Supermartingales in Quantum Resources Theories: Where do quantum resources go when you're watching?
arXiv:2607.26225
The paper connects quantum resource theories with probability theory by proving that resource measures form supermartingales under repeated free operations, and uses martingale theorems to bound the efficiency of post‑selection and adaptive strategies, showing that resources either vanish or freeze asymptotically.
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.
5 pages, 3 figures (+ 2 pages, 1 figure of Appendix + 8 pages of Supp. Mat.)