Modular Zauner's conjecture for Hilbert C*-modules

Let A\mathcal{A} be a unital C*-algebra, let dNd\in\mathbb{N}, and let Ad\mathcal{A}^d denote the standard Hilbert C*-module. A unit inner product frame has τj,τj=1\langle\tau_j,\tau_j\rangle=1 for every index, and a frame is γ\gamma-equiangular when the products of mutual inner products equal a fixed positive element γ\gamma. Modular Zauner's conjecture. For every dNd\in\mathbb{N}, there exists a 1d+1\frac{1}{d+1}-equiangular unit inner product frame {τj}j=1d2\{\tau_j\}_{j=1}^{d^2} for Ad\mathcal{A}^d; equivalently, there is a tight frame satisfying τj,τj=1\langle\tau_j,\tau_j\rangle=1 and

τj,τkτk,τj=1d+1,1j,kd2, jk.\langle\tau_j,\tau_k\rangle\langle\tau_k,\tau_j\rangle=\frac{1}{d+1},\qquad 1\leq j,k\leq d^2,\ j\neq k.

This extends Zauner's conjecture from Hilbert spaces to Hilbert C*-modules. The supplied text does not state whether the conjecture is open, solved, or refuted.

Sources & referencesView supporting material

Primary source

K. Mahesh Krishna, “Modular Welch Bounds with Applications”, arXiv:2201.00319 (2022).

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