Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond

We prove that the stabilizer fidelity is multiplicative for the tensor product of an arbitrary number of single-qubit states. We also show that the relative entropy of magic becomes additive if all the single-qubit states but one belong to a symmetry axis of the stabilizer octahedron. We extend the latter results to include all the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B1;</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi></mml:math> Rényi relative entropy of magic. This allows us to identify a continuous set of magic monotones that are additive for single-qubit states. We also show that all the monotones mentioned above are additive for several standard two and three-qubit states subject to depolarizing noise. Finally, we obtain closed-form expressions for several states and tighter lower bounds for the overhead of probabilistic one-shot magic state distillation.