SIC-POVM - Relation To MUBs

Relation To MUBs

In a d-dimensional Hilbert space, two distinct bases are said to be mutually unbiased if

This seems similar in nature to the symmetric property of SIC-POVMs. In fact, the problem of finding a SIC-POVM is precisely the problem of finding equiangular lines in ; whereas mutually unbiased bases are analogous to affine spaces. In fact it can be shown that the geometric analogy of finding a "complete set of mutually unbiased bases is identical to the geometric structure analogous to a SIC-POVM ". It is important to note that the equivalence of these problems is in the strict sense of an abstract geometry, and since the space on which each of these geometric analogues differs, there's no guarantee that a solution on one space will directly correlate with the other.

An example of where this analogous relation has yet to necessarily produce results is the case of 6-dimensional Hilbert space, in which a SIC-POVM has been analytically computed using mathematical software, but no complete mutually unbiased bases has yet been discovered.