Sample efficient identity testing and independence testing of quantum states

In this paper, we study the quantum identity testing problem, i.e., testing whether two given quantum states are identical, and quantum independence testing problem, i.e., testing whether a given multipartite quantum state is in tensor product form. For the quantum identity testing problem of D(C^d) system, we provide a deterministic measurement scheme that uses O(^d_ε2²) copies via independent measurements with d being the dimension of the state and ε being the additive error. For the independence testing problem D(C^d₁ ⊗ C^d2 ⊗ · · · ⊗ C^dm) system, we show that the sample complexity is Θ(^{~ Πmi=1}_ε2^di) via collective measurements, and O(^Πmi=1_ε2^d2i) via independent measurements. If randomized choice of independent measurements are allowed, the sample complexity is Θ(^d3_ε2^/2) for the quantum identity testing problem, and Θ(^{~ Πmi=1}_ε2^{d3 i/2}) for the quantum independence testing problem.