Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
arxiv(2024)
摘要
We show that spin chains in thermal equilibrium have a correlation structure
in which individual regions are strongly correlated at most with their near
vicinity. We quantify this with alternative notions of the conditional mutual
information defined through the so-called Belavkin-Staszewski relative entropy.
We prove that these measures decay super-exponentially, under the assumption
that the spin chain Hamiltonian is translation-invariant. Using a recovery map
associated with these measures, we sequentially construct tensor network
approximations in terms of marginals of small (sub-logarithmic) size. As a main
application, we show that classical representations of the states can be
learned efficiently from local measurements with a polynomial sample
complexity. We also prove an approximate factorization condition for the purity
of the entire Gibbs state, which implies that it can be efficiently estimated
to a small multiplicative error from a small number of local measurements. As a
technical step of independent interest, we show an upper bound to the decay of
the Belavkin-Staszewski relative entropy upon the application of a conditional
expectation.
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