Permutation Equivalence Classes of Kronecker Products of Unitary Fourier Matrices

Kronecker products of unitary Fourier matrices play important role in solving multilevel circulant systems by a multidimensional Fast Fourier Transform. They are also special cases of complex Hadamard (Zeilinger) matrices arising in many problems of mathematics and theoretical physics. The main result of the paper is splitting the set of all kronecker products of unitary Fourier matrices into permutation equivalence classes. The choice of permutation equivalence to relate the products is motivated by the quantum information theory problem of constructing maximally entangled bases of finite dimensional quantum systems. Permutation inequivalent products can be used to construct inequivalent, in a certain sense, maximally entangled bases.Comment: 26 page

Defect of a Kronecker product of unitary matrices

AbstractThe generalized defect D(U) of a unitary N×N matrix U with no zero entries is the dimension of the real space of directions, moving into which from U we do not disturb the moduli Ui,j as well as the Gram matrix U∗U in the first order. Then the defect d(U) is equal to D(U)-(2N-1), that is the generalized defect diminished by the dimension of the manifold {DrUDc:Dr,Dc unitary diagonal. Calculation of d(U) involves calculating the dimension of the space in RN2 spanned by a certain set of vectors associated with U. We split this space into a direct sum, assuming that U is a Kronecker product of unitary matrices, thus making it easier to perform calculations numerically. Basing on this, we give a lower bound on D(U) (equivalently d(U)), supposing it is achieved for most unitaries with a fixed Kronecker product structure. Also supermultiplicativity of D(U) with respect to Kronecker subproducts of U is shown

Defect and equivalence of unitary matrices. The Fourier case

Consider the real space D_U of directions moving into which from a unitary N x N matrix U we do not disturb its unitarity and the moduli of its entries in the first order. dim( D_U ) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov's theory where D_U is parametrized by the imaginary subspace of the eigenspace, associated with lambda = 1, of a certain unitary operator I_U on the N x N complex matrices, and where D(U) is the multiplicity of 1 in the spectrum of I_U. This characterization allows us to establish dependence of D(U_1 x ... x U_r) - where x stands for the Kronecker product - on D(U_k)'s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of I_U we study the 'permute and enphase' symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups.Comment: 92 pages. The paper has undergone hundreds of minor corrections and improvements. Some paragraphs have been completely changed, a few addded. Sections Introduction and Conclusions have been added, as well as a new abstrac

Mubs and Hadamards of Order Six

We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organise our results. Finally we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.Comment: 33 pages, 3 figures. References added in v

Birkhoff's polytope and unistochastic matrices, N=3 and N=4

The set of bistochastic or doubly stochastic N by N matrices form a convex set called Birkhoff's polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff's polytope. For N=3 we present fairly complete results. For N=4 partial results are obtained. An interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der Waerden matrix for N=3, while this is not the case for N=4.Comment: 30 pages, 4 figure

Defect of a unitary matrix

We analyze properties of a map B = f(U) sending a unitary matrix U of size N into a doubly stochastic matrix defined by B_{i,j} = |U_{i,j}|^2. For any U we define its DEFECT, determined by the dimensionality of the space being the image Df(T_U Unitaries) of the space T_U Unitaries tangent to the manifold of unitary matrices Unitaries at U, under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, gives an upper bound for the dimensionality of a smooth orbit (a manifold) of inequivalent unitary matrices V mapped into the same image, f(V) = f(U) = B, stemming from U. We demonstrate several properties of the defect and prove an explicit formula for the defect of a Fourier matrix F_N of size N. In this way we obtain an upper bound for the dimensionality of a smooth orbit of inequivalent unitary complex Hadamard matrices stemming from F_N. It is equal to zero iff N is prime and coincides with the dimensionality of the known orbits if N is a power of a prime. Two constructions of these orbits are presented at the end of this work.Comment: 46 page