Definition
Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used for the remainder of this article.
In general, a POVM over a finite d-dimensional Hilbert space is defined as a set of positive semidefinite operators on a Hilbert space H that sum to unity,
While a SIC-POVM will still satisfy this property, the POVM elements are now restricted to be subnormalized projectors. More specifically, if is a rank one projector in a d-dimensional Hilbert space, then the corresponding subnormalized projector is
Furthermore, SIC-POVMs add to the theory of general POVMs by demanding additional structure on the projector states. Indeed, for to be informationally complete it must consist of linearly independent projector operators so as to form a basis for the Hilbert-Schmidt space . This ensures that any mixed state can be uniquely represented in terms of the POVM elements by utilizing the Hilbert-Schmidt inner product to calculate the statistical coefficients of the pure state projectors.
An additional notion of symmetry is implied by demanding that the inner product of any two distinct elements be invariant. More precisely,
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