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

TL;DR

This paper investigates the additivity properties of various magic monotones based on quantum relative entropies for single-qubit and multi-qubit states, providing new insights into their behavior and applications in quantum resource theory.

Contribution

It proves multiplicativity of stabilizer fidelity for tensor products of single-qubit states and establishes conditions for additivity of the relative entropy of magic and related monotones.

Findings

Stabilizer fidelity is multiplicative for tensor products of single-qubit states.

Relative entropy of magic is additive when all but one state are on a symmetry axis.

Derived closed-form expressions and tighter bounds for magic state distillation.

Abstract

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 $\alpha$-$z$ R\'enyi 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.