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arXiv:2308.13935 · International Journal of Modern Physics A

Lars Brink and SIC-POVMs

Ingemar Bengtsson

Published Oct 17, 2024• 0 citations
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Abstract

The notion of SIC-POVMs comes from quantum theory, and they were not on the horizon when I was Lars Brink’s student in the early 1980s. In the summer of 2022, I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension [Formula: see text], and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars’ student, and of what his reaction to our construction was.

DOI
10.1142/s0217751x24470067

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The notion of SIC-POVMs comes from quantum theory, and they were not on the horizon when I was Lars Brink’s student in the early 1980s. In the summer of 2022, I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension [Formula: see text], and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars’ student, and of what his reaction to our construction was.
The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when 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 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 SIC-POVM in dimension 23.
Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd 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=0s=0s=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 SIC-POVM in dimension 23.