Volumes of Orthogonal Groups and Unitary Groups ∗ Lin Zhang Institute of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, PR China Abstract The matrix integral has many applications in diverse fields. This review article begins by presenting detailed key background knowledge about matrix integral. Then the volumes of orthogonal groups and unitary groups are computed, respectively. As a unification, we present Mcdonald’s volume formula for a compact Lie group. With this volume formula, one can easily derives the volumes of orthogonal groups and unitary groups. Applications are also presented as well. Specifically, The volume of the set of mixed quantum states is computed by using the volume of unitary group. The volume of a metric ball in unitary group is also computed as well. There are no new results in this article, but only detailed and elementary proofs of existing results. The purpose of the article is pedagogical, and to collect in one place many, if not all, of the quantum information applications of the volumes of orthogonal and unitary groups. arXiv:1509.00537v5 [math-ph] 3 Nov 2017 ∗ E-mail: [email protected]; [email protected] 1 Contents 1 Introduction 3 2 Volumes of orthogonal groups 5 2.1 Preliminary ..................................... ..... 5 2.2 Thecomputationofvolumes . ....... 17 3 Volumes of unitary groups 28 3.1 Preliminary ..................................... ..... 28 3.2 Thecomputationofvolumes . ....... 44 4 The volume of a compact Lie group 55 5 Applications 73 5.1 Hilbert-Schmidt volume of the set of mixed quantum states.............. 73 5.2 Areaoftheboundaryofthesetofmixedstates . ........... 77 5.3 Volumeofametricballinunitarygroup . .......... 79 6 Appendix I: Volumes of a sphere and a ball 85 7 Appendix II: Some useful facts 89 7.1 Matrices with simple eigenvalues form open dense sets of fullmeasure . 89 7.2 Resultsrelatedtoorthogonalgroups . ............ 93 7.3 Resultsrelatedtounitarygroups . ........... 96 8 Appendix III: Some matrix factorizations 107 9 Appendix IV: Selberg’s integral 112 2 1 Introduction Volumes of orthogonal groups and unitary groups are very useful in physics and mathematics [3, 4]. In 1949, Ponting and Potter had already calculated the volume of orthogonal and unitary group [27]. A complete treatment for group manifolds is presented by Marinov [17], who ex- tracted the volumes of groups by studying curved path integrals [18]. There is a general closed formula for any compact Lie group in terms of the root lattice [?]. Clearly the methods used previously are not followed easily. Zyczkowski˙ in his paper [38] gives a re-derivation on the volume of unitary group, but it is still not accessible. The main goal of this paper is to compute the volumes of orthogonal groups and unitary groups in a systematic and elementary way. The obtained formulae is more applicable. As an application, by re-deriving the volumes of orthogonal groups and unitary groups, the authors in [38] computes the volume of the convex (n2 1)-dimensional set D (Cn) of the density − matrices of size n with respect to the Hilbert-Schmidt measure. Recently, the authors in [30] give the integral representation of the exact volume of a metric ball in unitary group; and present diverse applications of volume estimates of metric balls in manifolds in information and coding theory. Before proceeding, we recall some notions in group theory. Assume that a group acts on G the underlying vector space via g x for all g and x . Let x be any nonzero vector in X | i ∈ G ∈X | i . The subset x := g x : g is called the -orbit of x . Denote X G| i { | i ∈ G} G | i∈X xG := g : g x = x . { ∈ G | i | i} We have the following fact: x /xG . G| i ∼ G If x = e , where e is a unit element of , then the action of on x is called free. In this case, G { } G G | i x / e x . G| i ∼ G { }⇐⇒G ∼G| i Now we review a fast track of the volume of a unitary group. Let us consider the unitary groups = (n + 1), n = 1,2,... To establish their structure, we look at spaces in which the group acts G U transitively, and identify the isotropy subgroup. The unitary group (n + 1) acts naturally in U the complex vector space = Cn+1 through the vector or "defining" representation; the image X of any nonzero vector x = ψ Cn+1 is contained in the maximal sphere S2n+1 of radius | i | i ∈ ψ since ψ = Uψ for U (n + 1), and in fact it is easy to see that it sweeps the whole k k k k k k ∈ U sphere when U runs though (n + 1), i.e. the group (n + 1) acts transitively in this sphere. U U The isotropy group of the vector n + 1 is easily seen to be the unitary group with an entry less, | i that is x = (n) 1. Indeed, the following map is onto: G U ⊕ ϕ : (n + 1) (n + 1) n + 1 = S2n+1. U −→ U | i 3 If we identity (n) with (n) 1, as a subgroup of (n + 1), then U U ⊕ U (n) n + 1 = n + 1 , U | i | i implying that ker ϕ = (n), and thus U (n + 1)/ker ϕ (n + 1) n + 1 = S2n+1 = ( (n + 1)/ker ϕ) n + 1 . U ∼ U | i U | i Therefore we have the equivalence relation (n + 1)/ (n)= S2n+1. (1.1) U U This indicates that vol ( (n + 1)) = vol S2n+1 vol ( (n)) , n = 1, 2, . (1.2) U · U That is, 1 3 2n 1 vol ( (n)) = vol S vol S vol S − . (1.3) U × ×···× We can see this in [8]. The volume of the sphere of unit radius embedded in Rn(n > 1), is calculated from the Gaussian integral (see also Appendix I for the details): +∞ t2 √π = e− dt. ∞ Z− Now +∞ n n t2 v 2 √π = e− dt = e−k k dv ∞ Rn Z− Z +∞ +∞ r2 r2 = e− dσdr = σn 1(r)e− dr, Sn 1 − Z0 Z − (r) Z0 Sn 1 where σn 1(r)= Sn 1 dσ is the volume of sphere − (r) of radius r. Since − − (r) R n 1 σn 1(r)= σn 1(1) r − , − − × it follows that +∞ n n 1 r2 √π = σn 1(1) r − e− dr, − × Z0 implying that n 2π 2 vol Sn 1 := . (1.4) − Γ n 2 4 Finally, we get the volume formula of a unitary group: n k n n(n+1) 2π 2 π 2 vol ( (n)) := ∏ = . (1.5) U Γ(k) 1!2! (n 1)! k=1 · · · − 2 Volumes of orthogonal groups 2.1 Preliminary The following standard notations will be used [19]. Scalars will be denoted by lower-case letters, vectors and matrices by capital letters. As far as possible variable matrices will be denoted by X, Y,... and constant matrices by A, B,.... n Let A = [aij] be a n n matrix, then Tr (A) = ∑j=1 ajj is the trace of A, and det(A) is the T× determinant of A, and over a vector or a matrix will denote its transpose. Let X = [xij] be a m n matrix of independent real entries x ’s. We denote the matrix of differentials by dX, i.e. × ij dx dx dx 11 12 · · · 1n dx dx dx 21 22 · · · 2n dX := [dxij]= . . .. dxm1 dxm2 dxmn · · · Then [dX] stands for the product of the m n differential elements × m n [dX] := ∏ ∏ dxij (2.1) i=1 j=1 and when X is a real square symmetric matrix, that is, m = n, X = XT, then [dX] is the product of the n(n + 1)/2 differential elements, that is, n n [dX] := ∏ ∏ dxij = ∏ dxij. (2.2) j=1 i=j i>j Throughout this paper, stated otherwise, we will make use of conventions that the signs will be ignored in the product [dX] of differentials of independent entries. Our notation will be the following: Let X = [x ] be a m n matrix of independent real entries. Then ij × [dX]= m n dx (2.3) ∧i=1 ∧j=1 ij 5 when [dX] appears with integrals or Jacobians of transformations; m n [dX]= ∏i=1 ∏j=1 dxij (2.4) when [dX] appears with integrals involving density functions where the functions are nonnegative and the absolute value of the Jacobian is automatically taken. As Edelman said [7] before, many researchers in Linear Algebra have little known the fact that the familiar matrix factorizations, which can be viewed as changes of variables, have simple Ja- cobians. These Jacobians are used extensively in applications of random matrices in multivariate statistics and physics. It is assumed that the reader is familiar with the calculation of Jacobians when a vector of scalar variables is transformed to a vector of scalar variables. The result is stated here for the sake of completeness. Let the vector of scalar variables X be transformed to Y, where x1 y1 . X = . and Y = . , x y n n by a one-to-one transformation. Let the matrix of partial derivatives be denoted by ∂Y ∂y = i . ∂X ∂x j The the determinant of the matrix ∂yi is known as the Jacobian of the transformation X going ∂xj to Y or Y as a function of X it is writtenh i as ∂y J(Y : X)= det i or [dY]= J(Y : X)[dX], J = 0 ∂x 6 j and 1 J(Y : X)= or 1 = J(Y : X)J(X : Y). J(X : Y) Note that when transforming X to Y the variables can be taken in any order because a permu- tation brings only a change of sign in the determinant and the magnitude remains the same, that is, J remains the same where J denotes the absolute value of J.