SIC-POVMs and the Stark conjectures

Five open problems in quantum information

“…Numerical solutions are known for all N ≤ 189 and for several much larger dimensions [6][7][8]11], including N = 844, 2208. However, in spite of the recent research effort [12][13][14][15], the general conjecture of Zauner remains unproven. Finding an infinite family of SICs, as requested in KCIK Problem 1, could become decisive step in this direction.…”

“…Finding an infinite family of SICs, as requested in KCIK Problem 1, could become decisive step in this direction. Furthermore, let us emphasize inspiring connections to some major open questions in algebraic number theory, including the 12th problem of Hilbert [10,12,16,17]. KCIK Problem 2.…”

Discrete Wigner Functions from Informationally Complete Quantum Measurements

“…These can of course be defined for any Wigner function. Of particular note is the case where the Wigner function is the principal Wigner function of a SIC, because the SIC triple products tr Π j Π k Π l are remarkable numbers [11,[33][34][35][36]. We have that…”

“…The larger the enclosed area, the greater the geometric phase. The SIC triple products, and thus by extension those of their associated Wigner functions, have rich number-and group-theoretic properties [30,31,[33][34][35][36]. What these properties imply for the Wigner functions derived from SICs is largely an open question.…”

Misreading EPR: Variations on an Incorrect Theme

“…Getting a clean and precise expression of this deviation requires finding the optimal reference measurements. To the surprise of the people who have worked on this topic, the question of finding the optimal reference measurements turns out to have intriguing connections with far-flung areas of mathematics [55][56][57][58][59][60].…”