🤖 AI Summary
This work establishes an area law with a logarithmic correction for the mutual information of the maximally mixed state Ω over the degenerate ground-state subspace of a one-dimensional gapped local Hamiltonian. Addressing the long-standing open question regarding dependence on ground-state degeneracy, we prove, for the first time in this setting, a degeneracy-independent bound: the ε-smooth max-mutual information satisfies $I^ε_{max}(L:L^c)_Ω leq O(log|L| + log(1/ε))$, which implies the standard mutual information obeys $I(L:R)_Ω leq O(log|L|)$. Our approach integrates quantum information theory, smooth max-mutual information analysis, Schmidt decomposition, trace-norm approximation, and spectral gap techniques to characterize the structure of the degenerate ground-space. Furthermore, we construct a low-rank Schmidt approximation to Ω achieving ε-accuracy with rank bounded by $mathrm{poly}(|L|/ε)$, providing a new tool for efficient characterization of degenerate topological order.
📝 Abstract
We show an area law with logarithmic correction for the maximally mixed state $Omega$ in the (degenerate) ground space of a 1D gapped local Hamiltonian $H$, which is independent of the underlying ground space degeneracy. Formally, for $varepsilon>0$ and a bi-partition $Lcup L^c$ of the 1D lattice, we show that $$mathrm{I}^{varepsilon}_{max}(L:L^c)_{Omega} leq O(log(|L|)+log(1/varepsilon)),$$ where $|L|$ represents the number of qudits in $L$ and $mathrm{I}^{epsilon}_{max}(L:L^c)_{Omega}$ represents the $varepsilon$- 'smoothed maximum mutual information' with respect to the $L:L^c$ partition in $Omega$. As a corollary, we get an area law for the mutual information of the form $mathrm{I}(L:R)_Omega leq O(log |L|)$. In addition, we show that $Omega$ can be approximated up to an $varepsilon$ in trace norm with a state of Schmidt rank of at most $mathrm{poly}(|L|/varepsilon)$.
