Robust Hadamard Matrices,Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces

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 or a symmetric conference matrix of this size. This is the case for any even \(n\le 20\), 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. In the case \(n=4\) we study geometry of the set \({\mathcal U}_4\) of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope \({\mathcal B}_4\).