Abstract
We introduce a new quantum Rényi divergenceDα#forα∈(1,∞)defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwichedα-Rényi divergence between quantum channels forα>1. Second it allows us to prove a chain rule property for the sandwichedα-Rényi divergence forα>1which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.
Highlights
Given nonnegative vectors P, Q ∈ RΣ, the α-Renyi divergence is defined as Dα(P Q) =1 α−1 log x∈Σ P (x)αQ(x)1−α if P ∞ else
We note that the matrix geometric mean is often defined for positive semidefinite operators as the limit as ε → 0 of the formula (7) applied to A + εI and B + εI and this clearly matches with the general approach of Kubo-Ando
Using equality (26) together with the explicit convergence bounds in Theorem 5.1 as well as the strong converse exponent established for states [27], we show that this bound is tight
Summary
When regularized, it gives the regularized sandwiched divergence between channels. We note that divergences are most interesting when the first argument is normalized, i.e., tr (ρ) = 1 but it is convenient to keep the definition general To define it for a general positive semidefinite operator ρ, we use a convention which is not standard. We will be using the property that the regularized measured divergence is equal to the sandwiched divergence, a property which clearly holds well for both conventions
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