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: Mathematical, Information-theory, Approximate-structure
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. Formally, what they prove is that the best fidelity of the best recovery channel from rho_AB to rho_ABC is lower bounded by the exponential of -1/2 times the CMI of rho_ABC.
The proof of the result uses the notion of "typical subspaces". Instead of directly attacking the main inequality of this paper, the authors consider the same inequality but with the fidelity taken instead between N copies of the state. They then argue that the fidelity is lower bounded by the desired quantity, up to dividing by some polynomial. Taking the Nth root of both sides and then taking N->infinity, one concludes the desired lower bound. This is a great proof structure for proving tight inequalities. The individual steps are allowed to be "loose", losing polynomial factors that can depend badly on the dimension of the underlying qudits, but since every polynomial grows slower than an exponential, everything is "forgiven" at the end of the day.
The key technical lemma in this proof is Lemma 4.2. It shows in a sense that every permutation-invariant operator on an N-layer system can be well-approximated by a tensor product of unitaries on single layers. Lemma 4.2 combined with the general typical subspace results makes for a powerful proof-technique, as demonstrated beautifully by this paper. Namely: you work on N layers, you show that a permutation-invariant operator satisfying nice bonds exists on the N layers (in this case, a certain typical subspace projection), and then you use Lemma 4.2. to push projection onto a one-layer unitary. This allows you to then take an Nth root, and then take N to infinity.
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.