Joint Hilbert-Schmidt Determinantal Moments of Product Form for the Two-Rebit and Two-Qubit and Higher-Dimensional Quantum Systems
JJoint Hilbert-Schmidt Determinantal Moments of Product Formfor Two-Rebit and Two-Qubit and Higher-Dimensional QuantumSystems
Paul B. Slater ∗ University of California,Santa Barbara, CA 93106 (Dated: December 3, 2018)
Abstract
We report formulas for the joint moments of the determinantal products (det ρ ) k (det ρ P T ) κ ( k =0 , , , . . . , N ; κ = 1 , . . . ,
12) of Hilbert-Schmidt (HS) probability distributions over the generic two-rebit and two-qubit density matrices ρ ( κ = 1 , . . . , P T denotes the partial transpositionoperation of quantum-information-theoretic central importance. Each formula is the product of theexpression for the HS moments of (det ρ ) k , k = 0 , , , . . . , N –special cases of results of Cappellini,Sommers and ˙Zyczkowski ( Phys. Rev. A , 062322 (1996))–and an adjustment factor. Thefactor is a biproper rational function, with its numerators and denominators both being 3 κ -degreepolynomials in k . We infer the structure that the denominators follow for arbitrary κ in boththe two-rebit and two-qubit cases, and the six leading-order coefficients of k of the numerators inthe two-rebit scenario. We also commence an analogous investigation of generic rebit-retrit andqubit-qutrit systems. This research was motivated, in part, by the objective of using the computedmoments to well reconstruct the HS probabilities over the determinant of ρ and of its partialtranspose, and to ascertain–at least to high accuracy–the associated (separability) probabilities of”philosophical, practical and physical” interest that (det ρ P T ) > PACS numbers: Valid PACS 03.67.Mn, 02.30.Cj, 02.30.Zz, 02.50.Sk ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] M a y e begin our investigation into certain statistical aspects of the ”geometry of quantumstates” [1, 2] by noting the two following special cases–which we will extend below–of thegeneral formulas [3][eq. (3.2)]: (cid:104)| ρ | k (cid:105) − rebit/HS = 945 (cid:16) − k Γ(2 k + 2)Γ(2 k + 4)Γ(4 k + 10) (cid:17) (1)and (cid:104)| ρ | k (cid:105) − qubit/HS = 108972864000 Γ( k + 1)Γ( k + 2)Γ( k + 3)Γ( k + 4)Γ(4( k + 4)) , (2) k = 0 , , , . . . The bracket notation (cid:104)(cid:105) is employed by us to denote expected value, while ρ indicates a generic (symmetric) two-rebit or generic (Hermitian) two-qubit (4 ×
4) densitymatrix. The expectation is taken with respect to the probability distribution determinedby the Hilbert-Schmidt/Euclidean/flat metric on either the 9-dimensional space of generictwo-rebit or 15-dimensional space of generic two-qubit systems [1, 4].We report below sixteen (twelve two-rebit and four two-qubit) non-trivial extensions ofthese formulas, involving now in addition to | ρ | , the quantum-theoretically important deter-minant | ρ P T | of the partial transpose of ρ . (The nonnegativity of | ρ P T | –by the celebratedPeres-Horodeccy results [5–7]–constitutes a necessary and sufficient condition for separabil-ity/disentanglement, when ρ is either a 4 × × (cid:104)| ρ | k | ρ P T |(cid:105) − rebit/HS = ( k − k (2 k + 11) + 16)32( k + 3)(4 k + 11)(4 k + 13) (cid:104)| ρ | k (cid:105) − rebit/HS , (3) (cid:104)| ρ | k | ρ P T | (cid:105) − rebit/HS = k ( k ( k ( k (4 k ( k + 12) + 203) + 368) + 709) + 2940) + 48601024( k + 3)( k + 4)(4 k + 11)(4 k + 13)(4 k + 15)(4 k + 17) (cid:104)| ρ | k (cid:105) − rebit/HS (4)and (cid:104)| ρ | k | ρ P T |(cid:105) − qubit/HS = k ( k ( k + 6) − − k + 9)(4 k + 17)(4 k + 19) (cid:104)| ρ | k (cid:105) − qubit/HS . (5)These three formulas were, first, established by ”brute force” computation–that is calcu-lating the first ( k = 0 , , , . . . ,
15 or so) instances, then employing the Mathematica com-mand FindSequenceFunction, and verifying any formulas generated on still higher values of k . ( Initially , although we had the specific values of (cid:104)| ρ | k | ρ P T | (cid:105) − rebit/HS for k = 0 , . . . , (cid:104)| ρ | k | ρ P T | (cid:105) − rebit/HS , we were not able to determine, in the same manner,encompassing expressions for them.) 2s 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 generictwo-rebit systems, the two determinants | ρ | and | ρ P T | comprise a pair of nine-dimensional orthogonal polynomials [9–11] with respect to Hilbert-Schmidt measure. (C. Dunkl haskindly pointed out that orthogonality here does not imply zero correlation .) In additionto this first ( k = 1) HS zero-moment of the (”equally-mixed”) product variable | ρ || ρ P T | inthe two-rebit case, we had been able to compute its higher-order moments, k = 2 , . . . , k = 2 can be obtained by direct application of (4). The feasible range ofthe variable is | ρ || ρ P T | ∈ [ − , ]–the lower bound of which − = − − − wedetermined by analyzing a general convex combination of a Bell state and the fully-mixedstate.)These five further moments of | ρ || ρ P T | , k = 2 , . . . ,
6, are all rational numbers. T If wetake the ratios of these first six moments of | ρ || ρ P T | to the first six even moments given by(1), that is the values (cid:104)| ρ | k (cid:105) − rebit/HS , k = 1 , . . . ,
6, we obtain the rather succinct sequence, (cid:104) ( | ρ || ρ P T | ) k (cid:105) − rebit/HS (cid:104)| ρ | k (cid:105) − rebit/HS = { , , , , , } (6) ≈ { , . , . , . , . , . } . (As to the two-qubit counterpart of this sequence, we had so far only been able to computeits very first term–turning out, quite remarkably, to be the negative value − .)Since these ratios (6) are so comparatively simple, it suggested to us that we might bemore able to progress in a series of analyses [12–19] (mainly devoted to the determination of separability probabilities), by making our initial goal the computation of these ratios for stillhigher-order moments–rather than the direct computation of the very small values, havinglengthy multi-digit denominators, of the moments (cid:104) ( | ρ || ρ P T | ) k (cid:105) − rebit/HS themselves. (In[8][eqs, (33)-(41)], we were able to report and analyze the first nine moments (cid:104)| ρ P T | k (cid:105) –thefirst two of which, − and , can be obtained by directly setting k = 0 in (3) and (4),respectively. However, to this point, we have not found any associated similarly compactsequences of moment ratios, as above.)Accordingly, in Fig. 1, we display the sequence of ratios (cid:104) ( | ρ || ρ PT | ) k (cid:105) − rebit/HS (cid:104)| ρ | k (cid:105) − rebit/HS , k = 1 , . . . , nu- IG. 1: The six-member two-rebit HS exact moment ratio sequence (6), supplemented by itsnumerical continuation, using extended-precision (60-digit) arithmetic. Three hundred and twentymillion random density matrices were employed. merical estimates yielded by this procedure of the six-member exact sequence (6)were {− . , . , . , . , . , . } .) Similarly, inFig. 2, we display the (quite differently-behaving) two-qubit sequence of moment ratios (cid:104) ( | ρ || ρ PT | ) k (cid:105) − qubit/HS (cid:104)| ρ | k (cid:105) − qubit/HS , k = 1 , . . . , , we were initially ableto exactly compute, and the remaining ninety-nine, numerically, using extended-precision.If we are, at some point, in the course of these extended analyses, able to develop formulasexplaining the full sequences of ratios in the two-rebit and two-qubit cases, we should beable to reconstruct the Hilbert-Schmidt univariate probability distributions over the productvariable | ρ || ρ P T | (cf. [3][Figs. 2-4]). From such reconstructed distributions, HS separability probabilities should be determinable to high accuracy.For the further edification of the reader, we present in Fig. 3 a contour plot of the jointHilbert-Schmidt (bivariate) probability distribution of | ρ | and | ρ P T | in the two-rebit case,and in Fig. 4, its two-qubit analogue. (A colorized grayscale output is employed, in whichlarger values appear lighter.) In Fig. 5 is displayed the difference obtained by subtracting thesecond (two-qubit) distribution from the first (two-rebit) distribution. (The black curves inall three contour plots appear to be attempts by Mathematica to establish the nonzero-zeroprobability boundaries–which, it would, of course, be of interest to explicitly determine, if4 IG. 2: The two-qubit analogue of the two-rebit sequence depicted in Fig. 1, with only the firstmember ( − ) having initially been exactly known, and the next ninety-nine computed numerically,using extended-precision (60-digit) arithmetic. Twenty-four million random density matrices wereemployed. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) FIG. 3: Contour plot of the joint Hilbert-Schmidt probability distribution of | ρ | (horizontal axis)and | ρ P T | in the two-rebit case. Larger values appear lighter. The variable ranges are | ρ | ∈ [0 , ]and | ρ P T | ∈ [ − , ]. One billion random density matrices were employed. .000 0.001 0.002 0.003 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) FIG. 4: Contour plot of the joint Hilbert-Schmidt probability distribution of | ρ | (horizontal axis)and | ρ P T | in the two-qubit case. Six hundred million random density matrices were employed. possible–of the joint domain of | ρ | and | ρ P T | .)These last three figures are based on Hibert-Schmidt sampling (utilizing Ginibre ensem-bles [3]) of random density matrices, using 10 ,
000 = 100 bins. In regard to the two-qubitplot, K. ˙Zyzckowski informally wrote: ”A high peak in the upper corner means that: a) amajority of the entangled states is ’little entangled’ (small det ( ρ T )) or rather, they are ’close’to the boundary of the set, so one eigenvalue is close to zero, and the determinant is small;b) as det ( ρ ) is also small, it means that these entangled states live close to the boundary ofthe set of all states (at least one eigenvalue is very small), but this is very much consistentwith the observation that the center of the convex body of the 2-qubit states is separable(so entangled states have to live ’close’ to the boundary). Similar reasoning has to hold inthe real case as well.”At a later point in our investigation, we realized that we might make further progress–despite limitations on the number of moments we could explicitly compute–by exploitingthe evident pattern followed by our newly-found formulas (3) and (4)–in particular, thestructure in their denominators. This encouragingly proved to be the case, as we were ableto establish that (cid:104)| ρ | k | ρ P T | (cid:105) − rebit/HS = A B (cid:104)| ρ | k (cid:105) − rebit/HS , (7)6 .000 0.001 0.002 0.003 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) FIG. 5: Difference obtained by subtracting the two-qubit HS probability distribution in Fig. 4 fromthe two-rebit probability distribution in Fig. 3. Darker colors indicate more negative values. where A = 8 k +180 k +1674 k +8559 k +29493 k +84291 k +136801 k − k − k − B = 32768( k + 3)( k + 4)( k + 5)(4 k + 11)(4 k + 13)(4 k + 15)(4 k + 17)(4 k + 19)(4 k + 21) . (9)So, it is now rather evident that we can write for general non-negative integer κ , (cid:104)| ρ | k | ρ P T | κ (cid:105) − rebit/HS = A κ B κ (cid:104)| ρ | k (cid:105) − rebit/HS , (10)where both the numerator A κ and the denominator B κ are 3 κ -degree polynomials (thus,forming a ”biproper rational function” [20]) in k (the leading coefficient of A κ being 2 κ ),and B κ = 128 κ ( k + 3) κ (cid:18) k + 112 (cid:19) κ , (11)where the Pochhammer symbol ( x ) n ≡ Γ( x + n )Γ( x ) = x ( x + 1) . . . ( x + n −
1) is employed. Furtherstill, moving upward to the next level ( κ = 4), we have determined that (cid:104)| ρ | k | ρ P T | (cid:105) − rebit/HS = A B (cid:104)| ρ | k (cid:105) − rebit/HS , (12)7hereˆ A = 16 k + 576 k + 9112 k + 84496 k + 525681 k + 2389416 k + 7805462 k + 13904508 k +(13)+6212189 k + 166748972 k + 1636873812 k + 5496485760 k + 6610161600 , and B is given by (11) with κ = 4. The real part of one of the roots of A is 2.999905,suggesting to us some possible interesting asymptotic behavior of the roots of these numer-ators, κ → ∞ . In our previous related study [8][sec. II.B.2], we were also able to discern thegeneral structure that the denominators of certain ”intermediate [rational] functions” usedin computing the (univariate) moments of (cid:104) ρ P T | κ (cid:105) − rebit/HS , κ = 1 , . . . , constant terms in the 3 κ -degree numerator A κ are − , , − κ = 1 , , ,
4. Since we had previously computed [8][eqs, (33)-(41)] the moments of (cid:104)| ρ P T | κ (cid:105) − rebit/HS , κ = 1 , . . . ,
9, we are also able to determine the next five members of thissequence {− , , − , } . However, no general rule for this sequence,which would directly allow us to obtain a formula for (cid:104)| ρ P T | κ (cid:105) − rebit/HS , has, to our dis-appointment, yet emerged for them. (With such a rule, we could address the separabilityprobability question through the reconstruction of a univariate probability distribution.)A simple algebraic exercise involving (1) shows that if we multiply the conversion factor c = √ π Γ(2 κ + 4)Γ(8 κ + 10)8Γ (cid:0) κ + (cid:1) Γ(4 κ + 2)Γ(4 κ + 10) (14)by the rational function factors A κ B κ found above that applied to (cid:104)| ρ | κ (cid:105) − rebit/HS yield (cid:104) ( | ρ || ρ P T | ) κ (cid:105) − rebit/HS , we obtain the κ -member of the sequence of moment ratios (6). Sincethe members of this sequence appear (Fig. 1) to asymptotically approach 1 (our numericalestimate for the 100- th term is 1.001542), it would seem that the conversion factor c and B κ A κ asymptotically approach one another.Certainly, it would be of interest to conduct analyses parallel to those reported abovefor metrics of quantum-information-theoretic interest other than the Hilbert-Schmidt, suchas the Bures (minimal monotone) metric [1, 21]. The computational challenges involved,however, might, at least in certain respects, be even more substantial.At this stage of our research, after posting the results above as a preprint, Charles Dunkldetailed a computational proposal that he had outlined to us somewhat earlier. The at-tractive feature of this proposal would be that it would–holding the exponent κ of | ρ P T | k , rather than having to doso for sufficient numbers of individual members of the sequence k = 1 , . . . , N , to be ableto successfully apply the Mathematica command FindSequenceFunction, as had been ourstrategy beforehand. The proposal of Dunkl (see Appendix) involved parameterizing 4 × | ρ | and the jacobian for the transformation to Choleskyvariables are simply monomials.) Using this approach, we were able to extend our single( κ = 1) two- qubit result (5) to the κ = 2 case, (cid:104)| ρ | k | ρ P T | (cid:105) − qubit/HS = (15) k ( k ( k ( k ( k ( k + 15) + 67) + 45) + 220) + 4260) + 1094464(2 k + 9)(2 k + 11)(4 k + 17)(4 k + 19)(4 k + 21)(4 k + 23) (cid:104)| ρ | k (cid:105) − qubit/HS . Additionally, in the following several arrays, we show ( κ = 1 , . . . ,
12) column-by-column,the (3 κ + 1) coefficients of the numerator polynomials in ascending order–the entries inthe first row corresponding to the constant terms,. . . –in the two-rebit case. For the cases9 = 1 , . . . , −
16 4860 − − − − − − − −
203 84291 6212189 − −
48 29493 13904508 29246867605 − − − − − − −
180 525681 883461210 1632448582425 − − − − − − − −
576 5660714 23817008856 − − −
16 575800 3786901675 − − − − − − − − − − − −
32 3143808 − − − − − − − − − − − − − − − (16)10or κ = 7 , − − − − − − − − − − − (17)11or κ = 9. − − − − − − − (18)12or κ = 10 − − (19)13or κ = 11, − − − − − − − − − − (20)14 κ = 12, − − − (21)16 (We are presently attempting extensions to the cases κ = 13 , C κ +1 = 2 κ ; C κ = 3 × κ − κ ( κ + 2); C κ − = 2 κ − κ ( κ ( κ (9 κ + 32) + 24) − C κ − = 2 κ − κ (cid:0) κ (cid:0) κ (cid:0) κ (cid:0) κ + 42 κ + 52 (cid:1) − (cid:1) − (cid:1) − (cid:1) . (23)From these four formulas, we are able to reconstruct ( κ = 1) all four entries in the firstcolumn of (16). Thus, it appears that, in general, C κ − i is a polynomial in κ of degree 2( i +1).(For i = 3 κ −
1, we obtain the constant term, of strong interest. With the knowledge of onlythis term, and none of the other coefficients, we can obtain (cid:104)| ρ P T | κ (cid:105) − rebit/HS .) Further, wehave found that C κ − = (24)175 2 κ − ( κ − (cid:0) κ + 855 κ + 1895 κ − κ − κ − κ + 32394 κ (cid:1) , and C κ − = 15 2 κ − ( κ − κ (25) κ ( κ ( κ ( κ ( κ (3 κ (3 κ (9 κ + 59) + 377) − − − − . The numerators of our four sets ( κ = 1 , , ,
4) of two- qubit results are expressible, insimilar fashion, as −
42 10944 − − − − −
67 27783 − −
15 5373 1080858 − − −
282 278478 − −
27 50991 − − − − − − − − − − − (26)Of course, the leading coefficients C κ +1 of all four numerators are 1, so they are monic incharacter, while the next-to-leading coefficients fit the pattern C κ = 3 κ ( κ + 3) / (cid:104) ( | ρ || ρ P T | ) k (cid:105) − rebit/HS (cid:104)| ρ | k (cid:105) − rebit/HS = (27) (cid:26) , , , , , , , , , , , (cid:27) ≈{ , . , . , . , . , . , . , . , . , . , . , . } . Further, we now have for the two-qubit analogue of this sequence, (cid:104) ( | ρ || ρ P T | ) k (cid:105) − rebit/HS (cid:104)| ρ | k (cid:105) − qubit/HS = (cid:26) − , , − , (cid:27) , (28)18 {− . , . , − . , . } It is also evident at this point, in striking analogy to the general two-rebit formula (10),that in the two-qubit scenario, (cid:104)| ρ | k | ρ P T | κ (cid:105) − qubit/HS = ˆ A κ ˆ B κ (cid:104)| ρ | k (cid:105) − qubit/HS , (29)where, again, both the numerator ˆ A κ and the denominator ˆ B κ are 3 κ -degree polynomials in k , and (cf. (11)) ˆ B κ = 2 κ (cid:18) k + 92 (cid:19) κ (cid:18) k + 172 (cid:19) κ . (30)In the course of this work, Charles Dunkl further communicated to us a result (followinghis joint work with K. ˙Zyzckowski reported in [22], where ”the machinery for producingdensities from moments of Pochhammer type” was developed) giving the univariate proba-bility distribution over t ∈ [0 ,
1] that reproduces the Hilbert-Schmidt moments of t = 2 | ρ | ,where ρ is a generic two-rebit density matrix. (It would be interesting to try to extend themethodology employed to the two-qubit and other higher-order cases. Dunkl commentedthat ”The formula is slightly misleading near t = 1, there the density is (1 − t ) times ananalytic function, I imagine a polynomial approximation is better for computation there,but it’s obviously the stuff near zero that’s important.”) This probability distribution tookthe form (cf. [3][eq. (4.3)])638 (cid:18)(cid:113) − √ t (cid:16) − t − √ t + 2 (cid:17) + 15 t log (cid:18)(cid:113) − √ t + 1 (cid:19) − t log( t ) (cid:19) (31)(see Appendix below for further details).Of course, one may also consider issues analogous to those discussed above for bipartitequantum systems of higher dimensionality. To begin such a course of analysis, we havefound for the generic real 6 × (cid:104)| ρ | k | ρ P T |(cid:105) rebit − retrit/HS = 4 k + 40 k + 95 k − k − k − k + 4)(3 k + 11)(3 k + 13)(6 k + 23)(6 k + 25) (cid:104)| ρ | k (cid:105) rebit − retrit/HS . (32)Increasing the parameter κ from 1 to 2, we obtained that the rational function adjustmentfactor for (cid:104)| ρ | k | ρ P T | (cid:105) rebit − retrit/HS is the ratio of16 k +336 k +2616 k +8496 k +12069 k +101979 k +903539 k +3316809 k +5620320 k +3715740(33)19o another ninth-degree polynomial331776( k + 5)(3 k + 11)(3 k + 13)(3 k + 14)(3 k + 16)(6 k + 23)(6 k + 25)(6 k + 29)(6 k + 31) . (34)Additionally, for the generic complex 6 × (cid:104)| ρ | k | ρ P T |(cid:105) qubit − qutrit/HS = k + 15 k + 37 k − k − k − k + 13)(3 k + 19)(3 k + 20)(6 k + 37)(6 k + 41) (cid:104)| ρ | k (cid:105) qubit − qutrit/HS . (35) Appendix A: Derivation by C. Dunkl of probability distribution (31) over t ∈ [0 , having the moments of t = 2 | ρ | Notes on moments, etc. C. Dunkl 4/11/11Cholesky decomposition:Let C be a real upper-triangular N × N matrix, entries c ij , c ij = 0 for i > j and c ii ≥ i . Let P = C t C , entries p ij = (cid:80) Nk =1 c ki c kj = (cid:80) min( i,j ) k =1 c ki c kj . Consider the Jacobianmatrix ∂p∂c where the dependent variables are p ij , i ≤ j . Claim: (cid:12)(cid:12)(cid:12)(cid:12) det ∂p∂c (cid:12)(cid:12)(cid:12)(cid:12) = 2 N N (cid:89) i =1 c N +1 − iii . Lemma: suppose y i = f i ( x , x , . . . , x i ), 1 ≤ i ≤ N then the matrix (cid:16) ∂y i ∂x j (cid:17) is lower-triangular(0 for j > i ) and det (cid:16) ∂y i ∂x j (cid:17) = (cid:81) Ni =1 ∂y i ∂x i .This applies to Cholesky: order the variables: c , c , . . . , c N , c , . . . , c N , c ,. . . c N − ,N − , c N − ,N , c NN . For i ≤ j , p ij = (cid:80) i − k =1 c ki c kj + c ii c ij ; the lemma shows (cid:12)(cid:12)(cid:12)(cid:12) det ∂p∂c (cid:12)(cid:12)(cid:12)(cid:12) = N (cid:89) i =1 N (cid:89) j = i ∂p ij ∂c ij = N (cid:89) i =1 (cid:0) c N − i +1 ii (cid:1) . Consider random variables, values in 0 ≤ t ≤
1. Moments for the Beta distribution: let α, β > B ( α, β ) (cid:90) t n t α − (1 − t ) β − dt = ( α ) n ( α + β ) n , n = 0 , , , . . . . D = det P where P is a random positive definite 4 × n = 0 , , , . . .E ( D n ) = (cid:0) (cid:1) n (2) n (cid:0) (cid:1) n (1) n (10) n = 2 − n (4) n − n (2) n n (5) n (cid:0) (cid:1) n = 12 n (4) n (2) n (5) n (cid:0) (cid:1) n . Let X = 2 D ; X is (equidistributed as) the product of two independent random variables X , X with E ( X n ) = (4) n (5) n = 44 + 2 n = 22 + n ,E ( X n ) = (2) n (cid:0) (cid:1) n . Clearly X has the density f ( t ) = 2 t, ≤ t ≤
1. The density of X is f ( t ) = 12 B (cid:0) , (cid:1) (cid:16) − √ t (cid:17) / , (cid:90) t n f ( t ) dt = 12 B (cid:0) , (cid:1) (cid:90) t n (cid:16) − √ t (cid:17) / dt = 1 B (cid:0) , (cid:1) (cid:90) s n s (1 − s ) / ds = (2) n (cid:0) (cid:1) n . The density f ( t ) of X X is given by f ( t ) = (cid:90) t f (cid:18) ts (cid:19) f ( s ) dss = 22 B (cid:0) , (cid:1) (cid:90) t ts (cid:0) − √ s (cid:1) / dss = 2 tB (cid:0) , (cid:1) (cid:90) √ t u − (1 − u ) / du = 63 t (cid:90) √ t u − (1 − u ) / du. u = 1 − s , du = − sds , f ( t ) = 63 t (cid:90) √ −√ t s (1 − s ) ds = 63 t (cid:40) − s (15 − s + 8 s )(1 − s ) + 152 ln (1 + s ) − s (cid:41) s = √ −√ ts =0 = 638 (cid:26)(cid:16) − √ t (cid:17) / (cid:16) − √ t − t (cid:17) + 15 t ln (cid:18) (cid:113) − √ t (cid:19) − t ln t (cid:27) . This can be easily plotted. Also f ( t ) = O (cid:16) (1 − t ) / (cid:17) near t = 1. Acknowledgments
I would like to express appreciation to the Kavli Institute for Theoretical Physics (KITP)for computational support in this research, and Christian Krattenthaler, Mihai Putinar,Robert Mnatsakanov, Mark Coffey and K. ˙Zyczkowski for various communications. SergeProvost, Jean Lasserre, Partha Biswas and Luis G. Medeiros de Souza provided guidance onreconstruction of probability distributions from moments. The earlier stages of the compu-tations were greatly assisted by the Mathematica expertise of Michael Trott, and the laterstages by the mathematical insights and suggestions of Charles Dunkl. [1] I. Bengtsson and K. ˙Zyczkowski,
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