In quantum information, Mutually Unbiased Bases (MUBs) correspond to sets of pairwise complementary observables. The maximal number of such bases in a system of dimension d is d+1, and construction of maximal sets achieving this bound are known only if the dimension is a prime power. For other dimensions, we have a lower bound of three bases, and for infinitely many dimensions, we do not know how to improve this lower bound. For specific constructions, we can show that they do not achieve the upper bound. On the other hand, even in prime power dimensions where the maximal number of MUBs can be constructed, there are unextendible sets of smaller size. The smallest possible example, a pair of MUBs based on Bachelor Hadamard Matrices, has been found in dimension 6. We present results about small sets in other dimensions, including prime powers and even primes, supporting the conjecture that there are unextendible triples of MUBs in every dimension.
Markus GRASSL received his diploma degree in Computer Science in 1994 and his doctoral degree in 2001, both from the Fakultät für Informatik, Universität Karlsruhe (TH), Germany. His dissertation was on constructive and algorithmic aspects of quantum error-correcting codes. From 1994 to 2007 he was a member of the Institut für Algorithmen und Kognitive Systeme, Fakultät für Informatik, Universität Karlsruhe (TH), Germany. From 2007 to 2008 he was with the Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences in Innsbruck. From 2009 to 2014, he was a Senior Research Fellow at the Centre for Quantum Technologies at the National University of Singapore. In 2014, Markus Grassl joined the Institute of Optics, Information and Photonics at the Universität Erlangen-Nürnberg and the Max Planck Institute for the Science of Light (MPL). Currently, he is co-group-leader at MPL.
His research interests include quantum computation, focusing on quantum error-correcting codes, and methods of computer algebra in algebraic coding theory. He maintains tables of good block quantum error-correcting codes as well as good linear block codes.