20 problems

All-orders relative-entropy conjecture. The equality

Christandl–Ferrara–Lancien's conjecture. There should exist a universal constant $\kappa$ and a function $g:[0,1]\to\mathbb{R}_+$ with

Let $(\mathcal{M}_n)_n$ be a family of sets of quantum states satisfying Axioms 1--5 in Section, and let $D^\infty_{\mathcal{M}}(\rho)$ denote the corresponding regularised relativ…

Centered gamma moment-generating-function conjecture. For every $n$, $k$, and $P$,

Concentration conjecture. There are positive constants $c_1$ and $c_2$ such that for every $n$, $k$, $P$, and $\varepsilon\geq 0$,

Let $P$ be a distribution on a finite alphabet of size $k$, let $\hat{P}_{n,k}$ be its empirical distribution from $n$ independent samples, and let $D(\hat{P}_{n,k} \| P)$ denote t…

Let $P$ be a distribution on a finite alphabet of size $k$, let $\hat{P}_{n,k}$ be its empirical distribution from $n$ independent samples, and let $D(\hat{P}_{n,k} \\| P)$ denote…

Universality conjecture. The theorem above should hold for any rational conformal net, with $r$ replaced by the central charge.

Winter's recoverability conjecture. One should have

Complex relative entropy principle. For each positive subsequence, the limit points of the normalized critical complex entropies are contained in the real parts of the critical val…

Föllmer drift information-spending conjecture.

Descent conjecture. An analogous descent property should hold for $\widehat{F}$: along solutions $p(\cdot)$ of the limiting equation, $\frac{d}{dt}\widehat{F}(p(t))\leq 0$, with eq…

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})_{++}$ and let $\mathcal{N}$ be a strict completely positive trace-preserving map. For $0\leq\alpha\leq\beta$, define…

Let $d\geq 2$ and let $M(\Delta,d)$ denote the penalty function defined by the minimization over pairs of distributions with entropy difference $\Delta$. For $\Delta\in[0,\log d]$,…

Let $\rho_{AB}$ and $\sigma_{AB}$ be quantum states on $\mathcal{H}_A\otimes\mathcal{H}_B$, with reduced states $\rho_A$, $\rho_B$, $\sigma_A$, and $\sigma_B$ on the corresponding…

Let $omega_1$ and $omega_2$ be states, let $S(omega_1\Vertomega_2)$ denote their quantum relative entropy, and let $\mathrm{M}(omega_1\precomega_2)$ be the set of measurement schem…

Conjecture 2. If the von Neumann entropy is continuous on $\mathcal{A}$, then there exists a state $\sigma\in\mathfrak{S}(\mathcal{H})$ such that the function…

Let $\N$ be a von Neumann algebra and let $\N_*^+$ denote its positive normal linear functionals. For $\gamma\in[0,1]$, let $D_\gamma(\omega,\phi)$ be the deviation functionals def…

The proposed family of relative entropy functionals is defined using the non-commutative flow of weights and associated non-commutative $L_p$ spaces, and generalises the quantum re…

Let $\rho$ and $\sigma$ be density matrices, let $S$ denote the von Neumann entropy, and define … Let $S(\sigma\|\rho)$ denote the Umegaki relative entropy. Di2si1os222, Feldmann a…