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Many body quantum eigenstates of generic Hamiltonians at finite energy density typically satisfy "volume law" of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume.
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The distinction between real wavefunctions (i.e. wavefunctions which take both positive and negative values, but are not complex) and complex wavefunctions turns out to be irrelevant for our discussion, both of whom thus possessing a non-trivial sign structure in our nomenclature
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