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

Abstract

Abstract. We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.Key words: unitary group, Haar measure, circular unitary ensemble, symbolic integration. Integer partition λ of a positive integer n is a weakly decreasing sequence λ = (λ 1 , λ 2 , …, λ l ) of positive integers, such that ∑ l i =1 λ i = jλj = n. To denote that λ is a partition of n, we write λ ` n. The length of a partition is denoted by l(λ). By λ t μ we denote a partition of n 1 + n 2 obtained by joining partitions λ ` n 1 and μ ` n 2 . Symbolic integration with respect to theEach permutation σ 2 S n can be uniquely decomposed into a sum of disjoint cycles, where the lengths of the cycles sum up to n. Thus the vector of the lengths of the cycles, after reordering, forms a partition λ ` n. The partition λ is called the cycle type of σ permutation. Moments of the U(d). Let us consider a polynomial p.From the linearity of an integral we have bstract. We present package for Mathematica computer algebra system. The presented package performs a symbolic integration of olynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which an be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package. ntegrals of the above type are known as moments of the U(d) nd are well-known in mathematical physics literature for a ong time. The problem of the integration of elements of uniary matrices was for the first time considered in the context of uclear physics in [2]. The asymptotic behaviour of the interals of the type (1) was considered by Weingarten in [3]. In this paper we describe a Mathematica package [4] or calculating polynomial integrals over U(d) with respect o the Haar measure. We describe a number of special cases hich can be used to optimize the calculation speed for some lasses of integrals. We also provide some examples of usge of the presented package including the applications in the tudy of the geometry of the quantum states. This paper is organised as follows. In Section 2 we introduce otation present mathematical background concerning polynoial integrals over unitary group. In Section 3 we describe ome special cases, in which the integration can be calculated ore efficiently. In Section 4 we provide the description of the package with the list of main functions. In Section 5 we how some examples of the usage. In Section 6 we provide a ummary of the presented results and give conclusions.Integer partition λ of a positive integer n is a weakly de-To denote that λ is a partition of n we write λ n. The length of a partition is denoted by l(λ ). By λ µ we denote a par...

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