22 problems

The classical entropy for a quantum-mechanical density matrix on $L^2(\mathbb{R})$ is defined through the associated Husimi function, obtained by evaluating the density matrix on G…

Lieb's generalized Wehrl conjecture. The analogue of Wehrl's conjecture should hold for at least a wide class of Lie groups.

Meijer $G$-function conjecture. The general form of the resolution measure is conjectured to be

Series conjecture. The same formula should hold for all $2K\geq 1$.

Let $Q$ and $D$ be the basic affine coherent-state operators, and let ${\cal H}$ be a self-adjoint Hamiltonian operator. For $0<\beta\leq 1/2$, suppose the propagator and the assoc…

Let the domain of convergence of a family of coherent states be a ring in the complex plane, and let the resolution of the identity involve a function $F$ supported on that ring. N…

Generalized Lieb–Wehrl conjecture. The minimum value of the R'enyi–Wehrl entropy for $q>0$ is attained by coherent states. This conjecture generalizes the Lieb–Wehrl conjecture fro…

The $Z_k$-graded supersymmetric oscillator has coherent states, and $Q$-deformed coherent states are understood as a separate class of coherent states. Coherent-state decomposition…

Sharp $L^p$ estimate conjecture. For any real $p\geq2$ and for any $h\in H$ there exists a constant $C(p)\leq1$, with $C(2)=1$, such that:

Let a representation of $SU(2)$ be irreducible, and consider the corresponding Wehrl-type entropy and coherent states. Lieb's conjecture. An analogous result to the Glauber coheren…

Connectedness and over-completeness conjecture. For every irreducible matrix configuration, ${\cal M}$ is connected, and the quasi-coherent states are over-complete, meaning

Let $\mathbb{C}_+=\{z\in\mathbb{C}:\operatorname{Im}z>0\}$ and let $F$ be analytic on $\mathbb{C}_+$. For $s\geq1$, consider the weighted integral involving…

Fix $\alpha>0$ and $s\geq1$. Let $U$ be the affine-group representation on $L^2(\mathbb{R}_+)$, let $\eta_\alpha(k)=C(\alpha)k^\alpha e^{-k}$, and define…

Consider coherent-state branches in the setting discussed for nonlinear wave equations, with the zero-energy boundary denoted by $\{E=0\}$. In the example with nonlinearity…

Lieb–Solovej's quantum-channel conjecture. The entropy conjecture extends to these channels: highest-weight vectors minimize the trace of concave functions of the channel output.

Lieb–Solovej's extended conjecture. The minimization property of the Wehrl entropy continues to hold when $f(x)=-x\ln x$ is replaced by any concave function: $S_f(v)$ is minimized…

Lieb–Solovej's conjecture. The quantity $S(v)$ is minimized when $v$ is any of the coherent state vectors $\Omega_R$.

Grassmannian Husimi second-moment conjecture. The maximum second moment is

Grassmannian Wehrl entropy conjecture. The entropy is minimized when $\psi$ is a coherent state, namely $|\psi\rangle=|Z'\rangle$ for some $Z'\in\mathbb{G}_2$, and the minimum is

Resolution-of-unity conjecture. For sufficiently small $t>0$, the displayed resolution of unity and the stated asymptotic scaling of $C_t$ hold for an arbitrary Lie group $G$. If…

Covariant-state description conjecture. Arbitrary classical string solutions $X^{\mu}_{\rm cl}$ are described in string theory by the covariant states, or by the corresponding ligh…

Coherent-state asymptotic transport conjecture. One has