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Nuclear Physics B

Symmetry packaging II: Packaged subspaces, packaged entanglement, and classification

Rongchao Ma

Published Aug 25, 2025• 0 citations
Publisher

Abstract

Symmetry packaging is the phenomenon whereby, upon particle creation, all the internal quantum numbers (IQNs) become locked into a single irreducible representation (irrep) block of the gauge group, as required by locality and gauge invariance. The resulting packaged quantum states exhibit characteristic symmetry constraints and entanglement patterns. We develop a group-theoretic framework to describe the symmetry packaging for a variety of concrete symmetries and to classify the corresponding packaged states:(1) We prove that for any finite or compact group G, there exist G-associated packaged subspaces, in which every vector is automatically a packaged state. In particular, in multi-particle systems, any nontrivial representation of G induces inseparable packaged entanglement that locks together all IQNs.(2) We apply this framework to symmetry packaging in finite groups (cyclic group ZN, charge conjugation C, fermion parity, parity P, time reversal T, and dihedral groups), compact groups (U(1), SU(N), SU(2), and SU(3)), p-form symmetries, and hybrid symmetries. In each case, gauge invariance and superselection rules forbid the factorization of the resulting states. We illustrate how Bell-type packaged entangled states, color confinement, and hybrid gauge-invariant configurations all arise naturally. These results yield a complete classification of packaged quantum states.(3) Finally, we extend the packaging principle to incorporate full spacetime symmetry and hybrid systems of local, global, and Lorentz/Poincaré charges.Our approach unifies tools from group theory, gauge theory, and topological classification. These results may be useful for potential applications in high energy physics, quantum field theory, and quantum technologies.

DOI
10.1016/j.nuclphysb.2025.117093

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Symmetry packaging is the phenomenon whereby, upon particle creation, all the internal quantum numbers (IQNs) become locked into a single irreducible representation (irrep) block of the gauge group, as required by locality and gauge invariance. The resulting packaged quantum states exhibit characteristic symmetry constraints and entanglement patterns. We develop a group-theoretic framework to describe the symmetry packaging for a variety of concrete symmetries and to classify the corresponding packaged states:(1) We prove that for any finite or compact group G, there exist G-associated packaged subspaces, in which every vector is automatically a packaged state. In particular, in multi-particle systems, any nontrivial representation of G induces inseparable packaged entanglement that locks together all IQNs.(2) We apply this framework to symmetry packaging in finite groups (cyclic group ZN, charge conjugation C, fermion parity, parity P, time reversal T, and dihedral groups), compact groups (U(1), SU(N), SU(2), and SU(3)), p-form symmetries, and hybrid symmetries. In each case, gauge invariance and superselection rules forbid the factorization of the resulting states. We illustrate how Bell-type packaged entangled states, color confinement, and hybrid gauge-invariant configurations all arise naturally. These results yield a complete classification of packaged quantum states.(3) Finally, we extend the packaging principle to incorporate full spacetime symmetry and hybrid systems of local, global, and Lorentz/Poincaré charges.Our approach unifies tools from group theory, gauge theory, and topological classification. These results may be useful for potential applications in high energy physics, quantum field theory, and quantum technologies.
Symmetry packaging is the phenomenon whereby, upon particle creation, all the internal quantum numbers (IQNs) become locked into a single irreducible representation (irrep) block of the gauge group, as required by locality and gauge invariance. The resulting packaged quantum states exhibit characteristic symmetry constraints and entanglement patterns. We develop a group-theoretic framework to describe the symmetry packaging for a variety of concrete symmetries and to classify the corresponding packaged states: \textbf{(1)} We prove that for any finite or compact group GGG, there exist GGG-associated packaged subspaces, in which every vector is automatically a packaged state. In particular, in multi-particle systems, any nontrivial representation of GGG induces inseparable packaged entanglement that locks together all IQNs. \textbf{(2)} We apply this framework to symmetry packaging in finite groups (cyclic group ZN\mathbb{Z}_NZN​, charge conjugation CCC, fermion parity, parity PPP, time reversal TTT, and dihedral groups), compact groups (U(1)\mathrm{U}(1)U(1), SU(N)\mathrm{SU}(N)SU(N), SU(2)\mathrm{SU}(2)SU(2), and SU(3)\mathrm{SU}(3)SU(3)), ppp-form symmetries, and hybrid symmetries. In each case, gauge invariance and superselection rules forbid the factorization of the resulting states. We illustrate how Bell-type packaged entangled states, color confinement, and hybrid gauge-invariant configurations all arise naturally. These results yield a complete classification of packaged quantum states. \textbf{(3)} Finally, we extend the packaging principle to incorporate full spacetime symmetry and hybrid systems of local, global, and Lorentz/Poincaré charges. Our approach unifies tools from group theory, gauge theory, and topological classification. These results may be useful for potential applications in high energy physics, quantum field theory, and quantum technologies.
We demonstrate that quantum field excitations can generate packaged entangled states, in which all internal quantum numbers (IQNs) (e.g., electric charge, flavor, and color) are inseparably entangled and constrained to irreducible representation (irrep) blocks. This is a consequence of local gauge invariance and superselection rules. The confinement restricts the net gauge charge to a single superselection sector, thereby excluding cross-sector superpositions but allowing entanglement within one sector. We establish theorems that: \textbf{(1)} Explain how these packaged entangled states naturally arise from quantum field excitations, \textbf{(2)} Show how they remain gauge invariant or transform covariantly within a fixed net-charge sector, and \textbf{(3)} Illustrate how external degrees of freedom (DOFs) (e.g., spin or momentum) can combine with packaged internal charges to yield gauge-invariant entanglement. Finally, we show that spin or momentum measurements on these hybrid states induce a collapse of the internal entanglement.
We develop a mathematical framework for the quantum simulation of lattice gauge theories using gauge-invariant packaged quantum states \cite{Ma2017,Ma2025}. In this formalism, every single excitation transforms as a complete \textbf{irreducible representation (irrep)} of the local gauge group, preventing any appearance of fractional or partial \textbf{internal quantum numbers (IQNs)}. Multi-particle excitations can form nontrivial packaged entangled states that are also gauge invariant, thereby forbidding partial or fractional IQNs. In other words, all IQNs of such packaged entangled states remain inseparably entangled. This ``packaging principle'' ensures that physical states remain confined to the correct gauge sector and excludes partial charges or colors, even when multiple excitations are entangled. We illustrate this approach for U(1)\mathrm{U}(1)U(1), SU(2)\mathrm{SU}(2)SU(2), and SU(3)\mathrm{SU}(3)SU(3) lattice gauge theories, discussing explicit constructions, Trotterized Hamiltonian evolution, and gauge-invariant measurements on a quantum simulator. We also outline how packaged states can mitigate gauge-violating errors and serve as natural building blocks for gauge-invariant coding schemes, while noting that standard quantum error correction is still required against typical local noise that respects gauge symmetry.
Packaged quantum states are gauge-invariant states in which all internal quantum numbers (IQNs) form an inseparable block. This feature gives rise to novel packaged entanglements that encompass all IQNs, which is important both for fundamental physics and for quantum technology. Here we develop a framework for gauge-invariant quantum information processing based on packaged quantum states. We propose the necessary and sufficient conditions for a valid packaged superposition state of a single particle and multi-particle. We then present the details of constructing gauge-invariant packaged qubits (or qudits), packaged gates, and packaged circuits (which commute with the total charge operator). These serve as alternative foundation for gauge-invariant quantum information science. We then adapt conventional quantum error-correction codes, quantum algorithms, and quantum communication protocols to the (d×Dd \times Dd×D)-dimensional hybrid-packaged subspace. This high-dimensional hybrid-packaged subspace is flexible for pruning and scaling to match available physics systems. Thus, packaged quantum information processing becomes feasible and testable. Our results show that the gauge-invariant packaged quantum states may provide a possible route toward robust, fault-tolerant, and secure quantum technologies.
The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g., with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely, (i) [Singh, Pfeifer, and Vidal, Phys. Rev. A 82, 050301 (2010)] and (ii) [Singh, Pfeifer, and Vidal, Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group G\mathcal{G}G that is compact, completely reducible, and multiplicity free, acting as a global internal symmetry. Then, in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper, we describe the implementation of non-Abelian group symmetries in great detail. For concreteness, we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multiscale entanglement renormalization ansatz and apply it to study the low-energy spectrum of a quantum spin chain with a global SU(2) symmetry.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of GGG, the so-called GGG-injective PEPS, and on global on-site symmetries coming from a finite group QQQ: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.