37 problems

Positive–PPT composition conjecture. The composition

Let $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$ be a finite-dimensional Hilbert space of dimension $D=d_A d_B$, and let $W$ be a normalized entanglement witness, so that…

Uniqueness conjecture. The decomposition

Let $\Phi:B(\mathcal{H}_d)\to B(\mathcal{H}_d)$ be a unital positive map. The Woronowicz conjecture. Every such map satisfies the strong Kadison inequality: for every…

Strengthened Sanpera–Bruß–Lewenstein conjecture. In dimension $d$, the Choi matrix of every map that is both completely positive and completely co-positive has Schmidt number at mo…

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…

Cross-positive decomposition conjecture. Every cross-positive map is a sum of a positive map and a map of the form

Two-thirds relaxation-rate conjecture. Every semigroup of Schwarz maps on a $d$-level quantum system satisfies

Minimal Rényi interval conjecture. The dichotomies are interconvertible if and only if their minimal quantum Rényi divergences agree throughout the interval:

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 $\mathcal{MU}(d)$ be the subgroup of $\mathcal{U}_d$ generated by all permutation matrices and all diagonal unitary matrices, and let $\pi_{\mathcal{MU}(d)}$ denote its fundame…

Let $\Phi:M_n(\mathbb{R})\rightarrow M_n(\mathbb{R})$ be a completely positive map whose Choi matrix is invariant under partial transposition. Such a map is an IPT map. A map break…

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.

Fillipov–Magadov conjecture. Every positive tensor square $P\otimes P$ of a positive map $P:\mathcal{M}_2\rightarrow\mathcal{M}_2$ is decomposable.

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 $\mathcal{M}_{d_i}$ 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…

Positivity criteria conjecture. If $\Phi_A$ is positive, then

Let $\phi$ be a positive map between matrix algebras, and consider the property described at the beginning of the section. The source identifies maps of the form given in equation…

Let $\mathcal K$ and $\mathcal H$ be Hilbert spaces, let $B(\mathcal K)$ and $B(\mathcal H)$ denote their algebras of bounded operators, and let…

Let $\Upsilon$ be a positive unital map on two qubits, and consider its tensor square $\Upsilon\otimes\Upsilon$. A map is decomposable if it can be written as the sum of a complete…

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…