A counting argument shows the translation-symmetric fixed-point strong-to-weak SSB mixed state is long-range entangled and cannot be expressed as a mixture of short-range entangled states.
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The chiral anomaly is extended to mixed states via symmetry-charge flow derived algebraically from symmetry and flux-insertion operators, restoring universality for Abelian symmetries in fermionic and bosonic systems.
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Translation symmetry-enforced long-range entanglement in mixed states
A counting argument shows the translation-symmetric fixed-point strong-to-weak SSB mixed state is long-range entangled and cannot be expressed as a mixture of short-range entangled states.
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Anomaly and symmetry-charge flow in mixed states
The chiral anomaly is extended to mixed states via symmetry-charge flow derived algebraically from symmetry and flux-insertion operators, restoring universality for Abelian symmetries in fermionic and bosonic systems.