$\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}$

Definition.

$\{f_1,\dots, f_n\}\subset \mathbb{K}^d$ is a Parseval frame if $\operatorname{Id}_{d\times d}=FF^*=f_1f_1^*+\dots+f_nf_n^*$.

The rows of $F$ form an orthonormal set in $\mathbb{K}^n$, so the space of all length-$n$ Parseval frames in $\mathbb{K}^d$ is the Stiefel manifold $\operatorname{St}_d(\mathbb{K}^n)$.

An equal-norm Parseval frame (ENP frame) is a Parseval frame $f_1,\dots , f_n$ with $\|f_i\|^2=\|f_j\|^2$ for all $i$ and $j$.

Theorem [Casazza–Kovačevic, Goyal–Kovačevic–Kelner, Holmes–Paulsen]

ENP frames are optimal for signal reconstruction in the presence of white noise and erasures.

Think of $f_if_i^*-\frac{1}{n}\mathrm{Id}_{d \times d} \in \mathscr{H}_0(d)$, the space of traceless Hermitian $d \times d$ matrices.

Theorem  [with Cantarella, Duplantier, and Uehara]

Rejection sampling differences of $d_i$’s from the hypercube yields random $n$-gons in expected time $\Theta(n^{5/2})$.

If $F \in \mathbb{C}^{d \times n}$ is an ETF, then $(f_1f_1^* - \frac{1}{n}\operatorname{Id}_{d \times d}, \dots , f_nf_n^* - \frac{1}{n}\operatorname{Id}_{d \times d})$ are the vertices of a regular $(n-1)$-simplex in $\mathscr{H}_0(d) \simeq \mathfrak{su}(d)^\ast$. In particular, this implies $n \leq d^2$.

(Weak) Zauner Conjecture

For all positive integers $d$, there exists an ETF $F \in \mathbb{C}^{d \times d^2}$.

Such ETFs are called maximal ETFs in the math literature, and Symmetric, Informationally-Complete, Positive Operator-Valued Measures (SIC-POVMs) in quantum information theory.