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Correlations, area laws, and stability of open and thermal many-body quantum systems

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Mathematical Challenges in Quantum Information

Investigating scaling laws of correlations and entanglement, stability and simulatability of quantum states on spin lattice systems is a central topic in Hamiltonian complexity theory. In this talk, we discuss open systems and thermal analogues of features of ground states of quantum many-body systems, using proof tools inspired by ideas of quantum information theory. For open systems, we establish a connection between mixing times – either captured by Liouvillian gaps or Log-Sobolev-constants independent of the system size – and the clustering of correlations and area laws. For Gibbs states, we prove that above a universal critical temperature only depending on local properties of the Hamiltonian’s interaction hypergraph, thermal quantum states of local Hamiltonians are stable against distant Hamiltonian perturbations. As a consequence, local expectation values can be approximated in polynomial time. The stability theorem also provides a definition of temperature as a local quantity. We prove our clustering result via a reduction to a cluster expansion originally used to approximate thermal states by matrix product operators.

This talk is part of the Isaac Newton Institute Seminar Series series.

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