MUBs and Hadamards

          @misc{ scivideos_PIRSA:08020035,
            doi = {10.48660/08020035},
            url = {https://pirsa.org/08020035},
            author = {Ericsson, Asa},
            keywords = {Quantum Foundations},
            language = {en},
            title = {MUBs and Hadamards},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2008},
            month = {feb},
            note = {PIRSA:08020035 see, \url{https://scivideos.org/pirsa/08020035}}
          }
          

Abstract

Mutually unbiased bases (MUBs) have attracted a lot of attention the last years. These bases are interesting for their potential use within quantum information processing and when trying to understand quantum state space. A central question is if there exists complete sets of N+1 MUBs in N-dimensional Hilbert space, as these are desired for quantum state tomography. Despite a lot of effort they are only known in prime power dimensions. I will describe in geometrical terms how a complete set of MUBs would sit in the set of density matrices and present a distance between bases–a measure of unbiasedness. Then I will explain the relation between MUBs and Hadamard matrices, and report on a search for MUB-sets in dimension N=6. In this case no sets of more than three MUBs are found, but there are several inequivalent triplets.