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

Grzegorz Rajchel, Adam Gąsiorowski, Karol Życzkowski
2018 Mathematics in Computer Science  
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 is a skew Hadamard matrix of a symmetric conference matrix of this size. This is the case for any even n ≤ 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
more » ... with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix U , such that Bij = |Uij| 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 × 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 U4 of unistochastic matrices, conjecture that this set is star-shaped and estimate its relative volume in the Birkhoff polytope B4.
doi:10.1007/s11786-018-0384-y fatcat:4czcwgt4brebpfoodod2w5uwmi