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Entanglement entropy bounds for pure states of rapid decorrelation

Michael Aizenman, Simone Warzel

TL;DR

This work provides a general framework for proving area-law entanglement bounds for pure states on quantum lattices by introducing conditional decoupling criteria and constructing low-complexity, high-fidelity approximations. It furnishes two complementary approximation schemes (reduced-state and Markovian) and derives fidelity and entropic bounds that lead to area-law scaling with log-type corrections governed by a decoupling length l0. The results yield exponential decay of mutual information and exponential clustering of local observables, with explicit model-driven instantiations for the quantum Ising model in the subcritical regime across dimensions. The approach unifies probabilistic and information-theoretic methods to extend area-law insights beyond strictly one-dimensional or exactly solvable cases, offering a pathway to quantify entanglement in broad non-critical settings.

Abstract

For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states' entanglement entropy. The condition is also shown to imply exponential decay of the state's mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model's subcritical ground states, valid in all dimensions and up to the model's quantum phase transition.

Entanglement entropy bounds for pure states of rapid decorrelation

TL;DR

This work provides a general framework for proving area-law entanglement bounds for pure states on quantum lattices by introducing conditional decoupling criteria and constructing low-complexity, high-fidelity approximations. It furnishes two complementary approximation schemes (reduced-state and Markovian) and derives fidelity and entropic bounds that lead to area-law scaling with log-type corrections governed by a decoupling length l0. The results yield exponential decay of mutual information and exponential clustering of local observables, with explicit model-driven instantiations for the quantum Ising model in the subcritical regime across dimensions. The approach unifies probabilistic and information-theoretic methods to extend area-law insights beyond strictly one-dimensional or exactly solvable cases, offering a pathway to quantify entanglement in broad non-critical settings.

Abstract

For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states' entanglement entropy. The condition is also shown to imply exponential decay of the state's mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model's subcritical ground states, valid in all dimensions and up to the model's quantum phase transition.
Paper Structure (21 sections, 14 theorems, 98 equations, 2 figures)

This paper contains 21 sections, 14 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entanglement entropy
  3. Mutual information
  4. States of quasi-local structure in a computational basis
  5. Example: the quantum Ising model
  6. State approximations of decreased complexity
  7. Reduced state approximation
  8. Markovian state approximation
  9. Fidelity bound for the Markovian state approximation
  10. Entropic implications of rapid decorrelation
  11. An entanglement bound based on a single buffer estimate
  12. An improved multiscale analysis
  13. Entropy approximations
  14. Mutual information bounds
  15. Bounds on conditional dependence
  16. ...and 6 more sections

Key Result

Theorem 1.1

For the quantum Ising model on $W \subset \mathbb{Z}^d$, $d\geq 1$, with the Hamiltonian QIM_H at $b > b_c(d)$ (i.e. throughout the ground states' subcritical regime):

Figures (2)

  • Figure 1: Sketch of the decomposition of $W \subset \mathbb{Z}^d$ into $A$, a buffer $B$, and the remaining set $C$.
  • Figure 2: Sketch of the geometric situation in Definition \ref{['def:expdec2']} and the proof of Theorem \ref{['thm:mutualinfo']}: two cubes $A_1 , A_2$ with their buffers $B_1 , B_2$ of width $l \approx d(A_1,A_2)/ 3$.

Theorems & Definitions (34)

  • Theorem 1.1: Bounds for QIM
  • Theorem 3.1: Fidelity bound
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • proof
  • Definition 4.5
  • ...and 24 more