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Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n -qubit channel to precision ϵ in ℓ ∞ using just O ( 1 / ϵ 2 ) log ⁡ ( n / ϵ ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O ( 1 / ϵ ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤ 1 / 4 .We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1 − η . In the regime of small η we extend our algorithm to achieve multiplicative precision 1 ± ϵ (i.e., additive precision ϵ η ) using just O ( 1 ϵ 2 η ) log ⁡ ( n / ϵ ) applications of the channel. 

In this work we are interested in 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 ϱ on Cd⊗Cd , and one wants to test whether ϱ is separable or ϵ -far from all separable states in trace distance. We prove that n=Ω(d2/ϵ2) copies are necessary to test separability, assuming ϵ is not too small, viz. ϵ=Ω(1/d−−√) . We also study completely positive distributions on the grid [d]×[d] , as a classical analogue of separable states. We analogously prove that Ω(d/ϵ2) samples from an unknown distribution p are necessary to decide whether p is completely positive or ϵ -far from all completely positive distributions in total variation distance.