Area Laws and Tensor Networks for Maximally Mixed Ground States

Abstract

We show an area law in the mutual information for the maximally-mixed state \(\Omega \) in the ground space of general Hamiltonians, which is independent of the underlying ground space degeneracy. Our result assumes the existence of a ‘good’ approximation to the ground state projector (a good AGSP), a crucial ingredient in previous area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any \(\epsilon >0\) and any bipartition \(L\cup L^c\) of the system,

$$\begin{aligned} \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega } \le {\textrm{O}}\left( \log (|L|\log (d))+\log (1/\epsilon )\right) , \end{aligned}$$

where |L| represents the number of sites in L, d is the dimension of a site and \( \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega }\) represents the \(\epsilon \)-smoothed maximum mutual information with respect to the \(L:L^c\) partition in \(\Omega \). From this bound we then conclude \( \textrm{I} \, \! \! \left( L : L^c \right) _\Omega \le {\textrm{O}}\left( \log (|L|\log (d))\right) \) – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that \(\Omega \) can be approximated in trace norm up to \(\epsilon \) with a state of Schmidt rank of at most \(\textrm{poly}(|L|\log (d)/\epsilon )\), leading to a good MPO approximation for \(\Omega \) with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.