August 4th-7th, 2014
Yukawa Institute for Theoretical Physics, Kyoto University
"A new quantum version of f-divergence"
Keiji Matsumoto (NII)
Quantum f-Divergence proposed by Petz (see [1] for comprehensive review) is widely studied,
especially due to its implication to perfect error correction.
In [2], the present author proposed a new quantum version of "generalized fidelity".
This is essentially a new version of quantum f-divergence (Dfmax, below)
when f is operator monotone decreasing.
In the definition, we supposed two states are strictly positive.
In this paper, the definition is generalized to include operator convex function f and,
to states which may have null eigenspace.
Not only the definition, also the explicit formula for Dfmax(ρ||σ)
when ρ and σ have null eigenspaces is given, and several properties of the quantity is studied
(convexity, monotonicity by CPTP maps.)
Also, the condition Dfmax(ρ||σ) =
Dfmax(Λ(ρ)||Λ(σ)), where Λ is CPTP map,
is studied. As is well-known, the analogous condition for Df implied reversibility of Λ,
or "sufficiency" of the map Λ.
Dfmax-version of the condition gives another type of "sufficiency" property of Λ.
[1] F. Hiai, M. Mosonyi, D. Petz, and C. Beny, Rev. Math. Phys. 23, 691 (2011).
[2] K. Matsumoto, arXiv:1311.4722.
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