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

36 pages, 8 figures; v2 changes: added summary of main assumptions and results; v3 changes: added additional references and reformatted for Quantum; v4 changes: added acknowledgement

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Cited by in corpus (3)

• Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements
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• Non-onsite symmetries and quantum teleportation in split-index matrix product states