30 problems

Let $|0\rangle$ and $|1\rangle$ represent horizontally and vertically polarized photons, respectively, and let $\theta$ be an angle such that … Define … Here $n$ is the number of q…

Munro–Milburn conjecture. The best way to see the quantum nature of correlations in entangled states is through measurement of the observable canonically conjugate to photon number…

Let $\mathcal H^k_a$ be the Hilbert space associated with the measurement procedure indexed by $k$ and $a$, let $\omega\in\mathfrak S^k_a$ be a state with associated measure…

Path-graph robustness conjecture. For all $n\geq 1$,

Skew-Hadamard matrix existence conjecture. The upper bound is saturated if and only if an $n\times n$ skew-Hadamard matrix exists, and such a matrix should exist whenever $n$ is di…

Born's rule conjecture. The result of Proposition 3.1 of Finster (2013) should apply to pull back the spacetime observable of $\mathcal{U}$ to an observable on $B$.

Let $\mathfrak{A}$ be a von Neumann algebra with unit $\mathbf{1}$, and let $\omega$ be a normal state on $\mathfrak{A}$. Generalized almost-commutation conjecture. The Main Result…

Let $\hat p$ be the physical density operator and $\hat q$ the pricing operator in the one-period quantum-financial model. In the absence of physical interaction with the environme…

Van Kampen's conjecture. The entropy of the Universe is not affected by measurements, in the sense that the mean entropy is conserved under measurement.

Reciprocal channel-compatibility conjecture. If these two POVMs are compatible for every pair of orthonormal bases $\mathbf e$ and $\mathbf f$, then $\Phi$ and $\Psi$ are compatibl…

Let $\mathcal{A}$ be a qubit observable, and let $\mathcal{A}^{\lambda}$ denote its unsharp version for $1\geq\lambda\geq 0$. Let…

Let $\mathscr{A}$ and $\mathscr{B}$ be SICs in dimension $3$, and let $F(\mathscr{A}^{\otimes 3})$ and $F(\mathscr{B}^{\otimes 3})$ denote their three-copy estimation fidelities. S…

Let $f$ and $g$ be test functions for smeared fields $\phi(f)$ and $\phi(g)$ in a real scalar quantum field theory. An ideal measurement of a smeared field is an ideal measurement…

Let $d_{\mu}$ denote the dimension associated with the attractor of the place-dependent iterated function system generated by a hidden Markov model, and let the attractors in Fig.…

Strong monitoring localization conjecture. In the double limit $\gamma\to\infty$ and $\hbar/m\to0$ at fixed $\varepsilon$, a solution of the quantum trajectory equation localizes a…

Let $\mathcal{H}_d$ be a $d$-dimensional Hilbert space, and let $S_{d,1}$ denote the set of pure states, represented by rank-1 density matrices. A POVM is pure-state informationall…

Let $\Sigma$ be an arbitrary Cauchy surface, and suppose it is approximated by a surface that is piecewise flat but non-horizontal. Piecewise-flat approximation conjecture. Such an…

Let $\mathsf{A}$ and $\mathsf{B}$ be either two nonorthogonal spin-$1/2$ components or Fourier-conjugated mutually unbiased bases in even prime-power dimensions. Consider the sets…

Spike ubiquity conjecture. The phenomenon of spikes is ubiquitous in tightly monitored quantum systems.

Ubiquity conjecture. Any continuously measured quantum system should display fluctuations of the same kind.

Let $Q$ and $P$ be the position and momentum observables, and for $\theta\in[0,2\pi]$ let … For an angle $\alpha\notin\mathbb{Q}\pi$, set…

Let $\mathcal{H}$ be a Hilbert space, and let $\mathcal{I}$ be an instrument for the algebra $\operatorname{B}(\mathcal{H})$ on a measurable outcome space $(S,\mathcal{F})$. The in…

A wave function evolves unitarily, and its possible outcomes are represented by branches of that wave function. Non-overlapping branches conjecture. Unitary evolution of a wave fun…

Periodic collapse conjecture. The state of the particle experiences a periodic collapse such that the width of the state following each collapse returns to the value $\sigma$.

Standard-deviation requirement. The explicit presence of the standard deviation in some way, such as $\sigma(M^{out})$ or the inaccuracy $\bar{\epsilon}(A)$, is essential for formu…