Simplified exact SICs

Abstract

In the standard basis exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of a fiducial SIC vector can result in a massive reduction in complexity. We rely on a conjectural number theoretic connection between SICs in dimension $d_j$ and SICs in dimension $d_{j+1}=d_j(d_j-2)$. We focus on the sequence 5, 15, 195, ... . We rewrite Zauner's exact solution for the SIC in dimension 5 to make its simplicity manifest, and use our adapted basis to convert numerical solutions in dimensions 15 and 195 to exact solutions. Comparing to the known exact solutions in dimension 15 we find that the simplification achieved is dramatic. The proof that the exact vectors are indeed SIC fiducial vectors, also in dimension 195, is a long calculation guided by the standard ray class hypothesis about the algebraic number fields generated by the SICs. We conjecture that our result generalizes to every dimension in the particular sequence we consider.

arXiv comment20 pages; one section improved in version 2

References in corpus (11)

• Symmetric Informationally Complete Quantum Measurements
• SIC-POVMs and the Extended Clifford Group
• SIC-POVMs: A new computer study
• The SIC Question: History and State of Play
• On SIC-POVMs and MUBs in Dimension 6
• Constructing exact symmetric informationally complete measurements from numerical solutions
• Fibonacci-Lucas SIC-POVMs
• Properties of the extended Clifford group with applications to SIC-POVMs and MUBs
• SICs: Extending the list of solutions
• Dimension towers of SICs. I. Aligned SICs and embedded tight frames
• Aligned SICs and embedded tight frames in even dimensions

Cited by in corpus (10)

• Entropic uncertainty relations from equiangular tight frames and their applications
• What are the minimal conditions required to define a SIC POVM?
• 9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers
• Five open problems in quantum information
• Bipartite quantum measurements with optimal single-sided distinguishability
• Rényi formulation of uncertainty relations for POVMs assigned to a quantum design
• Entanglement criterion via general symmetric informationally complete measurement
• SICs: Some explanations
• On Kirkwood--Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame
• Dimension towers of SICs. II. Some constructions