Lower bounds for testing complete positivity and quantum separability

In this work we are interested the problem of testing quantum entanglement. More specifically, we study the separability problem in quantum property testing, where one is given $n$ copies of an unknown mixed quantum state $\varrho$ on $\mathbb{C}^d \otimes \mathbb{C}^d$, and one wants to test whether $\varrho$ is separable or $\epsilon$-far from all separable states in trace distance. We prove that $n = \Omega(d^2/\epsilon^2)$ copies are necessary to test separability, assuming $\epsilon$ is not too small, viz.\ $\epsilon = \Omega(1/\sqrt{d})$. We also study completely positive distributions on the grid $[d] \times [d]$, as a classical analogue of separable states. We analogously prove that $\Omega(d/\epsilon^2)$ samples from an unknown distribution $p$ are necessary to decide whether $p$ is completely positive or $\epsilon$-far from all completely positive distributions in total variation distance. Towards showing that the true complexity may in fact be higher, we also show that learning an unknown completely positive distribution on $[d] \times [d]$ requires $\Omega(d^2/\epsilon^2)$ samples.