Some Operator and Trace Function Convexity Theorems

2014/09/01 by Carlen, Eric A., Frank, Rupert L., Lieb, Elliott H. · 1 citation
#15A99 #47A63 #94A17 #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.1409.0564

Abstract

We consider convex trace functions Φ_p,q,s = Trace[ (A^q/2B^p A^q/2)^s] where A and B are positive n× n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of A^q/2B^p A^q/2 and convexity/concavity of the closely related trace functional Trace[ A^q/2B^p A^q/2 C^r]. For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta Rényi entropy conjectures are proved for some cases.

Cited by

• Reliability Functions of Quantum Soft Covering and Privacy Amplification via a Mixed-Order Rényi Divergence

Related

• Convexity of a certain operator trace functional
• Concavity of certain matrix trace and norm functions
• Concavity of certain matrix trace and norm functions. II
• Lieb type convexity for positive operator monotone decreasing functions
• Some operator convex functions of several variables