Advances in Mathematical Physics
Volume 2018 (2018), Article ID 9365213, 55 pages
https://doi.org/10.1155/2018/9365213

Research Article

Formulas for Generalized Two-Qubit Separability Probabilities

Paul B. Slater

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

Correspondence should be addressed to Paul B. Slater; slater@kitp.ucsb.edu

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 .

Figure 1: Plots of the separability probability formulas over the range , for . For fixed , we have , if .

Figure 2: Plots of over the range , for . For fixed , , if .

3.1. Distinguished Function with 2 as an Upper Parameter in

In each of the eleven -specific formulas obtained, there is a distinguished generalized hypergeometric function, with the (“omnipresent,” we will find) argument of (cf. [27] [28, Ex. 8.6, p. 159]), having 2 as one of the seven upper parameters (cf. [17]).

3.1.1. The Six Lower Parameters

The lower (bottom) six parameters , , of the function conform for all eleven cases to the simple linear rule

The six entries are summed to .

3.1.2. The Six Upper Parameters

The six upper parameters (aside from the seventh -invariant constant of 2 already indicated), , can be broken into one set of two (the numerical parts summing to integers), incorporating consecutive fractions having 6’s in their denominators, and one set of four (the numerical parts also summed to integers), incorporating consecutive fractions having 5’s in their denominators. For the set of two, the smaller of the two upper entries abides by the rule

where the (integer-valued) floor function is employed, while the larger entry is given by

For integral values of , the same values of and are yielded by the interpolating functions

respectively.

For , for illustrative purposes, application of the two rules yields , and for , we have . (We have noted that is an integer. The sequence of these integers—for arbitrary integer or half-integer values of —is found in the On-Line Encyclopedia of Integer Sequences [https://oeis.org/ol.html] as A004523 [“two even followed by one odd”] and as A232007 [“maximal number of moves needed to reach every square by a knight from a fixed position on an chessboard, or −1 if it is not possible to reach every square”].)

For the complementary set of four upper parameters of the function, the entries in order of increasing magnitude are expressible as

For , for illustrative purposes, application of these four rules yields , and for , we have . For arbitrary , the sum of the four terms under discussion minus is an integer, namely, . Further, let us note that, for integral values of , has values

3.2. Distinguished Function with 1 as an Upper Parameter in

Each -specific formula we have found also incorporates a second function (again with argument , which is, to repeat, invariably the case throughout this paper), having all its thirteen parameters simply equalling 1 less than those in the function just described. (A basic transformation exists [consulting the HYP manual of C. Krattenthaler, available at http://mathematik.univie.ac.at], allowing one to convert the thirteen [twelve -dependent parameters, plus 1] of this function [i.e., add 1 to each of them] to those thirteen of the first distinguished previously described, plus other terms.)

3.3. The Remaining Functions, , in

Now, in addition to the two distinguished functions just presented, there are more hypergeometric functions , , for each , where

Each of these additional functions possesses, to begin with, the same seven upper parameters (that is, 2, plus those six indicated in (10), (11), and (13)) and the same six lower parameters (9), as in the first function detailed above (Section 3.1). Then, the seven upper parameters are supplemented by from 1 to 2’s, and the six lower parameters supplemented by 1 to 1’s.

3.4. Large -Free Terms Collapsing to 0

We now point out a rather remarkable property of the formulas for yielded by the FindSequenceFunction command. If we isolate those (often quite bulky) terms that do not involve any of the hypergeometric functions for each already described above, we find (to hundreds of digits of accuracy) that they collapse to zero. These terms, typically, do contain hypergeometric functions similar in nature to those described above, but with the crucial difference that the Dyson-index-like parameter does not occur among their upper and lower parameters. Thus, we are left, after this nullification of terms, with formulas that are simply sums of polynomial-weighted functions (of ), with .

3.5. Summary

To reiterate, for each , our formulas for all contain a single function of the form

There is another distinguished single function, with all its thirteen parameters being one less. Also there are additional functions, ,

with the number of upper 2’s running from 2 to and the number of lower 1’s, simultaneously running from 1 to .

4. Decomposition of into the Product

The formulas for that we have obtained can all be written—we have found—in the product form . The factor involves the summation of the hypergeometric functions indicated above, each such function weighted by a polynomial in , the degrees of the weighting polynomials diminishing as increases. Let us first discuss the other (hypergeometric-free) factor , involving ratios of products of Pochhammer symbols.

4.1. Hypergeometric-Function-Independent Factor

Some supplementary computations (involving an independent use of the FindSequenceFunction command) indicated that these (hypergeometric-free) factors can be written quite concisely, in terms of the upper and lower parameter sets, setting , , as

where the Pochhammer symbol (rising factorial) is employed. We note that, remarkably, , further apparent indication of the special/privileged status of the standard (complex, ) two-qubit states.

4.2. Hypergeometric-Function-Dependent Factor

4.2.1. Canonical Form

In Appendix A, for , we show the “canonical form” we have developed for the factors (cf. [17, Fig. 3]); the component hypergeometric parts of which we have discussed in Section 3.

5. Difference Equation Formulas for

It further appears that all the factors () (Appendix B) can be equivalently written as functions that satisfy first-order difference (recurrence) equations of the form

where ’s are polynomials in (cf. [29]). This finding was established by yet another application of the Mathematica FindSequenceFunction command.

That is, we generated—for each value of under consideration—a sequence () of the rational values yielded by the hypergeometric-based formulas for , to which the command was then applied. While we have limited ourselves in Appendix B to displaying our results for , and 4, we do have the analogous set of results in terms of the hypergeometric functions for the additional instances, , and 9 and presume that an equivalent set of difference equation results is constructible (though substantial efforts with have not to this point succeeded). The initial points in the six difference equations shown are—in the indicated order—,8348/. The next five members of this monotonically increasing sequence are , , . Since, as noted above, , these are the respective separability probabilities themselves. We would like to extend this sequence sufficiently, so that we might be able to establish an underlying rule for it. (However, since the sequence is increasing in value, the Legendre-polynomial density-approximation procedure of Provost converges more slowly as increases, so our quest seems somewhat problematical, despite the large number of moments incorporated [cf. [1, App. II]].)

If in the difference equation for , we replace the term by , then we can add

to the -specific values obtained from the so-modified equation to recover the values generated by the original difference equation.

5.1. Polynomial Coefficients in Difference Equations

5.1.1. The Polynomials

We have for the six () cases at hand (Appendix B) the proportionality relation

where ’s (and ’s)—as indicated in Section 3—are themselves functions of .

5.1.2. The Polynomials

For all six displayed cases,

5.1.3. The Polynomials

Further, for all six cases, the polynomial coefficients —constituting the inhomogeneous parts of the recurrences—are proportional to the product of a factor of the form

and an irreducible polynomial. These irreducible polynomials are, in the indicated order (),

and (for )

The irreducible polynomial for is also of degree 7; that is,

For , this auxiliary polynomial is now the product of times an irreducible polynomial of degree 7; that is,

The coefficients of the highest powers of in all six irreducible polynomials are factorable into the product of 37 and powers of 2 and 5.

6. Hypergeometric-Free Formulas for

In Appendix C we show formulas we have generated for the differences between the formulas for for successive values of . We note that these are hypergeometric-free. We will find below (64) that these obey the formula

7. Partial Separability Probability Asymptotics

7.1. -Specific Formulas

Now, as concerns the eleven formulas () we have obtained for , which have been the principal focus of the paper, we have computed the ratios of the probability for to the probability for . These ranged from 0.419810 () to 0.4204296 (). Let us note here that .

7.2. -Specific Formulas

We had available , and 2 computations for for this scenario. We found that, for each of the three values of , we could construct strongly linear plots—with unit-like slopes between 1.00177 and 1.00297—by taking times the ratio () of the separability probability to the th separability probability. (From this, it appears, simply, that , as .)

7.3. “Diagonal” Formulas

For values , we were able to construct a strongly linear plot by—similarly to the immediate last analysis—taking times the ratio of the separability probability to the th separability probability. Now, however, rather than a slope very close to 1, we found a slope near to one-half, that is, 0.486882. The ()-intercept of the estimated line was 0.894491.

8. Total Separability Probability Formulas

Efforts of our to conduct parallel sets of (-specific) analyses to those reported above for the total separability probabilities , corresponding to , rather than for that component part of the probabilities satisfying the determinantal inequality had been unsuccessful, in the following sense. We had computed what appeared to be appropriate sequences of rational values for and for , but the Mathematica FindSequenceFunction did not yield any underlying governing rules. (This can be contrasted with the results in [1], where such successes were reported in obtaining -specific [] formulas [ and ], including (4)–(6). However, we do eventually succeed in characterizing the nature of these two () sequences [cf. Appendix G].)

In Figure 3, we plot the logs of these seventy-four total separability probabilities (based on ). A least-squares linear fit to these points is , while in Figure 4, we show (based on ) the counterpart, with an analogous fit of . (We note that .) Although the slopes of these two linear fits are quite close, the -intercepts themselves are of different sign. The predicted probabilities at , the first of the fitted points, are 0.289019 and 0.602955, respectively. In statistical parlance, the “coefficients of determination” or for the two linear fits to the log-plots are both greater than 0.99995. Further, sampling at , we obtained an estimated, again, very well-fitting line of .

Figure 3: Plot of logs of total separability probability () for random induced measure with . A least-squares linear fit to these 74 points is .

Figure 4: Plot of logs of total separability probability () for random induced measure with . A least-squares linear fit to these 124 points is . We note that .

8.1. Total Separability Probability Asymptotics

8.1.1. -Specific Formulas

Dunkl, on the basis of our , analysis just above (and its companions), did advance the bold and (certainly, in our overall analytical context) elegant hypothesis of a -invariant () slope equal to , which does seem quite consistent with the numerical properties we have observed (i.e., with the direction in which the estimates of the slope tend as the number of points sampled increase).

As further support, we obtained for a , analysis, a slope estimate of , again converging in the direction of . (Let us remark, regarding the generalized two-qubit version of the [simpler, lower-dimensional] X-states model [20, 30, 31], that it has been shown that the slope of a [now, log-log] plot of versus tends to , as .)

8.1.2. -Specific Formulas

These interesting observations led us to reexamine, for their asymptotic properties, the “dual” formulas (4)–(6), given above and previously reported in [1]. We now find—through analytic means—that for each of , and , that, as , the ratio of the logarithm of the -st separability probability to the logarithm of the th separability probability is (cf. [32, sec. 7]). (Presumably, the pattern continues for larger , but the required computations have, so far, proved too challenging.)

For example, for , we have for the two-rebit total separability probability , as a function of , formula (6) given above. In Figure 5, we show a plot of versus . The slope of a least-squares-fitted line based on the 200 points is , while . (As we increase from but hold the number of points constant at 200, the approximation of the slope to this value slowly weakens.)

Figure 5: Plot of ) versus . The slope of a least-squares-fitted line is −0.523280, while .

9. “Concise Formulas” for

Let us remind the reader of the interesting “concise” (Hilbert-Schmidt []) generalized two-qubit result—applying Zeilberger’s (“creative telescoping”) algorithm [33]—of Qing-Hu Hou, reported in [17, eqs. ]. This—in our present notation—takes the form

where

We divide the originally reported formula by one-half [9], since we have moved here from the () Hilbert-Schmidt original scenario to its counterpart. Using our earlier results above, Hou has further been able to construct the analogue of the “concise formula” above (a Maple worksheet of his is presented in Appendix D [cf. [17, Figs. 5, 6]). That is,

where

(These results correspond to the variable “dif” in Appendix D.) Thus, in passing from the (symmetric ) Hilbert-Schmidt setting to the random induced scenario, the degree of “conciseness” somewhat diminishes. The polynomials and in this pair of formulas are the same as the difference equation (19) polynomials and , given in (25) and (26).

At this point in our research, we were able to employ the Mathematica-based HolonomicFunctions package of Christoph Koutschan of the Research Institute for Symbolic Computation (RISC) of Johannes Kepler University. With it, we were readily able to derive the result

where

We see that the polynomial above is, in expanded form, the same as given in (24).

For the standard trio of Dyson indices , and 2, this formula for yields , and , respectively, while lead to . (Also, gives , and yields .) Additionally, gives , where is Baxter’s four-coloring constant for a triangular lattice; that is, . (Also, gives .) Continuing with this “zoo” of remarkable results (suggested largely by use of WolframAlpha), gives , where is Gauss’s constant, that is, the reciprocal of the arithmetic-geometric mean of 1 and , equalling . Now, for , we get , where is the Lemniscate constant; that is, . To continue, gives us , where is a known constant of interest (cf. [17, sec. 3.2.1]).

Further, employing the RISC package, we obtained

where

is a degree 7 polynomial in .

For the standard trio of Dyson indices , and 2, this formula for yields and , respectively.

It would clearly be of interest to find such “concise” expressions for , encompassing the four () examples above, as well as values . (We have so far encountered certain difficulties in applying the RISC HolonomicFunctions program to the scenario.)

10. Series of Exact -Values for Certain and Associated Formulas

10.1. Series

We have previously noted , where is Baxter’s four-coloring constant for a triangular lattice; that is, . For the succeeding values , we obtain , , , , , , , , .

For the series () with , we obtain , , . Here, all the denominators () are simply increasing powers of 2.

For the series () with , we obtain , , , , , where is the indicated Lemniscate constant; that is, .

10.2. Formulas

This last series has the explanatory rule ()

where the regularized hypergeometric function is indicated. For , the formula yields , while our prior computations indicate a value of .

Also (now agreeing for ),

where

Absorbing the Lemniscate constant , we obtain, equivalently,

We see some obvious parallels between the formulas for and . (We note that , where Gauss’s constant is indicated.)

In fact, we can subsume both these last two formulas () into

Building upon (45), we found

Strikingly simply, we have the result (valid for all eleven values for which we have computations)

having a root at . So, using formula (6) above, we find that the complementary separability probability, that is, associated with the determinantal inequality , is

Also, we have found (agreeing with the earlier formulas for all eleven ) that

for , with the results for of differing from the prediction of given by the early formulas given above.

Further, we have

having a root at .

To continue (with a root at ),

Further,

having a root at , agreeing with our earlier formulas for all eleven (as well as and 10).

Our formulas give that is equal to for both and 1 and equal to for . Here, presumably corresponds to a classical/nonquantum scenario.

Dunkl has observed that, for integral values of , the arguments of the gamma functions in the numerators are of the form and in the denominators of the form . He further noted that the leading coefficient (highest power) in the polynomial takes the form . Also, the second leading coefficient (normalizing the leading coefficient of the polynomial to 1) follows the rule

Similarly, the so-normalized leading third coefficient takes the form

We have been able to generate a considerable number (including ) of such formulas, a limited number of which we present in Appendix E.

Each half-integral formula contains a gamma function in its numerator with an argument of the form and in its denominator a gamma function with an argument of the form .

10.3. Sets of Consecutive Negative Roots

All the formulas we have (Appendix E), for nonnegative half-integer and integer values of , have roots (in unit steps) from downwards to . So, there are

associated roots. (The formulas displayed in Appendix E with negative values of match our computations only above certain [nonnegative] values of .)

11. Hypergeometric Formula for

Based on the information presented above, including that, in an extended form of Appendix E, Dunkl developed the following formula, succeeding in reproducing our computations for :

where

(In explaining how this formula was obtained, Dunkl stated that the key insights were that factors nicely and .) If we let both and be free and perform the indicated summation in (56), we obtain a hypergeometric-based formula that appears to reproduce not only the formulas in Appendix E for integer , but also half-integer and other nonnegative fractional values (such as ) of .

Dunkl argued that for and

Taking the limit as

thus

(Let us point the reader to an interesting partial matching between entries of the hypergeometric function and arguments of the gamma functions.) The resultant master formula takes the form

The value from which these terms are subtracted itself has an interesting provenance. It was obtained by conducting the sum indicated in (56), not over from 0 to as indicated there, but over from 0 to ; that is, . (The formula can then be recovered by subtracting the sum over from to ; that is, .) This resulted in the expression (cf. http://math.stackexchange.com/questions/1872364/prove-that-a-certain-hypergeometric-function-assumes-either-the-value-frac1)

For this gives us the indicated value of . Let us note that, for both this function and the immediately preceding, the sums of the denominator entries minus the sums of the numerator parameters equal , while if these differences had been 1, the two functions could be designated as “-balanced” [34].

In the notation of this section (cf. (30)),

11.1. Implications for Formula

Let us note that, for the Hilbert-Schmidt () case, apparently [9], , where

Thus, any presumed “master formula” for (Section 12) should reduce to for (cf. eqs. (31)–(33)). We have been investigating the use of as an initial candidate for , then padding out the six upper and five lower entries of the function with additional pairs of entries, identical for , but different for . Then, for , the initial candidate would be recovered. (The somewhat interesting “-balanced” property, mentioned above, or some -free counterpart of it would, then, be lost.) Initial limited numerical investigations along these lines have been somewhat disappointing, as they appeared to indicate that the best fits would be obtained for pairs of padded entries with equal coefficients of . Also, fits to values of did not seem to be improved through the padding strategy.

However, another considerably more interesting approach along similarly motivated lines was, then, developed. We mapped the parameter in the function to , so that for the original function would be recovered, no matter the specific value of . We evaluated the transformed functions by seeing how well they fit the series of (known) eight values , . For the original , the figure-of-merit for the fit was 0.7703536. This figure rather dramatically decreases/improves as increases, reaching a near minimum of 0.0479732 for (and 0.108008 for and 0.153828 for .) The implications of this phenomenon will be further investigated. Perhaps it might be of value to combine the last two (padding and scaling of ) strategies.

11.2. Conjectured Identity

In relation to (63), Dunkl formulated the conjecture

To avoid zero denominators, it is necessary that . For , the value is 1, while the sum is rational for , , .

In response to this conjecture, C. Koutschan wrote: “The sum fits into the class of identities that can be done with Zeilberger’s algorithm. I attach a Mathematica notebook with some computations. More precisely, using the creative telescoping method, my program finds a linear recurrence equation that is satisfied by the sum. It is a trivial calculation to verify that also the right-hand side satisfies the same recurrence. As you remark, both sides give 1 for . We can conclude that the identity holds for all in .” However, cases where is neither an integer nor half-integer still require attention. (Gasper has commented that the function is not a special case of the formulas in his paper with Rahman [35].)

12. Master Formula Investigation for

Appendix  A in [1] considered the possibility of developing a master formula for the total separability probability , associated with the determinantal inequality (cf. (4)–(6)). It now clearly seems appropriate to reexamine those results (Appendix F) in terms of the striking hypergeometric-based formula (Section 11) we have obtained for the partial separability probability , associated with the determinantal inequality .

In the earlier study [1], the formulas took the form of 1 minus terms involving polynomials in and gamma functions, while above the interesting such terms have been subtracted from . So, conjecturally there exists a tightly related analogue of the results reported in Section 11 for . (Dunkl did note the qualitative difference that “the ratio tends to 1 as but tends to .”)

In investigating these matters, we have found that, for our set of computed , , the number and location of the consecutive negative roots (Section 10.3) are precisely the same (55) as for (Section 10.3). (There strangely appears to be a sole exception to this rule for , where there are five such roots for and six such for , with anomalously equalling 0.) However, in the situation, the component polynomials are of degree , while in the setting the corresponding polynomials are of considerably smaller degree , so we are faced with a greater number of coefficients to determine.

Here, is the equation we have solved to determine—based on [1, Appendix  A]—formulas for for . The c’s are (nonnegative integer) coefficients we fitted to exact values obtained using the Legendre-polynomial density-approximation routine of Provost [18]. (The first 15,761 of the moments (8) were employed.)

12.1. The Ratios

In Appendix G, we show a number of formulas we have generated for the differences between the formulas for for successive values of , in relation to the earlier -based formulas shown in Appendix C. (A stark contrast occurs, with the formulas initially yielding [“biproper”] rational functions—with equal-degree numerators and [zero constant term] denominators [the degrees satisfying a certain difference equation]—and, then, difference equations for .) So, it appears that the quest for a general formula could be successfully addressed by employing the same framework as in the case, by modifying the function to incorporate the new terms shown in Appendix C and their extensions to , in general. We see an evident relation between the coefficients of the terms in the difference equations in Appendix G and the six hypergeometric upper parameters described in Section 3.1.2 in the pattern of two 6’s and four 5’s. Also, the coefficients of the terms appear related to the six hypergeometric lower parameters described in Section 3.1.1.

Further, in Appendix H we show the ratios as functions of , rather than of .

12.1.1. Solution of Difference Equation for

We have been successfully able to solve the second difference equation recorded (in two forms) in Appendix G. The initial solution consisted of a large (multipage) output with numerous hypergeometric functions (again with argument ). (In Appendix I, we show the Maple counterpart, provided by Carl Love (http://math.stackexchange.com/questions/1903720/what-solution-does-maple-give-to-this-difference-equation), of our Mathematica solution. There is an implicit [unperformed] summation in it.) The solution naturally broke into the sum of two parts. For the first part—using high-precision numerics, rationalizations, and the FindSequenceFunction command—we were able to obtain the (hypergeometric-free) formula

Remarkably, when this term was multiplied by the function (which comprises the denominator of the ratio), examples of which are shown in Appendix C and formulated in (64),

the product simplified to the form . So, we can consider this term to be the first of two parts of a formula for . Now, in quest of the remaining term, when we formed a new difference equation for just the second part, we obtained a new solution, again naturally breaking into the sum of two parts. Now, the first part—previously given by (68)—was zero, and the new second part was given by precisely the same difference equation as original, but for the single change of the initial value (at ) from to .

12.2. X-States Counterpart

In Appendix J we show the analogue of the formulas for the “toy” model of X-states [30, 31]. One feature to be immediately noted is that the arguments of the indicated hypergeometric functions are −1. Another is that, for half-integer ’s, yields rational values, while yields value of the form 1 minus rational numbers divided by .

12.3. Use of Consecutive Negative Roots

We have noted that both and have roots at consecutive negative values of (Section 10.3). If we examine the (limiting) values of for immediately (one) below the end of the consecutive series, we find that they satisfy the relation

(This might serve as a “starting point” analogous to the use ((56), (57)) of ). For the analogous set of ’s, the real parts appear to be for even and for odd , with the imaginary parts given by

Dunkl has observed that the sequence generated by (70) is really two interspersed sequences, one for odd and one for even values of . They can be represented as and

12.4. Setting so That the Parameters in the Formula Are Zero

It appeared to be an exercise of interest to set in so that, in turn, one of the five variable upper and lower parameters in the function in the formula for would equal zero. We now enumerate those such scenarios, for which we were able to construct formulas.

For , we found that

As already observed, since we have consecutive roots descending downward from , for , we have .

Further, we found that, in the limit ,

Also, for , for even

and for odd

Additionally, along similar investigative lines, we have the formulas in Appendix L, particularly elegantly (),

12.5. Formula for

Also, eventually (after having computed for ), we were able to obtain the formula (not as explicit as that for ) shown in Appendix M for .

12.6. Two Formulas Involving the Lerch Transcendent

In the limit , we found for even ,

Here, the Lerch transcendent . In the same limit, we have for odd ,

Next, in the limit , we found for odd ,

While for even ,

We note the similarities in integer coefficients between the two Lerch-based formulas and then by now familiar occurrences (Sections 3.1 and 12.1.1, Appendix A) of simple fractions with denominators that are multiples of five and six.

12.7. Rules for Leading Coefficients of the Polynomials

In Appendix N we show for , the first of the rules we have developed for the leading coefficients of the polynomials given in the formula above (1) for , having been normalized to monic form (the original leading degree-() coefficient being ). (For convenience, we drop this term, leaving us with degree- polynomials.) We note that these resultant polynomials are of degree . Now, we can make the interesting observation (essentially putting the polynomial in Horner form) that their leading (highest power) coefficients are given (in descending order) by the rules:

Also, is the product of

Further, is the product of

Continuing, is the product of

while is the product of

So, an obvious important challenge would be to find the common formula generating these results. (The pattern of [negative] integer exponents of 2, that is, , is yielded by sequence A004134 “denominators in expansion of are ” of the On-Line Encyclopedia of Integer Sequences.)

Let us make the observation that the constant (lowest-order) coefficient in the polynomial in the formula for in (1) is equal to .

13. Concluding Remarks

The asymptotic analyses reported here and those in studies of Aubrun et al. [4, 36, 37] both employ Hilbert-Schmidt and (more generally) random induced measures (cf. [38]). However, contrastingly, we chiefly consider asymptotics as the Dyson-index-like parameter (cf. [39, 40]), while they implicitly are concerned with the standard case of and large numbers of qubits. Perhaps some relation exists, however, between their high-dimensional findings and the quite limited set of asymptotics we have presented above (Sections 7.2, 7.3, and 8.1.2), pertaining to the dimensional index .

A strong, intriguing theme in the analyses presented above has been the repeated occurrence of the interesting constant . Let us note that Guillera in his article “A New Ramanujan-Like Series for ,” applying methods related to Zeilberger’s algorithm [33], obtained a hypergeometric identity involving a sum over from 0 to of terms involving factors of the form [27, sec. 3] (cf. [32, sec. 8]).

Further, in a study of products of Ginibre matrices of Penson and Życzkowski, the Fuss-Catalan distribution is represented as a sum of generalized hypergeometric functions , somewhat analogous to that given in (Fig. 3 in [17]), since only functions are employed. These functions have hypergeometric arguments , where is a nonnegative integer, and have support [41, eq. ]. So, for , . (We had inquired of Hou whether the telescoping procedure might be profitably applied in such a context. He replied “the method I used only works for with a concrete integer ” [cf. [41, eqs. ]].) As an item of further curiosity, we note that, in the MathWorld entry on hypergeometric functions, the identity , the argument being , is noted. (Also, cf. (57) above.)

Appendix

A. Hypergeometric Forms of the Factors

B. Difference Equation Forms of the Factors

   (the  Hilbert-Schmidt  case)

C. Hypergeometric-Free Formulas for : Term-by-Term Differences

D. Maple Worksheet of Qing-Hu Hou for “Concise” Formula (34)

 > Use the package APCI > with  (APCI);

 > Input 1 and 2 >

 > >

 > Use command Zeil to compute recurrence relation (on alpha) of sum_i  > >  re ≔ Zeil  (, “cert”): > The first part is the recurrence relation >  re[l];

 > The second part is the function such that lhs(re[1])=Delta_i   >  : > For the sum_, we need the 0th term and the infinity term ;

 ;

 ;

 > > We need initial values. It seems that But we do not have the proof. >  + ;

 It seems that . But we do not have the proof. > ;

 With the initial values, we find a formula for (alpha) > ;

 ;

  ;

 ;

 > ;

 ;

 The right most fraction equals  ;

E. Collected Formulas

F. Collected Formulas

G. Formulas for the Ratios as Functions of : Term-by-Term Differences

H. Formulas for the Ratios as Functions of : Term-by-Term Differences

I. Maple Solution, Provided by Carl Love, of Difference Equation (Appendix H) for

 restart: rsolve(

  62022240 + 545995032    n + 2056791388    2 + 4333244560    3 + 5587600700    4 + 4517982000    5 + 2238010000    6 + 621200000    7

 +

 74000000    8 + (−19027008 − 158454120    n − 566231672    2 − 1135130960    3 − 1397526400    4 − 1082880000    5 − 516080000    6 − 138400000    7

 −

 16000000    8)    y(n) + 3    (2 + 5    n)    (3 + 5    n)2    (4 + 5    n)    (6 + 5    n)   (5 + 6    n)    (7 + 6    n)    (31 + 20    n)    y(1 + n) = 0, y(1) = 158/31 ,

 );

 lprint(%); − (3 + 5    n)    Pi(1/2)    GAMMA(n + 13/10)    GAMMA(n + 11/10)    GAMMA(n + 9/10)    GAMMA(n + 7/10)    GAMMA(n + 3/2)    (−cos((1/5)    Pi)+2    cos((1/5)    Pi)2 − 1)   64n   27(−n)    GAMMA(1 + n)    (Sum((4/3)    (cos((1/5)    Pi) + 1)    (2    cos((1/5)    Pi) − 1)    GAMMA(n1 + 7/5)    GAMMA(n1 + 9/5)    GAMMA(n1 + 11/6)    GAMMA(n1 + 11/5)    GAMMA(n1 + 13/6)    GAMMA(n1 + 8/5)    (2    n1 + 3)    (9250000    n17+63775000    n16+184088750    n15+288614625    n14 +265528150    n13+143363345    n12 + 42053906    n1 + 5168520)/(GAMMA(2 + n1)   27(− n1 − 1)  64(n1+1)    (2    cos((1/5)    Pi)+1)    (cos((1/5)    Pi) − 1)    GAMMA(n1 + 5/2)    GAMMA(n1 + 17/10)    GAMMA(n1 + 19/10)    GAMMA(n1 + 21/10)    GAMMA(n1 + 23/10)    Pi(1/2)    (8 + 5    n1)    (6    n1 + 5)    (5    n1 + 2)    (5    n1 + 4)    (5    n1 + 6)    (6    n1 + 7)    (5    n1 + 3)2), n1 = 1⋯n − 1) +7900/1287)/((20    n + 11)    GAMMA(n + 3/5)    GAMMA(n + 7/6)    GAMMA(n + 6/5)    GAMMA(n + 5/6)    GAMMA(n + 4/5)    GAMMA(n + 2/5)    (cos((1/5)    Pi)+2    cos((1/5)    Pi)2 − 1)).

J. Formulas

K. Formula for

L. Further Formulas: Additional Formulas

M. Formula for

N. Formulas for Leading Coefficients of

O. “Exterior” Separability Probabilities

O.1. Inspheres

The convex set of two-qubit states possesses an “insphere” of maximum radius. The states within in it are all separable [7, 13]. So, one can ask what is the Hilbert-Schmidt separability probability outside of it, presuming the apparent total separability probability of . Using the formulas in [2], we have for the total volume of the two-qubit states, for the radius of this insphere, and thus for its 15-dimensional volume. This yields an exterior separability probability of

Let us proceed similarly for the two-rebit states. We use, again, the pertinent formulas [2, sec. 7], obtaining a total volume of , a radius of the insphere of , and a 9-dimensional insphere volume of . This yields a separability probability (ever so slightly less than the presumed value of ) exterior to the insphere of

O.2. Absolutely Separable States

Next, let us observe that these inspheres are themselves contained within the sets of absolutely separable states [42], those states that can not be entangled through unitary transformations. In [43, eq. ], the result was reported for the two-rebit absolute separability probability. This leads to an exterior separability probability of

Also, a considerably more complicated two-qubit formula [43, eq. ] was given. The corresponding absolutely separable probability is approximately 0.00365826. This yields, proceeding similarly, to .

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to express appreciation to Charles Dunkl for his many, many expert contributions and interactions in this research program in the past few years and specifically for his important insights reported in Section 11. Qing-Hu Hou has been very generous in his assistance also. Christioph Koutschan helped with the implementation of the fast Zeilberger algorithm in the RISC package. Christian Krattenthaler provided advice at early stages of the research reported. Robert Israel, Brendan Godfrey, and Michael Love have helpfully responded to a number of questions posted on the Mathematics and Mathematica Stack Exchanges.

References

P. B. Slater and C. F. Dunkl, “Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States,” Journal Advances in Mathematical Physics, vol. 2015, Article ID 621353, 9 pages, 2015.

K. Życzkowski and H.-J. Sommers, “Hilbert–Schmidt volume of the set of mixed quantum states,” Journal of Physics, vol. 36, Article ID 10115, 2003.

K. Życzkowski and H.-J. Sommers, “Induced measures in the space of mixed quantum states,” Journal of Physics, vol. 34, p. 7111, 2001.

G. Aubrun, S. J. Szarek, and D. Ye, “Entanglement Thresholds for Random Induced States,” Communications on Pure and Applied Mathematics, vol. 67, no. 1, pp. 129–171, 2014.

F. J. Dyson, “Correlations between eigenvalues of a random matrix,” Communications in Mathematical Physics, vol. 19, no. 3, pp. 235–250, 1970.

I. Dumitriu, A. Edelman, and G. Shuman, “MOPS: multivariate orthogonal polynomials (symbolically),” Journal of Symbolic Computation, vol. 42, p. 587, 2007.

K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, “Volume of the set of separable states,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 58, p. 883, 1998.

D. Petz and C. Sudár, “Geometries of quantum states,” Journal of Mathematical Physics, vol. 37, no. 6, p. 2662, 1996.

P. B. Slater and C. F. Dunkl, “Generalized two-qubit whole and half Hilbert–Schmidt separability probabilities,” Journal of Geometry and Physics, vol. 90, p. 42, 2015.

R. Augusiak, M. Demianowicz, and P. Horodecki, “General construction of noiseless networks detecting entanglement with the help of linear maps,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 77, no. 5, 2008.

A. Peres, “Separability Criterion for Density Matrices,” Physical Review Letters, vol. 77, p. 1413, 1996.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Physics Letters A, vol. 223, pp. 1–8, 1996.

S. Szarek, I. Bengtsson, and K. Zyczkowski, “On the structure of the body of states with positive partial transpose,” Journal of Physics A: Mathematical and Theoretical, vol. 39, p. L119, 2006.

A. Lovas and A. Andai, “Invariance of separability probability over reduced states in 4 × 4 bipartite systems,” Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 29, 2017.

O. Gamel, “Entangled bloch spheres: bloch matrix and two-qubit state space,” Physical Review A, vol. 93, Article ID 062320, 2016.

P. B. Slater and C. F. Dunkl, Journal of Physics A: Mathematical and Theoretical, vol. 45, Article ID 095305, 2012.

P. B. Slater, “A concise formula for generalized two-qubit Hilbert-Schmidt separability probabilities,” Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 44, Article ID 445302, 2013.

S. B. Provost, Journal of Mathematical Analysis and Applications, vol. 9, 2005.

R. Askey, I. Schoenberg, and A. Sharma, “Hausdorff's moment problem and expansions in Legendre polynomials,” Journal of Mathematical Analysis and Applications, vol. volume86, no. 1, p. 237, 1982.

K. Bartkiewicz, B. Horst, K. Lemr, and A. Miranowicz, “Entanglement estimation from Bell inequality violation,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 88, no. 5, 2013.

J. Fei and R. Joynt, “Numerical Computations of Separability Probabilities,” Reports on Mathematical Physics, vol. 78, no. 2, 2016.

C. M. Caves, C. A. Fuchs, and P. Rungta, “Entanglement of formation of an arbitrary state of two rebits,” Foundations of Physics Letters, vol. 14, p. 199, 2001.

H. Aslaksen, “Quaternionic determinants,” The Mathematical Intelligencer, vol. 18, p. 57, 1996.

A. Peres, “Proposed test for complex versus quaternion quantum theory,” Physical Review Letters, 1979.

G. Najarbashi, B. Seifi, and S. Mirzaei, “Two-and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection,” Quantum Information Processing, vol. 15, no. 1, p. 509, 2016.

P. B. Slater, “Octonionic two-qubit separability probability conjectures,” 2016.

J. Guillera, “A new Ramanujan-like series for 1/p 2,” The Ramanujan Journal, vol. 26, p. 369, 2011.

W. Koepf, “Hypergeometric Identities,” in Hypergeometric Summation, Advanced Lectures in Mathematics, pp. 11–30, Vieweg+Teubner Verlag, Wiesbaden, 1998.

M. Petkovšek, “Hypergeometric solutions of linear recurrences with polynomial coefficients,” Journal of Symbolic Computation, vol. 14, no. 2, p. 243, 1992.

P. Mendonça, M. A. Marchiolli, and D. Galetti, “Entanglement universality of two-qubit X-states,” Annals of Physics, vol. 351, p. 79, 2014.

C. F. Dunkl and P. B. Slater, “Separability probability formulas and their proofs for generalized two-qubit X-matrices endowed with Hilbert–Schmidt and induced measures,” Random Matrices: Theory and Applications, vol. 4, no. 4, Article ID 1550018, 2015.

W. Chu and W. Zhang, “Accelerating Dougall's 5F4-sum and infinite series involving p,” AMS :: Mathematics of Computation - American Mathematical Society, vol. 83, no. 285, p. 475, 2014.

D. Zeilberger, “A fast algorithm for proving terminating hypergeometric identities,” Discrete Mathematics, vol. 80, p. 207, 1990.

C. Wenchang, “Inversion Techniques and Combinatorial Identities: Balanced Hypergeometric Series,” Rocky Mountain J. Math, vol. 32, no. 3, p. 561, 2002, http://dx.doi.org/10.1216/rmjm/1030539687.

G. Gasper and M. Rahman, “On necessary multiplier conditions for Laguerre expansions,” CJM: Canadian Journal of Mathematics, vol. 42, no. 1, p. 27, 1991.

S. Szarek, “Volume of separable states is super-doubly-exponentially small in the number of qubits,” Physical Review A, vol. 72, Article ID 032304, 2005.

G. Aubrun and S. Szarek, “Tensor products of convex sets and the volume of separable states on N qudits,” Physical Review A, vol. 72, Article ID 022109, 2006.

K. Szymański, B. Collins, T. Szarek, and K. Zyczkowski, “Convex set of quantum states with positive partial transpose analysed by hit and run algorithm,” Journal of Physics A: Mathematical and Theoretical, 2016.

I. Dumitriu and A. Edelman, “Matrix models for beta ensembles,” Journal of Mathematical Physics, vol. 43, p. 5830, 2002.

I. Dumitriu and A. Edelman, “Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics,” Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, vol. 41, no. 6, pp. 1083–1099, 2005.

K. A. Penson and K. Zyczkowski, “Product of Ginibre matrices: Fuss-Catalan and Raney distributions,” Physical Review E, vol. 83, Article ID 061118, 2011.

F. Verstraete, J. Dehaene, and B. DeMoor, “Local filtering operations on two qubits,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 64, no. 1, 2001.

P. B. Slater, “Eigenvalues, separability and absolute separability of two-qubit states,” Journal of Geometry and Physics, vol. 59, no. 1, pp. 17–31, 2009.