A new path to quantum entanglement built from counting, not symmetry

Long-range entanglement in mixed many-body states has so far been tied to symmetry anomalies or long-range correlations. This work shows a third route: dimensional mismatch. In a one-dimensional ring restricted to translation-invariant states, short-range entangled states span only a polynomially large subspace while the full translation-invariant space is exponentially larger—so the maximally mixed state over that space is forced to be long-range entangled. The resulting state displays logarithmically growing conditional mutual information, strong-to-weak symmetry breaking, and Rényi-index-dependent entanglement, with a concrete Lindbladian construction that stabilizes it.