Citation: Fawzi, Omar, and Renato Renner. "Quantum conditional mutual information and approximate Markov chains." Communications in Mathematical Physics 340.2 (2015): 575-611.
Web: https://arxiv.org/abs/1410.0664
Tags: Information-theory, MTC-reconstruction
This paper proves that approximately conditionally mutually independent states have an approximate recovery map. That is, there is a recovery map for which any two locally indistinguishable states will be sent to approximately the same state. This is a fantastic result.
The main application I have in mind when reading this paper is the pursuit of the entanglement bootstrap program at finite correlation length. Here, the axioms of entanglement bootstrap will only be approximately satisfied. Hence, we need all of the tools of entanglement bootstrap to give approximately correct answers when given approximately satisfying data. In particular, we need an approximate recovery map!
Since this paper came out, there has been more work in the study of approximately conditionally mutually independent states, listed chronologically below:
> Sutter, David, Omar Fawzi, and Renato Renner. "Universal recovery map for approximate Markov chains." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472.2186 (2016): 20150623.
> Junge, Marius, et al. "Universal recovery maps and approximate sufficiency of quantum relative entropy." Annales Henri Poincaré. Vol. 19. No. 10. Cham: Springer International Publishing, 2018.
> Carlen, Eric A., and Anna Vershynina. "Recovery map stability for the data processing inequality." Journal of Physics A: Mathematical and Theoretical 53.3 (2020): 035204.