Data processing inequality for the generalized Bures–Wasserstein distance in dimension two

Determine whether the generalized Bures–Wasserstein distance induced by the generalized fidelity satisfies the data processing inequality for arbitrary qubit density matrices and quantum channels.

Background

The paper studies the generalized fidelity F_R(P,Q) and the associated generalized Bures–Wasserstein distance B_R(P,Q). It proves that the data processing inequality fails in dimensions n greater than or equal to 3 by constructing counterexamples.

For dimension two, the paper derives an explicit formula for the real part of the generalized fidelity in terms of pairwise Uhlmann fidelities and establishes two sufficient conditions under which the data processing inequality holds. However, these sufficient conditions do not resolve whether the inequality holds in full generality for qubit states and channels.