22 problems

Let $(\mathcal{M},\rho)$ be a quantum probability space, and let $A_1,\ldots,A_k$ be self-adjoint elements with mean $0$ and symmetric covariance matrix. For each $t$, write … and…

Purity conjecture. Any strongly clustering entangled Markov chain should generate a pure state on

Let $E_\eta(M(\sigma))$ be the multiple integral defined above, associated with the permutation $\sigma$, and let $d(\sigma)$ denote the quantity assigned to that permutation in th…

Consider the four quantum central limit results discussed in the source: the theorem for a pair of conjugate variables, its generalization to arbitrary canonical commutation relati…

Let $\mathcal{M}$ be a von Neumann algebra, let $\rho$ be a normal state on $\mathcal{M}$, and let $A_1,\ldots,A_a$ be bounded self-adjoint elements of $\mathcal{M}$. Let…

Let $T'$ and $T$ be operators defining the positivity operators $P_n(T')$ and $P_n(T)$ for the corresponding Wick representations, and suppose that $T'\leq T$. Monotonicity conject…

For $2\leq p<8$, let $t'_p$ denote the optimal time and let $\beta$ be the parameter of the quantum Ornstein–Uhlenbeck semigroup. There are positive constants…

Consider a quantum diffusive system with $N$ sites and local number operators $4hat n_j$ for $1\leqslant j\leqslant N$. Let $4\varrho_{\rm eq}$ be a random density matrix whose pro…

Let $\mathcal{H}_a$ be a Hilbert space of dimension $d_a$, let $\mathcal{H}_b$ be a finite-dimensional Hilbert space of dimension $d_b$, and let $\rho$ be a density matrix on…

The proposal concerns quantum stochastic processes, whose probabilities evolve according to quantum dynamics, and the semi-classical limit in which gravitational dynamics is recove…

Sum identities conjecture. For each fixed $\rho\in\mathbb{Z}_+$, the following identities hold:

Let $\sigma$ denote the loop correlation functions of the quantum symmetric simple exclusion process (Q-SSEP) invariant measure, where ${\bm x}=(x_1,\ldots,x_P)$ consists of distin…

Let $E_0(u)$ be a strongly continuous semigroup arising from a unitary cocycle, and let its generator be a $\tau$-measurable operator on $L^2(\mathcal{A}_0,\tau)$. Stone-extension…

Let $b$ be an additive cocycle, let $u$ be its corresponding unitary cocycle, and let $E_0(u)$ denote the associated semigroup. The symmetric variance is $E_0(b^*b+bb^*)$. Relaxati…

Let a Hadamard matrix model be a matrix model arising from a complex Hadamard matrix, and call it stationary on its image when its associated integration state has the stationarity…

Let $G\subset S_N^+$ be a transitive closed subgroup. Let $\chi=\sum_i u_{ii}$ be its main character, and let $\mu$ be the distribution of $\chi$. The universal flat model of…

In algebraic probability spaces, a notion of stochastic independence is an algebraically defined rule describing how random variables or subalgebras combine. Schürmann's conjecture…

Let $L$ and $R$ be the transition operators … so that $L^*L+R^*R=I$. Let the initial density matrix be … with $x_i\in\mathbb{R}$, and let $k_{k1}(\rho)$ denote the mean hitting tim…

Let $\mathfrak{h}$ be the system Hilbert space, let $\mathcal{F}$ be the Fock space, let $\mathsf{E}_{\alpha\beta}$ be bounded operators on $\mathfrak{h}$ with…

Let $X\in\mathcal{A}\otimes\mathcal{B}$ be a selfadjoint operator with $\langle X\rangle_{ss}^{out}=0$. Let $\mathbb{F}_n(X)$ denote the associated fluctuations operator, and let…

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…

The paper considers quantum filtering models in the finite-dimensional setting of section 6, with stability understood in the asymptotic sense for the associated quantum filters. A…