Hypothesis tests for large density matrices of quantum systems based on Pauli measurements

For a quantum system, its density matrix usually has a size growing exponentially with the number of particles in the system, and quantum state tomography techniques often encounter an exponential complexity problem for recovering the density matrix based on experimental data. Recent statistical methods for estimating a large density matrix have been developed for the cases that (i) the entries of the density matrix with respect to the Pauli basis are sparse, or (ii) the density matrix has a low rank, and its eigenvectors are sparse. Their performances depend on the assumed structures, and it is important to test for the structures and choose appropriate estimation methods accordingly. This paper investigates hypothesis tests for sparsity. Specifically, we propose hypothesis test procedures and establish asymptotic theories for the proposed tests. Numerical studies are conducted to check the finite sample performances of the proposed hypothesis tests.