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SIC-POVMs: A new computer study

A. J. Scott, M. Grassl

arXiv:0910.5784v2quant-phmath.COmath.FA

TL;DR

The paper addresses whether maximal sets of d^2 equiangular lines exist in every finite complex dimension, objects that define SIC-POVMs in quantum theory. It conducts a computer study, finding numerical solutions through d ≤ 67 and a putatively complete Weyl-Heisenberg-covariant list through d ≤ 50, alongside new algebraic solutions. The results substantially extend the known computational and algebraic record, but do not provide a proof of Zauner’s conjecture.

  • Problem

    The open problem is whether maximal sets of d^2 equiangular lines exist in every finite complex dimension; these sets define SIC-POVMs.

  • Method

    The study numerically searches for fiducial vectors generating Weyl-Heisenberg covariant SIC-POVMs and analyzes symmetries in the resulting solutions.

  • Results

    Numerical solutions are provided for all d ≤ 67, except that d = 66 remains inconclusive; a putatively complete list is given for d ≤ 50, with new algebraic solutions in d = 24, 35, and 48.

  • Takeaways & Limitations

    The study provides a substantial computational resource and extends the catalogue of algebraic Weyl-Heisenberg covariant SIC-POVM solutions.

  • Takeaways & Limitations

    The study does not establish a proof of Zauner’s conjecture, and the automorphism group of Weyl-Heisenberg covariant SIC-POVMs remains incompletely understood.

Abstract

from arXiv · show

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.

1. INTRODUCTION

The paper studies whether the maximum possible number, d^2, of equiangular lines exists in every finite complex dimension. This longstanding question has drawn growing attention across quantum physics, design theory, and frame theory.

  • The central question is whether d^2 equiangular lines exist in all finite complex dimensions d.The paper describes this as one of the most urgent unanswered questions in the area.
  • Zauner’s affirmative conjecture had already been confirmed exactly in several dimensions, including d = 2, 3, 4, 5, 6, 7, 8, 9, . . . , 13, and 15.

2. SIC-POVMS

SIC-POVMs are equivalent to maximal sets of d^2 equiangular lines in complex projective space. They also admit equivalent descriptions as tight complex projective 2-designs and maximally equiangular tight frames.

  • A SIC-POVM maps d^2 measurement outcomes to d^2 subnormalised rank-one projectors on the Hilbert space C^d.Its defining property is equiangularity under the Hilbert-Schmidt inner product.
  • A SIC-POVM is equivalent to a set of d^2 equiangular lines through the origin of C^d.The corresponding outcome set is viewed as a subset of complex projective space CP^(d−1).
  • The absolute bound gives |C| ≤ d^2 for equiangular lines, with the maximum requiring a common angle.
  • SIC-POVMs are precisely tight complex projective 2-designs that meet the absolute bound on size.All 2-designs with |D| = d^2 are necessarily sets of equiangular lines.
  • In frame theory, the unit vectors specifying a SIC-POVM form a maximally equiangular tight frame.Projection into traceless Hermitian operators maps a SIC-POVM to a tight frame, specifically a simplex.
  • SIC-POVMs provide a standard informationally complete measurement whose state-inversion formula establishes informational completeness.They are also described as robust minimally informationally complete POVMs against statistical error.

3. WEYL-HEISENBERG SIC-POVMS AND THE CLIFFORD GROUP

The paper develops the Weyl-Heisenberg and Clifford-group framework used to construct and classify SIC-POVMs. A fiducial vector generates a Weyl-Heisenberg covariant orbit, while Clifford and complex-conjugation symmetries organize equivalent solutions.

  • Weyl-Heisenberg covariance: The main construction route translates a fiducial vector under Weyl displacement operators to generate a Weyl-Heisenberg covariant SIC-POVM.The conjecture is that a suitable fiducial vector exists in every finite dimension.
  • Weyl-Heisenberg covariance: The displacement operators generate a variant of the Heisenberg group, whose quotient by its center is isomorphic to Z_d^2.
  • Zauner symmetry: Zauner’s conjecture requires a fiducial vector that is an eigenvector of the order-three matrix Z.The eigenspaces are labelled Z_k according to eigenvalue e^(2πik/3).
  • Clifford group: Conjugation by Z preserves the Heisenberg group, placing Z in its normaliser and connecting it to the Clifford group.
  • Clifford group: The Clifford group describes automorphisms of H(d) that fix its center pointwise, with displacement operators as inner automorphisms and metaplectic operators as outer automorphisms.
  • Clifford group: In even dimensions, the construction adapts the symplectic matrices to SL_2(Z_2d) because metaplectic operators can otherwise introduce sign changes.The paper follows Appleby’s approach for this even-dimensional modification.
  • Extended symmetry: Complex conjugation maps any Weyl-Heisenberg fiducial vector to another fiducial vector, motivating the extended Clifford group of unitary and anti-unitary normalisers.

4. NUMERICAL COMPUTER SOLUTIONS

The computer study finds fiducial vectors in every dimension d ≤67 and reports a putatively complete list of Weyl-Heisenberg solutions for d ≤50. Analysis of these solutions identifies recurring order-2 and order-3 symmetries, including new algebraic solutions and special cases.

  • Numerical method: The t-design reformulation converts the SIC condition into an equality condition and enables numerical searches by parameterising the set and minimising the associated sum.For Weyl-Heisenberg covariance, the resulting equality is equivalent to the fiducial-vector condition.
  • Numerical search: The numerical search found fiducial vectors in all dimensions d ≤67, with values listed to 38 quoted digits.Each solution generates an orbit of related fiducial vectors under the extended Clifford group.
  • Numerical search: The list is considered complete for d ≤50 because the computer search was exhaustive, and the listed solutions generate unique orbits.The orbit structure accounts for related fiducial vectors without listing every member individually.
  • Orbit counting: Table I counts unique Weyl-Heisenberg SIC-POVMs by fiducial-vector orbits, whose lengths are reduced when fiducials have nontrivial stabilisers.Orbit size is |PEC(d)|/|S(φ)|, and an example in dimension 15 contains 24 SIC-POVMs with three stabilised vectors each.
  • Symmetry analysis: For d > 3, most stabilisers are cyclic groups whose order is a multiple of 3 and whose symmetry is Zauner’s order-3 unitary [Fz|0].Exceptions with the order-3 unitary [Fa|0] occur in dimensions d = 9k + 3, including 12, 21, 30, 39 and 48.
  • Symmetry analysis: All tested order-3 canonical unitaries stabilise a fiducial vector; when 9 divides d − 3, two canonical conjugacy classes occur, represented by [Fz|0] and [Fa|0].Otherwise, the canonical elements are conjugate to Zauner’s unitary.
  • Symmetry analysis: Additional order-2 permutation and anti-unitary symmetries occur in listed dimension families and assist the discovery of new analytical solutions, although their reason remains unknown.The permutation [Fb|0] appears for d = k^2 −1, while the anti-unitary [Fc|0] appears for d = (3k ± 1)^2 + 3.

5. SYMBOLICAL COMPUTER SOLUTIONS

The paper develops algebraic SIC-POVM solutions by solving polynomial systems within prescribed symmetry eigenspaces, extending known dimensions and identifying relations between solutions through field automorphisms.

  • New algebraic SIC-POVM solutions are added for dimensions d = 24, 35, and 48, alongside solutions through d = 15 and for d = 19.
  • The general strategy solves polynomial equations for fiducial vectors constrained to eigenspaces of prescribed symmetries.Smaller eigenspaces reduce the number of variables and can improve the prospects of computation.
  • Complex fiducial-vector coordinates are replaced by real variables so that inner-product and squared-modulus conditions can be expressed polynomially.Only solutions with all resulting real components are retained.
  • Magma computations produced at least one fiducial vector in each dimension d = 4, . . . , 15, 19, 24, 35, and 48.The solutions lie in number fields with solvable Galois groups and can therefore be represented using radicals in the reported cases.
  • Algebraic representations reveal that the two d = 9 extended-Clifford orbits share a number field and are connected by a field automorphism.The corresponding triple-product sets are not invariant under that automorphism, so the two orbits are not related by a unitary or anti-unitary transformation.
  • The paper does not determine when field automorphisms relate different orbits because the relevant number field is known only after a solution is found.

6. CONCLUSION

The conclusion reports broad numerical and algebraic progress on Zauner’s conjecture while emphasizing that a proof remains out of reach and the symmetry structure is incompletely understood.

  • Numerical Weyl-Heisenberg covariant solutions are provided for all d ≤67 except d = 66, where the result is inconclusive.
  • A putatively complete list of solutions is given for d ≤50 as a resource for SIC-POVM research.
  • A preliminary symmetry analysis produced new algebraic solutions in dimensions d = 24, 35 and 48.
  • Zauner’s conjecture remains open despite increased confidence in its truth and extensive numerical evidence.
  • The automorphism group of Weyl-Heisenberg covariant SIC-POVMs is incompletely understood, limiting interpretation of relationships among known orbits.

7. NOTES

The notes state Zauner’s eigenspace conjecture, describe numerical optimization and refinement, and specify a probabilistic criterion for declaring orbit lists complete.

  • Zauner conjectured that a specified eigenspace of the matrix Z contains fiducial vectors for Weyl-Heisenberg covariant SIC-POVMs.
  • For d = 3m + 2, the eigenspace with eigenvalue e2πi/3 is also conjectured to contain fiducial vectors.
  • Solutions were found by minimizing the cost-function left-hand side until the stated bound was reached, then refining them to 38 digits.
  • The completeness test requires 30(n + 1) consecutive random trials to recover known orbits, giving a missing-orbit probability no more than e−30 under equal-probability sampling.
  • Under this criterion, the orbit list is complete for d ≤47, while computations for d = 48, . . . , 50 were still ongoing.

APPENDIX A: NUMERICAL SOLUTIONS

The appendix presents numerical complex-coordinate entries for candidate fiducial vectors, including both real-valued and complex-valued components.

  • The numerical appendix contains entries with explicitly real components, including zero imaginary parts.
  • Other entries contain nonzero real and imaginary components, indicating complex coordinates.
  • The listed numerical entries vary substantially in their displayed coordinate values across the appendix.

APPENDIX B: SYMBOLICAL SOLUTIONS

The appendix records symbolic solution data through radical expressions, polynomial-root definitions, and algebraic coordinate formulas involving auxiliary variables.

  • Several coordinate expressions combine these auxiliary quantities with real and imaginary terms to specify symbolic vectors.
  • The appendix includes long polynomial expressions in s1, s2 and s3 for symbolic solution coordinates.
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