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

Abstract

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.

arXiv comment5 pages, 3 figures (+ 2 pages, 1 figure of Appendix + 8 pages of Supp. Mat.)