19 problems

Let a quantum measurement be a POVM on a finite-dimensional quantum system. A measurement uses a single auxiliary qubit if its implementation requires only one ancillary two-level…

For $k\in\mathbb N$, let $s_{\mathbb R}(2,2,(2,k+1))$ and $s_{\mathbb C}(2,2,(2,k+1))$ be the minimum compatibility degrees for one dichotomic and one $(k+1)$-outcome measurement o…

Let $s_{\mathbb C}(2,4)$ denote the minimum compatibility degree of four dichotomic measurements on a complex qubit. Four-dichotomic-qubit compatibility-degree conjecture. … The va…

Let $k\in\mathbb N$ and $d\geq 2$. The minimum compatibility degree $s_{\mathbb C}(d,2,(2,k+1))$ concerns one dichotomic measurement and one measurement with $k+1$ outcomes on a…

Let $g\geq 2$. For each $d$, let $P_1,\ldots,P_g$ be $g$ independent Haar-random projections of rank $d/2$ on $\mathbb C^d$, and let $A_1,\ldots,A_g$ be their associated dichotomic…

Let GEAMs denote generalized equiangular measurements, and let $S$, $\mathcal{C}_{\max}$, $\kappa_+$, and $\kappa_-$ be the measurement parameters used in the paper. Two-constant d…

Gentleness conjecture. The measurement consisting of the operators $|M_y|$ has at least the same gentleness as the measurement given by the operators $M_y$.

Optimality conjecture. The value $t\leq 1/3$ in the preceding result is optimal: pairs of noisy binary measurements with noise parameter $t>1/3$ cannot, in general, be simulated by…

Convergence conjecture. The state $\phi$ converges in probability to the set of Fock states, which correspond to the computational basis in $\mathcal{S}_N$.

Let $M$ and $N$ be quantum measurements, and let $D$ denote the divergence used to define the entropic orderings $D_N\leq D_M$ and $S_N\geq S_M$. Write $D^{\max}$ for the maximal d…

Nonlinear majorization monotone conjecture. If there exists a blockwise bistochastic matrix $\bm{B}\in\mathcal{B}_{n,d}$ such that $\bm{Q}=\bm{B}\*\bm{P}$, then for any ordering of…

Let $\mathbf{M}=(M_1,\ldots,M_n)$ be an arbitrary extremal rank-one POVM on $\mathbb{C}^d$. A partition $\{X_\gamma\}_{\gamma=1}^\alpha$ of $[n]$ satisfies $|X_\gamma|\leq d-1$ whe…

Non-PMP and non-weak-Gibbs conjecture. For all $\Delta\in{\cal I}$, the unraveling $\mathbb{P}$ is not PMP and is not weak Gibbs.

General random-qubit incompatibility conjecture. The incompatibility probability of a pair of random qubit measurements $\textsf{A}$ and $\textsf{B}$, generated by…

Random qubit measurement incompatibility conjecture. The incompatibility probability for a pair of general qubit measurements is

The CGLMP inequality is considered for quantum systems of dimension $2$. Qubit CGLMP-bound conjecture. The qubit bound $0.2071$ obtained using projective measurements is optimal: a…

An exact $t$-design is a collection of vectors reproducing the relevant $t$-th moments of the uniform distribution, and Kuperberg's construction uses only…

A physical reality is understood to have an initial state and a final state connected by a measurement. Universal measurement conjecture. If there is one physical reality, then the…

Groenewold's conjecture. The entropy reduction is nonnegative for quantum measurements of the von Neumann–Lüders type.