15 problems

Finite-rank approximation conjecture. Every WPS is the limit in the trace norm of a sequence of finite-rank WPS from $\mathcal D_+^N$.

Let $Wf$ be a Wigner function on $\mathbb{R}^{2}$ whose negative part is nonempty and contained in a disc. For a Wigner function $Wf$, denote its negative part by…

Let $Wf(z)$ be the Wigner distribution of some function $f\in L^2(\mathbb{R})$, with bounded nodal set $\mathcal{N}(Wf)$, and suppose that $Wf$ is centered at $z=0$. For a Hermite…

Majorization conjecture along the family $\widetilde{W}_u$. There is a Wigner majorization order along the family $\widetilde{W}_u$ only when $u>1/2$.

Majorization conjecture for $\hat{\rho}_0$. Any mixed state with a positive Wigner function is majorized by $\hat{\rho}_0$ under the Wigner majorization ordering.

Van Herstraeten–Jabbour–Cerf conjecture. Every Wigner-positive state is majorized by a coherent state under the Wigner majorization ordering.

Van Herstraeten–Cerf Rényi bound conjecture. For $d=1$,

Van Herstraeten–Cerf Rényi-entropy conjecture. For positive Wigner functions, $h_{\alpha}[W\rho]$ is bounded below, and its minimum can only be attained by Gaussian pure states.

Van Herstraeten–Cerf conjecture. The functional is bounded below on the convex set of Wigner-positive states, and for $d=1$ its lower bound is

Let $E_j(\hbar)$ be the eigenvalues, let $\psi_j$ be the corresponding eigenfunctions, and let $W_{\psi_j,\psi_j}(x,\xi)$ denote their Wigner functions. Let $E$ be the center of a…

Let $\mathcal{B}_N$ be the Birkhoff polytope of $N\times N$ bistochastic matrices, let $\mathcal{U}_N\subset\mathcal{B}_N$ be the set of unistochastic matrices, and let…

Let $V$ be the covariance matrix and let $A^{\pm}_{\rm mix}$ denote the matrix appearing in the condition for Wigner-function negativity. The negativity volume is the integral of t…

Wigner-function convergence conjecture. For every initial Wigner function $W_0=W^{\rho_0}$ with finite energy,

Roy–Singh's N+1 marginal theorem. The $N+1$ marginals of $W$ can be reproduced by positive phase-space densities, but no more than $N+1$ marginals can be so reproduced for arbitrar…

Let the Wigner function represent the evolving density operator of a quantum Markovian system, and consider the intermediate stage between the initial loss of quantum coherence and…