Area law for the maximally mixed ground state in degenerate 1D gapped systems

• URL: http://arxiv.org/abs/2310.19028v1
• Date: Sun, 29 Oct 2023 14:36:30 GMT
• Title: Area law for the maximally mixed ground state in degenerate 1D gapped systems
• Authors: Itai Arad, Raz Firanko, Rahul Jain
• Abstract summary: 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 $H$. We also get an area law for the mutual information of the form $mathrmI(L:R)_Omega leq O(log |L|)$.
• Score: 5.088702935364181
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $L\cup L^c$ 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 $\mathrm{poly}(|L|/\varepsilon)$.

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