226 problems

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…

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…

Consider a Hilbert space ${\mathcal{H}}$ of finite dimension $d=|{\mathcal{H}}|$. Let $\rho$ and $\sigma$ be states on $\mathcal{H}$. Define … The function $f_d(\varepsilon,\nu)$ i…

Let $(a,b)$ and $(a',b')$ lie in $\mathbf{U}_3\cup\mathbf{V}_3\cup\mathbf{W}_3$, and consider the bridge graph with bond dimension $3$ and boundary dimensions $a,b,b',a'$. Three-bo…

Let $w$ be a positive integer and let $(a,b)$ and $(a',b')$ lie in $\mathbf{U}_w\cup\mathbf{V}_w\cup\mathbf{W}_w$, with $ab'\leq a'b$. Consider the corresponding bridge graph with…

Let $w$ be a positive integer and let $(a,b)$ and $(a',b')$ lie in the union of the regions $\mathbf{U}_w$, $\mathbf{V}_w$, and $\mathbf{W}_w$. Consider the bridge graph with bond…

A gate in $U(2^n)$ is generalized semi-Clifford if it can be expressed as … where $C_1,C_2$ are $n$-qubit Clifford gates, $P$ is a permutation matrix in $U(2^n)$, and the $Z_j$ are…

Let $(\tau,(W_i))$ be a $d$-dimensional superdense coding protocol. Let $\mathcal{H}_{A'}$, $\mathcal{H}_{A”}$, $\mathcal{H}_{B'}$, and $\mathcal{H}_{B”}$ be the Hilbert spaces in…

Rank-$n+1$ hypergraph conjecture. If an $n$-party inequality contracts on $(n+1)$-graphs, then it is a valid inequality on all hypergraphs of finite rank.

Real quadratic ray class field conjecture. The statement that, for the dimensions listed in Proposition 1, there is a SIC orbit $E$ with $\mathbb{Q}(E)=\mathfrak{R}$, while every o…

Clifford bound conjecture. The cardinality of any unitary 2-design is at least

Equality conjecture. For each fixed $g\in{\cal G}$,

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

WYD scalar-curvature conjecture. There exists $\varepsilon>0$ such that, for every $p\in I$, the scalar curvature of the WYD$(p)$ metric is a Schur-increasing function.

Noncommutative alpha-geometry conjecture. Suppose $n\geq2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then…

Let $D_1$ and $D_2$ be states, and write $D_1\prec D_2$ when $D_1$ is more mixed than $D_2$ in the majorization order. Let $\operatorname{Scal}(D)$ denote the scalar curvature of t…

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…

For each finite NPA level $k$, let $a_k$ be the coefficient in the small-$s$ expansion … where the coefficients satisfy $a_k\downarrow0$ as $k$ increases. Positivity conjecture.…

Relative quantumness data-processing conjecture. Under these assumptions,

Global monotonicity conjecture. The Kubo–Mori update can only increase the log-likelihood:

Generic-fiber conjecture. Equality holds in the dimension bound

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…