Supermartingales in Quantum Resources Theories: Where do quantum resources go when you're watching?

We establish a connection between quantum resource theory and probability theory, under repeated application of free operations.

We show that strong monotonicity of a resource measure implies that the resource is a supermartingale.

We then use the optional stopping and martingale convergence theorems to derive bounds on the efficiency of post-selection, and other free adaptive strategies.

We also describe the asymptotic dynamics of the conditional state, where resource fluctuations are necessarily absent, showing one of two distinct phenomena occurs: resource vanishing or resource freezing.