Symmetry enforced entanglement in maximally mixed states

TL;DR

This paper demonstrates that in quantum systems with certain symmetries, the maximally mixed state can exhibit significant entanglement, especially under non-Abelian continuous symmetries, challenging the typical view of mixed states as unentangled.

Contribution

It reveals that symmetry constraints can induce entanglement in maximally mixed states, providing exact calculations and scaling laws for entanglement in symmetric quantum channels.

Findings

Maximally mixed states can be entangled under strong symmetry conditions.

Entanglement of formation is exactly computable and equal to distillation for these states.

For non-Abelian continuous symmetries, entanglement scales logarithmically with system size.

Abstract

Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. $SU(2)$), the bipartite entanglement of formation for the MMIS scales logarithmically $\sim \log N$ with the number of qudits $N$.