Efficient witnessing and testing of magic in mixed quantum states

Nonstabilizerness, or ‘magic’, is a crucial resource for quantum computation, but quantifying the magic of mixed states has been a notoriously difficult task. We introduce efficient magic witnesses based on stabilizer Rényi entropy that both robustly indicate magic and quantitatively estimate magic monotones. Building on these witnesses, we design testing algorithms that distinguish high- and low-magic states under entropy constraints and apply them to certify the number of noisy T-gates for a broad class of noise models. Using the IonQ quantum computer, we experimentally verify magic in noisy random circuits and find that magic remains robust, persisting even under depolarizing noise with probability exponentially close to one. Our witnesses are efficiently computable for matrix product states, showing that subsystems of many-body states can host extensive magic even when the system is entangled. Finally, we show that mimicking high-magic states with minimal magic requires an extensive amount of entropy, implying that entropy is a necessary cryptographic resource for hiding magic from eavesdroppers. Our results provide practical tools for characterizing the complexity of noisy quantum systems.