contrastingly (“dually”) with respect to the approach indicated in [1], we will find -specific formulas ( ) as a function of , that is, , for the indicated one ( ) of the two component determinantal inequality parts of . We utilized an exceptionally large number of the first set of moments above in the routine of Provost [18], helping to reveal—to extraordinarily high accuracy—the rational values that the corresponding desired (partial) separability probabilities strongly appear to assume. Sequences ( ) of such rational values, then, served as input to the FindSequenceFunction command of Mathematica, which then yielded the initial set of -specific (hypergeometric-based) formulas for . (This apparently quite powerful [but “black-box”] command of which we have previously and will now make copious use has been described as attempting “to find a simple function that yields the sequence when given successive integer arguments.” It can, it seems, succeed too, at times, for rational-valued inputs, and perhaps even ones of a symbolic nature.) We, then, decompose into the product form .

#### 3. Common Features of the -Specific Formulas

For each , the FindSequenceFunction command yielded what we can consider as a large, rather cumbersome (several-page) formula, which we denote by . These expressions, in fact, faithfully reproduce the rational-valued (separability probability) sequences that served as input. This fidelity is indicated by numerical calculations to apparently arbitrarily high accuracy (hundreds of digits). (The difference equation results below [Section 5] will provide a basis for our observation as to the rational-valuedness [fractional nature] of these separability probabilities.)

In Figure 1, we show plots of the formulas obtained over the range , for . For fixed , we have , if . In Figure 2, we show a companion plot, exhibiting strongly log-linear-like behavior, for .