255 problems

For each integer $d\ge 2$, determine whether there exists a classical protocol using a finite amount of classical communication and unlimited shared randomness that exactly reprodu…

For every separable Hilbert space, every $p\in[0,1]$, and all positive trace-class operators $A\leq B$ satisfying $\operatorname{Tr}A=p$ and $\operatorname{Tr}B=1$, prove the sharp…

For every rank-$r$ admissible tuple, the rank-$r$ twisted convolution identity for the modular quantum dilogarithm holds. The supplied sources do not state the identity's explicit…

Consider a quantum thermalization process that, for a set of distinct Hamiltonians, prepares states close to the corresponding thermal ensembles at temperature $T$. The Planckian t…

Let $\rho^{A_1\cdots A_m}$ be an $m$-party quantum state and let $\varepsilon>0$. Writing $\rho^K$ for the marginal on every nonempty subset $K\subseteq\{1,\ldots,m\}$, the conject…

For all finite graphs $G$ and $H$, the quantum chromatic number of their categorical product satisfies $\chi_q(G\times H)=\min\{\chi_q(G),\chi_q(H)\}$, where $G\times H$ has vertex…

For every number of qubits $n$, every level $k\geq 1$ of the Clifford hierarchy, and every $n$-qubit unitary $U\in\mathcal{C}_k$, there exist Clifford unitaries $C_1,C_2$, a comput…

For every nonlocal game $G$, if $G$ self-tests a strategy among strategies consisting of a pure state of full Schmidt rank and projective measurements, then $G$ self-tests the same…

Let $\mathcal{NS}_{2,2,2}$ denote the nonsignalling conditional distributions $P(a,b\mid x,y)$ with $a,b,x,y\in\{0,1\}$, and let $\mathcal{Q}_{2,2,2}$ denote the quantum-correlatio…

Given arbitrary nonempty families of finite-dimensional quantum channels $\mathcal{N}_0$ and $\mathcal{N}_1$, characterize the optimal asymptotic Stein exponent for discriminating…

Given a quantum error-correcting code encoding $k$ logical qubits, characterize the codes and fusion strategies for which the induced logical fusion succeeds for every pattern in w…

Every gapped ground state satisfies approximate Haag duality.

Let $|\Psi\rangle$ be a ground state in a two-dimensional topological phase, let $A$ be a spatial subsystem, and let $\rho_A=\operatorname{Tr}_{\bar A}|\Psi\rangle\langle\Psi|$ be…

For every pair of constants $\varepsilon,\delta\ge 0$ satisfying $\varepsilon+\delta<1$, every $(\varepsilon,\delta)$-disentangler with output space $K$ and input space $H$ must sa…

For a generic chaotic finite-dimensional quantum system evolving unitarily from a low-complexity state, let $C(t)$ denote its quantum circuit complexity and let $t_{\mathrm{eq}}$ b…

Given a finite energy bound $E<\infty$, does there exist a prepare-and-measure semi-device-independent protocol such that the observed data certify strictly positive extractable ra…

Let $\mathcal{P}_n$ denote the set of $n$-qubit Pauli observables. For every $n$-qubit quantum state $\varrho$ and every $\epsilon\in(0,1]$, define…

Let $G$ be any finite two-player, one-round entangled game with entangled value $\omega^*(G)=1-\varepsilon<1$. The conjecture asserts that there exists a constant $c=c(G)>0$ such t…

Classify, up to standard equivalence, all matrices $H\in\mathbb{C}^{6\times 6}$ satisfying $|H_{ij}|=1$ for every $i,j$ and $HH^{*}=6I_{6}$. The conjecture asserts that every such…

For every $n\ge 1$, every unital stochastic map $\Phi:M_n\to M_n$, and every normalized monotone Riemannian metric specified by an operator-monotone function $\kappa$, the Riemanni…

Let $\mathsf{R}$ be a retrodiction, or quantum Bayesian inversion, assignment for normal quantum processes between von Neumann algebras equipped with faithful states, satisfying th…

For integers $n,k\ge 2$, let $C_{qa}(n,k)$ and $C_{qc}(n,k)$ denote the corresponding approximately quantum and commuting quantum correlation sets, and let…

Triangle inequality conjecture. For any pure states $\textup{\textbf{x}},\textup{\textbf{y}},\textup{\textbf{z}}\in\mathbb{P}(\mathbb{C}^n)$,

Let $S=\{v_i\}_{i=1}^g\subseteq\matrices$ be a collection of matrices, and define its Wielength by … where $S^k=\{v_\sigma:\sigma\in\{1,\dots,g\}^k\}$,…

Quantum Fourier entropy-influence conjecture. There is a constant $c$ independent of $n$ such that