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arXiv:2310.19028

Area laws and tensor networks for maximally mixed ground states

Itai Arad, Raz Firanko, Rahul Jain

Published Oct 29, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2310.19028

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We show an area law in the mutual information for the maximally-mixed state ΩΩΩ 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 ε>0\varepsilon>0ε>0 and any bi-partition L∪LcL\cup L^cL∪Lc 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∣|L|∣L∣ represents the number of sites in LLL, ddd is the dimension of a site and Imax⁡ε(L:Lc)Ω\mathrm I_{\max}^\varepsilon (L:L^c)_ΩImaxε​(L:Lc)Ω​ represents the ε\varepsilonε-\emph{smoothed maximum mutual information} with respect to the L:LcL:L^cL:Lc partition in ΩΩΩ. From this bound we then conclude I(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d)))\mathrm I (L:L^c)_Ω \le \mathrm O\big(\log(|L|\log(d))\big)I(L:Lc)Ω​≤O(log(∣L∣log(d))) -- an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that ΩΩΩ can be approximated in trace norm up to ε\varepsilonε with a state of Schmidt rank of at most poly(∣L∣/ε)\mathrm{poly}(|L|/\varepsilon)poly(∣L∣/ε), leading to a good MPO approximation for ΩΩΩ with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.
We show an area law in the mutual information for the maximally-mixed state Ω\Omega 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 ϵ>0\epsilon >0ϵ>0 and any bipartition L∪LcL\cup L^cL∪Lc of the system, Imax⁡ϵ  ⁣ ⁣(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d))+log⁡(1/ϵ)),\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}Imaxϵ​(L:Lc)Ω​≤O(log(∣L∣log(d))+log(1/ϵ)),​ where |L| represents the number of sites in L, d is the dimension of a site and Imax⁡ϵ  ⁣ ⁣(L:Lc)Ω \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega }Imaxϵ​(L:Lc)Ω​ represents the ϵ\epsilon ϵ -smoothed maximum mutual information with respect to the L:LcL:L^cL:Lc partition in Ω\Omega Ω . From this bound we then conclude I  ⁣ ⁣(L:Lc)Ω≤O(log⁡(∣L∣log⁡(d))) \textrm{I} \, \! \! \left( L : L^c \right) _\Omega \le {\textrm{O}}\left( \log (|L|\log (d))\right) I(L:Lc)Ω​≤O(log(∣L∣log(d))) – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that Ω\Omega Ω can be approximated in trace norm up to ϵ\epsilon ϵ with a state of Schmidt rank of at most poly(∣L∣log⁡(d)/ϵ)\textrm{poly}(|L|\log (d)/\epsilon )poly(∣L∣log(d)/ϵ) , leading to a good MPO approximation for Ω\Omega Ω with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.
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 former area-law proofs. Such approximations have been explicitly derived for 1D gapped local Hamiltonians and 2D frustration-free locally-gapped local Hamiltonians. As a corollary, we show that in 1D gapped local Hamiltonians, for any є>0 and any bi-partition L∪ Lc of the system,
We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error є by a degree O(√nlog(є−1)) multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special case of commuting Hamiltonian terms. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation in the vicinity of the boundary of the bipartition of interest. This AGSP has sufficiently low entanglement and error to establish the area law using a known technique.
We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$. Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).
An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of nnn spins the ground states of a local Hamiltonian with energy gap ϵ\epsilonϵ are constant-fold degenerate. Then, the Renyi entanglement entropy Rα(0<α<1)R_\alpha(0<\alpha<1)Rα​(0<α<1) of any ground state across any cut is upper bounded by O~(α−3/ϵ)\tilde O(\alpha^{-3}/\epsilon)O~(α−3/ϵ), and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension 2O~(ϵ−1/4log⁡3/4n)2^{\tilde O(\epsilon^{-1/4}\log^{3/4}n)}2O~(ϵ−1/4log3/4n).
The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.