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 $0<α<1$ . 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 $ε$ across a planar cut of cross-section $A$ scales as $\exp[\tildeΘ((A\logε^{-1})^{2/3})]$ when $\logε^{-1}\gtrsim A$ . Thus $α=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.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 $0<α<1$. 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 $ε$ across a planar cut of cross-section $A$ scales as $\exp[\tildeΘ((A\logε^{-1})^{2/3})]$ when $\logε^{-1}\gtrsim A$. Thus $α=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.

quant-pharXiv:2610.15199Oct 13, 20260 citations

From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians

Anurag Anshu, Aram W. Harrow, Mehdi Soleimanifar

In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an "area law" behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an "area law" behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.

arXiv:2004.15009Apr 30, 20202 citations

Entanglement area law for long-range interacting systems

Zhexuan Gong, Michael Foss‐Feig, Fernando G. S. L. Brandão +1 more

Bulletin of the American Physical SocietyMar 18, 20160 citations

Area law in one dimension: Degenerate ground states and Renyi entanglement entropy

Yichen Huang

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 $n$ spins the ground states of a local Hamiltonian with energy gap $ε$ are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0An 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 $n$ spins the ground states of a local Hamiltonian with energy gap $ε$ are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0

arXiv:1403.0327Mar 3, 201422 citations

Entanglement area law in thermodynamically gapped spin systems

Jaeyoon Cho

Apr 30, 20142 citations

The Rényi area law conjecture is false below

α=1

in high dimensions

0<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

ε

across a planar cut of cross-section

A

scales as

\exp[\tildeΘ((A\logε^{-1})^{2/3})]

when

\logε^{-1}\gtrsim A

. Thus

α=1

is the sharp threshold for a universal Rényi area law within this adiabatically connected class.

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

n

spins the ground states of a local Hamiltonian with energy gap

ε

are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0