Non-onsite symmetries and quantum teleportation in split-index matrix product states

David T. Stephen

doi:10.22331/q-2025-05-12-1738

Non-onsite symmetries and quantum teleportation in split-index matrix product states

David T. Stephen

Department of Physics and Center for Theory of Quantum Matter, University of Colorado Boulder, Boulder, CO, USA
Department of Physics and Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA

Published:	2025-05-12, volume 9, page 1738
Editor:	Felix Huber
Eprint:	arXiv:2404.15883v4
Doi:	https://doi.org/10.22331/q-2025-05-12-1738
Citation:	Quantum 9, 1738 (2025).

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Abstract

We describe a class of spin chains with new physical and computational properties. On the physical side, the spin chains give examples of symmetry-protected topological phases that are defined by non-onsite symmetries, i.e., symmetries that are not a tensor product of single-site operators. These phases can be detected by string-order parameters, but notably do not exhibit entanglement spectrum degeneracy. On the computational side, the spin chains represent a new class of states that can be used to deterministically teleport information across long distances, with the novel property that the necessary classical side processing is a non-linear function of the measurement outcomes. We also give examples of states that can serve as universal resources for measurement-based quantum computation, providing the first examples of such resources without entanglement spectrum degeneracy. The key tool in our analysis is a new kind of tensor network representation which we call split-index matrix product states (SIMPS). We develop the basic formalism of SIMPS, compare them to matrix product states, show how they are better equipped to describe certain kinds of non-onsite symmetries including anomalous symmetries, and discuss how they are also well-suited to describing quantum teleportation and constrained spin chains.

Featured image: Every MPS can instead be represented as a SIMPS, revealing new physical and computational properties of the state.

Popular summary

The paper introduces a new kind of tensor network representation called split-index matrix product states (SIMPS). The SIMPS represent a generalization of the familiar matrix product states (MPS) which have been firmly established as the ultimate tool for studying one-dimensional quantum systems such as quantum spin chains. Compared to MPS, SIMPS provide an alternative representation that can more clearly reveal certain physical and computational properties of the system.

On the physical side, SIMPS are distinguished by the ability to more naturally encode "non-onsite" symmetries, i.e., symmetries that are not a tensor-product of single-spin operators. Such generalized symmetries have recently seen immense interest in condensed matter and high energy physics. The SIMPS representation allows the construction and analysis of new topological phases of matter protected by such symmetries.

On the computational side, SIMPS lead to the identification of new resources for long-range quantum teleportation and measurement-based quantum computation. It is known that certain quantum spin chains can be used to shuttle quantum information across long distances. In some cases, we can also modify this quantum information while it is being shuttled, leading to a simple form of quantum computation. The SIMPS formalism uncovers new mechanisms that enable these protocols and also challenges existing beliefs such as the necessity of entanglement spectrum degeneracy.

Finally, SIMPS are adept at encoding local constraints, such as the constraintfound in in atomic Rydberg arrays that forbids neighboring excitations. While such constraints require non-linear relations on the entries of an MPS tensor, they can be enforced via linear relations on the entries of a SIMPS tensor, simplifying the construction of variational families of constrained states.

Throughout the paper, the power of the SIMPS representation is demonstrated by constructing new families of spin chains with novel physical and computational properties and by analyzing examples that previously appeared in the literature, uncovering new structures therein.

► BibTeX data

@article{Stephen2025nononsitesymmetries,
  doi = {10.22331/q-2025-05-12-1738},
  url = {https://doi.org/10.22331/q-2025-05-12-1738},
  title = {Non-onsite symmetries and quantum teleportation in split-index matrix product states},
  author = {Stephen, David T.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1738},
  month = may,
  year = {2025}
}

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[3] Da-Chuan Lu, Fu Xu, and Yi-Zhuang You, "Strange correlator and string order parameter for non-invertible symmetry protected topological phases in 1+1d", arXiv:2505.00673, (2025).

[4] Yifan Hong, David T. Stephen, and Aaron J. Friedman, "Quantum teleportation implies symmetry-protected topological order", Quantum 8, 1499 (2024).

[5] Adrián Franco-Rubio, Arkadiusz Bochniak, and J. Ignacio Cirac, "Symmetry defects and gauging for quantum states with matrix product unitary symmetries", arXiv:2502.20257, (2025).

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