Abstract
Fawzi and Fawzi [Quantum 5, 387 (2021)] recently defined the sharp Rényi divergence, Dα#, for α ∈ (1, ∞), as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit α → 1 of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin–Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences for which we prove various properties, including the data-processing inequality.
Highlights
GEOMETRIC AND SHARP RÉNYI DIVERGENCESLet H be a complex finite-dimensional Hilbert space and B(H ) be the set of linear operators on H
For α ≥ 0, this definition can be extended to more general A, B ∈ P(H ), B ≪ A, by restricting the Hilbert space to the support of A [this corresponds to directly using pseudo-inverses in (1)]
We show in Theorem 7 that the limit α → 1 of the sharp divergence is the Belavkin–Staszewski relative entropy
Summary
Let H be a complex finite-dimensional Hilbert space and B(H ) be the set of linear operators on H. In Ref. 1, Fawzi and Fazwi defined the sharp trace function for α ∈ (1, ∞), Q#α(ρ∥σ) mA≥in0 {TrA They defined the sharp Rényi divergence of order α in terms of it as follows: D#α(ρ∥σ). Note that the A in the above expressions are, in general, unnormalized states and we use the definitions of the geometric trace function and the geometric Rényi divergence as in (2) and (4) without additional normalization factors. Recall that the sharp Rényi divergence is only defined for α > 1, and so all we really need here is the limit from above (28) Establishing that this is equal to the limit from below makes things slightly simpler, as we are able to directly use the chain rule later in (32).
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