12 problems

Let $\mathcal{H}_1, \mathcal{H}_2$ and $\mathcal{H}_3$ be finite-dimensional Hilbert spaces. For a density matrix $\rho_{123}$ on…

Let $\mathcal H^k_a$ be the Hilbert space associated with the measurement procedure indexed by $k$ and $a$, let $\omega\in\mathfrak S^k_a$ be a state with associated measure…

Dairyko et al.'s conjecture. Among connected graphs with fixed order $n$, the star graph $S_n$ minimizes the von Neumann entropy.

Let $|0\rangle$ and $|1\rangle$ denote the spin-up and spin-down states of a spin-$\frac{1}{2}$ particle, and let … where $0\leq s\leq \frac{1}{2}$. For each subset…

Let $\Phi$ and $\Psi$ be PFE-channels with infinite-dimensional input systems. A channel belongs to class $\mathfrak{C}$ if it is a PFE-channel that belongs to neither class…

Let $\mathcal{E}_{2,2}$ denote the Hilbert–Schmidt ensemble of $2\times2$ density matrices, let $\rho_1,\rho_2,\ldots$ be independent identically distributed random states from thi…

Star graph entropy minimization conjecture. The star graph minimizes the von Neumann entropy among all connected graphs.

Let $\Phi$ and $\Psi$ be quantum channels, and let a PCE-channel be a channel preserving continuity of the von Neumann entropy. Suppose that one of the channels, say $\Phi$, belong…

Small Incremental Mixing conjecture. For every such ensemble,

Conjecture 2. If the von Neumann entropy is continuous on $\mathcal{A}$, then there exists a state $\sigma\in\mathfrak{S}(\mathcal{H})$ such that the function…

Conjecture 1. If the von Neumann entropy is continuous on $\mathcal{A}$, then there is a positive operator $H$ with $g(H)=0$ and some $h>0$ such that…

Characterization conjecture. If