10 problems

Let $q$ be the alphabet size and let $t$ be an arbitrary fixed number of transpositions. Consider codes of block length $n$ correcting $t$ transpositions, and let the optimal code…

Consider the noncommutative polynomial problem defining the unitary capacity in Sections 4 and 5, with its tensor-product and commuting operator values. Equality conjecture. The te…

Let $G$ be a graph, and let $\Theta_q(G)$ and $\Theta_*(G)$ denote its quantum and entanglement-assisted Shannon capacities, respectively. Quantum capacity conjecture. For every gr…

Let $G$ be a graph, and let $\Theta(G)$, $\Theta_q(G)$, and $\Theta_*(G)$ denote its classical, quantum, and entanglement-assisted Shannon capacities, respectively. Quantum–entangl…

Cascade-channel capacity conjecture. If $|\mathcal S|<\infty$, so that the channel is finite-state, then the Shannon capacity of the $m$-fold cascade channel converges to…

One-shot entanglement-assisted Lovász-number conjecture. The one-shot entanglement-assisted zero-error capacity is equal to

For a graph $G$, let $\alpha^{\star}(G)$ denote its entangled independence number and let $\alpha_q(G)$ denote its quantum independence number. Roberson's conjecture. For every gra…

Entanglement-assisted capacity conjecture. The entanglement-assisted capacity is equal to the Lovász $\vartheta$-function for classical channels.

Let $G$ be a graph with at least three vertices, and let $H(G)$ be the graph constructed from $G$ whose vertices represent events consistent with the stabilizer group of the graph…

Let a classical channel have a representation by unit vectors $\{\lvert\tilde{\psi}_x\rangle\}$, and let $\vartheta(\rho)$ denote the relevant generalized Lovász value. Lovász' opt…