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Entanglement with Centers

Chen-Te Ma

TL;DR

This work develops a center-based framework for entanglement entropy in gauge theories, addressing the gauge-invariance challenge by extending the Hilbert space and employing non-tensor-product decompositions. It establishes both Hamiltonian and Lagrangian formalisms to compute entanglement with centers, revealing a duality structure in massless $p$-form theories and connecting codimension-two surface terms to non-trivial center decompositions. The analysis spans free scalar and $p$-form theories, Einstein gravity with holographic perspectives, and a strong-coupling lattice study of SU$(N)$ gauge theory, demonstrating area-law-like contributions and confinement signatures. The results offer a rigorous algebraic and variational footing for center-induced entanglement, with implications for holography, quantum gravity, and nonperturbative gauge dynamics, while highlighting open questions about universal center effects and their observability.

Abstract

Entanglement is a physical phenomenon that each state cannot be described individually. Entanglement entropy gives quantitative understanding to the entanglement. We use decomposition of the Hilbert space to discuss properties of the entanglement. Therefore, partial trace operator becomes important to define the reduced density matrix from different centers, which commutes with all elements in the Hilbert space, corresponding to different entanglement choices or different observations on entangling surface. Entanglement entropy is expected to satisfy the strong subadditivity. We discuss decomposition of the Hilbert space for the strong subadditivity and other related inequalities. The entanglement entropy with centers can be computed from the Hamitonian formulations systematically, provided that we know wavefunctional. In the Hamitonian formulation, it is easier to obtain symmetry structure. We consider massless $p$-form theory as an example. The massless $p$-form theory in ($2p+2)$-dimensions has global symmetry, similar to the electric-magnetic duality, connecting centers in ground state. This defines a duality structure in centers. Because it is hard to exactly compute the entanglement entropy from partial trace operator, we propose the Lagrangian formulation from the Hamitonian formulation to compute the entanglement entropy with centers. From the Lagrangian method and saddle point approximation, the codimension two surface term (leading order) in the Einstein gravity theory or holographic entanglement entropy should correspond to non-tensor product decomposition (center is not identity). Finally, we compute the entanglement entropy of the $SU(N)$ Yang-Mills lattice gauge theory in the fundamental representation using the strong coupling expansion in the extended lattice model to obtain spatial area term in total dimensions larger than two for $N>1$.

Entanglement with Centers

TL;DR

This work develops a center-based framework for entanglement entropy in gauge theories, addressing the gauge-invariance challenge by extending the Hilbert space and employing non-tensor-product decompositions. It establishes both Hamiltonian and Lagrangian formalisms to compute entanglement with centers, revealing a duality structure in massless -form theories and connecting codimension-two surface terms to non-trivial center decompositions. The analysis spans free scalar and -form theories, Einstein gravity with holographic perspectives, and a strong-coupling lattice study of SU gauge theory, demonstrating area-law-like contributions and confinement signatures. The results offer a rigorous algebraic and variational footing for center-induced entanglement, with implications for holography, quantum gravity, and nonperturbative gauge dynamics, while highlighting open questions about universal center effects and their observability.

Abstract

Entanglement is a physical phenomenon that each state cannot be described individually. Entanglement entropy gives quantitative understanding to the entanglement. We use decomposition of the Hilbert space to discuss properties of the entanglement. Therefore, partial trace operator becomes important to define the reduced density matrix from different centers, which commutes with all elements in the Hilbert space, corresponding to different entanglement choices or different observations on entangling surface. Entanglement entropy is expected to satisfy the strong subadditivity. We discuss decomposition of the Hilbert space for the strong subadditivity and other related inequalities. The entanglement entropy with centers can be computed from the Hamitonian formulations systematically, provided that we know wavefunctional. In the Hamitonian formulation, it is easier to obtain symmetry structure. We consider massless -form theory as an example. The massless -form theory in (-dimensions has global symmetry, similar to the electric-magnetic duality, connecting centers in ground state. This defines a duality structure in centers. Because it is hard to exactly compute the entanglement entropy from partial trace operator, we propose the Lagrangian formulation from the Hamitonian formulation to compute the entanglement entropy with centers. From the Lagrangian method and saddle point approximation, the codimension two surface term (leading order) in the Einstein gravity theory or holographic entanglement entropy should correspond to non-tensor product decomposition (center is not identity). Finally, we compute the entanglement entropy of the Yang-Mills lattice gauge theory in the fundamental representation using the strong coupling expansion in the extended lattice model to obtain spatial area term in total dimensions larger than two for .

Paper Structure

This paper contains 33 sections, 16 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Entanglement Entropy with Center
  3. Center
  4. Properties of the Entanglement with Center
  5. Partial Trace Operator
  6. Decomposition of the Hilbert Space
  7. Entanglement Inequalities
  8. Computation Methods in the Entanglement Entropy with Center
  9. The Hamitonian Formulation in a $p$-Form Theory
  10. Scalar Field Theory
  11. $p$-form Abelian Yang-Mills Gauge Theory
  12. The Lagrangian Formulation
  13. Lagrangian
  14. Replica Trick and Conical Method
  15. The Einstein Gravity Theory
  16. ...and 18 more sections

Key Result

Lemma 1

Let $\rho_{12}$ be a pure state density matrix on $H_{12}$ isomorphic to $\bigoplus_k H_1^k\otimes H_2^k$. Let $f( \cdot )$ be a real valued function, and $f(0)=0$. Then we obtain In particular, $S_1=S_2$.

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof : Proof
  • Lemma 2
  • Theorem 1
  • proof : Proof
  • Corollary 1
  • proof : Proof
  • Theorem 2
  • ...and 26 more