Structured Hadamard matrices and quantum information
Karol Zyczkowski, Jagiellonian University, Poland
Two classes of complex Hadamard matrices with certain special properties found recently applications in quantum physics. Consider a four index tensor $T_{ijkl}$ of size M. It can be reshaped into a square matrix $A_{\mu,\nu}$ of size $M^2$ with three different choices of composed indices e.g. $\mu=(i,j); \nu=(k,l)$ or $\mu=(i,k); \nu=(j,l)$, or $\mu=(i,l); \nu=(j,k)$. A tensor T is called perfect if all three matrices A, A' and A'' generated in this way are unitary. A matrix A is called multiunitary if it remains unitary after suitable reshuffling of their entries. Examples of multiunitary complex Hadamard matrices of size 9 are presented -- they correspond to absolutely maximally entangled states and quantum error correction codes. We discuss also skew complex Hadamard matrices, address the question for which size they do exist and show a relation to the problem of unistochasticity and construction of iso-entangled bases.