Generic Entanglement Entropy for Quantum States with Symmetry

TL;DR

This paper investigates how different symmetries affect the typical entanglement properties of quantum states, revealing that some symmetries reduce entanglement while others do not, with implications for quantum information and physics.

Contribution

It extends the concentration formula for entanglement entropy to symmetric subspaces and characterizes how various symmetries influence generic entanglement levels.

Findings

States with axial symmetry are still highly entangled, but less than asymmetric states.

Permutation symmetry significantly reduces entanglement.

Translation symmetry preserves high entanglement levels similar to generic states.

Abstract

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.