Quantum teleportation implies symmetry-protected topological order

Yifan Hong; David T. Stephen; Aaron J. Friedman

doi:10.22331/q-2024-10-10-1499

Quantum teleportation implies symmetry-protected topological order

Yifan Hong¹, David T. Stephen^1,2, and Aaron J. Friedman¹

¹Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA
²Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

Published:	2024-10-10, volume 8, page 1499
Eprint:	arXiv:2310.12227v4
Doi:	https://doi.org/10.22331/q-2024-10-10-1499
Citation:	Quantum 8, 1499 (2024).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated after initial publication: This publication was updated to version v4 after the initial publication. The authors left the following comment on the arXiv:

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

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.

► BibTeX data

@article{Hong2024quantum,
  doi = {10.22331/q-2024-10-10-1499},
  url = {https://doi.org/10.22331/q-2024-10-10-1499},
  title = {Quantum teleportation implies symmetry-protected topological order},
  author = {Hong, Yifan and Stephen, David T. and Friedman, Aaron J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1499},
  month = oct,
  year = {2024}
}

► References

[1] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
https://doi.org/10.1103/PhysRevLett.70.1895

[2] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).
https://doi.org/10.1103/PhysRevLett.81.5932

[3] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999).
https://doi.org/10.1038/46503

[4] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
https://doi.org/10.1103/PhysRevLett.86.5188

[5] M. Popp, F. Verstraete, M. A. Martí n-Delgado, and J. I. Cirac, Phys. Rev. A 71 (2005).
https://doi.org/10.1103/PhysRevA.71.042306

[6] F. Verstraete, M. Popp, and J. I. Cirac, Phys. Rev. Lett. 92, 027901 (2004a).
https://doi.org/10.1103/PhysRevLett.92.027901

[7] F. Verstraete, M. A. Martín-Delgado, and J. I. Cirac, Phys. Rev. Lett. 92, 087201 (2004b).
https://doi.org/10.1103/PhysRevLett.92.087201

[8] B.-Q. Jin and V. E. Korepin, Phys. Rev. A 69, 062314 (2004).
https://doi.org/10.1103/PhysRevA.69.062314

[9] J. K. Pachos and M. B. Plenio, Phys. Rev. Lett. 93, 056402 (2004).
https://doi.org/10.1103/PhysRevLett.93.056402

[10] L. C. Venuti and M. Roncaglia, Phys. Rev. Lett. 94, 207207 (2005).
https://doi.org/10.1103/PhysRevLett.94.207207

[11] S. O. Skrøvseth and S. D. Bartlett, Phys. Rev. A 80, 022316 (2009).
https://doi.org/10.1103/PhysRevA.80.022316

[12] J. Y. Lee, W. Ji, Z. Bi, and M. P. A. Fisher, Decoding measurement-prepared quantum phases and transitions: from ising model to gauge theory, and beyond (2022), arXiv:2208.11699 [cond-mat.str-el].
https://doi.org/10.48550/arXiv.2208.11699
arXiv:2208.11699

[13] G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Phys. Rev. Lett. 131, 200201 (2023).
https://doi.org/10.1103/PhysRevLett.131.200201

[14] F. Eckstein, B. Han, S. Trebst, and G.-Y. Zhu, Robust teleportation of a surface code and cascade of topological quantum phase transitions (2024), arXiv:2403.04767 [quant-ph].
https://doi.org/10.48550/arXiv.2403.04767
arXiv:2403.04767

[15] P. Sala, S. Murciano, Y. Liu, and J. Alicea, PRX Quantum 5, 030307 (2024).
https://doi.org/10.1103/PRXQuantum.5.030307

[16] R. Raussendorf, S. Bravyi, and J. Harrington, Phys. Rev. A 71, 062313 (2005).
https://doi.org/10.1103/PhysRevA.71.062313

[17] M. Van den Nest, A. Miyake, W. Dür, and H. J. Briegel, Phys. Rev. Lett. 97, 150504 (2006).
https://doi.org/10.1103/PhysRevLett.97.150504

[18] D. Gross and J. Eisert, Phys. Rev. Lett. 98, 220503 (2007).
https://doi.org/10.1103/PhysRevLett.98.220503

[19] D. Gross, S. T. Flammia, and J. Eisert, Phys. Rev. Lett. 102, 190501 (2009).
https://doi.org/10.1103/PhysRevLett.102.190501

[20] M. J. Bremner, C. Mora, and A. Winter, Phys. Rev. Lett. 102, 190502 (2009).
https://doi.org/10.1103/PhysRevLett.102.190502

[21] A. C. Doherty and S. D. Bartlett, Phys. Rev. Lett. 103, 020506 (2009).
https://doi.org/10.1103/PhysRevLett.103.020506

[22] D. Gross and J. Eisert, Phys. Rev. A 82, 040303 (2010).
https://doi.org/10.1103/PhysRevA.82.040303

[23] D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Phys. Rev. Lett. 108 (2012).
https://doi.org/10.1103/PhysRevLett.108.240505

[24] A. Prakash and T.-C. Wei, Phys. Rev. A 92, 022310 (2015).
https://doi.org/10.1103/PhysRevA.92.022310

[25] J. Miller and A. Miyake, NPJ Quant. Inf. 2, 16036 (2016).
https://doi.org/10.1038/npjqi.2016.36

[26] D. T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf, Phys. Rev. Lett. 119, 010504 (2017).
https://doi.org/10.1103/PhysRevLett.119.010504

[27] I. Marvian, Phys. Rev. B 95, 045111 (2017).
https://doi.org/10.1103/PhysRevB.95.045111

[28] T.-C. Wei, Adv. Phys. X 3, 1461026 (2018).
https://doi.org/10.1080/23746149.2018.1461026

[29] Y. Bao, M. Block, and E. Altman, Phys. Rev. Lett. 132, 030401 (2024).
https://doi.org/10.1103/PhysRevLett.132.030401

[30] G. Q. AI and Collaborators, Nature 622, 481 (2023).
https://doi.org/10.1038/s41586-023-06505-7

[31] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011).
https://doi.org/10.1103/PhysRevB.83.035107

[32] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 10.1103/PhysRevB.81.064439 (2010).
https://doi.org/10.1103/PhysRevB.81.064439

[33] N. Schuch, D. Pérez-García, and I. Cirac, Phys. Rev. B 84, 10.1103/physrevb.84.165139 (2011).
https://doi.org/10.1103/physrevb.84.165139

[34] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013).
https://doi.org/10.1103/PhysRevB.87.155114

[35] A. Miyake, Phys. Rev. Lett. 105, 040501 (2010).
https://doi.org/10.1103/PhysRevLett.105.040501

[36] M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989).
https://doi.org/10.1103/PhysRevB.40.4709

[37] T. Kennedy and H. Tasaki, Phys. Rev. B 45, 304 (1992).
https://doi.org/10.1103/PhysRevB.45.304

[38] F. Pollmann and A. M. Turner, Phys. Rev. B 86, 125441 (2012).
https://doi.org/10.1103/PhysRevB.86.125441

[39] R. Raussendorf, W. Yang, and A. Adhikary, Quantum 7, 1215 (2023).
https://doi.org/10.22331/q-2023-12-28-1215

[40] E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972).
https://doi.org/10.1007/BF01645779

[41] D. Poulin, Phys. Rev. Lett. 104, 190401 (2010).
https://doi.org/10.1103/PhysRevLett.104.190401

[42] A. J. Friedman, C. Yin, Y. Hong, and A. Lucas, arXiv 10.48550/arXiv.2206.09929 (2022a), arXiv:2206.09929 [quant-ph].
https://doi.org/10.48550/arXiv.2206.09929
arXiv:2206.09929

[43] H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001).
https://doi.org/10.1103/PhysRevLett.86.910

[44] D. Pérez-García, M. M. Wolf, M. Sanz, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 10.1103/PhysRevLett.100.167202 (2008).
https://doi.org/10.1103/PhysRevLett.100.167202

[45] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).
https://doi.org/10.1017/CBO9780511976667

[46] D. Barberena and A. J. Friedman, Overview of projective quantum measurements (2024), arXiv:2404.05679 [quant-ph].
https://doi.org/10.48550/arXiv.2404.05679
arXiv:2404.05679

[47] A. J. Friedman, O. Hart, and R. Nandkishore, arXiv 10.48550/arXiv.2210.1111 (2022b), arXiv:2210.11111 [quant-ph].
https://doi.org/10.48550/arXiv.2210.1111
arXiv:2210.11111

[48] W. F. Stinespring, Proc. Amer. Math. Soc. 6, 211 (1955).
https://doi.org/10.2307/2032342

[49] M.-D. Choi, Lin. Alg. Appl. 10, 285 (1975).
https://doi.org/10.1016/0024-3795(75)90075-0

[50] M. Takesaki, Theory of Operator Algebras I (Springer New York, NY, 2012).
https://doi.org/10.1007/978-1-4612-6188-9

[51] D. Gottesman, Stabilizer codes and quantum error correction (1997), arXiv:quant-ph/9705052 [quant-ph].
https://doi.org/10.48550/arXiv.quant-ph/9705052
arXiv:quant-ph/9705052

[52] M. Rossi, M. Huber, D. Bruß, and C. Macchiavello, New J. Phys. 15, 113022 (2013).
https://doi.org/10.1088/1367-2630/15/11/113022

[53] M. Gachechiladze, O. Gühne, and A. Miyake, Phys. Rev. A 99, 052304 (2019).
https://doi.org/10.1103/PhysRevA.99.052304

[54] Y. Hong, D. T. Stephen, and A. J. Friedman, arXiv , to appear (2024).

[55] M. B. Hastings, Locality in quantum systems (2010), arXiv:1008.5137 [math-ph].
https://doi.org/10.48550/arxiv.1008.5137
arXiv:1008.5137

[56] Z. Wang and K. R. Hazzard, PRX Quantum 1, 010303 (2020).
https://doi.org/10.1103/PRXQuantum.1.010303

[57] M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gorshkov, Phys. Rev. Lett. 114, 157201 (2015).
https://doi.org/10.1103/PhysRevLett.114.157201

[58] C.-F. Chen and A. Lucas, Phys. Rev. Lett. 123, 250605 (2019), arXiv:1907.07637 [quant-ph].
https://doi.org/10.1103/PhysRevLett.123.250605
arXiv:1907.07637

[59] T. Kuwahara and K. Saito, Phys. Rev. X 10, 031010 (2020), 1910.14477.
https://doi.org/10.1103/PhysRevX.10.031010
arXiv:1910.14477

[60] M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas, Phys. Rev. X 10, 031009 (2020).
https://doi.org/10.1103/PhysRevX.10.031009

[61] C.-F. A. Chen, A. Lucas, and C. Yin, Reports on Progress in Physics 86, 116001 (2023a).
https://doi.org/10.1088/1361-6633/acfaae

[62] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s theorem, quantum theory and conceptions of the universe (Springer, 1989) pp. 69–72.

[63] L. Piroli, G. Styliaris, and J. I. Cirac, Phys. Rev. Lett. 127, 220503 (2021).
https://doi.org/10.1103/PhysRevLett.127.220503

[64] M. A. Nielsen, Phys. Lett. A 308, 96 (2003).
https://doi.org/10.1016/s0375-9601(02)01803-0

[65] D. W. Leung, Two-qubit projective measurements are universal for quantum computation (2002), arXiv:quant-ph/0111122 [quant-ph].
https://doi.org/10.48550/arXiv.quant-ph/0111122
arXiv:quant-ph/0111122

[66] R. Raussendorf, D.-S. Wang, A. Prakash, T.-C. Wei, and D. T. Stephen, Phys. Rev. A 96, 012302 (2017).
https://doi.org/10.1103/PhysRevA.96.012302

[67] S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary (1998).
https://doi.org/10.48550/ARXIV.QUANT-PH/9811052

[68] P. Herringer and R. Raussendorf, Quantum 7, 1041 (2023).
https://doi.org/10.22331/q-2023-06-12-1041

[69] I. G. Berkovich and E. Zhmud, Characters of finite groups, Vol. 1 (American Mathematical Soc., 1998).

[70] Y. Huang and X. Chen, Phys. Rev. B 91, 195143 (2015).
https://doi.org/10.1103/PhysRevB.91.195143

[71] B. Zeng and X.-G. Wen, Phys. Rev. B 91, 125121 (2015).
https://doi.org/10.1103/PhysRevB.91.125121

[72] D. T. Stephen, W. W. Ho, T.-C. Wei, R. Raussendorf, and R. Verresen, Phys. Rev. Lett. 132, 250601 (2024).
https://doi.org/10.1103/PhysRevLett.132.250601

[73] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987).
https://doi.org/10.1103/PhysRevLett.59.799

[74] R. Verresen, R. Moessner, and F. Pollmann, Phys. Rev. B 96 (2017).
https://doi.org/10.1103/PhysRevB.96.165124

[75] D. T. Stephen, Non-onsite symmetries and quantum teleportation in split-index matrix product states (2024), arXiv:2404.15883 [quant-ph].
https://doi.org/10.48550/arXiv.2404.15883
arXiv:2404.15883

[76] E. H. Chen, G.-Y. Zhu, R. Verresen, A. Seif, E. Bäumer, D. Layden, N. Tantivasadakarn, G. Zhu, S. Sheldon, A. Vishwanath, S. Trebst, and A. Kandala, Realizing the nishimori transition across the error threshold for constant-depth quantum circuits (2023b), arXiv:2309.02863 [quant-ph].
https://doi.org/10.48550/arXiv.2309.02863
arXiv:2309.02863

[77] S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, Phys. Rev. A 96, 022306 (2017).
https://doi.org/10.1103/PhysRevA.96.022306

[78] R. Jozsa, J. Mod. Opt. 41, 2315 (1994).
https://doi.org/10.1080/09500349414552171

Cited by

[1] Rahul Sahay and Ruben Verresen, "Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements", PRX Quantum 6 1, 010329 (2025).

[2] Sanket Chirame, Fiona J. Burnell, Sarang Gopalakrishnan, and Abhinav Prem, "Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise", Physical Review Letters 134 1, 010403 (2025).

[3] David T. Stephen, "Non-onsite symmetries and quantum teleportation in split-index matrix product states", Quantum 9, 1738 (2025).

[4] Gal Shkolnik, Sarang Gopalakrishnan, David A. Huse, Snir Gazit, and J. H. Pixley, "Infinitely fast critical dynamics: Teleportation through temporal rare regions in monitored quantum circuits", Physical Review B 112 22, 224312 (2025).

[5] Finn Eckstein, Bo Han, Simon Trebst, and Guo-Yi Zhu, "Robust Teleportation of a Surface Code and Cascade of Topological Quantum Phase Transitions", PRX Quantum 5 4, 040313 (2024).

[6] Yifan Zhang, Sarang Gopalakrishnan, and Georgios Styliaris, "Characterizing Matrix-Product States and Projected Entangled-Pair States Preparable via Measurement and Feedback", PRX Quantum 5 4, 040304 (2024).

[7] Diego Barberena and Aaron J. Friedman, "Overview of projective quantum measurements", arXiv:2404.05679, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 03:54:49) and SAO/NASA ADS (last updated successfully 2026-08-08 03:54:50). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.