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arXiv · 1707.09911

Dimension towers of SICs. I. Aligned SICs and embedded tight frames

Marcus Appleby·Ingemar Bengtsson·Irina Dumitru·Steven Flammia
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Abstract

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.

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Marcus Appleby, Ingemar Bengtsson, Irina Dumitru, Steven Flammia. 2017-07-31. Dimension towers of SICs. I. Aligned SICs and embedded tight frames. https://doi.org/10.1063/1.4999844

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