Stabilizer Rényi entropy on qudits

Magic is an important part of quantum resource theory. The quantitative study of magic is a difficult but important task, which means a suitable magic measure is needed. We generalize the magic measure introduced by Leone et al. (Phys Rev Lett 128:050402, 2022) and extend the Pauli operators acting on qubits to the discrete Heisenberg–Weyl operators (i.e., the generalized Pauli operators) acting on qudits. We show that it is a suitable magic measure. We use this measure to calculate some related qutrit magic states and compare it with other measures. Subsequently, a tighter upper bound of this measure is obtained by using the group covariant symmetric informationally complete (SIC) states. Moreover, we extend this measure to mixed states and calculate the corresponding mixed magic states. Then, it is shown that there are different fundamental limits for the distillation of four qutrit magic states by using the quantum hypothesis testing at the end of this paper.