Translation symmetry-enforced long-range entanglement in mixed states

TL;DR

The paper demonstrates that certain translation symmetry-breaking states in mixed quantum states are inherently long-range entangled, beyond what symmetric short-range entangled states can describe, revealing subtle forms of entanglement.

Contribution

It introduces a counting argument showing the existence of long-range entangled mixed states that cannot be decomposed into symmetric short-range entangled states.

Findings

Symmetric SRE eigenstates do not span the zero momentum sector.

Translation symmetry-breaking states exhibit long-range entanglement.

Long-range entanglement in mixed states is undetectable by correlation functions.

Abstract

We show by a counting argument that even though translation symmetry admits symmetric short-range entangled (SRE) eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.