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

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.

24 pages, 2 figures

References in corpus (4)

• The SIC Question: History and State of Play
• Constructing exact symmetric informationally complete measurements from numerical solutions
• SICs and Algebraic Number Theory
• SICs: Extending the list of solutions

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• Simplified exact SICs
• SICs: Some explanations
• Quantum Theory as Symmetry Broken by Vitality
• Aligned SICs and embedded tight frames in even dimensions
• Optimal arrangements of classical and quantum states with limited purity
• Fibonacci Fervour in Linear Algebra and Quantum Information Theory
• SICs and the Triangle Group $(3,3,3)$
• Dimension towers of SICs. II. Some constructions