Smolin–Verstraete–Winter asymptotic quantum Birkhoff conjecture

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 the unitary channels on $\mathcal{H}$. For channels $\Phi$ and $\Psi$, let $D(\Phi,\Psi)$ be the diamond norm of $\Phi-\Psi$, and write $\operatorname{Conv}(\mathbb{U}(\mathcal{H}))$ for the convex hull of the unitary channels. Asymptotic quantum Birkhoff conjecture. If $\Phi\in\mathrm{T}(\mathcal{H})$ is unital, then $\Phi^{\otimes n}$ can be approximated by a mixture of unitary channels from $\mathbb{U}(\mathcal{H}^{\otimes n})$ with arbitrary precision, namely

$$\lim_{n\to\infty}D\bigl(\Phi^{\otimes n},\operatorname{Conv}(\mathbb{U}(\mathcal{H}^{\otimes n}))\bigr)=0.$$

The ordinary quantum Birkhoff conjecture fails in dimensions at least three, so this asymptotic formulation asks whether tensor powers of every unital channel become arbitrarily close to mixtures of unitary channels. It was identified as a major open problem in quantum information theory.