Stabilizer entropies and nonstabilizerness monotones

Abstract

We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for Rényi index 0 ≤ n < 2, the SEs are not monotones with respect to stabilizer protocols which include computationalbasis measurements, not even when restricting to pure states (while the question remains open for n ≥ 2). Next, we show that, for any Rényi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of manybody states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them. Finally, we discuss exact or efficient tensornetwork numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the Rényi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.