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SUMMARY:Equality condition for the data processing inequality of the quant
 um relative entropy - Felix Leditzky (University of Cambridge)
DTSTART:20160217T160000Z
DTEND:20160217T170000Z
UID:TALK64577@talks.cam.ac.uk
CONTACT:Mr. Cambyse Rouzé
DESCRIPTION:Relative entropies (or divergences) play a crucial role in bot
 h Classical and Quantum Information Theory. Their defining property\, the 
 data processing inequality\, states that a relative entropy of two probabi
 lity distributions/quantum states cannot increase when applying certain tr
 ansformations to the system. In the quantum setting\, one of the most impo
 rtant examples of a relative entropy satisfying the data processing inequa
 lity is the quantum relative entropy\, which was first defined by Umegaki 
 and proved to be the 'correct' quantum generalization of the well-known Ku
 llback-Leibler divergence in the classical theory. The transformations con
 sidered in the quantum setting are completely positive\, trace-preserving 
 linear maps (also called quantum channels).\n\nIn this talk\, we will revi
 ew a seminal result by Petz that gives a necessary and sufficient conditio
 n for equality in the data processing inequality for the quantum relative 
 entropy. More precisely\, we derive a necessary and sufficient condition o
 n the quantum channel and two states such that their quantum relative entr
 opy remains unchanged after application of the quantum channel. Time permi
 tting\, we will also review this result in the light of strong subadditivi
 ty\, an important and highly non-trivial property of the von Neumann entro
 py. In this case\, the equality condition identifies states satisfying a q
 uantum Markov chain condition. \n\nNote that no prior knowledge of quantum
  mechanics is needed to follow the talk\, as we will quickly review the ma
 thematical foundation of quantum mechanics from the viewpoint of C*-algebr
 as in the finite-dimensional setting.
LOCATION:MR14\, Centre for Mathematical Sciences
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