From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)

In this paper we study the joint convexity/concavity of the trace functions $$\Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R},$$ where $A$ and $B$ are positive definite matrices and $K$ is any fixed invertible matrix. We will give full range of $(p,q,s)\in\mathbb{R}^3$ for $\Psi_{p,q,s}$ to be jointly convex/concave for all $K$. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of $(\alpha,z)$ for $\alpha$-$z$ R\'enyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of $\Psi_{p,0,1/p}$ for $0<p<1$ which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions $$\Psi_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1,$$ using a variational method.Comment: 14 pages, 1 figure. Some errors and typos corrected. Title changed. Main results improved: a unified and simple proof of the convexity/concavity of a large family of trace functions $\Psi_{p,q,s}$ using a variational method and the convexity/concavity (due to Ando/Lieb) of $\Psi_{p,1-p,1}$. To appear in Adv. Mat

Complete gradient estimates of quantum Markov semigroups

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative $2$-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.Comment: 29 page

Noncommutative Bohnenblust--Hille inequalities

Bohnenblust--Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree \cite{defant2019fourier}. Such inequalities have found great applications in learning low-degree Boolean functions \cite{eskenazis2022learning}. Motivated by learning quantum observables, a qubit analogue of Bohnenblust--Hille inequality for Boolean cubes was recently conjectured in \cite{RWZ22}. The conjecture was resolved in \cite{CHP}. In this paper, we give a new proof of these Bohnenblust--Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr's radius phenomenon on quantum Boolean cubes.Comment: 20 pages. Revised based on the referee's repor

Hypercontractivity of heat semigroups on free quantum groups

In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.Comment: 19 page

A dimension-free Remez-type inequality on the polytorus

Consider $f:\Omega^n_K\to\mathbf{C}$ a function from the $n$-fold product of multiplicative cyclic groups of order $K$. Any such $f$ may be extended via its Fourier expansion to an analytic polynomial on the polytorus $\mathbf{T}^n$, and the set of such polynomials coincides with the set of all analytic polynomials on $\mathbf{T}^n$ of individual degree at most $K-1$. In this setting it is natural to ask how the supremum norms of $f$ over $\mathbf{T}^n$ and over $\Omega_K^n$ compare. We prove the following Remez-type inequality: if $f$ has degree at most $d$ as an analytic polynomial, then $\|f\|_{\mathbf{T}^n}\leq C(d,K)\|f\|_{\Omega_K^n}$ with $C(d,K)$ independent of dimension $n$. As a consequence we also obtain a new proof of the Bohnenblust--Hille inequality for functions on products of cyclic groups. Key to our argument is a special class of Fourier multipliers on $\Omega_K^n$ which are $L^\infty\to L^\infty$ bounded independent of dimension when restricted to low-degree polynomials. This class includes projections onto the $k$-homogeneous parts of low-degree polynomials as well as projections of much finer granularity.Comment: 21 pages. Largely revise