Operational Tube-Sector Theory of Quantum State Distinguishability Under Generalized Symmetries

Abstract: A variational principle for quantum-state distinguishability is established in many-body systems with generalized symmetries, including noninvertible cases described by fusion categories. Standard fidelity and symmetry-resolved diagnostics emerge as coarse-grained limits of a more refined operational structure. When symmetry actions terminate at entanglement cuts, distinguishability is governed by boundary tube algebras within a symmetry-constrained measurement resource theory. The physically admissible instruments are characterized by complete positivity, entanglement-cut locality, boundary-module covariance, and sequential stability. The resulting optimal measurement structure is uniquely fixed by the center of the boundary tube algebra, $\mathcal{A}{\mathrm{phys}} = Z!\left(\mathrm{Tube}{\mathcal{C}}(\mathcal{M}_A)\right)$, whose primitive idempotents define tube-sector probabilities that refine fidelity-based and symmetry-resolved descriptions. The associated tube positive-operator-valued measures (POVM) are extremal and yield optimal one-shot hypothesis-testing distinguishability under symmetry constraints. The construction is universal across fusion categories and independent of microscopic realization.

• The paper develops an operational criterion for quantum-state distinguishability by leveraging boundary tube algebras and fusion categories to optimize measurement protocols.
• It formulates a variational principle that uniquely determines optimal one-shot state discrimination within symmetry-restricted measurement cones based on tube algebra centers.
• Application to the doubled-Ising Kramers-Wannier model demonstrates enhanced conditional tube resolution compared to conventional fidelity or symmetry-resolved methods.

Operational Tube-Sector Theory of Quantum State Distinguishability

Context and Motivation

Quantum-state distinguishability is fundamental to quantum information theory and the characterization of many-body systems. Traditionally, fidelity and symmetry-resolved diagnostics compress distinguishability to scalar values or coarse sector labels, but these measures can miss finer distinctions, especially in systems exhibiting generalized or noninvertible symmetries. The paper "Operational Tube-Sector Theory of Quantum State Distinguishability Under Generalized Symmetries" (2606.19678) develops a variational principle for operational distinguishability that incorporates the measurement restrictions imposed by generalized symmetry actions, notably those described by fusion categories. This approach generalizes the symmetry-resolved entanglement framework by introducing boundary tube algebras and their centers, allowing for a categorical refinement of quantum state comparison at entanglement cuts.

Technical Framework

Symmetry Constraints and Measurement Admissibility

Generalized symmetries, including noninvertible cases characterized by fusion categories, introduce defect lines whose endpoints on boundaries or entanglement cuts carry boundary-module data beyond standard group-theoretical labels. Measurement processes in these settings are constrained—not only must instruments be completely positive and trace preserving, but they must also be localized at the entanglement cut, covariant under boundary-module actions, and stable under sequential composition and classical postprocessing.

The operational structure is formalized by the boundary tube algebra $\operatorname{Tube}_C(M_A)$, defined for a symmetry category $C$ and an entangling-cut boundary module $M_A$. Physically admissible measurement outcomes correspond to primitive central idempotents of the center $Z(\operatorname{Tube}_C(M_A))$. These define projectors $P_a$ onto tube sectors that refine the conventional symmetry-resolved description.

Variational Principle and Resource Theory

The paper imposes a resource-theoretic formulation: measurement instruments form a cone $\mathcal{M}_{\text{adm}}$ within which classical sector readouts must obey the aforementioned symmetry and locality constraints. The optimal one-shot hypothesis-testing distinguishability between states is shown to be uniquely fixed by the center of the boundary tube algebra, with every admissible measurement protocol reducing (up to free operations) to a coarse graining of the tube-sector readout. Specifically, for any family of tube-separating preparation protocols, the maximizer of the restricted discrimination functional $F_p(A)$ is $A_{\text{phys}} = Z(\operatorname{Tube}_C(M_A))$, and any further refinement is forbidden by physical locality and covariance constraints.

Hierarchy of Distinguishability

The framework induces a strict hierarchy:

• Tube-sector resolution: Allows access to conditional probability distributions within symmetry fibers.
• Symmetry-resolved diagnostics: Coarse grains tube outcomes, possibly missing conditional information.
• Scalar overlap (fidelity): Collapses all sector information, blind to any conditional fiber structure.

This hierarchy is formalized via stochastic pushforward maps $T: T \to Q$, mapping tube labels to coarse symmetry labels. No postprocessing at the coarse level can reconstruct the conditional tube information lost under this map, establishing irreducibility.

Realization in Doubled-Ising Product Kramers-Wannier Model

The paper demonstrates its main claims using the doubled-Ising product Kramers-Wannier (KW) defect model, the minimal case separating tube-sector and group-theoretical resolution. In this system, the KW defect $N$ refines the $C$0 group resolution only within the $C$1-trivial sector, splitting it into $C$2 based on endpoint signs. The endpoint morphism is fundamentally a boundary-module Hom-space element, not a naive source-target Hilbert space operator. Measurements that keep or forget the endpoint record change the conditional tube distribution without affecting the coarse symmetry-resolved or scalar data, as evidenced by strict operational difference in hypothesis-testing success rate.

Numerical checks confirm that the physically prescribed endpoint morphism behaves as a rectangular isometry, erasing off-tube coherence and leaving $C$3 empty for coherent endpoint protocols. Balanced, unread endpoint-sign channels exhibit nontrivial tube-sector discrimination even with fixed scalar and $C$4-resolved weights.

Theoretical Implications

Measurement Algebra Selection and Universality

The selection of $C$5 as the maximal commutative measurement algebra is categorical and independent of microscopic implementation. For any finite fusion category, this construction is universal: coarse-grained measurements are strictly quotients of the tube-sector readout. Boundary-module equivalences preserve tube weights, but changing the boundary module or symmetry category alters the measurement algebra, not merely the representation basis.

Irreducibility and No-Go Theorem

Any admissible readout not in the center of the tube algebra fails at least one physical constraint. The tube-sector readout is irreducible relative to any symmetry-compatible coarse graining; conditional probability distributions in tube fibers may be operationally accessed for state discrimination but cannot be reconstructed from coarse data.

Entropy Chain Rule and Operational Refinement

The entropy refinement follows the chain rule:

$C$6

where the increase in entropy relative to the coarse resolution corresponds to conditional uncertainty within tube fibers. Rényi-family extensions confirm that the refinement is spectral, not Shannon-specific.

Concrete Implementation

Tube-sector POVM measurements are implementable via defect matrix-product operators (MPOs) terminating on boundary modules, with endpoint degrees of freedom resolved by idempotent networks. Replica or swap estimators measure the sector weights $C$7. Unread endpoint-sign protocols correspond to classical forgetting of the tube outcome, while full readout utilizes the maximal algebra.

Practical and Future Directions

This operational tube-sector theory provides a sharp criterion for when quantum-state discrimination can be enhanced by categorical symmetry resolution. The approach extends symmetry-resolved quantum information beyond group-theoretical settings, relevant for systems with noninvertible symmetries (e.g., Fibonacci anyonic chains, Tambara-Yamagami and three-state Potts duality defects). The prescription is sharp and falsifiable: one identifies the boundary module, constructs the tube idempotents, and tests operational discrimination via conditional tube distributions.

Future developments include:

• Systematic benchmarking in broader fusion categories (e.g., beyond Ising and KW defects).
• Application to quantum circuits and topological quantum computation, where measurement constraints play an operational role.
• Exploration of tube-sector refinement in quantum reference frame problems and resource theories of measurement incompatibility.

Conclusion

The paper establishes a closed operational principle for quantum-state distinguishability under generalized symmetries, anchored in the measurement algebra of the center of the boundary tube algebra. The tube-sector resolution is strictly irreducible and operationally optimal within admissible symmetry-compatible measurement protocols. This furnishes a categorical hierarchy of distinguishability, extending the reach of symmetry-resolved information theory and providing a universal and testable framework for quantum state discrimination in noninvertible symmetry settings (2606.19678).