1. Definition

2. Properties
    Symmetry  Superoperator 

3. Finding SIC sets
    Group covariance   General group covariance  The case of Z_d × Z_d   Zauner's conjecture  Partial results 

4. Relation to spherical t-designs

5. Relation to MUBs

6. References

7. See also

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^[1] and quantum cryptography,^[2] and a possible connection has been discovered with Hilbert's twelfth problem.^[3]

Definition

A POVM over a -dimensional Hilbert space is a set of positive-semidefinite operators on the Hilbert space that sum to the identity:

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,

defines a minimal IC-POVM called a SIC-POVM.

Properties

Symmetry

The condition that the projectors defined above have equal pairwise inner products actually fixes the value of this constant. Recall that and set . Then

implies that . Thus,

This property is what makes SIC-POVMs symmetric; with respect to the Hilbert–Schmidt inner product, any pair of elements is equivalent to any other pair.

Superoperator

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:

where

From here, the left inverse can be calculated^[4] to be , and so with the knowledge that

,

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 sets

Group covariance

General group covariance

A SIC-POVM is said to be group covariant if there exists a group with a -dimensional unitary representation such that

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 SIC-POVM is then the set generated by the group action of on .

The case of Z_d × Z_d

So far, most SIC-POVM's have been found by considering group covariance under .^[5] 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,^[6] and is useful for numerical calculation of SIC sets.

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,^[6] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.

More specifically,

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 ^[8]

For any dimension , let be an orthonormal basis for , and define

Then such that the set is a SIC-POVM

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 .^{[6][8][7][8][9][10][11]} Furthermore, using the Heisenberg group covariance on , numerical solutions have been found for all integers up through .^{[5][8][8][12][13][14]}

The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,^[15] 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 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^[16] 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.

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. 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.^[17]

In dimension , the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.^[18] 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.^[19] 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.^[20][21]

References

See also

• Measurement in quantum mechanics
• Mutually unbiased bases
• POVM
• QBism

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