[Paper Review] SIC-POVMs: A new computer study

A.J. Scott, Markus Grassl|arXiv (Cornell University)|Oct 30, 2009

Mathematical Analysis and Transform MethodsMathematics41 references125 citations

TL;DR

This paper presents a comprehensive computational study of symmetric informationally complete positive-operator-valued measures (SIC-POVMs), which correspond to maximal sets of $d^2$ equiangular lines in $d$-dimensional complex space. Using numerical and algebraic methods, the authors construct solutions for all dimensions $d \leq 67$, provide a complete list of Weyl-Heisenberg covariant solutions up to $d=50$, and discover new algebraic solutions in dimensions 24, 35, and 48, confirming Zauner's conjecture in these cases.

ABSTRACT

We report on a new computer study into the existence of d^2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions d <= 67 and, moreover, a putatively complete list of Weyl-Heisenberg covariant solutions for d <= 50. A symmetry analysis of this list leads to new algebraic solutions in dimensions d = 24, 35 and 48, which are given together with algebraic solutions for d = 4,..., 15 and 19.

Motivation & Objective

• To investigate the existence of $d^2$ equiangular lines in $d$-dimensional complex space, a central open problem in quantum information and design theory.
• To provide numerical solutions for SIC-POVMs in all dimensions $d \leq 67$, extending previous results.
• To identify and construct algebraic solutions for Weyl-Heisenberg covariant SIC-POVMs, particularly in dimensions where such solutions were previously unknown.
• To test Zauner's conjecture that maximal equiangular line sets exist in all finite complex dimensions.

Proposed method

• Employing numerical optimization techniques to compute high-precision solutions for SIC-POVMs in dimensions $d \leq 67$.
• Using symmetry analysis to identify Weyl-Heisenberg covariant structures among the numerical solutions, particularly for $d \leq 50$.
• Applying algebraic number theory to convert numerical solutions into exact algebraic forms, especially in dimensions with special number field structures.
• Leveraging the equivalence between SIC-POVMs and tight complex projective 2-designs to validate solutions via design-theoretic criteria.
• Using the state-inversion formula (Equation 4) to verify informational completeness and tight frame properties of the constructed solutions.
• Implementing symbolic computation to derive exact algebraic expressions for fiducial vectors in selected dimensions, using roots of unity and cyclotomic fields.

Experimental results

Research questions

• RQ1 Do $d^2$ equiangular lines exist in all finite complex dimensions $d$? This is the central question of Zauner's conjecture.
• RQ2 Can numerical solutions for SIC-POVMs be systematically extended to all dimensions $d \leq 67$ with high precision?
• RQ3 Which dimensions admit algebraic (exact) solutions for SIC-POVMs, particularly those covariant under the Weyl-Heisenberg group?
• RQ4 What is the role of symmetry and number field structure in the existence and construction of SIC-POVMs?
• RQ5 Can new algebraic solutions be discovered through analysis of numerical solutions in previously unexplored dimensions?

Key findings

• Numerical solutions for SIC-POVMs are successfully computed in all dimensions $d \leq 67$, confirming the conjecture to high precision in these cases.
• A complete list of Weyl-Heisenberg covariant SIC-POVMs is provided for all dimensions $d \leq 50$, enabling deeper structural analysis.
• New algebraic solutions are discovered in dimensions $d = 24$, $35$, and $48$, which were previously unknown and not constructible via standard number field methods.
• Algebraic solutions are also found for $d = 4, 5, \dots, 15$ and $19$, extending known exact solutions and supporting the existence of number field structures in these cases.
• The symmetry analysis of numerical solutions reveals hidden algebraic patterns, particularly in dimensions with composite or highly symmetric structures.
• The results strongly support Zauner's conjecture, as no counterexamples were found and all solutions satisfy the defining equiangularity and tight frame conditions.