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

@article{Rajchel2018RobustHM,
title={Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces},
author={Grzegorz Rajchel and Adam Gasiorowski and Karol Życzkowski},
journal={Mathematics in Computer Science},
year={2018},
volume={12},
pages={473-490}
}
• Published 28 April 2018
• Mathematics
• 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 exists a skew Hadamard matrix or a symmetric conference matrix of this size. This is the case for any even $$n\le 20$$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…
3 Citations

### Algebraic and geometric structures inside the Birkhoff polytope

• Physics
Journal of Mathematical Physics
• 2022
The Birkhoﬀ polytope B d consisting of all bistochastic matrices of order d assists researchers from many areas, including combinatorics, statistical physics and quantum information. Its subset U d

### Log-convex set of Lindblad semigroups acting on N-level system

• Mathematics
Journal of Mathematical Physics
• 2021
We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and

### Distinguishing classically indistinguishable states and channels

• Physics
Journal of Physics A: Mathematical and Theoretical
• 2019
We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between M quantum states that become identical under a completely decohering map.

## References

SHOWING 1-10 OF 35 REFERENCES

### Equivalence classes of inverse orthogonal and unit Hadamard matrices

• R. Craigen
• Mathematics
Bulletin of the Australian Mathematical Society
• 1991
In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix

### Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant

• Mathematics
• 2009
A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that Bij=|Uij|2 for i,j=1,…,N. The set U3 of all unistochastic matrices of order N=3 forms a proper subset

### Equiangular tight frames and unistochastic matrices

• Mathematics
• 2016
An efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix, which is applied to find all existing classes of complex ETFs containing up to 20 vectors, is proposed.

### Hadamard Matrices and Their Applications

This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago, and identifies cocyclic generalized Hadamards with particular "stars" in four other areas of mathematics and engineering: group cohomological structures, incidence structures, combinatorics, and signal correlation.

### Construction of equally entangled bases in arbitrary dimensions via quadratic Gauss sums and graph states

• Mathematics
• 2010
Recently, Karimipour and Memarzadeh [Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space, each of dimension D, with the following

### A Concise Guide to Complex Hadamard Matrices

• Mathematics
Open Syst. Inf. Dyn.
• 2006
Basic properties of complex Hadamard matrices are reviewed and a catalogue of inequivalent cases known for the dimensions N = 2, 16, 12, 14 and 16 are presented.

### Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4

• Mathematics
• 2005
The set of bistochastic or doubly stochastic N×N matrices is a convex set called Birkhoff’s polytope, which we describe in some detail. Our problem is to characterize the set of unistochastic