Area laws and tensor networks for maximally mixed ground states
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| Format: | Preprint |
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2023
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| author | Arad, Itai Firanko, Raz Jain, Rahul |
| author_facet | Arad, Itai Firanko, Raz Jain, Rahul |
| contents | We show an area law in the mutual information for the
maximally-mixed state $Ω$ 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
$\varepsilon>0$ and any bi-partition $L\cup L^c$ of the system,
\begin{align*}
\mathrm I_{\max}^\varepsilon (L:L^c)_Ω \le \mathrm O \big( \log (|L|\log(d))+\log(1/\varepsilon)\big),
\end{align*}
where $|L|$ represents the number of sites in $L$, $d$ is the dimension of a site and
$ \mathrm I_{\max}^\varepsilon (L:L^c)_Ω $ represents the $\varepsilon$-\emph{smoothed
maximum mutual information} with respect to the $L:L^c$ partition
in $Ω$. From this bound we then conclude $\mathrm I (L:L^c)_Ω
\le \mathrm O\big(\log(|L|\log(d))\big)$ -- an area law for the mutual information in
1D systems with a logarithmic correction. In addition, we show
that $Ω$ can be approximated in trace norm up to $\varepsilon$ with
a state of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$, leading to
a good MPO approximation for $Ω$ with polynomial bond
dimension. Similar corollaries are derived for the mutual
information of 2D frustration-free and locally-gapped local
Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19028 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Area laws and tensor networks for maximally mixed ground states Arad, Itai Firanko, Raz Jain, Rahul Quantum Physics Other Condensed Matter Computational Complexity Information Theory We show an area law in the mutual information for the maximally-mixed state $Ω$ 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 $\varepsilon>0$ and any bi-partition $L\cup L^c$ of the system, \begin{align*} \mathrm I_{\max}^\varepsilon (L:L^c)_Ω \le \mathrm O \big( \log (|L|\log(d))+\log(1/\varepsilon)\big), \end{align*} where $|L|$ represents the number of sites in $L$, $d$ is the dimension of a site and $ \mathrm I_{\max}^\varepsilon (L:L^c)_Ω $ represents the $\varepsilon$-\emph{smoothed maximum mutual information} with respect to the $L:L^c$ partition in $Ω$. From this bound we then conclude $\mathrm I (L:L^c)_Ω \le \mathrm O\big(\log(|L|\log(d))\big)$ -- an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that $Ω$ can be approximated in trace norm up to $\varepsilon$ with a state of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$, leading to a good MPO approximation for $Ω$ with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians. |
| title | Area laws and tensor networks for maximally mixed ground states |
| topic | Quantum Physics Other Condensed Matter Computational Complexity Information Theory |
| url | https://arxiv.org/abs/2310.19028 |