Abstract
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
Highlights
The problem of symmetric, informationally complete quantum measurements [1,2,3,4] stands at the confluence of multiple areas of physics and mathematics
We find in general that numerical optimization finds a local minimum quickly, but a local minimum might only imply inner products between the vectors that are correct to a few decimal digits
By postulating that the symmetric informationally complete (SIC) fiducial we are looking for is a Zauner eigenvector, we can significantly reduce the effective size of the search space
Summary
The problem of symmetric, informationally complete quantum measurements [1,2,3,4] stands at the confluence of multiple areas of physics and mathematics. An intriguing feature of the SIC problem is that some numerical solutions, if extracted to sufficiently high precision, can be converted to exact ones [8,25] Most recently, this technique was used to derive an exact solution in dimension d = 48. One reason computational research is valuable, beyond extending the list of dimensions in which SICs are known, is that it provides what is likely a complete picture for many values of the dimension This is important for understanding the subtle connection between SICs and algebraic number theory [5], a connection that brings a new angle of illumination to Hilbert’s twelfth problem [6,7]. A SIC provides a frame—an equiangular tight frame—for the vector space Cd. Given a finite-dimensional Hilbert space H with an inner product h·, ·i, a frame for H is a set of vectors. J } have interesting properties with regard to Lie algebra theory [11] and the study of quantum probability [20,40]
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