39 problems

The paper parameterizes conditional entropies by a real parameter and a probability measure, and derives sufficient conditions on these parameters ensuring that the resulting quant…

Let $t$ be a real parameter and let $\tau$ be a probability measure parameterizing the conditional entropy $H_{t,\tau}$. The paper gives sufficient conditions on $t$ and $\tau$ for…

Let $\rho_A$ be the reduced density matrix of a region $A$, let $S_n(A)=\frac{1}{1-n}\log\operatorname{tr}\rho_A^n$ be its Rényi entropy, and write the spectral trace as … Here $d$…

Hyperbolic-group continuity conjecture. The function $h_\alpha$ should be continuous at $\alpha=1$ for any random walk on a hyperbolic group.

Van Herstraeten–Cerf Rényi bound conjecture. For $d=1$,

Van Herstraeten–Cerf Rényi-entropy conjecture. For positive Wigner functions, $h_{\alpha}[W\rho]$ is bounded below, and its minimum can only be attained by Gaussian pure states.

Seshadreesan–Lami–Wilde's conjecture. The implication

Let $M$ denote the system size, and let $h_{sigma_{\varepsilon}}$ be the conformal weight of the excited twist field appearing in the Potts-model one-point function. Next-to-sublea…

Continuity conjecture. For every stationary ergodic process $Z$,

Let $\rho,\sigma\in M_n^{++}(\mathbb{C})$ be density matrices, let $\alpha>0$ with $\alpha\neq 1$, and define the sandwiched Rényi relative entropy by … For a completely positive u…

Let $(X,Y)$ be a symmetric log-concave random vector in $\mathbb{R}^2$; in particular, $X$ and $Y$ may be independent symmetric real random variables. Write $N^{\frac12}$ for the R…

Madiman–Kontoyannis conjecture. The entropy satisfies

Strengthened varentropy conjecture. The function $K$ is log-concave: $\log K(t)$ is concave on $(-\gamma,+\infty)$.

Varentropy conjecture. The function $F(t)$ is concave on $(0,+\infty)$.

Let $h_\alpha$ denote the binary conditional Rényi-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.3863$ such that … is convex for…

Let $k_\alpha^A$ denote the Arimoto conditional-entropy function and let $\ast$ denote binary convolution. There exists a value $1<\hat\alpha<1.5783$ such that … is convex for…

For $\beta>0$, let the generalized Gaussian have density … where $\beta$ and $alpha$ are chosen so that it integrates to one. Let $X_j$, $j=1,\dots,n$, be independent random variab…

Let $X$ and $Y$ be independent random variables uniformly supported on $A=\{0,1\}^n\subseteq\mathbb Z^n$, with their distributions allowed to differ. For $0\le\alpha\le 1$, let…

Let $A$ be a subsystem of a colored Fredkin or Motzkin spin chain of total size $n$, with $A$ having general length, possibly much smaller than $O(n)$. Rényi entropy transition con…

Cross-section body conjecture. If $s\in[-1/n,\infty]$, then $C_p(f)$ is a symmetric convex body. This is the geometric reformulation of the reverse Rényi entropy power norm conject…

Let $(X,Y)$ be a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$, and suppose that $X$ and $Y$ have equal $p$-Rényi entropy. Convexity conjecture for Rényi entropy. The…

Recall that a probability measure on $\mathbb{R}^n$ is $\kappa$-concave if it satisfies the Brunn–Minkowski-type inequality associated with $\kappa$; a random vector is $\kappa$-co…

All-n Rényi trace conjecture. For the symmetric case $\alpha=\beta$,

Fix $\kappa\in[-\infty,\frac1d]$. Let $(U,V)$ be the coordinates of a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$. Symmetric $\kappa$-concave Busemann conjecture. Fo…

Let $(U,V)$ be the coordinates of a symmetric log-concave random vector in $\mathbb{R}^2$. Ball–Nayar–Tkocz's conjecture. … This inequality is equivalent to the triangle inequality…