16 problems

Let $\mathcal{M}_d$ denote the algebra of complex $d\times d$ matrices, and let $T:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a linear map. A map is completely positive if all its a…

Let $M_A$ be a matrix algebra, and let $\varrho_1,\varrho_2\in M_A\otimes M_A$ be states. Let $\varrho_1\square\varrho_2\in M_A\otimes(M_A\otimes M_A)$ denote their block Schur pro…

Let $\Phi_1,\Phi_2\in\mathcal{T}_n(\mathbb{C})$ be linear maps. A map $\Phi$ is PPT when both $\Phi$ and $\Phi\circ\mathsf{T}$ are completely positive, and it is entanglement break…

Positive–PPT composition conjecture. The composition

Let $\matrices$ denote the matrix algebra in the paper. A linear map $\psi:\matrices\to\matrices$ is PPT if it is completely positive and completely copositive, meaning that both…

Minimal Rényi sufficiency conjecture. The family $D^{\alpha}_{\min}$ provides a sufficient family of divergences for interconversion of finite-dimensional density matrices via posi…

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$, and let $\Phi:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$ be an IPT map, meaning that its Choi matrix is invariant under partial trans…

Tensor stable positivity conjecture. The set of yes-instances of tensor stable positivity, denoted $\textsc{tsp}$, is not recursively enumerable.

Let $\mathcal{M}_{d_i}$ denote the algebra of complex $d_i\times d_i$ matrices. Let $T:\mathcal{M}_{d_1}\rightarrow\mathcal{M}_{d_2}$ be completely positive and completely copositi…

Let $\mathcal{M}_{d_i}$ denote the algebra of complex $d_i\times d_i$ matrices. Let … be linear maps, each completely positive and completely copositive. For a positive matrix…

Let $\mathcal{M}_d$ denote the algebra of complex $d\times d$ matrices, and let $T:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a linear map. Its Choi matrix is denoted by $C_T$, and…

Kye's conjecture. Every 2-positive, respectively 2-copositive, linear map in $B(M_3(\mathbb{C}),M_3(\mathbb{C}))$ is decomposable.

Let ${\mathcal E}_{0:T}^\dagger$ be a positivity-improving Kraus map, meaning a Kraus map satisfying strict positivity, and let $\rho_0$ and $\rho_T$ be density matrices. Let…

Let ${\mathfrak D}$ denote the set of block-positive symmetries under consideration, and let $s\in{\mathfrak D}$ be a partial symmetry for $n=3$. Partial-symmetry conjecture. The r…

Optimality conjecture for $\Phi[\alpha]$. For $\frac{\pi}{3}\leq\alpha\leq\frac{5}{3\pi}$, the positive maps $\Phi[\alpha]$ are optimal.

Let $W$ be a normalized entanglement witness with ${\rm Tr}\,W=1$, and define … The structural physical approximation (SPA) of $W$ is an operator $\widetilde{W}(p)$ satisfying…