Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces

Abstract

We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a $2$-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order $n$ exists, if there exists a skew Hadamard matrix of this size. This is the case for any even dimension $n\le 20$, and for these dimensions we demonstrate that a bistochastic matrix $B$ located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix $U$, such that $B_{ij}=|U_{ij}|^2$, is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order $n \times n$. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis.