15 problems

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 $\mathcal N_0^{A_0\to B_0B_1}$ 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…

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…

Strong operator law of large numbers. Under these assumptions, $e^{A_1t/n}\ldots e^{A_nt/n}$ converges almost surely in the strong operator topology of $\mathcal{T}_p$ to…

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 ……

Capacity bounds conjecture. For a quantum channel ${\cal N}$,

Let $d\geq 3$ be an integer, let $|x\rangle$ be a vector on the unit sphere in $\mathbf R^{d-1}$, and construct matrices $A_m$ using $V_m=2|x\rangle\langle x|-I_{d-1}$ for every…

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 $Q:\mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$ be a positive quantum channel, and let $\rho_0,\rho_T$ be density matrices. Let $\mathrm{H}_{n,++}$ denote the positive defi…

Let $a,f\in[0,1]$ and let $\mathcal{Q}_{\mathbb{C}}(a,f)$ denote the set of qubit channels over the classical channel … For a qubit channel $Q$, let $\eta^{\operatorname{Tr}}(Q)$ b…

Optimality conjecture. Equality holds in the three displayed bounds above.

Let $\mathcal{H}$ be a finite-dimensional Hilbert space, let $\mathrm{T}(\mathcal{H})$ denote the set of quantum channels on $\mathcal{H}$, and let $\mathbb{U}(\mathcal{H})$ denote…

Let a quantum channel be a linear completely positive trace-preserving map $\Phi:\mathcal{M}_n(\mathbb{C})\to\mathcal{M}_k(\mathbb{C})$. Its minimum output entropy is … where…

Let $\mathcal{U}$ be the set of functions that are additive for every pair of unitarily covariant quantum channels $(\Phi,\Omega)$. A function is operator convex when it satisfies…