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SUMMARY:Renyi relative entropies and noncommutative L_p-spaces - Anna Jen
 čová (Slovak Academy of Sciences)
DTSTART:20180726T150000Z
DTEND:20180726T154500Z
UID:TALK108403@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>The standard quantum Renyi relative entropies belong to 
 the class of Petz quantum&nbsp\;f-divergences and have a number of applica
 tions in quantum information theory\, including and operational interpreta
 tion as error exponents in quantum hypothesis testing. In the last couple 
 of years\, the sandwiched version of Renyi relative entropies gained atten
 tion for their applications in various strong converse results. While the 
 Petz&nbsp\;f-divergences are defined for arbitrary von Neumann algebras\, 
 the sandwiched version was introduced for density matrices. In this contri
 bution\, it is shown that these quantities can be extended to infinite dim
 ensions. To this end\, we use the interpolating family of non-commutative&
 nbsp\;L_p-spaces with respect to a state\, defined by Kosaki. This definit
 ion provides us with tools for proving a number of properties of the sandw
 iched Renyi entropies\, in particular the data processing inequality with 
 respect to normal unital (completely) positive maps. It is also shown that
  this definition coincides with the previously introduced Araki-Masuda div
 ergences by Berta et. al.<br></span><br>The notion of sufficient (or rever
 sible) quantum channels was introduced and studied by Petz. One of the fun
 damental results in this context is the fact that equality in the data pro
 cessing inequality for the quantum relative entropy is equivalent to suffi
 ciency of the channel. We extend this result for sandwiched Renyi relative
  entropies. See arXiv:1609.08462 and arXiv:1707.00047 for more details.
LOCATION:Seminar Room 1\, Newton Institute
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