8 problems

Let $p$ satisfy $1\leq p\leq2$, and let $\mathcal{N}_1$ and $\mathcal{N}_2$ be quantum channels. Their maximal $p$-norms are denoted by $\nu_p(\mathcal{N}_1)$ and…

Local property conjecture. $n$-convexity is a local property, like $n$-monotonicity.

Let $A,B\geq 0$. Let $f$ be a nonnegative and nondecreasing function on an interval $\mathcal{J}$ such that $\operatorname{spec}(A)\subseteq\mathcal{J}$. Reverse trace inequality.…

Compatibility conjecture. For $\alpha\in[2,\infty)$, the function $-\Psi_\alpha$ is $(\mathbb{H}^n_+,(2\alpha-1)/3)$-compatible with the domain $\mathbb{H}^n_+\times\mathbb{H}^n_+$…

Let $M_n^+(\mathbb{C})$ denote the positive semidefinite $n\times n$ complex matrices. Fix $B\in M_n^+(\mathbb{C})$ and $m\in\mathbb{N}$. Lieb's conjecture. The function … is conca…

For a density matrix $\rho$ and a fixed self-adjoint operator $K=K^*$, the quasi-relative entropy associated with the power function satisfies … Here $I_p(\rho,K)$ is the Wigner–Ya…

Carlen–Lieb conjecture. For $1<p<2$, this map is convex. The source immediately states that the conjecture was proved, so the claim is no longer open.

Exhaustion conjecture. The matrix entropies given by Theorem exhaust the class of matrix entropies.