If a POVM consists of at least operators which span, it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly elements are called minimal. A set of rank-1 projectors which have equal pairwise Hilbert–Schmidt inner products,
This operator acts on a SIC-POVM element in a way very similar to identity, in that
But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:
From here, the left inverse can be calculated to be , and so with the knowledge that
where is the Dirac notation for the density operator viewed in the Hilbert space . This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state is given by
So far, most SIC-POVM's have been found by considering group covariance under . To construct the unitary representation, we map to , the group of unitary operators on d-dimensions. Several operators must first be introduced. Let be a basis for , then the phase operator is
where is a root of unity
and the shift operator as
Combining these two operators yields the Weyl operator which generates the Heisenberg-Weyl group. This is a unitary operator since
It can be checked that the mapping is a projective unitary representation. It also satisfies all of the properties for group covariance, and is useful for numerical calculation of SIC sets.
Given some of the useful properties of SIC-POVMs, it would be useful if it was positively known whether such sets could be constructed in a Hilbert space of arbitrary dimension. Originally proposed in the dissertation of Zauner, a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.
For every dimension there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator under the Weyl–Heisenberg group. What is more, commutes with an element T of the Jacobi group . The action of T on modulo the center has order three.
Utilizing the notion of group covariance on , this can be restated as 
For any dimension , let be an orthonormal basis for , and define
Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is . Furthermore, using the Heisenberg group covariance on , numerical solutions have been found for all integers up through .
The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question, but is an ongoing field of research in the quantum mechanics community.
A spherical t-design is a set of vectors on the d-dimensional generalized hypersphere, such that the average value of any -order polynomial over is equal to the average of over all normalized vectors . Defining as the t-fold tensor product of the Hilbert spaces, and
as the t-fold tensor product frame operator, it can be shown that a set of normalized vectors with forms a spherical t-design if and only if
It then immediately follows that every SIC-POVM is a 2-design, since
which is precisely the necessary value that satisfies the above theorem.
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.
^ abG. Zauner, Quantendesigns – Grundzüge einer nichtkommutativen Designtheorie. Dissertation, Universität Wien, 1999.
^ abcdRenes, Joseph M.; Blume-Kohout, Robin; Scott, A. J.; Caves, Carlton M. (2003). "Symmetric Informationally Complete Quantum Measurements". Journal of Mathematical Physics. 45 (6): 2171. arXiv:quant-ph/0310075. doi:10.1063/1.1737053.
^A. Koldobsky and H. K¨onig, “Aspects of the Isometric Theory of Banach Spaces,” in Handbook of the Geometry of Banach Spaces, Vol. 1, edited by W. B. Johnson and J. Lindenstrauss, (North Holland, Dordrecht, 2001), pp. 899–939.