Quantum teleportation implies symmetry-protected topological order

Abstract

We constrain a broad class of teleportation protocols using insights from locality. In the “standard” teleportation protocols we consider, all outcome-dependent unitaries are Pauli operators conditioned on linear functions of the measurement outcomes. We find that all such protocols involve preparing a “resource state” exhibiting symmetry-protected topological (SPT) order with Abelian protecting symmetry Gk = (Z2 ×Z2)k. The k logical states are teleported between the edges of the chain by measuring the corresponding 2k string order parameters in the bulk and applying outcome-dependent Paulis. Hence, this single class of nontrivial SPT states is both necessary and sufficient for the standard teleportation of k qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state.