515 research outputs found
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
where and are positive definite matrices and is any
fixed invertible matrix. We will give full range of
for to be jointly convex/concave for all . 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
for - 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 for 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 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 using a variational
method and the convexity/concavity (due to Ando/Lieb) of . 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 -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 and the free permutation quantum group . 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 a function from the -fold product of
multiplicative cyclic groups of order . Any such may be extended via its
Fourier expansion to an analytic polynomial on the polytorus ,
and the set of such polynomials coincides with the set of all analytic
polynomials on of individual degree at most .
In this setting it is natural to ask how the supremum norms of over
and over compare. We prove the following Remez-type
inequality: if has degree at most as an analytic polynomial, then
with independent
of dimension . 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
which are bounded independent of dimension when
restricted to low-degree polynomials. This class includes projections onto the
-homogeneous parts of low-degree polynomials as well as projections of much
finer granularity.Comment: 21 pages. Largely revise
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