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 $\alpha$-$z$ 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.

中文翻译：

---

基于单量子比特态及更高量子比特态的量子相对熵的魔幻单调的混合状态可加性特性


我们证明，稳定器保真度对于任意数量的单量子比特状态的张量积是乘法的。我们还表明，如果除一个之外的所有单量子比特状态都属于稳定器八面体的对称轴，则魔术的相对熵会变得加法。我们将后一个结果扩展为包括所有 $\alpha$-$z$ Rényi 的魔法相对熵。这使我们能够识别一组连续的魔幻单调，这些单调是单量子比特状态的加法。我们还表明，上面提到的所有单调对于受去极化噪声影响的几个标准二量子比特和三量子比特状态都是加法。最后，我们获得了几种状态的闭式表达式和更严格的概率一次性魔术状态蒸馏开销的下限。