A symmetric, informationally complete, positive operator valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement," utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography.
Contents
Definition
Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used throughout this article to represent elements in a Hilbert space.
A POVM over a
Symmetry
The condition that the projectors
Superoperator
In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map
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
an expression for a state
where
General group covariance
A SIC-POVM
The search for SIC-POVMs can be greatly simplified by exploiting the property of group covariance. Indeed, the problem is reduced to finding a normalized fiducial vector
The SIC-POVM is then the set generated by the group action of
The case of Zd × Zd
So far, most SIC-POVM's have been found by considering group covariance under
and the shift operator as
Combining these two operators yields the Weyl operator
It can be checked that the mapping
Zauner's conjecture
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.
More specifically,
For every dimension
Utilizing the notion of group covariance on
For any dimension
Then
Partial results
Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is
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.
Relation to spherical t-designs
A spherical t-design is a set of vectors
as the t-fold tensor product frame operator, it can be shown that a set of normalized vectors
It then immediately follows that every SIC-POVM is a 2-design, since
which is precisely the necessary value that satisfies the above theorem.
Relation to MUBs
In a d-dimensional Hilbert space, two distinct bases
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
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.