3 research outputs found
Renyi generalizations of the conditional quantum mutual information
The conditional quantum mutual information of a tripartite state
is an information quantity which lies at the center of many
problems in quantum information theory. Three of its main properties are that
it is non-negative for any tripartite state, that it decreases under local
operations applied to systems and , and that it obeys the duality
relation for a four-party pure state on systems . The
conditional mutual information also underlies the squashed entanglement, an
entanglement measure that satisfies all of the axioms desired for an
entanglement measure. As such, it has been an open question to find R\'enyi
generalizations of the conditional mutual information, that would allow for a
deeper understanding of the original quantity and find applications beyond the
traditional memoryless setting of quantum information theory. The present paper
addresses this question, by defining different -R\'enyi generalizations
of the conditional mutual information, some of which we can
prove converge to the conditional mutual information in the limit
. Furthermore, we prove that many of these generalizations
satisfy non-negativity, duality, and monotonicity with respect to local
operations on one of the systems or (with it being left as an open
question to prove that monotoniticity holds with respect to local operations on
both systems). The quantities defined here should find applications in quantum
information theory and perhaps even in other areas of physics, but we leave
this for future work. We also state a conjecture regarding the monotonicity of
the R\'enyi conditional mutual informations defined here with respect to the
R\'enyi parameter . We prove that this conjecture is true in some
special cases and when is in a neighborhood of one.Comment: v6: 53 pages, final published versio