Quantum teleportation implies symmetry-protected topological order

Understand

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 $\mathcal{G}_{k}= (\mathbb{Z}_2 \times \mathbb{Z}_2)^k$.
• 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.