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In a celebrated recent result, Fawzi and Renner [CMP 340 (2015)] showed that for a given quantum state, and a tripatition ABC of a quantum system, the conditional mutual information I(A:C|B) is small if and only if there exists a map which approximately recovers the erasure of subsystem C, without acting on system A. Such states are called Markov states, and the maps are called local recovery maps. I will consider applications of this important new result to the analysis of many-body systems. In particular, I will show that if the thermal (Gibbs) state of a local Hamiltonian is Markov, and satisfies some physically reasonable assumption on the long range correlations, then (i) expectation values of local observables can be evaluated efficiently, and (ii) the Gibbs state can be sampled (prepared) efficiently on a quantum computer. Using similar methods, I will argue that ground states of certain local gapped Hamiltonians can be prepared in sub-linear time (on a quantum computer) if and only if the topological entanglement entropy is zero. Furthermore, a slight strengthening of these methods provide checkable conditions for the parent Hamiltonian of a PEPS to be gapped, as long as the Li-Haldane type conjecture holds. I'll finally comment on further applications of local recovery maps, including approximate error correcting codes and toy models for the AdS/CFT correspondence, and argue that these tools from quantum information theory will play an important role in analysis of many body systems.

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Michael Kastoryano, Niels Bohr International Academy
Seminar Room 0.03, ETP
Contact: Achim Rosch