The information content of quantum field theory states depends on which operator algebra is available to an observer. Casini, Huerta, Magán, and Pontello developed entropic order parameters based on relative entropy to quantify which algebras can access superselection-sector information and established entropic complementarity relations for complementary algebra choices. This work computes that framework exactly in the θ-vacuum manifold of the massless Schwinger model, providing a fully solvable quantum field theory realization of the entropic structure associated with sector information. The neutral-algebra restriction of a θ-state is shown exactly to be the uniform mixture of flux sectors and completely independent of θ. Consequently, the relative entropy between different θ-sectors on the neutral algebra vanishes: the neutral algebra carries no information about the vacuum angle. The entropic order parameter is computed in closed form. For equal sector weights, the relative entropy order parameter is exactly ln(N), while for general sector weights it becomes the Shannon entropy H(p) of the sector distribution. The conjugate pinching in the θ-basis provides the complementary disorder parameter, and the pair realizes the entropic complementarity relation, including saturation of the Maassen–Uffink bound at the extreme limits. Combining these results with the companion vacuum decomposition results gives an exact information-theoretic realization of the principle in this model: the θ-sector information is globally present and perfectly distinguishable, absent from the neutral algebra, absent from null-surface data within the stated universality framework, and carried by the charged/boundary sector while requiring no energy cost. This work provides a fully solvable benchmark for entropic superselection frameworks in quantum field theory. The declared limits are explicit: the calculation is performed for a finite sector structure in the 1+1 dimensional Schwinger model; continuum algebraic results are cited rather than re-derived; no universal theorem and no gravitational claim are made. All symbolic derivations and numerical checks are accompanied by reproducibility materials, verification scripts, audit procedures, and recorded outputs.
The information content of quantum field theory states depends on which operator algebra is available to an observer. Casini, Huerta, Magán, and Pontello developed entropic order parameters based on relative entropy to quantify which algebras can access superselection-sector information and established entropic complementarity relations for complementary algebra choices. This work computes that framework exactly in the θ-vacuum manifold of the massless Schwinger model, providing a fully solvable quantum field theory realization of the entropic structure associated with sector information. The neutral-algebra restriction of a θ-state is shown exactly to be the uniform mixture of flux sectors and completely independent of θ. Consequently, the relative entropy between different θ-sectors on the neutral algebra vanishes: the neutral algebra carries no information about the vacuum angle. The entropic order parameter is computed in closed form. For equal sector weights, the relative entropy order parameter is exactly ln(N), while for general sector weights it becomes the Shannon entropy H(p) of the sector distribution. The conjugate pinching in the θ-basis provides the complementary disorder parameter, and the pair realizes the entropic complementarity relation, including saturation of the Maassen–Uffink bound at the extreme limits. Combining these results with the companion vacuum decomposition results gives an exact information-theoretic realization of the principle in this model: the θ-sector information is globally present and perfectly distinguishable, absent from the neutral algebra, absent from null-surface data within the stated universality framework, and carried by the charged/boundary sector while requiring no energy cost. This work provides a fully solvable benchmark for entropic superselection frameworks in quantum field theory. The declared limits are explicit: the calculation is performed for a finite sector structure in the 1+1 dimensional Schwinger model; continuum algebraic results are cited rather than re-derived; no universal theorem and no gravitational claim are made. All symbolic derivations and numerical checks are accompanied by reproducibility materials, verification scripts, audit procedures, and recorded outputs.
We approach the physical implications of the noncommutative nature of complementary observable algebras (COAs) from an information theoretic perspective. In particular, we derive a general entropic certainty principle stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, where the COAs are formed by the charge-anticharge local operators (intertwiners) and the unitary representations of the symmetry group (twists), and in theories with local symmetries, where the COAs are formed by Wilson and t'Hooft loops. In general, the entropic certainty principle naturally captures the physics of order and disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.
In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.
We prove an entropic c-theorem in two dimensions by constructing a c-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular p-nuclearity), and a finite ultraviolet bound on the resulting c-functions. We also note that, formally, the c-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.
Abstract We prove a recent conjecture by Harlow and Ooguri concerning a universal formula for the charged density of states in QFT at high energies for global symmetries associated with finite groups. An equivalent statement, based on the entropic order parameter associated with charged operators in the thermofield double state, was proven in a previous article by Casini, Huerta, Pontello, and the present author. Here we describe how the statement about the entropic order parameter arises, and how it gets transformed into the universal density of states. The use of the certainty principle, relating the entropic order and disorder parameters, is crucial for the proof. We remark that although the immediate application of this result concerns charged states, the origin and physics of such density can be understood by looking at the vacuum sector only. We also describe how these arguments lie at the origin of the so-called entropy equipartition in these type of systems, and how they generalize to QFT’s on non-compact manifolds.
Quantum information is usually characterized by a state available at a given time and by entropic or channel-theoretic quantities derived from it. We develop a broader operational framework in which entropic content is one coordinate of physical information ratherthan its exhaustive causal description. The primitive structure is an accessibility restriction CX together with an admissible intervention class F. Relative to it we distinguish entropic content S; total and quantum generative capacities GI and GE ; globally produced and regionally accessible causal ecacy gall and gX ; causal shielding fX ; and the local, distributed, recoverable and blocked records of a specied physical history. We retain and unify the mathematically stable results of the preceding programme: S does not determine G in a Rindler family; accessible causal ecacy has an asymmetry-resource structure and can be locked in correlations; horizons can suppress quantum generative structure while preserving entropic or classical content; and local QFT yields complete exterior shielding for genuinely localized, gauge-neutral interventions under the stated assumptions. Rooted causal blocks are introduced as history-labelled points in bres of the restriction map over a local carrier state. The Causal-Fibre Shielding and Recovery Theorem links local equivalence, complete shielding, distributed historical record and future recovery: equal local carrier states imply RX =gX =0, while global distinction implies positive external historical record, quantitatively lower-bounded by binary distinguishability, and any later common map that restores local distinction certies a recoverable fraction of that record. This makes precise the original photon hypothesis: local photon identity need not imply historical-causal identity. We further prove a Bekenstein causal non-exhaustiveness result: for a bounded carrier with xed local Hamiltonian and radius, equal local states have equal entropy, energy expectation and Bekenstein bound, yet may belong to distinct rooted blocks with dierent distributed record and causal futures. This does not violate the Bekenstein entropy bound; it rejects only its interpretation as a complete taxonomy of causal information. The horizon programme is restored through the generative causal residue and causal quality index q, treated as history diagnostics rather than established geometric hair. A structured Causal-Block Completion Hypothesis then introduces a possible universal sector C. In a minimal local scalar realization, Kallen-Lehmann positivity, response-noise consistency, coherent history writing and fth-force constraints turn completion into a positive-measure feasibility problem with a dual no-go certicate. Finally, we dene a microscopic historical causal tension as the extra eld energy required to make a rooted history compatible with present constraints. Its time integral supplies a candidate action scale. A Historical Tension Principle and Historical Backreaction are retained as stronger hypotheses: in a consistent two-boundary completion, the globally realized physical history could depend jointly on past and future boundary data. No alteration of the past is derived by the forward-time theorems. The resulting programme is a falsiable hierarchy from established information identities to a constrained candidate new eld and a separate two-boundary conjecture.
We compute the entanglement entropy and the entanglement spectrum of the vacuum state in the massive Schwinger model at a finite θ angle. By computing two-point correlation functions we show that the correlation length diverges in the critical region. We clarify the physical origin of entanglement entropy enhancement at θ=π by relating it to the competition between distinct electric-flux vacuum branches. We show that the peak near θ=π persists across the range of masses studied and corresponds to the point of maximal competition between distinct vacuum branches with opposite electric-field orientation, where quantum fluctuations due to fermion pair creation are maximized. While this entropy enhancement is generic, a pronounced compression of the low-lying entanglement spectrum at θ=π distinguishes the near-critical case (m/g≃0.33). Using the Bisognano--Wichmann (BW) theorem, we construct a lattice BW entanglement Hamiltonian and compare it with the exact modular Hamiltonian obtained from the reduced density matrix. We observe agreement in the infrared sector, indicating that the entanglement Hamiltonian is well approximated by a spatially weighted Hamiltonian. Finally, we discuss possible applications to entanglement in topological insulators and quantum wires. In this work, we used the chirally rotated formulation of the Schwinger model Hamiltonian. We identify an additive vacuum-energy artifact induced by the discretization of the chirally rotated lattice and derive its exact free-lattice form to subtract it in the continuum limit.