Constructing exact symmetric informationally complete measurements from numerical solutions

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 and 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 a SIC. Using this method we have calculated 69 new exact solutions, including 9 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 we address two questions suggested by the previous work.

22 pages + 19 page appendix with many data tables. v2: published version

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Cited by in corpus (26)

• The SIC Question: History and State of Play
• Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems
• Fibonacci-Lucas SIC-POVMs
• A family of multipartite separability criteria based on correlation tensor
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Joint measurability of quantum effects and the matrix diamond
• Dimension towers of SICs. I. Aligned SICs and embedded tight frames
• What are the minimal conditions required to define a SIC POVM?
• Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations
• Communication of partial ignorance with qubits
• Entanglement properties of multipartite informationally complete quantum measurements
• Five open problems in quantum information
• Experimental quantum tomography assisted by multiply symmetric states in higher dimensions
• Simplified exact SICs
• Construction of general symmetric-informationally-complete-positive-operator-valued measures by using a complete orthogonal basis
• Bipartite quantum measurements with optimal single-sided distinguishability
• Learning entanglement breakdown as a phase transition by confusion
• SIC-POVMs from Stark units: Prime dimensions n^2+3
• Tight frames over the quaternions and equiangular lines
• SICs: Some explanations
• The Poincaré half-plane for informationally complete POVMs
• Constructing high order spherical designs as a union of two of lower order
• Quantum computing with Bianchi groups
• A variational characterisation of projective spherical designs over the quaternions
• Putatively optimal projective spherical designs with little apparent symmetry
• Dimension towers of SICs. II. Some constructions