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arXiv:1701.05200 · Foundations of Physics

SICs and Algebraic Number Theory

Marcus Appleby, Steven T. Flammia, Gary McConnell, Jon Yard

Published Apr 24, 2017• 71 citations
arXivHTMLPDF
DOI
10.1007/s10701-017-0090-7

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Abstract Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs to their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here, we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of an SIC. Using this method, we have calculated 69 new exact solutions, including nine new dimensions, where previously only numerical solutions were known—which more than triples the number of known exact solutions. In some cases, the solutions require number fields with degrees as high as 12 288. We use these solutions to confirm that they obey the number-theoretic conjectures, and address two questions suggested by the previous work.
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.