14 problems

Let $X$ be an $n$-dimensional random vector, let $Z_t$ be an independent Gaussian random vector with covariance matrix $tI$, and set $X_t=X+Z_t$ for $t>0$. Write $H(X_t)$ for the d…

Conjecture on failure for $p>2$. For some closed Riemannian manifolds and some $p>2$, this inequality fails.

Guha et al.'s entropy photon number inequality. The inequality

Let a $5$-variable polymatroid be a polymatroid for which the two families of non-Shannon inequalities proved in Theorem … hold for polymatroids with an $n$-copy. An $n$-copy is th…

The entropy polynomial root conjecture. The polynomial $p_{k,r}(x)$ has exactly two real roots in $(0,1)$, counting multiplicity. The paper explains that this root statement would…

The entropy inequality for real exponents. For $0\leq x\leq 1$,

Let ${\mathtt{p}}=(p_1,\ldots,p_L)$ be the probability vector under consideration, and let $H_1({\mathtt{p}})$ and $H_s({\mathtt{p}})$ denote its Shannon entropy and Tsallis entrop…

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 a beam splitter receiving two independent input states, each with a prescribed entropy. Thermal Gaussian input states are states whose entropy is determined by their therm…

Let $X_1$ and $X_2$ be binary classical random variables with quantum side-information systems $B_1$ and $B_2$, respectively, and let $\rho^{X_1B_1}$ and $\rho^{X_2B_2}$ be classic…

Gaussian–differential inequality identity conjecture. The same identity between Gaussian vector inequalities and general differential inequalities should hold for any number of par…

Let $g(N)=(N+1)\ln(N+1)-N\ln N$ be the entropy of a single-mode thermal state with mean photon number $N$, and let $N(\rho)=g^{-1}(S(\rho)/n)$ denote the mean photon number per mod…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For $1<p\leq 2$, consider the sharp Riemannian $L^p$-entropy inequality … for all $u\in H^{1,p}(M)$ such that…

Four-atom conjecture. The lowest possible Ingleton score is approximately $-0.089373$. It is attained by a four-variable distribution with alphabet size two, given by