42 problems

Weyl–Heisenberg covariant SIC-POVM conjecture. There exists a normalized $\lvert\phi\rangle\in\mathbb{C}^d$ such that the set $\{D_{jk}\lvert\phi\rangle\}_{j,k=1}^{d}$ is a SIC-POV…

A SIC-POVM is a set $P$ of $d^{2}$ normalized vectors $\lvert\phi_k\rangle$ in $\mathbb{C}^d$ satisfying … Its POVM elements are the subnormalized projectors…

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…

Almost-flat SIC-POVM conjecture. This equation holds for all almost flat SIC fiducial vectors in odd dimension. The conjecture concerns an observed identity in exact SIC-POVM solut…

Fix $d>3$, let $\Delta=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and for each positive divisor $f'$ of $f$ let $\mathcal O'=\mathcal O_{(f')^2\Delta_0}$.…

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 a 2-design in dimension $d$ be a quantum state 2-design, and let a SIC-POVM denote a symmetric informationally complete positive-operator-valued measure and a maximal set of MU…

Let $d$ be the Hilbert-space dimension, and suppose the SIC conditions are expressed as $d^2$ quantities $G(i,k)$ indexed by the Weyl--Heisenberg displacement labels. Three-d verif…

Let $D$ be a square-free integer, and let $(d_\ell)_{\ell=1}^\infty$ be a dimension tower satisfying … for suitable integers $f$. A SIC is a set of equiangular quantum states, and…

For an integer $d\geq 2$, let $\Phi_t:\mathbb{C}^{d\times d}\to\mathbb{C}^{d\times d}$ be the entanglement-breaking quantum channel … for $t\in[-\frac{1}{d^2-1},\frac{1}{d+1}]$, an…

Let a SIC be an orbit under the Weyl–Heisenberg group in dimension $d$, and let the Clifford group act on its vectors. A symplectic unitary is a unitary representative of a symplec…

Let the SIC overlaps be the overlap quantities associated with a SIC, and let their Galois group act on these overlaps. Galois-group quotient conjecture. The Galois group acting on…

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 $d=n^2+3$, and consider the ray class SIC in dimension $d$, which has a unitary symmetry. Anti-unitary symmetry conjecture. The ray class SIC should also have an anti-unitary s…

Existence conjecture. Real fiducial states exist in every odd dimension of the form $d=4n^2+3$, with $n\in\mathbb{Z}$. The existence of real fiducial states would impose an additio…