21 problems

Let $d>3$, and consider the polynomial equations defining a SIC-POVM that is covariant under the action of the Weyl–Heisenberg group. Their solution set is an algebraic variety. Ze…

Let a SIC-POVM in $\mathbb{C}^d$ have normalized overlaps $e^{i\theta_{jl}}$, where the angles $\theta_{jl}$ are defined by the normalized inner products of its fiducial vectors. A…

Fix $d\geq4$, write $\Delta=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and let $f'\mid f$. Let $\mathcal M$ satisfy the extended order-to-multiplet conject…

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, write $\Delta=\Delta_d=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and let $\mathcal M(f')$ be the multiplet indexed by…

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, set $\Delta=\Delta_d=(d+1)(d-3)$ and $K=\Q(\sqrt{\Delta})$. Let $\mathcal O_\Delta$ and $\mathcal O_K$ denote the quadratic order of discriminan…

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, let $\Delta=(d+1)(d-3)$, and let $\mathcal O_\Delta$ be the quadratic order of discriminant $\Delta$. Geometric Count Conjecture. The number of…

Fix $d\geq1$, $d\neq3$, and let $S$ be a $d$-dimensional Weyl–Heisenberg line-SIC with associated fields $\fieldtrip{S}$, $\fieldproj{S}$, and $\fieldeproj{S}$. Galois-field conjec…

Let $d\geq1$, $d\neq3$, and let a Weyl–Heisenberg line-SIC consist of projective-algebraic lines in $\C^d$ when its vectors can be represented by algebraic coordinates. Weyl–Heisen…

For each $d\geq1$, except $d=3$, let $\WHSIC{d}$ denote the set of Weyl–Heisenberg SICs and identify scalar-equivalent SICs when they share a fiducial vector up to a nonzero scalar…

For $d\geq1$, let a Weyl–Heisenberg SIC be a set of $d^2$ equiangular complex lines in $\C^d$ generated by a Weyl–Heisenberg fiducial vector. Strong Zauner Conjecture. For every di…

Fix a positive integer $d\neq 3$, let $\Delta_d=(d+1)(d-3)$, and write $\Delta_d=f^2\Delta_0$, where $\Delta_0=\operatorname{disc}(\mathcal O_{K_d})$ and $K_d=\Q(\sqrt{\Delta_d})$.…

Let $d\geq 4$, set $\Delta_d=(d+1)(d-3)$, and let $K_d=\Q(\sqrt{\Delta_d})$. For a Weyl–Heisenberg line-SIC $S$ in dimension $d$, let $\fieldvec{S}$ denote its SIC field, and let…

Let $d=n^2+3=p$ be a prime in the specified sequence, let $K$ be the real quadratic field used in the construction, and let $e^{i\vartheta_j}$ be the phases in the almost-flat fidu…

Let $d=n^2+3=p$ be a prime in the specified sequence, let $K$ be the real quadratic base field, and let $\hat{\Psi}$ be the un-normalized SIC fiducial with components $x_j$. The tw…

Let $d=n^2+3=p$ be a prime in the specified sequence of dimensions, and let $obreakhat{\Psi}=(x0,x1,\ldots,x{d-1})^{rm T}$ be an un-normalized SIC fiducial. **Almost-flat fiducial c…

Let $S$ be the symmetry group of the overlap map of a fiducial, let $C(S)$ be its centralizer in $\mathrm{GL}_2(\mathbb{Z}_{\bar d})$, and let $M$ be a subgroup of $C(S)$ such that…

Let $d\geq 2$, let $\mathbb{Z}_d=\{0,1,\ldots,d-1\}$, and let $W_{i,j}=U^iV^j$ be the discrete Weyl matrices on $\mathbb{C}^d$, with $U$ the cyclic shift and $V$ the diagonal matri…

For each positive integer $d\geq 2$, a SIC POVM is a set of $d^2$ rank-one positive operators on $\mathbb{C}^d$ that forms an informationally complete symmetric positive operator-v…

Type-a Galois-group conjecture.

Maximal-Abelian-subgroup conjecture.

Zauner's conjecture. In all finite dimensions there exists a fiducial vector for a Weyl–Heisenberg covariant SIC that is an eigenvector of $Z$.