Magic Monotone for Faithful Detection of Nonstabilizerness in Mixed States

We introduce a monotone to quantify the amount of nonstabilizerness in mixed quantum states. The monotone gives a necessary and sufficient criterion for detecting the presence of nonstabilizerness for both pure and mixed states. The monotone is based on determining the boundaries of the stabilizer polytope in the space of Pauli string expectation values. The boundaries can be described by a set of hyperplane inequations, where violation of any one of these gives a necessary and sufficient condition for nonstabilizerness. The monotone is constructed by finding the hyperplane with the maximum violation and is a type of Minkowski functional. We also introduce a faithful witness based on similar methods. The approach is more computationally efficient than existing faithful mixed state monotones such as robustness of magic due to the smaller number and discrete nature of the parameters to be optimized.