Citation: Kim, Isaac H. "On the informational completeness of local observables." arXiv preprint arXiv:1405.0137 (2014).
Web: https://arxiv.org/abs/1405.0137
Tags: Information-theory, MTC-reconstruction
This paper discusses whether or not long range entanglement can be detected locally. The setting of the problem is a multibody quantum system on a 2D lattice, whose states satisfy a TEE area law. The author proves that if \rho and \sigma are two density matrices and the overall system which are equal on every local region, then there is a bound on the distance between \rho and \sigma. The constant in this bound is dependent on the correction term in the TEE area law. As the area law becomes more exact, the trace distance between \rho and \sigma becomes smaller.
There are several difficulties interpreting this result, and to what extent it says that "all long-range entangled states with an area can be distinguished locally". There are issues relating to boundary, to non-trivial topology on the physical space, and (most importantly) finite correlation length effects.
Regardless, the central technical proposition of the paper is good to be aware of. It gives an upper bound on the trace distance between two density matrices in terms of locally computable data. The proof goes roughly as follows. The first step is to use an explicit strengthening of the concavity of von Neumann entropy, which was proved in
> Kim, Isaac H. "Modulus of convexity for operator convex functions." Journal of Mathematical Physics 55.8 (2014).
and extended in
> Kim, Isaac, and Mary Beth Ruskai. "Bounds on the concavity of quantum entropy." Journal of Mathematical Phys
The next step is to use the chain rule for entanglement to break up to computations of entropy, and then restrict to local pieces at each step of the chain-rule decomposition. This is the Markov entropy decomposition, as introduced in
> Poulin, David, and Matthew B. Hastings. "Markov entropy decomposition: a variational dual for quantum belief propagation." Physical review letters 106.8 (2011): 080403.
Using one final application of weak-monotonicity to make all of the remaining terms local, the formula is established.
This proof is very clever, and the author claims it is robust to perturbations.