SIC-POVM
In the context of quantum mechanics and quantum information theory, symmetric, informationally complete, positive operator-valued measures (SIC-POVMs) are a particular type of generalized measurement (POVM).
SIC-POVMs are particularly notable thanks to their defining features of (1) being informationally complete; (2) having the minimal number of outcomes compatible with informational completeness, and (3) being highly symmetric.
In this context, informational completeness is the property of a POVM of allowing to fully reconstruct input states from measurement data.
The properties of SIC-POVMs make them an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism[citation needed].
SIC-POVMs have several applications in the context of quantum state tomography[1] and quantum cryptography,[2] and a possible connection has been discovered with Hilbert's twelfth problem.
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: From here, the left inverse can be calculated[4] to be
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
the equations that define the SIC-POVM can be solved by hand, yielding the vectors which form the vertices of a regular tetrahedron in the Bloch sphere.
For higher dimensions this is not feasible, necessitating the use of a more sophisticated approach.
-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
It also satisfies all of the properties for group covariance,[6] and is useful for numerical calculation of SIC sets.
Given some of the useful properties of SIC-POVMs, it would be useful if it were positively known whether such sets could be constructed in a Hilbert space of arbitrary dimension.
Originally proposed in the dissertation of Zauner,[7] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.
there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator
The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,[6] but is an ongoing field of research in the quantum information community.
Exact expressions for SIC sets have been found for Hilbert spaces of all dimensions from
[b] There exists a construction that has been conjectured to work for all prime dimensions of the form
on the d-dimensional generalized hypersphere, such that the average value of any
as the t-fold tensor product of the Hilbert spaces, and as the t-fold tensor product frame operator, it can be shown that[8] a set of normalized vectors
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.
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.
, the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.
[21] 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.
