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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:Equality condition for the data processing inequal
ity of the quantum relative entropy - Felix Leditz
ky (University of Cambridge)
DTSTART;TZID=Europe/London:20160217T160000
DTEND;TZID=Europe/London:20160217T170000
UID:TALK64577AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/64577
DESCRIPTION:Relative entropies (or divergences) play a crucial
role in both Classical and Quantum Information Th
eory. Their defining property\, the data processin
g inequality\, states that a relative entropy of t
wo probability distributions/quantum states cannot
increase when applying certain transformations to
the system. In the quantum setting\, one of the m
ost important examples of a relative entropy satis
fying the data processing inequality is the quantu
m relative entropy\, which was first defined by Um
egaki and proved to be the 'correct' quantum gener
alization of the well-known Kullback-Leibler diver
gence in the classical theory. The transformations
considered in the quantum setting are completely
positive\, trace-preserving linear maps (also call
ed quantum channels).\n\nIn this talk\, we will re
view a seminal result by Petz that gives a necessa
ry and sufficient condition for equality in the da
ta processing inequality for the quantum relative
entropy. More precisely\, we derive a necessary an
d sufficient condition on the quantum channel and
two states such that their quantum relative entrop
y remains unchanged after application of the quant
um channel. Time permitting\, we will also review
this result in the light of strong subadditivity\,
an important and highly non-trivial property of t
he von Neumann entropy. In this case\, the equalit
y condition identifies states satisfying a quantum
Markov chain condition. \n\nNote that no prior kn
owledge of quantum mechanics is needed to follow t
he talk\, as we will quickly review the mathematic
al foundation of quantum mechanics from the viewpo
int of C*-algebras in the finite-dimensional setti
ng.
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:Mr. Cambyse RouzĂ©
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