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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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G<\/mml:mi><\/mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msubsup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z<\/mml:mi><\/mml:mrow><mml:mn>2<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"\/><\/mml:msubsup><mml:mo>&amp;#x00D7;<\/mml:mo><mml:msubsup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z<\/mml:mi><\/mml:mrow><mml:mn>2<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"\/><\/mml:msubsup><mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:msup><\/mml:math>. The <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> logical states are teleported between the edges of the chain by measuring the corresponding <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>2<\/mml:mn><mml:mi>k<\/mml:mi><\/mml:math> 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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state.<\/jats:p>","DOI":"10.22331\/q-2024-10-10-1499","type":"journal-article","created":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T10:35:13Z","timestamp":1728556513000},"page":"1499","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Quantum teleportation implies symmetry-protected topological order"],"prefix":"10.22331","volume":"8","author":[{"given":"Yifan","family":"Hong","sequence":"first","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA"}]},{"given":"David T.","family":"Stephen","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA"},{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Aaron J.","family":"Friedman","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,10,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"C. 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