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Equations for the overlaps of a SIC

Len Bos, Shayne Waldron

TL;DR

The paper develops a holomorphic, projective-Fourier-transform framework to characterize Weyl-Heisenberg SIC fiducials from overlap data $c_{jk}$. It proves that the fiducial projector $vv^*$ can be reconstructed as $vv^*=Tc$, where $T$ is the projective Fourier transform, and that the condition $\mathrm{trace}((Tc)^4)=1$ together with standard overlap constraints guarantees a rank-one fiducial. It establishes that $\sqrt{d}T$ is unitary with finite order $(\sqrt{d}T)^{6d}=(-1)^{\frac{d(d-1)}{2}}I$, and discusses an alternative $z$-transform description of the overlaps that yields equivalent SIC equations. The paper also analyzes the special case $d=3$, finding a continuum of SIC overlaps parameterized by a hypocycloid, and notes connections to Zauner symmetry and Clifford-group actions. Overall, it provides a holomorphic, finite-order transform-based criterion for SIC existence and reconstruction beyond traditional conjugate-variable equations.

Abstract

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .

Equations for the overlaps of a SIC

TL;DR

The paper develops a holomorphic, projective-Fourier-transform framework to characterize Weyl-Heisenberg SIC fiducials from overlap data . It proves that the fiducial projector can be reconstructed as , where is the projective Fourier transform, and that the condition together with standard overlap constraints guarantees a rank-one fiducial. It establishes that is unitary with finite order , and discusses an alternative -transform description of the overlaps that yields equivalent SIC equations. The paper also analyzes the special case , finding a continuum of SIC overlaps parameterized by a hypocycloid, and notes connections to Zauner symmetry and Clifford-group actions. Overall, it provides a holomorphic, finite-order transform-based criterion for SIC existence and reconstruction beyond traditional conjugate-variable equations.

Abstract

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .
Paper Structure (7 sections, 13 theorems, 117 equations, 2 figures)

This paper contains 7 sections, 13 theorems, 117 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The reconstruction operator
  3. Properties of the projective Fourier transform
  4. Equivalent equations for Heisenberg frames
  5. The Special Case of $d=3$
  6. Closing Comment
  7. Acknowledgement

Key Result

Lemma 2.1

Let $T$ be given by (Tcdefn). Suppose that $c=(c_{jk})\in\mathbb{C}^{\mathbb{Z}_d\times\mathbb{Z}_d}$ satisfies Then $Tc$ is Hermitian, and its eigenvalues $\lambda_1,\ldots,\lambda_d$ satisfy i.e., its characteristic polynomial has the form

Figures (2)

  • Figure 1: The Hypocycloid
  • Figure 2: Points A and B on the Hypocycloid

Theorems & Definitions (15)

  • Lemma 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Example 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Definition 4.1
  • Theorem 4.1
  • ...and 5 more