Quantum inequalities and their applications

TL;DR

This paper reviews quantum inequalities related to quantum symmetries, exploring their mathematical foundations, applications in category theory and subfactor theory, and highlighting fundamental differences from non-commutative inequalities.

Contribution

It provides a comprehensive introduction to quantum inequalities on subfactors and discusses their applications, emphasizing the role of complete positivity and algebraic structures.

Findings

Quantum inequalities include Hausdorff-Young, Young's, and entropic inequalities.

Applications demonstrate the importance of complete positivity in quantum symmetries.

The review highlights fundamental differences between quantum and non-commutative inequalities.

Abstract

In recent years, various quantum inequalities have been established on quantum symmetries in the framework of quantum Fourier analysis. We provide a detailed introduction to quantum inequalities including Hausdorff-Young inequality, Young's inequality, uncertainty principles, entropic convolution inequalities etc on subfactors, an important type of quantum symmetries. We cite several applications of the complete positivity of the comultiplication in category theory and subfactor theory, which indicate the fundamental differences between quantum inequalities and non-commutative inequalities. We also review the Perron-Frobenius theorem together with the algebraic structures of eigenvector spaces.