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, and a possible connection has been discovered with Hilbert's twelfth problem.
|Unsolved problem in mathematics:|
Do SIC-POVMs exist in all dimensions?(more unsolved problems in mathematics)
The condition that the projectors defined above have equal pairwise inner products actually fixes the value of this constant. Recall that and set . Then
In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map . This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the 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:
an expression for a state can be created in terms of a quasi-probability distribution, as follows:
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
Finding SIC setsEdit
General group covarianceEdit
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 such that
The case of Zd × ZdEdit
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
Then such that the set is a SIC-POVM
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.
Relation to spherical t-designsEdit
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
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 MUBsEdit
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. Wootters points out that a complete set of unbiased bases yields a geometric structure known as a finite projective plane, while a SIC-POVM (in any dimension that is a prime power) yields a finite affine plane, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged. In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.
In dimension , the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM. The 9 vectors of the SIC-POVM, together with the 12 vectors of the mutually unbiased bases, form a set that can be used in a Kochen–Specker proof. However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete set of mutually unbiased bases has yet been discovered, and it is widely believed that no such set exists.
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