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.
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, Aphys=Z(TubeC(MA)), 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.
Ocneanu's tube algebra provides a finite algorithm to compute the Drinfeld center of a fusion category. In this work we reveal the universal property underlying the tube algebra. Take a base category V which is strongly concrete, bicomplete, and closed symmetric monoidal. For physical applications one takes V=Vect the category of vector spaces. Given a V-enriched rigid monoidal category C (not necessarily finite or semisimple) we define the tube category XC using coends valued in V. Our main theorem established the relation between (the category of representations of) the tube category XC and the Drinfeld center Z(C): Z(C)↪Fun(XCop,V)≅Z(C↪Fun(Cop,V))↪Z(Fun(Cop,V)). Physically, besides viewing the tube category as a version of TFT with domain being the tube, we emphasize the "Wick-rotated" perspective, that the morphisms in XC are the local tensors of fixed-point matrix product operators which preserves the symmetry C in one spatial dimension. We provide a first-principle flavored construction, from microscopic quantum degrees of freedom and operators preserving the symmetry, to the macroscopic universal properties of the symmetry which form the Drinfeld center. Our work is thus a proof to the 1+1D topological holography in a very general setting.
We propose a framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra B of a quasi-local algebra A, there is a canonical fusion category C that acts on A by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that B is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category C with no fiber functor there always exist gapless pure symmetric states on an anyon chain.
This paper develops the state and measurement layer of Variable-Signature QuantumTheory and Obstruction Theory. Building on the projected positive-sector geometry ofVSQT-OT I, it defines θ-density states, algebraic states, effects, finite POVMs, sectorcompletely positive maps, and passive instruments intrinsically on the physical sectorEθ = Pθ K, Hphys,θ = (Eθ , ⟨·, ·⟩θ ).The paper fixes the sector trace, θ-adjoint, sector positivity, amplified sector positivity, andsector complete-positivity conventions, carefully distinguishing abstract sector endomorphismsfrom ambient representatives. Its main results prove compact convexity of the θ-density statespace, Born positivity and normalization for sector POVMs, selected-operator-system states,algebraic-state restriction and positive-extension representation on finite-dimensional ∗θ-subalgebras, finite-dimensional sector Kraus representation, Schrödinger–Heisenberg dualityfor sector CP maps, passive instrument normalization, instrument-to-POVM compatibility,admissibility of stratumwise measurement families, pointwise equivalence with ordinary finite-dimensional measurement theory, a fixed-Hilbert obstruction criterion, and fixed-Hilbertrecovery in Born-pair, POVM, and instrument forms. The paper does not introduce a newfinite-dimensional Born rule or a new Kraus theorem; pointwise sector measurement theoryis ordinary finite-dimensional measurement theory written in the sector Hilbert structure.Its contribution is the sector-typed formulation of the standard measurement calculus overvariable positive sectors, together with a first obstruction class detecting when the sectorbundle cannot be globally reduced to one admissible fixed-Hilbert carrier, gauge-compatibleadmissibility rules for stratumwise families, a formal admissible comparison class, and arecovery criterion identifying when that layer is ordinary after compatible admissible transport.The selected-observable layer is an operator-system layer by default, with selected ∗θ -algebrastreated as a stronger optional case. All operator-algebraic statements are finite-dimensionaland algebraic; no infinite-dimensional, unbounded, spectral, semigroup, or von Neumannalgebraic operator theory is claimed. The paper is finite-dimensional, stratumwise, andnon-dynamical. Active inter-sector instruments are isolated as a separate typed skeletonrather than claimed as a consequence of passive measurement calculus
A comprehensive symmetry resolution of the entanglement entropy in ( 1 + 1 )-d rational conformal field theories with categorical noninvertible symmetries is presented. This amounts to symmetry resolving the entanglement with respect to the generalized twisted and anyonic charged sectors of the theory. The anyonic sectors label the irreducible representations of a modular fusion category defining the symmetry and can be understood through the ( 2 + 1 )-d symmetry topological field theory that encodes the symmetry features of the conformal field theory. Using this, we define the corresponding generalized boundary-dependent charged moments necessary for the symmetry resolution of the entanglement entropy and find that, contrary to the case of invertible symmetries, the entanglement equipartition between different charged sectors at the next-to-leading order in the ultraviolet cutoff, is broken. Published by the American Physical Society 2024
This paper proves a finite-dimensional structural theorem identifying canonical projective quantum measurement as a boundary-channel projection. For a finite projection-valued measurement context P = {Π₁, …, Πₘ}, the central sectorwise phase action Uθ = ∑ⱼ e^{iθⱼ}Πⱼ has normalized Haar average E_P(X) = ∫ Uθ X Uθ* dθ = ∑ⱼ Πⱼ X Πⱼ. Its trace-dual state-side map is the Lüders nonselective channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The theorem characterizes this Lüders, or minimal-disturbance, channel: a projection-valued measurement fixes the outcome weights, but in degenerate sectors it does not by itself determine every compatible instrument. The additional structure used here is the sectorwise phase symmetry together with preservation of the full fixed block algebra. The fixed observable algebra is B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique weights representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except for the trivial one-sector context, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; it therefore defines a boundary-channel operation rather than full reversible Heisenberg–Schrödinger collapse. The results are interpretation-neutral and isolate the operator-theoretic structure that any finite projective measurement account using the canonical Lüders update must preserve. License note: Distributed under CC BY-NC-ND 4.0.
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024