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Communications in Mathematical Physics

Area Laws and Tensor Networks for Maximally Mixed Ground States

Itai Arad, Raz Firanko, Rahul Jain

Published Feb 5, 2026• 0 citations
Publisher

Abstract

We show an area law in the mutual information for the maximally-mixed state $$\Omega $$ 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 $$\epsilon >0$$ and any bipartition $$L\cup L^c$$ of the system, $$\begin{aligned} \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega } \le {\textrm{O}}\left( \log (|L|\log (d))+\log (1/\epsilon )\right) , \end{aligned}$$ where |L| represents the number of sites in L, d is the dimension of a site and $$ \textrm{I}^\epsilon _{\max } \, \! \! \left( L : L^c \right) _{\Omega }$$ represents the $$\epsilon $$ -smoothed maximum mutual information with respect to the $$L:L^c$$ partition in $$\Omega $$ . From this bound we then conclude $$ \textrm{I} \, \! \! \left( L : L^c \right) _\Omega \le {\textrm{O}}\left( \log (|L|\log (d))\right) $$ – an area law for the mutual information in 1D systems with a logarithmic correction. In addition, we show that $$\Omega $$ can be approximated in trace norm up to $$\epsilon $$ with a state of Schmidt rank of at most $$\textrm{poly}(|L|\log (d)/\epsilon )$$ , leading to a good MPO approximation for $$\Omega $$ with polynomial bond dimension. Similar corollaries are derived for the mutual information of 2D frustration-free and locally-gapped local Hamiltonians.

DOI
10.1007/s00220-026-05554-z

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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 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 $ \mathrm I_{\max}^\varepsilon (L:L^c)_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 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.
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.
The Rényi area law conjecture is false below α=1α=1α=1 in high dimensions
Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed 0<α<10<α<10<α<1. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error εεε across a planar cut of cross-section AAA scales as exp⁡[Θ~((Alog⁡ε−1)2/3)]\exp[\tildeΘ((A\logε^{-1})^{2/3})]exp[Θ~((Alogε−1)2/3)] when log⁡ε−1≳A\logε^{-1}\gtrsim Alogε−1≳A. Thus α=1α=1α=1 is the sharp threshold for a universal Rényi area law within this adiabatically connected class.
One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is O~(nM(n)){\tilde{O}(nM(n))}O~(nM(n)) , where M(n) is the time required to multiply two n × n matrices.