Fluctuations in extensive observables decay as 1/N for states with bounded multipartite entanglement and exponentially only when entanglement grows with N, so entanglement is not needed for typicality in macroscopic systems.
Since∆ A ≤2∥A∥for any HermitianA, his inequality immediately implies ∆A2≤4∥A∥ 2 Tr[ϱ2]
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Necessity of entanglement for the typicality argument in statistical mechanics
Fluctuations in extensive observables decay as 1/N for states with bounded multipartite entanglement and exponentially only when entanglement grows with N, so entanglement is not needed for typicality in macroscopic systems.