Symbolic integration with respect to the Haar measure on the unitary groups

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1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 65, No., 207 DOI: 0.55/bpasts Symbolic integration with respect to the Haar measure on the unitary groups Z. PUCHAŁA* an J.A. MISZCZAK Institute of Theoretical an Applie Informatics, Polish Acaemy of Sciences, 5 Bałtycka Str., Gliwice, Polan Abstract. We present IntU package for Mathematica computer algebra system. The presente package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalize Haar measure. We escribe a number of special cases which can be use to optimize the calculation spee for some classes of integrals. We also provie some examples of usage of the presente package. Key wors: unitary group, Haar measure, circular unitary ensemble, symbolic integration.. Introuction The integration over unitary group is an important subject of stuies in many areas of science, incluing mathematical physics, ranom matrix theory an algebraic combinatorics. In 2006 Collins an Śniay [] prove a formula for calculating monomial integrals with respect to the Haar measure on the unitary group U IJ U I J * z.puchala@iitis.pl U i j...u in j n U i j...u i n j n U. () Integrals of the above type, calle moments of the, have been known in mathematical physics literature for a long time. The problem of the integration of elements of unitary matrices was consiere for the first time in the context of nuclear physics in [2]. The asymptotic behaviour of the integrals of the type () was consiere by Weingarten in [3]. In this paper we escribe IntU, a Mathematica package [4] for calculating polynomial integrals over with respect to the Haar measure. We escribe a number of special cases which can be use to optimize the calculation spee for some classes of integrals. We also provie some examples of usage of the presente package, incluing the applications in the stuy of the geometry of the quantum states. This paper is organise as follows. In Section 2, we introuce notation an present the mathematical backgroun of polynomial integrals over unitary group. In Section 3, we escribe some special cases, in which the integration can be calculate more efficiently. In Section 4 we provie the escription of IntU package with the list of main functions. In Section 5 we show some examples of the usage. In Section 6 we provie a summary of the presente results an conclusions. 2. Mathematical backgroun 2.. Basic concepts. By M n we enote square matrices of size n. The compact group of unitary matrices is enote as. We equip the above group with unique normalize Haar measure enote by U. Ranom elements istribute with measure U form the so-calle circular unitary ensemble. Integer partition λ of a positive integer n is a weakly ecreasing sequence λ = (λ, λ 2,, λ l ) of positive integers, such that l i= λ i = jλj = n. To enote that λ is a partition of n, we write λ ` n. The length of a partition is enote by l(λ). By λ t μ we enote a partition of n + n 2 obtaine by joining partitions λ ` n an μ ` n 2. Each permutation σ 2 S n can be uniquely ecompose into a sum of isjoint cycles, where the lengths of the cycles sum up to n. Thus the vector of the lengths of the cycles, after reorering, forms a partition λ ` n. The partition λ is calle the cycle type of σ permutation Moments of the. Let us consier a polynomial p. From the linearity of an integral we have p(u) c(i,j,i,j ) U IJ U I J U, (2) I,J,I,J where I, J, I 0, J 0 are multi-inices an c are the coefficients of p. The value of such monomial integrals is given as [] U IJ U I J U i j...u in j n U i j...u i n j n = δ i,i...δ i σ() n,i δ j σ(n), j...δ (3) j τ() n, j Wg(τσ,), τ(n) σ,τ S n ( where Wg is the Weingarten function iscusse below. In the case of the multi-inices iffering in length, i.e. n 6= n 0, we have Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM 2

2 Z. Puchała an J.A. Miszczak U IJ U I J (4) = U i j...u in j n U i j...u i 0. n j n (4) The integrals of the above type are known as moments of the Weingarten function. The Weingarten function [] is efine for σ 2 S n an positive integer as Wg(σ,)= (n!) 2 λ n l(λ) χ λ (e) 2 s λ, () χλ (σ), (5) ( (,,, ) where the sum is taken over all integer partitions of n with length l(λ), s λ, () is the Schur polynomial s λ evaluate at an χ λ is an irreucible character of the symmetric group S n inexe by partition λ The imension of irreucible representation of. The value of the Schur polynomial at the point }{, ( (,,, ) i.e. the imension of irreucible representation of corresponing to partition λ, is equal to e.g. Theorem 6.3 in [5] s λ, ()=s λ (,,...,)= }{{} i< j ( (,,, ) λ i λ j + j i. (6) j i Irreucible character of S n. The irreucible character of S n inexe by partition λ, χ λ (σ) epens on a conjugacy class of permutation σ. Two permutations are in the same conjugacy class if an only if they have the same cycle type, thus it is common to write χ λ (σ) = χ λ (μ), where μ is an integer partition corresponing to the cycle type of σ. In the case of ientity permutation the cycle type is given by a trivial partition, e =, an the character value }{{ n} is equal to the imension of the irreucible representation of S n inexe by λ. In this case it is given by the celebrate hook length formula (see Eq 4.2 in [5]) χ λ (e)= λ! i, j h λ, (7) i, j λ where jλj = λ + λ λ l(λ), an hi, λ j is the hook length of the cell (i, j) in a Ferrers iagram corresponing to partition λ [6]. In the case of a non-trivial partition the character of symmetric group χ λ (σ) = χ λ (μ) can be evaluate with the use of Murnaghan-Nakayama rule (see Theorem in [7]), which escribes a combinatorial way of calculating the character; [8] is use. From the above consierations one can euce that the Weingarten function epens only on a cycle type of a permutation σ an thus it is constant on a conjugacy class represente by σ. Thus we may efine the Weingarten function as Wg(µ,)= ( µ!) 2 λ µ l(λ) χ λ (e) 2 s λ, () χλ (µ), (8) where μ is an integer partition, which is a cycle type of σ. 3. Special cases In this section we present some special cases of integrals with respect to the Haar measure on the unitary group. In these cases the value of the integral can be calculate without the irect usage of (3), which requires processing of i k i! j l j! permutations, where k i (l j ) enotes the number of i( j) in multi-inices I(J), respectively. The presente special cases have been implemente in the package to increase its efficiency. This goal has been achieve by minimizing the number of calculations of the Weingarten function. 3.. First two moments of. In the case of monomials of rank equal to 2 an 4, from [9] we have u ij u i j δ ii δ jj, (9) an u i j u i2 j 2 u i j u i 2 j 2 δ δ δ δ δ = δ i i δ i 2 i δ 2 j j δ j 2 j + δ 2 i i δ 2 i 2 i δ j j δ 2 j 2 j 2 + (0) δ i i δ i 2 i δ 2 j j 2 δ j 2 j δ + δ i i δ 2 i 2 i δ j j δ j 2 j 2 δ ( 2, ) allowing calculation of the values for polynomial integrals of egree less than 5 without the irect usage of (3). This optimization gives only a minor improvement of efficiency, as in these cases the irect calculation of Weingarten function is very fast Elements from one row (column). The next optimization is base on the fact that the istribution of ranom vector consisting of squares of absolute values of a row (or a column) of unitary matrix istribute with the Haar measure, is uniform on a stanar -simplex Δ [9] { u i, 2, u i,2 2,..., u i, 2 } U( ), () where U(A) enotes the normalize uniform measure (proportional to Lebesgue measure) on a set A ½ R, an stanar -simplex Δ is efine as, Δ = {λ 2 R : λ i 0, i= λ i = }. 22 Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM

3 Symbolic integration with respect to the Haar measure on the unitary group Using Beta integral, one obtains that for a fixe row i 0 an a vector p with non-negative entries p j, we have the following j= u i0, j 2p j (2) = Γ() Γ(p + ) Γ(p + ). Γ(p + + p + ) ( The above, as a special case, gives: u i0, j 2k ( )!k! ( + k)!, (3) u i0, j 2 u i0,k 2 ( + ), (4) which can be foun in literature [9, 0]. This optimization allows for an enormous improvement in efficiency thanks to avoiing ( p i )! p i! executions of Weingarten function neee in the case of the irect usage of formula (3) Even powers of iagonal element absolute values. In this subsection we consier the integrals of the type u i, j 2p u k,l 2q U, (5) where p, q are non-negative integers an i 6= k, j 6= l. In the case of p = q = this integral is known [9] u i, j 2 u k,l 2 2. (6) For general non-negative integers p, q the following proposition is true. Proposition. Let p, q be non-negative integers, for i 6= k an j 6= l, we have u i, j 2p u k,l 2q p!q! k λ k µ Wg(λ µ,), (7) λ p µ q where the above sum is taken over all integer partitions of p an q. Symbol k ν enotes a carinality of conjugacy class for a permutation with cycle type given by partition ν ` r [7] k ν = r! m m!2 m 2 m2! r m rm r!, (8) where m i enotes the number of i in partition ν. The above is a special case of a general fact. Proposition 2. For any permutation π 2 S an any non-negative integers p, p 2,, p, we have u j,π( j) 2p j u j, j 2p j j= j= ( )( ) ( ( = p j!... k λ j ) j= λ p λ 2 p 2 λ p j= Wg(λ λ 2 λ,). (9) Proof. If we apply the formula from Eq. (3) to integral (9), then the non-vanishing permutations of inices are those which permute within the blocks of sizes {p, p 2,, p }, i.e. σ = σ σ 2 σ, where σ j 2 S pj, permuting inices in j th block of size p j. The same situation hols in the case of the secon inices. Thus the permutation τσ is also in this form. Each permutation of the above type will be present in the sum j= p j! times. The cycle type of such permutations is given by a partition which is obtaine by joining cycle types of small permutations. Since Weingarten function epens only on a cycle type of permutation, the size of conjugacy class is calculate for each partition, instea of evaluating Weingarten function multiple times. The formula (9) is far less computationally emaning than the irect usage of (3). One can notice that Proposition 2 allows us to avoi at least j= p j! executions of Weingarten function comparing to the irect usage of (3). However, the exact time-efficiency epens also on the carinality of conjugacy classes for partitions of p,, p Cycle permutations. In this section we consier another special type of integral for positive integer k an a permutation σ of {, 2,, m} being a cycle ( ) k u i,i u i,σ(i) U.. (20) ) ( m i= wing proposition. We have the following proposition. Proposition 3. Let m, k is a positive integer an σ 2 S m be a cycle, i.e. the cycle type is given by partition {m}. Then ( m i= u i,i u i,σ(i) ) k (2) =(k!) 2m k λ Wg(mλ,). λ k (2 Proof. One can notice that cycle lengths of permutations in this setting must be ivisible by m. The number of permutations Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM 23

4 Z. Puchała an J.A. Miszczak with cycle type given by a particular partition can be easily obtaine by the usual counting argument. Using (2) one obtains an enormous improvement in efficiency by avoiing more than (k!) 2m executions of Weingarten function as compare to the case of the irect usage of (3). 4. Package escription Below we escribe the functions implemente in IntU package. The functions are groupe in three categories: functions implementing the main functionality, functions relate to the calculation of Weingarten function an helper functions. 4.. Main functionality. The main functionality of IntU package is provie be the IntegrateUnitaryHaar an IntegrateUnitaryHaarInices functions. The first one operates irectly on polynomial expressions, while the secon one accepts four-tuple of inices. The examples of the usage are given in Section 5. IntegrateUnitaryHaar[integran,{var,im}] gives the efinite integral on unitary group with respect to the Haar measure, accepting the following arguments integran the polynomial type expression of variable var with inices place as subscripts, can contain any other symbolic expression of other variables, var the symbol of variable for integration, im the imension of a unitary group, must be a positive integer. This function is presente in the examples escribe in Sections 5., 5.3 an 5.5. IntegrateUnitaryHaar[f,{u,},{v,2},...] gives multiple integral U V.... (22) U( ) U( ) presente in the example This function is presente in the example escribe in Section 5.4. IntegrateUnitaryHaarInices[{I,J,I2,J2}, im] calculates the integral in (3) for given multi-inices I,J,I2,J2 an the imension im of the unitary group. This function is presente in the example escribe in Section Weingarten function. The main functions implemente in the package, IntegrateUnitaryHaar an IntegrateUnitaryHaarInices, utilize the following functions to fin the value of the integral. Weingarten[type,im] returns the value of the Weingarten function given in Eq. (8) an accepts the following arguments type an integer partition which correspons to cycle type of permutation (see Section 2.3), im the imension of a unitary group, which must be a positive integer. CharacterSymmetricGroup[part,type] gives the character of the symmetric group χ part (type) (see Section 2.3.2). Parameter type is optional. The efault value is set to a trivial partition an in this case the function returns the imension of the irreucible representation of symmetric group inexe by part, given by (7). If type is specifie, the value of the character is calculate by Murnaghan-Nakayama rule using MNInner algorithm provie in [8]. SchurPolynomialAt[part,im] returns the value of the Schur polynomial s part at point, i.e. the im ( (,,, ) imension of irreucible representation of U(im) corresponing to part, see (6) Helper functions. PermutationTypePartition[perm] gives the partition which represents the cycle type of the permutation perm (see Section 2.). MultinomialBeta[p] for a given -imensional vector of non-negative numbers p, p 2,, p returns the value of multinomial Beta function efine as B(p)= i= Γ(p i) Γ( i= p i). (23) This function is use in the optimization escribe in Section 3.2. ConjugatePartition[part] gives a conjugate of a partition part (see [5]). This function is use for calculating hook-length formula given by (7). CarinalityConjugacyClassPartition[part] gives a carinality of a conjugacy class for the permutation with cycle type given by partition part (see [7, Eq..2]). This function is use in the optimization escribe in Section 3.3. BinaryPartition[part] gives a binary representation of a partition part. This function is neee for the implementation of MNInner algorithm. 5. Examples of use In orer to present the main features of the escribe package, we provie a series of examples. 5.. Elementary integrals. Let us assume that = 3. We want to calculate the following integrals u, 2 U,, (24) u, 2 u 2,2 2 U,, (25) u, u 2,2 u,2 u 2, U. (26) 24 Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM

5 Symbolic integration with respect to the Haar measure on the unitary group Let us start by initializing the package which is equivalent to In[]:= IntU` In[6]:= IntegrateUnitaryHaar Next, we calculate the integrals. In[2]:= 3; In[3]:= Out[3]= In[4]:= Out[4]= IntegrateUnitaryHaar Abs u, ^2, u, 3 IntegrateUnitaryHaar Abs u, u 2,2 ^2, u, 8 In[5]:= IntegrateUnitaryHaar u, u 2,2 Conjugate u,2 u 2,, u, Out[5]= Operations on inices. Let us take the following set of multi-inices I = f,,, 2, 2g, J = f2, 2,,, g (27) I 0 = f,,, 2, 2g, J 0 = f2,,, 2, g (28) Out[6]= Abs u, u,2 u 2, ^2 u,2 u 2, Conjugate u, u 2,2, u, Matrix expressions. IntU package allows us to integrate matrix expressions, for example let us take = 2 an integrate In[2]:= 2; U Array u, 2 &,, ; In[5]:= U 2 Ū 2 U.. (3) bolic matrices U We efine symbolic matrices U 2 an U2 = U U 2 2 U( 2 ) as U2 KroneckerProuct U, U ; an construct the integran as integran KroneckerProuct U2, Conjugate U2 ; By using {IntegrateUnitaryHaar} function an set = 6. The above is equivalent to expression In[6]:= IntegrateUnitaryHaar integran, u, u,2 u,2 u, u 2, u 2, u,2 u, u, u 2,2 u 2, (29) with symbolic variable u, which we aim to integrate over. After simplification the expression is equal to u, 2 u,2 2 u 2, 2 u,2 u 2, u, u 2,2. (30) After initializing the package an efining appropriate inices In[2]:= I,,, 2, 2 ; J 2, 2,,, ; I2,,, 2, 2 ; J2 2,,, 2, ; 6; we calculate the integral using provie function IntegrateUnitaryHaarInices as we learn that the integral in (3) is equal to (32) In[5]:= Out[5]= IntegrateUnitaryHaarInices I, J, I2, J2, Mean value of local unitary orbit. In this example we calculate the mean value of a local unitary orbit of a given matrix X 2 M 2 E[(U V )X(U V ) ]. (33) Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM 25

6 Z. Puchała an J.A. Miszczak We assume that U an V are stochastically inepenent ranom unitary matrices of size istribute with the Haar measure. In this case we have E[(U V )X(U V ) ]= = (U V )X(U V ) UV. (34) (34) In this example we take = 3. We efine symbolic matrices X of size 2 an U, V 2 as In[2]:= 3; X Array x, 2 &, ^2, ^2 ; U Array u, 2 &,, ; V Array v, 2 &,, ; UV KroneckerProuct U, V ; Using IntegrateUnitaryHaar function with two variable specifications, we calculate the ouble integral In[7]:= int IntegrateUnitaryHaar UV.X.ConjugateTranspose UV, u,, v, an fin out that the expectation value in (33) is equal to E[(U V )X(U V ) ]= 2 trx l. (35) Similarly one can calculate the covariance tensor of the local unitary orbit of a given matrix X 2 M 2. If z = (U V) X(U V) then the covariance tensor is given by [] E[{z ij z kl } i jkl ]= =E[(U V )X(U V ) (U V )X(U V ) ]. (36) (36) Using IntegrateUnitaryHaar one can check that for the fixe imension, the integral agrees with the calculations presente in [] Moments of maximally entangle numerical shaow. If U is a ranom unitary matrix istribute accoring to the Haar measure [2], then the pure state obtaine by vectorization jξi = vec(u) is maximally entangle on H A H B = C C. Moreover, state jξ i has a istribution invariant to multiplication by local unitary matrices. We enote the corresponing probability measure on pure states of size as μ. The numerical shaow of operator X 2 M 2 with respect to this measure (maximally entangle numerical shaow) is efine as P X (z)= δ(z ψ X ψ ) µ(ψ). (37) C2 The corresponing probability measure is enote by μx E. For efinition an basic facts concerning numerical shaows see [, 3, 4]. The first two moments of μx E are calculate in [], an are given by an C zµ X E (z)= 2 trx, (38) ( zzµ C X (z) ( ( ) E = trx 2 2 ( 2 + X 2 ) HS) ( ( tra 3 ( 2 (X) 2 HS + tr B (X) 2 ) HS, ) (39) where tr A an tr B enote partial traces over a specifie sub-system an k k HS is a Hilbert-Schmit norm given by kx k HS = trxx.. In orer to calculate (39) an (39) efine symbolic matrices In[2]:= 3; X Array x, 2 &, ^2, ^2 ; In[8]:= In[9]:= U Array u, 2 &,, ; Ξ Sqrt Flatten U ; z Ξ.X.Conjugate Ξ ; zz Simplify z Conjugate z ; Now we calculate the first moment IntegrateUnitaryHaar z, u, an the secon moment int IntegrateUnitaryHaar zz, u, After some algebraic manipulations one can see that the above agrees with Eqs. (38) an (39). IntU package has been applie successfully in the context of quantum entanglement in [5], where it was use to calculate the moments of the three-tangle. 6. Summary We escribe IntU package for Mathematica computing system for calculating polynomial integrals over with respect to Haar measure. We escribe a number of special cases which can be use to optimize the calculation spee for some classes of integrals. We also provie examples of using of the presente package, incluing the applications in the geometry of quantum states. 26 Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM

7 Symbolic integration with respect to the Haar measure on the unitary group Calculation time of the package strongly epens on a egree of the integran. For polynomials of small egree, the package is able to calculate the value of integral using the irect formula (3). For polynomials of large egree, the calculation time grows rapily an the calculation is possible only if one of the special cases (optimizations) is use. Nevertheless, the presente package can be very useful in the investigations involving circular unitary ensemble an the geometry of quantum states an quantum entanglement. Acknowlegements. The work of Z. Puchała was partially supporte by the Polish National Science Centre uner research project N N , while the work of J. A. Miszczak was partially supporte by the Polish National Science Centre uner research project N N Authors woul like to thank K. Życzkowski an P. Gawron for motivation an interesting iscussions. References [] B. Collins an P. Śniay, Integration with respect to the Haar measure on unitary, orthogonal an symplectic group, Commun. Math. Phys. 264, (2006). [2] N. Ullah an C. Porter. Expectation value fluctuations in the unitary ensemble, Physical Review 32 (2), 948 (963). [3] D. Weingarten, Asymptotic behavior of group integrals in the limit of infinite rank, Journal of Mathematical Physics 9, 999 (978). [4] Z. Puchała an J. A. Miszczak, IntU package for Mathematica (20). Software available at [5] W. Fulton an J. Harris, Representation Theory: A First Course Grauate Texts in Mathematics vol. 29, Springer Verlag (99). [6] G. James an A. Kerber, The representation theory of the symmetric group, Encyclopaeia of Mathematics, vol.6 (98). [7] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, an Symmetric Functions, Springer Verlag (200). [8] D. Bernstein, The computational complexity of rules for the character table of S n, Journal of Symbolic Computation 37 (6), (2004). [9] F. Hiai an D. Petz, The semicircle law, free ranom variables an entropy, Amer. Mathematical Society 77 (2006). [0] C. Donati-Martin an A. Rouault, Truncations of Haar unitary matrices, traces an bivariate Brownian brige. Arxiv preprint, arxiv: (200). [] Z. Puchała, J. A. Miszczak, P. Gawron, C. F. Dunkl, J. A. Holbrook, an K. Życzkowski, Restricte numerical shaow an the geometry of quantum entanglement, Journal of Physics A: Mathematical an Theoretical 45 (4), (202). [2] J. Miszczak, Generating an using truly ranom quantum states in Mathematica, Comput. Phys. Commun. 83 (), 8 24 (202). [3] C. F. Dunkl, P. Gawron, J. A. Holbrook, Z. Puchała, an K. Życzkowski, Numerical shaows: measures an ensities on the numerical range, Linear Algebra Appl. 434, (20). [4] C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak, Z. Puchała, an K. Życzkowski, Numerical shaow an geometry of quantum states, J. Phys. A: Math. Theor. 44 (33), (20). [5] M. Enríquez, Z. Puchała, an K. Życzkowski, Minimal Rényi Ingaren Urbanik entropy of multipartite quantum states, Entropy 7 (7), 5063 (205). Bull. Pol. Ac.: Tech. 65() 207 Downloa Date 2/23/7 :52 AM 27