Maximal sets of mutually unbiased quantum states in dimension 6

Abstract

We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the moduli of their scalar products are equal to zero, one, or 1/ √ d. These sets will be called a MU constellation, and if four MU bases were to exist for d = 6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.

Constructing mutually unbiased bases in dimension six

“…• strong numerical evidence against the existence of various MU constellations (corresponding to subsets of four MU bases) has been obtained, making the existence of a complete set highly unlikely [12].…”

Least-bias state estimation with incomplete unbiased measurements

“…(2.11) involves only the measured bases, it is also well defined for systems where M MUB exist, but this set cannot be extended to d + 1 bases. This is the case, e.g., for d = 6 and M = 3 where only sets of at most three MUB have been found thus far [43][44][45][46][47][48]. There are also nonextendable sets for prime-power dimensions [49], although ULIN estimators for these sets suffer from the same problems as the ULIN estimators for subsets of full sets (see Sec.…”

Practical implementation of mutually unbiased bases using quantum circuits

“…It has been shown that the number of MUBs is at most d + 1 [5], and that such a complete set exists whenever d is a prime or power of a prime [39]. Remarkably, there is no known answer for any other values of d, although there have been some attempts to find a solution to this problem in some simple cases, such as d = 6 [40][41][42][43][44][45] or when d is a non-prime integer squared [46,47].…”