Translation symmetry forces long-range entanglement that correlations cannot detect

When a translation symmetry is broken in the weak sense across a statistical mixture of quantum states, the resulting mixed state cannot be decomposed into short-range entangled components — even though such components individually exist. The argument is combinatorial: there simply are not enough short-range entangled eigenstates to span the relevant symmetry sector. This reveals a class of long-range entanglement in mixed states that is invisible to standard diagnostics like connected correlation functions, suggesting current tools for characterizing quantum phases of open systems are incomplete.