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[2310.19028] Area law for the maximally mixed ground state in degenerate 1D gapp...
source link: https://arxiv.org/abs/2310.19028
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Quantum Physics
[Submitted on 29 Oct 2023]
Area law for the maximally mixed ground state in degenerate 1D gapped systems
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).
| Comments: | 23 pages, version 1 |
| Subjects: | Quantum Physics (quant-ph); Other Condensed Matter (cond-mat.other); Computational Complexity (cs.CC); Information Theory (cs.IT) |
| Cite as: | arXiv:2310.19028 [quant-ph] |
| (or arXiv:2310.19028v1 [quant-ph] for this version) | |
| https://doi.org/10.48550/arXiv.2310.19028 |
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