Sparsity of the stabilizer projector decomposition of a density matrix and robustness of magic

We extend the stabilizer rank of state vectors to mixed states and define the rank(minimal l_0 norm) for stabilizer projector decomposition of a density matrix and show its advantage over the rank of Pauli decomposition. Both improvements on the scaling over standard orthonomal basis(computational basis for state vector and Pauli basis for density matrix) come from the fact that stabilizer states form a densely overcomplete basis. In comparison with Monte Carlo simulation that scales with Robustness of Magic(minimal l_1 norm), we analyse the strong simulation cost of noisy Clifford+T circuits with respect to the rank. Using results from compressed sensing, we explore the sparsity condition where the minimal l_0 and l_1 norm are reached at the same decomposition.