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Mathematics > Number Theory

arXiv:1807.05877 (math)
[Submitted on 16 Jul 2018 (v1), last revised 16 Dec 2018 (this version, v2)]

Title:SIC-POVMs and the Stark conjectures

Authors:Gene S. Kopp
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Abstract:The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields.
We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact solution to the SIC-POVM problem in dimension 23.
Comments: 15 pages
Subjects: Number Theory (math.NT); Quantum Physics (quant-ph)
MSC classes: 11R37, 11R42, 81P15, 81R05
Cite as: arXiv:1807.05877 [math.NT]
  (or arXiv:1807.05877v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1807.05877

Submission history

From: Gene Kopp [view email]
[v1] Mon, 16 Jul 2018 14:06:31 UTC (2,250 KB)
[v2] Sun, 16 Dec 2018 20:09:39 UTC (2,250 KB)
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