88 problems

Additivity conjecture. Quantum entanglement does not enhance the channel capacity, and the classical capacity of a quantum channel is additive.

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be finite-dimensional Hilbert spaces, and let $\mathcal{T}_A$ and $\mathcal{T}_B$ be super-operators on $\mathcal{H}_A$ and $\mathcal{H}_B$,…

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…

Gaussian optimizer conjecture. The optimizers belong to the class of pure Gaussian states.

Let ${\mathcal{S}}$ denote the set of completely positive trace-preserving maps on ${\mathcal{M}}_2$, and let $T_2$ be the partial trace over the second factor. For a unitary matri…

Let $\Phi$ be a stochastic map, let $\mathcal M=\{E_b\}$ be a measurement, and let $\rho$ be a density matrix. Define … where…

Let $\Phi_1$ and $\Phi_2$ be quantum channels. For $p\geq 1$, define the maximal output $p$-norm by … and define the minimal output von Neumann entropy by … where the extrema range…

Let $\Phi_1,\ldots,\Phi_n$ be quantum channels, and let the quantities characterizing them include capacities for classical information and maximal output purity. Tensor products o…

Let $P\in \operatorname{Herm}(\mathbbm{C}^{d_1}\otimes \mathbbm{C}^{d_2})$ be the Choi matrix of a completely positive trace-preserving map…

Relative quantumness data-processing conjecture. Under these assumptions,

Let $\mathcal N:\mathcal L(\mathbb C^2)\to\mathcal L(\mathbb C^2)$ be a unital qubit channel with Bloch representation $r\mapsto Tr$, where $T\in\mathbb R^{3\times 3}$ has singular…

Let $mathcal N_0^{A_0\to B_0B_1}$ and $mathcal N_1^{A_1\to B_0B_1}$ be the partially erasing channels defined by fixed states $sigma_0\forin\fomathcal D(B_0)$ and…

PPT squared conjecture. If the Choi matrices of $\mathcal{L}_1$ and $\mathcal{L}_2$ are PPT, then the Choi matrix of

Let $\overline{\mathcal{M}_L}$ and $\overline{\mathcal{M}_R}$ be the algebraic closures of operators obtained by evolving operators that are local in the left subsystem $L$ and rig…

Let $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ be finite-dimensional complex Hilbert spaces, let $\rho \in \mathsf{D}(\mathcal{X}\otimes \mathcal{Z})$ and…

Let $A$ and $B$ be density matrices, let $\phi$ be a quantum channel, and let $\mathcal{R}$ be the Petz recovery map associated with $B$ and $\phi$, given by … Here…

All-$p$ strong converse conjecture. The strong converse rate can be improved to

Relative contraction conjecture. For all $\gamma_1,\gamma_2\in(0,1)$,

Relative expansion conjecture. For all $\gamma_1,\gamma_2\in(0,1)$,

Let $\mathcal A_{\gamma}$ denote the amplitude damping channel with damping parameter $\gamma$, and for two channels define the complete contraction and expansion coefficients by ……

Let $\mathcal N^{A\to B_1}$ and $\mathcal M^{A\to B_2}$ be degradable quantum channels with common input, with complementary output systems $E_1$ and $E_2$. Define the non-negative…

Consider quasi-local quantum dynamics with a steady state and Gibbs states of local Hamiltonians. A Gibbs state of a Hamiltonian is strictly local when the Hamiltonian contains onl…

Let a quantum system have local channels on its edges, with no dissipators and a single unitary channel on each edge. In this setting, ergodicity refers to the ability of products…

Mutual information is considered for quantum channels generated by finite quantum groups. Fragility conjecture. Fragility of mutual information should be broad: it should hold for…

Let $A$ and $B$ be $n\times n$ density matrices with $\operatorname{supp}(A)\subset\operatorname{supp}(B)$. Let $\phi:M_n(\mathbb{C})\to M_k(\mathbb{C})$ be a completely positive t…