8 problems

Product-state conjecture. For every state $\rho$, there exists a product state $\rho_1\otimes\rho_2$ with the same pair of entropies.

Let $\mathcal{E}$ be a Hilbert $C^*$-module over a commutative unital $C^*$-algebra $\mathcal{A}$. Let $\{\tau_j\}_{j=1}^\infty$ and $\{\omega_k\}_{k=1}^\infty$ be two Parseval fra…

Let $\c2$ be a doubly stochastic matrix, and let $\mu$ and $\lambda$ be the parameters defining its induced norm. Write $\sigma_2$ for the second largest singular value of $\c2$. E…

Real-real symmetric-state conjecture. There exists a real-real symmetric state $\sigma$ with the same entropy pair as $\rho$.

Independence conjecture. If $\rho$ is optimal for $U$ and one dual pair $(\alpha,\beta)$, then $\rho$ is also optimal for every other dual pair.

Fourier-matrix decomposition conjecture. Replacing $U_d^F$ by $U_{d_1}^F\otimes U_{d_2}^F$ does not change the entropy diagram.

Englert's conjecture. The states above trace out the curve of minimal entropies. The conjecture is correct for $d=2$, appears to hold numerically for $d=3$, and fails for $d=4$, wh…

Unextendible-set/EUR conjecture. The connection between the existence of unextendible sets of mutually unbiased bases and the tightness of the corresponding entropic uncertainty re…