Quantum Event Learning and Gentle Random Measurements

Understand

We prove the expected disturbance caused to a quantum system by a sequence of randomly ordered two-outcome projective measurements is upper bounded by the square root of the probability that at least one measurement in the sequence accepts.

• We call this bound the Gentle Random Measurement Lemma.
• We then consider problems in which we are given sample access to an unknown state $\rho$ and asked to estimate properties of the accepting probabilities $\text{Tr}[M_i \rho]$ of a set of measurements $\{M_1, M_2, \ldots , M_m\}$.
• We call these types of problems Quantum Event Learning Problems.