Modular zero modes and sewing the states of QFT

TL;DR

This paper demonstrates that in quantum field theory, any collection of local states in separate regions can be derived from a global pure state, contrasting with lattice models where such a state may not exist due to eigenvalue constraints.

Contribution

The authors show that unlike lattice models, QFT allows for constructing a global pure state from local states via local unitaries, highlighting a fundamental difference in entanglement structure.

Findings

In QFT, local states in separate regions can originate from a global pure state.

Local unitaries can transform local states arbitrarily close to any other local state.

Lattice models require eigenvalue inequalities for the existence of a global pure state.

Abstract

We point out an important difference between continuum relativistic quantum field theory (QFT) and lattice models with dramatic consequences for the theory of multi-partite entanglement. On a lattice given a collection of density matrices $\rho^{(1)},\rho^{(2)}, \cdots, \rho^{(n)}$ there is no guarantee that there exists an $n$-partite pure state $|\Omega\rangle_{12\cdots n}$ that reduces to these marginals. The state $|\Omega\rangle_{12\cdots n}$ exists only if the eigenvalues of the density matrices $\rho^{(i)}$ satisfy certain polygon inequalities. We show that in QFT, as opposed to lattice systems, splitting the space into $n$ non-overlapping regions any collection of local states $\omega^{(1)},\omega^{(2)},\cdots \omega^{(n)}$ come from the restriction of a global pure state. The reason is that rotating any local state $\omega^{(i)}$ by unitary $U_i$ localized in the $i^{th}$ region we come arbitrarily close to any other local state $\psi^{(i)}$. We construct explicit examples of such local unitaries using the cocycle.