An infinite chain of atoms spaced at equal intervals. Shift the entire chain by one step, and the picture doesn't change. That's translational symmetry: the world looks the same wherever you look. You'd think such a simple rule would give rise to simple states, like a crystal lattice assembled from identical building blocks.
But the authors counted those blocks anew. It turns out that states resembling independent pieces (known as short-range entangled states) are simply insufficient in number. To fill the entire space of possible options, you have to use states where parts far apart are invisibly linked, as if each brick in a wall remembered a neighbor a hundred meters away. Such a link isn't visible in ordinary measurements: correlations between distant parts remain zero, but the entanglement hasn't gone anywhere.
This discovery adds a new facet to our understanding of quantum systems. Previously, it was thought that if long-range correlations are absent, then entanglement is short-range. Now it's clear: there exist mixed states where entanglement hides deeper, changing the very classification of phases of matter. Thus, crystal-clear simplicity turns into complexity that has yet to be fully grasped.
as written by the authors arXiv:2605.15200
We show by a counting argument that even though translation symmetry admits symmetric short-range entangled (SRE) eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.
Ordinary crystals, like table salt, possess translational symmetry: their atoms repeat at regular intervals. But in the quantum world, even such order can turn out to be an illusion, behind which lies irreducible entanglement.