Topological quantum order

The local-pair condition for topological quantum order asks for an orthogonal partner state with local indistinguishability on supports of diameter at most a fixed fraction of the system diameter, with a fixed error smaller than one. This criterion captures a form of macroscopic information hidden from local observables; it is not a complete definition of every topological phase.

Undo a short evolution of two orthogonal locally indistinguishable states. Orthogonality is preserved exactly. A Lieb-Robinson bound localizes every backward-evolved observable within an expanded region and bounds its norm error. After normalizing the approximate observable, the initial expectation difference is at most $\epsilon (1+\delta )+2\delta$. Polynomial volume growth in the system diameter controls the localization prefactor and yields preservation for sufficiently small linear-time coefficient.