Symmetry defects and gauging for quantum states with matrix product unitary symmetries

Adrián Franco-Rubio; Arkadiusz Bochniak; J. Ignacio Cirac

doi:10.22331/q-2026-10-22-2243

Symmetry defects and gauging for quantum states with matrix product unitary symmetries

Adrián Franco-Rubio^1,2,3, Arkadiusz Bochniak^1,2, and J. Ignacio Cirac^1,2

¹Max-Planck-Institut für Quantenoptik,Hans-Kopfermann-Str. 1, Garching, 85748, Germany
²Munich Center for Quantum Science and Technology, Schellingstraße 4, München, 80799, Germany
³University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria

Published:	2026-10-22, volume 10, page 2243
Editor:	Luca Tagliacozzo
Eprint:	arXiv:2502.20257v2
Doi:	https://doi.org/10.22331/q-2026-10-22-2243
Citation:	Quantum 10, 2243 (2026).

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Abstract

In this work, we examine the consequences of the existence of a finite group of matrix product unitary (MPU) symmetries for matrix product states (MPS). We generalize the well-understood picture of onsite unitary symmetries, which give rise to virtual symmetry defects given by insertions of operators in the bonds of the MPS. In the MPU case, we can define analogous defect tensors, this time sitting on lattice sites, that can be created, moved, and fused by local unitary operators. We leverage this formalism to study the gauging of MPU symmetries. We introduce a condition, block independence, under which we can gauge the symmetries by promoting the symmetry defects to gauge degrees of freedom, yielding an MPS of the same bond dimension that supports a local version of the symmetry given by commuting gauge constraints. Whenever block independence does not hold (which happens, in particular, whenever the symmetry representation is anomalous), a modification of our method which we call state-level gauging still gives rise to a locally symmetric MPS by promotion of the symmetry defects, at the expense of producing gauge constraints that do not commute on different sites.

Featured image: The action of a truncated symmetry operator on a finite interval creates symmetry defects at its endpoints.

Popular summary

Symmetries help physicists understand the laws of nature and distinguish different forms of matter. In this work, we explore how particular symmetries shape one-dimensional quantum systems, known as quantum spin chains, and how we can use them to discover new connections between these systems.

We focus on symmetries that can be built from many small pieces, much like assembling a structure from building blocks. This allows us to study what happens when a symmetry is applied only to part of a system, creating a symmetry defect—an object that behaves somewhat like a particle. We show how these defects can be created in pairs, moved along the chain, and combined with one another. We also explain how to turn them into new quantum degrees of freedom, effectively expanding the original system.

This process, called gauging, transforms a symmetry that acts on the system as a whole (a global symmetry) into one that acts locally. By doing so, it provides a way to construct new quantum systems and better understand the relationships between different phases of quantum matter.

► BibTeX data

@article{FrancoRubio2026symmetrydefects,
  doi = {10.22331/q-2026-10-22-2243},
  url = {https://doi.org/10.22331/q-2026-10-22-2243},
  title = {Symmetry defects and gauging for quantum states with matrix product unitary symmetries},
  author = {Franco-Rubio, Adri{\'{a}}n and Bochniak, Arkadiusz and Cirac, J. Ignacio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2243},
  month = oct,
  year = {2026}
}

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Cited by

[1] Sahand Seifnashri, Shu-Heng Shao, and Xinping Yang, "Gauging non-invertible symmetries on the lattice", SciPost Physics 19 2, 063 (2025).

[2] Bram Vancraeynest-De Cuiper, José Garre-Rubio, Frank Verstraete, Kevin Vervoort, Dominic J. Williamson, and Laurens Lootens, "From gauging to duality in one-dimensional quantum lattice models", SciPost Physics 20 4, 113 (2026).

[3] Georgios Styliaris, Rahul Trivedi, and J. Ignacio Cirac, "Quantum Circuits for Matrix-Product Unitaries", Physical Review Letters 135 26, 260602 (2025).

[4] Kansei Inamura, "Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions", arXiv:2602.12053, (2026).

[5] Giorgio Stucchi, J. Ignacio Cirac, Rahul Trivedi, and Georgios Styliaris, "Structure and Classification of Matrix Product Quantum Channels", Physical Review Letters 137 12, 120401 (2026).

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