6 research outputs found
Trace inequalities in nonextensive statistical mechanics
In this short paper, we establish a variational expression of the Tsallis
relative entropy. In addition, we derive a generalized thermodynamic inequality
and a generalized Peierls-Bogoliubov inequality. Finally we give a generalized
Golden-Thompson inequality
Tensor Multivariate Trace Inequalities and their Applications
We prove several trace inequalities that extend the Araki Lieb Thirring (ALT)
inequality, Golden Thompson (GT) inequality and logarithmic trace inequality to
arbitrary many tensors. Our approaches rely on complex interpolation theory as
well as asymptotic spectral pinching, providing a transparent mechanism to
treat generic tensor multivariate trace inequalities. As an example application
of our tensor extension of the Golden Thompson inequality, we give the tail
bound for the independent sum of tensors. Such bound will play a fundamental
role in high dimensional probability and statistical data analysis
Multivariate Trace Inequalities
We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities