States, Measurements, and Information in Type-III QFT via a Categorical Density Matrix
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States, Measurements, and Information in Type-III QFT via a Categorical Density Matrix Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We introduce a categorical density matrix (CDM) for local quantum field theory that replaces trace - class density operators—absent in type - III von Neumann algebras—by functorial state data carried either as Haagerup L1 ( M )densities or standard - form vectors in the natural cone. In the symmetric monoidal category of W∗ -algebras and normal CP maps, the CDM makes restriction, measurement (selective/non - selective), coarse - graining, and center/edge prescriptions explicit as arrows, and reduces exactly to ordinary density matrices in type - I/II settings. We develop an operational calculus on local algebras: sequential measurement statistics, optimal local discrimination, accessible information (Holevo-type bounds without S(ρ)), local quantum Fisher information, recovery maps, and conditional modular - energy/one - shot work bounds—all computable directly in type - III factors. We illustrate with (i) Rindler - wedge physics in free fields (KMS/Unruh, modular Hamiltonians) and (ii) gauge - theory subregions with electric/magnetic centers, where edge contributions and sector-resolved information are produced transparently by the CDM. I. INTRODUCTION Quantum states without density matrices. In textbook quantum mechanics every normal state is represented by a density operator ρ≥ 0with Tr ρ = 1 acting on a Hilbert space H ; observables are bounded operators and expectations are hAi = Tr ( ρA ). This picture silently presupposes a type I setting, where the algebra of observables is B ( H )(or a direct sum/integral of matrix algebras), so that a faithful (semi)finite normal trace exists and minimal projections (rank–1 projectors) abound. Already at this level, Penrose emphasized a conceptual shortcoming: a density operator does not uniquely specify an “underlying ensemble” and therefore is not, in general, the quantum state in an intrinsic sense [ 1 , 2 ]. In relativistic quantum field theory (QFT) the challenge is sharper: the von Neumann algebras A ( O ) associated to bounded regions O are (typically) type III factors. Type III algebras admit no faithful normal semifinite tracial state and have no minimal projections; the familiar density matrix formalism and partial trace simply do not exist in this arena [5, 57, 87]. Nevertheless, physics demands an operational calculus of probabilities, measurements, conditioning, information flow, and thermodynamic response inside local algebras. Two broad responses and their limits. One response is to reduce type III behavior to type I/II surrogates by introducing cutoffs, discretizations, or split inclusions. On lattices, or via the split property interposing a type I factor N between nested regions, one recovers traces and thus ordinary density matrices as regularized objects. More radical proposals (e.g. deterministic or cellular–automaton underpinnings) also aim, in spirit, to replace continuum type III features with effectively type I/II kinematics at fundamental scales [3]. Such moves can be insightful, including for numerics and conceptual models, but they bypass the algebraic obstruction rather than solve it within the native type III framework. A second response is to embrace the type III world and work with its canonical structure: Tomita– Takesaki modular theory [87]. Modular theory attaches to any faithful normal state ϕon Ma modular operator ∆ ϕ , modular conjugation Jϕ , and an intrinsic one–parameter automorphism group σϕ t ; faithful cyclic and separating vectors (e.g. the vacuum restricted to a region, by Reeh–Schlieder) ensure these objects exist in local QFT. This structure underlies KMS thermality (e.g. Bisognano–Wichmann for wedges), relative entropy (Araki), and many deep results [ 58 , 76 ]. However, modular theory is not by itself a calculus of measurements,selective updates,channels, and operational information—the very tasks handled effortlessly by the density matrix in type I. Our solution: a categorical density matrix. We propose and develop a categorical density matrix (CDM) that upgrades “ ρ ” to a functorial state object that exists for any von Neumann algebra (type I/II/III) and reduces to the ordinary density matrix when a trace is present. Concretely, we work in the symmetric monoidal category W∗ AlgnCP :objects M(von Neumann algebras), arrows Φ : M→N(normal, unital, CP maps), with the spatial tensor product as monoidal product and preadjoints Φ ∗ : N∗→M∗ acting on normal functionals (Schrödinger picture). A state is an arrow ω : C→M . For each local algebra M = A ( O )we
2 represent ωin either of two equivalent, trace–free carriers: (CDM–L1)hω,O∈L1(M)+,khω,Ok1= 1,hhω,O, Ai=ω(A),(1) (CDM–cone) ξω,O∈ P\ M(natural cone in the standard form), ω(A) = hξω,O, A ξω,Oi.(2) Here L1 ( M )is Haagerup’s noncommutative L1 space, canonically isometric to the predual M∗ for any M (including type III) [ 86 ]; P\ M is the self–dual positive cone in the standard form of M , which yields a unique representing vector for each normal state [ 87 ]. Inclusions i : M ,→N are arrows; functoriality/naturality reads hω,M =i∗hω,N and hξω,M , A ξω,M i=hξω,N , i(A)ξω,N i, A ∈M. Thus the CDM is a section of state data over the net O 7→ A(O)that composes with arrows. Reduction, centers, and arrows. In gauge theories, subregion algebras typically have nontrivial centers generated by gauge–invariant boundary data (e.g. functions of normal electric flux or boundary holonomy). To speak about “the” algebra of region A one must choose which central subalgebra C to include; this yields an inclusion ιC : AC ( A ) ,→ M into an ambient algebra M . The act of “reducing to A with center C” is precisely a Heisenberg arrow rC:M −→ AC(A),with rC,∗(ω) = ω◦ιC.(3) Different legitimate center prescriptions ( Celec , Cmag , hybrids) correspond to different rC , making the dependence explicit rather than implicit [ 69 , 70 ]. When a state–preserving conditional expectation Eϕ : M → AC ( A )exists (Takesaki’s criterion: modular invariance under σϕ t ), we may take rC = Eϕ , which is canonical relative to the reference state ϕ. Measurements and updates without traces. An operational theory requires instruments and updates. An instrument on M with outcome space ( X, B )is a σ –additive map X7→ IX from measurable sets to normal CP maps M→M such that X7→ IX ( frm [ o ] −− )is a POVM. The CDM supports outcome probabilities and both non–selective and selective updates in type III algebras: probability: pω(X) = hhω,IX(frm[o]−−)i=hξω,IX(frm[o]−−)ξωi,(4) non–selective: h7→ IX,∗(h),selective (on X): h7→ IX,∗(h) hh, IX(frm[o]−−)i,(5) with the cone formulation analogous. Equations (4) – (5) are the trace–free analogues of the familiar Kraus calculus: they exist and compose in any type, so they fill the operational gap left by modular theory alone. Information and thermality in the CDM. Relative entropy and thermality are expressed within the same framework. Given faithful normal ϕ, ψ on M , the relative modular operator ∆ ψ|ϕ and Araki relative entropy S(ψkϕ) = −hξψ,log ∆ψ|ϕξψi(6) are intrinsic, finite in many QFT cases, and monotone under all normal CP maps M→N (data processing) [ 76 ]. The Connes cocycle ( Dψ : Dϕ ) t = ∆ it ψ|ϕ ∆ −it ϕ intertwines modular flows and implements a noncommutative Radon–Nikodym derivative. The CDM makes these objects computational: one can propagate states by arrows and evaluate (6) directly in type III, thereby obtaining operational distinguishability, recovery bounds, and metrological limits. For dynamics/thermality, the modular flow σω t of a wedge region coincides with Lorentz boosts (Bisognano–Wichmann), so the “modular Hamiltonian” K=−log ∆ωreproduces Unruh–type responses; our CDM attaches this data to the state of the region without ever introducing a Gibbs density [58]. How this work is different and better. Our contribution is twofold. First, we formalize the CDM as a functorial state object on a net of type III algebras, simultaneously in the Haagerup L1 and standard–form representations, and we treat all operational primitives (restriction, measurement, coarse–graining, center choice) as arrows in W∗ AlgnCP . This turns the ambiguity of edges/centers into explicit, composable choices rC . Second, we compute physical, operational quantities that modular theory alone does not provide in a direct, local way: • Sequential, selective measurement statistics and post–measurement states in a type III local algebra, via (5).
3 • Optimal local discrimination (Helstrom program) and accessible information (Holevo–type bounds) with von Neumann entropies replaced by relative entropies that are well–defined: PA succ =1 2(1 + kp0ω0−p1ω1kM∗), χrel({pi, ωi}) = X i piS(ωik¯ω). • Local quantum Fisher information as the second variation of symmetric relative entropy (Bogoliubov– Kubo–Mori/Petz metric), yielding Cramér–Rao bounds for parameter estimation in a region [ 68 , 79 ]. • Recovery maps (Petz and rotated Petz) constructed from modular data and arrows, quantifying reversibility of local channels. • Conditional modular–energy/one–shot work bounds after selective updates, something not expressible in modular theory alone without a state–update calculus. • In gauge theories, edge/center–resolved entropies and tests: the center–sector (classical) contribution appears transparently as a Shannon term from the central direct integral of h(C) = rC,∗ ( h ); switching centers is a CP arrow θC→C0[69, 70]. Each of these is carried out inside type III algebras with no need to “reduce to type I/II” by hand.[ 100 ] Relation to prior viewpoints (Penrose and ’t Hooft). Penrose’s critique that density operators are not the quantum state—since distinct ensembles can share the same ρ —is resolved here by replacing ρ with intrinsic, functorial state data ( hω or ξω )attached to the algebra, eliminating ensemble talk altogether [ 1 , 2 ]. At the same time, efforts to render QFT effectively type I/II via discretization, cellular automata, or cutoffs (as advocated in some of ’t Hooft’s foundational work) are complementary in spirit: they engineer a traceful setting where the matrix calculus is valid [ 3 ]. Our approach shows that such engineering is not necessary for operational physics: the CDM plus modular theory already gives a complete measurement–and–information calculus within type III. When traces do exist (type I/II), our construction collapses exactly to the textbook density matrix and all familiar formulas. A brief map of the paper. Section 2 recalls von Neumann types, standard forms, Haagerup Lp , and Haag–Kastler nets. Section 3 defines the CDM precisely in both L1 and cone versions and proves functoriality. Section 4 develops instruments, selective updates, and channels on local algebras without traces. Section 5 establishes information–theoretic quantities (relative entropy, discrimination, accessible information, QFI, recovery) in the CDM. Section 6 provides worked computations in type III QFT: wedge physics (Unruh/Bisognano–Wichmann) and gauge subregions with edges. Section 7 compares with what modular theory alone can do and articulates the conceptual gain. Section 8 shows the reduction to type I/II, recovering the ordinary density matrix formalism. Appendices collect technical proofs (standard form; Haagerup Lp; instrument dilations) and calculation details. Notation. M, N denote von Neumann algebras acting on separable Hilbert spaces; M0 is the commutant. Z ( M )is the center. The standard form of M is ( M, HM, J, P\ M ). Haagerup L1 ( M )is identified with the predual M∗ ; pairings are written hh, Ai . Normal, unital, completely positive maps are the arrows of W∗ AlgnCP ; their preadjoints act on M∗ and L1 ( M ). For a region A with a chosen central subalgebra C , the reduction arrow rC : M → AC ( A )is defined by (3) . Relative entropy is always Araki’s (6) . We use units c=~=kB= 1. II. BACKGROUND AND PRELIMINARIES This section fixes notation and recalls the operator–algebraic structures that underlie our construction of the categorical density matrix (CDM). We aim to balance formal precision with an intuitive narrative that clarifies why each object is the correct replacement for density matrices in type III settings. A. Von Neumann algebras, projections, traces, and types Von Neumann algebras. A von Neumann algebra M⊆B ( H )is a unital ∗ –subalgebra that is closed in the weak operator topology; equivalently, M = M00 by the bicommutant theorem. Its commutant is M0={X∈B(H) : [X, A]=0∀A∈M}, and its center is Z(M) = M∩M0.
4 Projections and their comparison. Aprojection is p = p∗ = p2∈M . We compare projections in two complementary ways: •Order: p≤qiff pq =qp =p(equivalently, Ran(p)⊆Ran(q)). • Murray–von Neumann equivalence: p∼q if there exists a partial isometry v∈M with v∗v = p and vv∗=q. We write p-qif p∼r≤q. A nonzero projection p is minimal if 0 6 = q≤p⇒q = p . In a factor (i.e. Z ( M ) = Cfrm [ o ] −− ) minimal projections play the role of atomic sharp outcomes: there is no nontrivial yes/no test strictly below them. Finite and (properly) infinite projections; traces. A projection p is finite if it is not equivalent to any proper subprojection of itself ( p6∼ q < p ). Otherwise it is infinite. A faithful normal (semi)finite tracial weight on M is a map τ : M+→ [0 ,∞ ]linear on commuting elements, normal, faithful, tracial ( τ ( x∗x ) = τ ( xx∗ )), finite on a large cone (finite on 1in the finite case). When such a τ exists, it induces adimension function d(p) = τ(p)on projections. Type classification of factors (Murray–von Neumann, Connes). For factors Mone has: •Type I: minimal projections exist. Up to isomorphism, M≃B ( H )or a matrix algebra Mn ( C ). The canonical trace on B(H)is semifinite. •Type II: no minimal projections, but a faithful normal (semi)finite trace exists. II 1 : finite trace with τ(1) = 1 (continuous dimension in [0,1]). II∞: semifinite, with d(p)∈[0,∞). •Type III: no nonzero finite projections; hence no nonzero faithful normal semifinite trace. Connes refines this to IIIλ(0≤λ≤1) via modular spectrum–based invariants [8]. Intuition and physics. Type I is the Hilbert–space (matrix) world of textbook QM, where density operators ρ and partial traces are well–defined, and rank–1 projectors realize pure normal states. Type II retains a trace but loses atoms: every nonzero test can be split indefinitely while keeping track of “size” via τ —a good model for infinite, tracial many–body limits. Type III (typical of local QFT algebras) has no trace and no minimal projections; informally, every nonzero projection is “as big as two disjoint copies of itself.” This algebraic fact encodes the ultraviolet entanglement that obstructs region density matrices and canonical partial traces. Examples. B ( H )is type I ∞ ; Mn ( C )is type I n . A hyperfinite II 1 factor R (closure of an increasing sequence of matrix algebras with normalized trace) has no minimal projections but a faithful finite trace. Local algebras A(O)in relativistic QFT are typically (hyperfinite) type III1factors [5, 57]. B. States, GNS representations, standard form, and modular theory States and GNS. A (normal) state on M is a positive, norm–one linear functional ω : M→C that is σ –weakly continuous. By the GNS theorem there exists a triple ( πω,Hω, Ω ω )with πω : M→B ( Hω )a normal representation and a cyclic vector Ωωsuch that ω(A) = hΩω, πω(A) Ωωi, A ∈M. If ωis faithful, then Ωωis separating for πω(M). Cyclic and separating vectors (Reeh–Schlieder in QFT). A vector Ωis cyclic if MΩ = H , and separating if A Ω = 0 ⇒A = 0 for A∈M . In local QFT, for each bounded region O , the vacuum is cyclic and separating for M=A(O)(Reeh–Schlieder), ensuring the applicability of modular theory [57]. Tomita–Takesaki modular theory. Given a von Neumann algebra M on H and a faithful cyclic and separating vector Ω, define the (closable) Tomita operator S by S ( A Ω) = A∗ Ωfor A∈M . Its polar decomposition S = J ∆ 1/2 yields the modular conjugation J (antiunitary) and the modular operator ∆ (positive selfadjoint). The Tomita–Takesaki theorem asserts [87]: 1. J M J =M0. 2. σΩ t(A) := ∆itA∆−it defines a one–parameter group of ∗–automorphisms of M(the modular flow). 3. The vector state ωΩis KMS at inverse temperature 1for σΩ. In QFT, for wedge regions the modular flow implements Lorentz boosts (Bisognano–Wichmann), giving a geometric meaning to the “modular Hamiltonian” K=−log ∆ [58].
5 Relative modular theory and relative entropy (Araki). For faithful normal states ϕ, ψ on M with standard–form vectors ξϕ, ξψ (see below), the relative Tomita operator Sψ|ϕ is defined on Mξϕ by Sψ|ϕ ( Aξϕ ) = A∗ξψ . Its closure yields the relative modular operator ∆ ψ|ϕ := S∗ ψ|ϕSψ|ϕ . Araki’s relative entropy is S(ψkϕ) := −hξψ,log ∆ψ|ϕξψi ∈ [0,∞], which reduces to the Umegaki expression when M is type I. It is monotone under all normal CP maps (data processing) [ 76 ]. The Connes cocycle ( Dψ : Dϕ ) t := ∆ it ψ|ϕ ∆ −it ϕ∈M is a unitary 1–cocycle intertwining the modular flows: σψ t(A)=(Dψ :Dϕ)tσϕ t(A) (Dψ :Dϕ)∗ t. Standard form and the natural cone. Every von Neumann algebra M admits a canonical standard form (M, HM, J, P\ M)with J M J =M0and a closed, self–dual cone P\ M⊂ HMsuch that: For every normal state ω∈M∗ there exists a unique vector ξω∈ P\ M with ω ( A ) = hξω, A ξωi . This representation bypasses density matrices and puts all normal states into a single Hilbert space HM with a fixed positivity structure [ 87 ]. In type I, identifying HM with the Hilbert–Schmidt space realizes ξω=ρ1/2. C. Haagerup noncommutative Lp(M): a trace–free space of densities Crossed product and canonical trace. Let φ be a faithful normal semifinite weight on M and σφ its modular automorphism group. The crossed product N := MoσφR is a von Neumann algebra that always carries a faithful normal semifinite trace TrNand a dual action bσ:R y N. Definition of Lp ( M ).Haagerup defines Lp ( M )(1 ≤p≤ ∞ ) as the space of TrN –measurable operators x affiliated with N satisfying a covariance under bσ : bσt ( x ) = e−t/p x . This construction is independent of the choice of φup to canonical isometry [86]. Identifications and pairing. There is a canonical isometric order–isomorphism M∗∼ =L1 ( M )sending a normal state ωto a positive hω∈L1(M)with khωk1= 1 such that ω(A) = hhω, Ai:= TrN(hωA) (A∈M⊂N). Similarly, L2 ( M )identifies with the standard–form Hilbert space HM . Thus L1 ( M )is a trace–free but fully canonical replacement for trace–class operators: it exists for all types, including type III. CP maps on L1 .Any normal CP map Φ : M→N has a preadjoint Φ ∗ : N∗→M∗ that extends to a positive contraction Φ ∗ : L1 ( N ) →L1 ( M ), giving the Schrödinger picture of state evolution in our CDM. Intuition. Think of L1 ( M )as the “space of densities” relative to the core trace TrN rather than a trace on M itself. In type I, L1 ( B ( H )) = S1 ( H )and hω is the usual ρ ; in type II, L1 ( M ) = L1 ( M, τ ) with respect to the tracial state; in type III, L1(M)persists even though Mhas no trace. D. Nets of local algebras, split property, and centers Haag–Kastler nets. A local QFT in the algebraic sense assigns to each suitable spacetime region O a von Neumann algebra A(O)on a common Hilbert space, subject to: •Isotony: O1⊆ O2⇒ A(O1)⊆ A(O2). •Locality (Einstein causality): if O1⊆ O0 2, then [A(O1),A(O2)] = 0. •Covariance: a (projective) unitary representation implements spacetime symmetries on the net. • Vacuum and spectrum: a Poincaré invariant vector Ωwith spectrum condition; typically Ωis cyclic and separating for A(O). For many models one has Haag duality and that each A(O)is a (hyperfinite) type III1factor [5, 57]. Split property. If O1bO2 (proper inclusion with a positive collar), there exists a type I factor N such that A(O1)⊂N⊂ A(O2). This split inclusion provides an approximate tensor–product structure and a place where traces and density matrices live; it is crucial for constructing regularized entropies and for comparing with type I formulas [53].
6 Centers and edge data (gauge theories). In gauge theories, the gauge–invariant algebra localized in a subregion A generally has a nontrivial center generated by gauge–invariant boundary observables (e.g. functions of normal electric flux or boundary holonomy). Selecting which central subalgebra C to include produces a concrete region algebra AC ( A )and an inclusion arrow ιC : AC ( A ) ,→ M into an ambient algebra M (global or larger local). This is the origin of the reduction arrows rC used throughout our framework, and it is where “edge mode” prescriptions enter as explicit choices [69, 70]. Why density matrices fail and what replaces them. Because local A ( O )are type III, there is no faithful semifinite trace and no canonical partial trace: the familiar assignment ρ7→ Tr ¯ A ( ρ )has no analogue intrinsic to M. The correct replacements are: •States as functionals (or as elements of L1(M), or as vectors in the natural cone). • Processes as arrows (normal CP maps) M→N , covering restriction, coarse–graining, measurements (instruments), and center choices (rC). • Modular objects ( J, ∆ , σω )and relative quantities (Araki entropy, Connes cocycle) for thermality and information. Our CDM sits precisely at this junction: it is a functorial way to carry state data across the net and through arrows, valid for all types and reducing to ρwhen a trace is present. Standing assumptions for the rest of the paper. Unless otherwise stated, M, N denote von Neumann algebras that are factors (typically type III 1 in local QFT examples). States are normal; instruments and channels are normal, unital, completely positive (nCP) maps. For region reductions we either specify a center C and use rC as in (3) , or (when available) a Takesaki state–preserving conditional expectation. When split inclusions are used, we indicate the interposed type I factor N explicitly to make regulator dependences manifest. III. THE CATEGORICAL DENSITY MATRIX (CDM) In this section we build the categorical density matrix (CDM) as a functorial replacement for trace–class density operators that works for all von Neumann algebras, in particular type III local algebras of QFT. We present two equivalent carriers of state data (Haagerup L1 “densities” and standard–form vectors in the natural cone), formulate reduction to subregions (with or without centers) as arrows in the category of W∗–algebras and normal CP maps, and prove functoriality, uniqueness, and basic calculus rules. We conclude with physically motivated examples and a summary of operational gains. A. Ambient category and dual pictures of states The category W∗ AlgnCP .Objects are von Neumann algebras M, N, . . . . Arrows are normal, unital, completely positive (n.u.c.p.) maps Φ : M→N . Composition is the usual composition of maps; identities are the identity maps. The monoidal product is the spatial tensor product of von Neumann algebras, written ¯ ⊗ . Given Φ : M→N and Ψ : P→Q , the monoidal arrow is Φ ¯ ⊗ Ψ : M¯ ⊗P→N¯ ⊗Q , again normal and u.c.p. Schrödinger vs. Heisenberg pictures. Every von Neumann algebra M has a Banach predual M∗ (normal functionals). An arrow Φ : M→N has a preadjoint Φ ∗ : N∗→M∗ defined by (Φ ∗ω )( A ) = ω (Φ( A )). If Φis u.c.p. then Φ∗is completely positive, norm–contractive kΦ∗k= 1, and Φ∗preserves states. Two carriers of state data (CDM). For each state ω∈M∗ we fix either of the following canonical representatives: 1. CDM–L1.A unique hω∈L1(M)+with khωk1= 1 such that ω(A) = hhω, Ai:= Trcore(hωA), A ∈M. (7) Here L1 ( M )is Haagerup’s noncommutative L1 , canonically identified with M∗ (Sec. II C); Trcore is the canonical trace on the crossed product core. 2. CDM–cone. In the standard form (M, HM, J, P\ M), a unique vector ξω∈ P\ Msuch that ω(A) = hξω, A ξωi, A ∈M. (8)
7 Either choice will be called the categorical density matrix of ω on M ; we write CDMM ( ω )to denote hω or ξωdepending on context. Proposition III.1 (Existence, uniqueness, and functoriality of CDM) . Let M be a von Neumann algebra and ω∈M∗a normal state. Then: 1. (Existence/uniqueness) There exists a unique hω∈L1 ( M ) + with khωk1 = 1 satisfying (7) ; and a unique ξω∈ P\ Msatisfying (8). 2. (Functoriality) For any n.u.c.p. Φ : M→N, CDMM(Φ∗ω) = Φ∗CDMN(ω) in the L1picture, and in the cone picture hξΦ∗ω, A ξΦ∗ωi=hξω,Φ(A)ξωi(∀A∈M). Proof sketch. (1) The L1 statement is the canonical identification M∗∼ =L1 ( M )of Haagerup, with positivity and norm preservation; uniqueness follows from duality with M . The cone statement is the standard–form theorem: each normal state has a unique representing vector in the natural cone (Takesaki). (2) For L1 , use h Φ ∗h, Ai = hh, Φ( A ) i by definition of preadjoint; uniqueness enforces CDMM (Φ ∗ω ) = Φ ∗hω . For the cone, define ξΦ∗ω by Riesz representation via hξΦ∗ω, AξΦ∗ωi = hξω, Φ( A ) ξωi and invoke uniqueness in the cone. Physical reading. The CDM provides an intrinsic carrier of the state that requires no trace on M . Expectations and probabilities are computed by the pairings (7) – (8) . Processes (channels, restrictions, measurements) act as arrows Φ, pushing CDMs by Φ ∗ . This restores the everyday operational calculus of quantum mechanics in type III. B. Regionalization, centers, and reduction arrows Let Mbe an ambient algebra (global or larger local), and let Abe a spacetime region. Choosing the region algebra. In gauge theories one must specify a center choice C (electric, magnetic, or hybrid). This determines the region algebra AC ( A )generated by interior gauge–invariant observables and the chosen central boundary subalgebra C, together with the inclusion arrow ιC:AC(A),→ M (normal ∗–monomorphism). Reduction at the Schrödinger level (always available). Given ω∈ M∗ , the reduced state on AC ( A )is the restriction ω(C) A:= ω◦ιC. In CDM–L1terms: h(C) ω,A := (ιC)∗hω,M∈L1(AC(A))+,(9) and analogously in the cone picture. This needs no conditional expectation or trace; it exists for all types. Heisenberg reduction arrows (when they exist or by choice). Sometimes one wants an explicit n.u.c.p. map rC : M→AC ( A )(Heisenberg picture) such that rC,∗ ( ω ) = ω◦ιC for a distinguished class of ω ’s. We distinguish three cases: 1. Takesaki expectation (state–preserving, canonical relative to ϕ). If a faithful normal state ϕ on M has the property that AC ( A )is globally invariant under σϕ t , then there exists a unique normal conditional expectation Eϕ : M→AC ( A )with ϕ◦Eϕ = ϕ (Takesaki). Set rC := Eϕ . This is canonical once ϕis fixed (e.g. vacuum for wedges). 2. Split–based reduction (choice–dependent but type–I). If Ab˜ A and the split property holds, there is a type–I factor N with AC ( A ) ⊂N⊂ A ( ˜ A )and an isomorphism θ : N→B ( HA ) ¯ ⊗R . Choosing a normal state ψRon Rdefines r(N,ψR) C:= (id ⊗ψR)◦θ◦incl : M→AC(A),(10) a normal u.c.p. map. Different (N, ψR)are different, but the dependence is explicit.
8 3. Arrow–by–dual prescription (always at the level of states). Even if no CP right–adjoint exists for ιC , (9) gives a state–space reduction functorially (Schrödinger side). In many computations this is sufficient. Remark III.2 (Center change as a channel).If C⊆C0 are two center choices, there is a normal u.c.p. map θC→C0 : AC ( A ) → AC0 ( A )that adjoins the extra central observables. Then ω(C0) A = ω(C) A◦θC→C0 and h(C0) ω,A =θC→C0,∗h(C) ω,A. Thus center/edge prescriptions are explicit arrows. Physical usefulness. Reduction arrows let you compute in a subregion: expectations hAi = hh(C) ω,A, Ai , probabilities for effects 0 ≤E≤ 1, and, crucially, post–measurement states and sequential protocols (Sec. IV). All of this is available in type III without density matrices and with center choices made explicit. C. CDM calculus: expectations, effects, and arrows Expectations and effects. Given an effect E∈M (0 ≤E≤frm [ o ] −− ), the probability of “yes” in state ωis pω(E) = hhω, Ei=hξω, Eξωi. For a POVM X7→ EXwith values in M,pω(X) = hhω, EXi. Channels and contractivity. If Φ : M→Nis n.u.c.p., then for any two normal states ψ, ϕ on N, S(ψkϕ)≥S(Φ∗ψkΦ∗ϕ)(data processing), and k Φ ∗hkL1(M)≤ khkL1(N) . Thus pushing CDMs forward by arrows never increases relative entropy and never creates trace–norm (base–norm) distance; this is the operational backbone for discrimination, metrology, and recovery. Composition with reduction arrows. For a reduction rC : M → AC ( A )and a local channel Φ : AC(A)→ AC0(A0), the effective arrow on the ambient algebra is Φ◦rC. On CDMs: hω,AC0(A0)= Φ∗rC,∗hω,M. Hence regionalization and processing are just composition of arrows. D. Consistency with matrices and tracial settings Proposition III.3 (Collapse to ordinary density matrices).1. If M = B ( H )(type I), then L1 ( M ) = S1 ( H )(trace–class). For any state ω there is a unique density matrix ρ with Tr ρ = 1 such that hω = ρ and ω ( A ) = Tr ( ρA ). If Φ( A ) = PiK† iAKi , then Φ ∗ ( ρ ) = PiKiρK† i . If M = MA¯ ⊗MB with MB=B(HB), the reduction arrow r= id ⊗τByields hA= TrB(ρ). 2. If M is type II with faithful normal trace τ , then L1 ( M ) = L1 ( M, τ )and any ω∈M∗ has a density hω∈L1(M, τ)+with ω(A) = τ(hωA). Proof. (1) Standard identifications: Haagerup L1(B(H)) = S1(H); the preadjoint action matches Kraus action. Partial trace is id ⊗τB . (2) By definition of L1 ( M, τ )and Radon–Nikodym for normal functionals relative to τ. Physical upshot. Whenever the matrix formalism is legitimate, the CDM is the matrix formalism. Outside that regime (type III) nothing is lost: the same algebra of arrows and the same state carrier compute expectations, probabilities, and information–theoretic quantities without any appeal to trace–class operators on M. E. Relative modular theory through the CDM Let ϕ, ψ be faithful normal states on M . In the cone picture, their relative modular operator ∆ ψ|ϕ and Araki relative entropy S(ψkϕ) = −hξψ,log ∆ψ|ϕξψi are defined intrinsically. In the L1 picture, one may equivalently work in the core with hϕ, hψ and recover S ( ψkϕ )from relative cocycles and Connes’ spatial derivatives; in type I this reduces to S ( ρψkρϕ ) = Tr(ρψ(log ρψ−log ρϕ)).
9 Proposition III.4 (Data processing for CDMs) . For any n.u.c.p. Φ : M→N and faithful normal ϕ, ψ ∈N∗, S(ψkϕ)≥S(Φ∗ψkΦ∗ϕ), with equality iff there exists a (rotated) Petz recovery map RΦ ϕ : N→M satisfying RΦ ϕ,∗◦ Φ ∗ψ = ψ and similarly for ϕ. Proof sketch. Standard monotonicity of Araki relative entropy under normal CP maps; equality conditions via Petz–type recovery constructed from modular data (implementable as an arrow because we work in W∗–algebras). Physical usefulness. Relative entropy controls distinguishability, bounds error probabilities in hypothesis testing (Chernoff/Hoeffding exponents), and yields first–law type identities with modular Hamiltonians in QFT. Through the CDM, these quantities are computable locally in type III and propagate correctly under regionalization and channels. F. Instruments and selective updates in CDM form (We develop the full measurement calculus in Sec. IV; here we record the CDM translation.) Definition III.5 (Instrument) . An instrument on M with measurable outcome space ( X, B )is a map X7→ IX taking measurable sets to n.u.c.p. maps M→M , countably additive in the strong operator topology, with X7→ IX(frm[o]−−)a POVM. Given a CDM hω(or ξω) and an event X∈ B, pω(X) = hhω,IX(frm[o]−−)i, hω|X=IX,∗(hω) pω(X). These are the trace–free analogues of Kraus update rules and exist in any type. Sequential protocols are built by composing the corresponding arrows; regionalized measurements use rC to pull the instrument to the ambient algebra or to push the state to the region. Physical usefulness. Type III algebras admit no density matrices, yet detectors do click, and experiments do involve conditioning on outcomes and sequential control. CDMs make this operational calculus rigorous and local: probabilities, post–measurement states, and their propagation through channels are all defined and computable. G. Worked templates: wedge physics and gauge edges We illustrate how CDMs enter real computations; full details appear in Sec. VI. (i) Rindler wedge (type III 1 ). Let M = A ( W )be the right–Rindler wedge algebra and ω the vacuum. The CDM ξω,W determines the modular flow (boosts) and modular Hamiltonian KW = −log ∆ ω . For a coherent excitation ψgenerated by a Weyl operator localized in W, S(ψkω)W=hKWiψ−hKWiω, computed directly from ∆ ψ|ω using ξ ’s (no ρ ). A localized instrument I measuring a smeared field yields outcome distributions p ( x ) = hξω,W ,Ix ( frm [ o ] −− ) ξω,W i and selective updates ξ7→ ξx , after which one evaluates conditional modular energies hKWiξx. (ii) Gauge subregion with electric center. Let ACelec ( A )be the region algebra with electric center; define rCelec as in (9) or (10) . The reduced CDM h(Celec) A = rC,∗hglobal admits a central (direct–integral) decomposition into edge sectors with weights p ( q ). Sector–resolved quantities (Shannon edge term H ( {p ( q ) } ), conditional bulk relative entropies) are therefore computed transparently, and switching to magnetic center is the CP arrow θE→B.
16 B. Mutual information and conditional mutual information In general type III settings, von Neumann entropies S ( ρA )are not available; yet mutual information retains a precise meaning via relative entropy. Two commuting subalgebras. Let MA, MB⊂M be commuting von Neumann subalgebras ([ MA, MB ] = 0), and let ω be a normal state on M . Assume the split property: there exists a type–I factor N with MA⊂N⊂(MB)0.Then the Araki mutual information is Iω(A:B) := Sω|MA∨MBω|MA⊗ω|MB,(23) where the product state is formed inside N¯ ⊗N0 using the split inclusion (the value does not depend on the particular split N by monotonicity and directedness of splits) [ 53 , 88 ]. In type I this reduces to S(ρABkρA⊗ρB) = S(ρA) + S(ρB)−S(ρAB). Conditional mutual information (CMI). Given a triplet MA, MB, MC of commuting subalgebras with suitable split inclusions AbAB bABC, define Iω(A:C|B) := Iω(A:BC)−Iω(A:B)≥0,(24) which equals (by relative entropy chain rules) S ( ωABkωA⊗ωB ) −S ( ωABC kωA⊗ωBC ) ≥ 0. Nonnegativity is equivalent to strong subadditivity in the split setting. A small CMI implies approximate recoverability of ωABC from ωAB via a localized recovery channel (Sec. V F), extending [66] to the W∗setting. Physical message. Iω ( A : B )quantifies total correlations accessible by measurements in A and B ; Iω ( A : C|B )quantifies correlations between A and C not already present in B and bounds the error of locally Markovian reconstructions. C. Binary discrimination and Stein–Chernoff asymptotics Let ω0, ω1∈M∗ be normal states with priors p0, p1 . A one–shot decision test localized in M is an instrument with two outcomes. The optimal success probability is [36, 63, 88] PM succ =1 21 + kp0ω0−p1ω1kM∗,(25) where k·kM∗ is the base norm on the predual M∗ . The optimal instrument is a two–outcome instrument whose effects are built from the Jordan decomposition of the signed functional p0ω0−p1ω1. Proposition V.1 (Pinsker–type bound).For any faithful ϕand states ψ, ϕ on M, 1 2kψ−ϕk2 M∗≤S(ψkϕ). Consequently, PM succ ≤1 21 + √1−e−S(ω0kω1). Proof sketch. The inequality follows from Hiai–Petz for general von Neumann algebras [38]. The bound on Psucc is immediate from (25). Asymptotics (Stein’s lemma, Chernoff bounds). For i.i.d. samples under a fixed localized instrument (or tensor product copies in type I/II approximants), the optimal type II error exponent at fixed type I level is S ( ω0kω1 ); Chernoff exponents are expressed by the Petz–Rényi divergences. These statements can be formulated directly in the W∗setting using Araki’s relative entropy and its Rényi variants [88]. Physical message. With the CDM one computes best–possible local discrimination performance (and error exponents) inside a type–III region algebra, something density matrices cannot deliver there. D. Accessible information and Holevo–type bounds without S(ρ) An information source emits labels i with probabilities pi , preparing normal states ωi on M ; a receiver performs an instrument I on M and records outcomes y with distribution q ( y|i ) = ωi ( Iy ( frm [ o ] −− )). The classical mutual information I(i:y)is bounded by a relative Holevo quantity: I(i:y)≤χrel({pi, ωi}) := X i piSωik¯ω,¯ω:= X i piωi.(26)
17 Proposition V.2 (Holevo bound via DPI).For any instrument Ion M,(26) holds. Proof sketch. Consider the quantum–classical (qc) state on the direct sum algebra c M = LiM , b Ω = Lipiωi , and its coarse version b¯ Ω = Lipi¯ω . Apply the lifted instrument b I = LiI (Sec. IV E) and the data processing inequality (21) to S ( b Ωkb¯ Ω ); the left–hand side reduces to I ( i : y )while the right–hand side equals χrel [36, 64, 88]. Physical message. Even when S ( ρ )is undefined for a region (type III), the locally accessible classical information is still bounded by a computable quantity χrel in terms of Araki relative entropies, and the CDM furnishes all ingredients. E. Quantum Fisher information (QFI) and metrology in regions Let θ7→ ωθ be a smooth family of faithful normal states on M (e.g. a coupling constant, a source strength, or time under a local Hamiltonian). The Bogoliubov–Kubo–Mori (BKM/Petz) metric is defined by the second variation of (symmetrized) relative entropy: FBKM(ωθ) := d2 dθ2θ=0 S(ωθkω0) + S(ω0kωθ)≥0.(27) It equals the Kubo–Mori inner product on tangent directions X= ˙ω0: hX, XiBKM,ω0=Z1 0 ds ω0 σω0 −is(LX)LX, where LX is the unique selfadjoint element in the GNS commutant representing the tangent functional, and σω0 t is the modular flow of ω0 [ 78 , 79 ]. Among monotone quantum metrics, BKM is singled out by information–geometric and thermodynamic properties; in type I it coincides with the symmetric logarithmic derivative (SLD) QFI. Proposition V.3 (Cramér–Rao bound in a region) . For any unbiased estimator ˆ θ obtained from outcomes of a localized instrument on Mapplied to ωθ, Var(ˆ θ)≥1 FBKM(ωθ). Moreover, FBKM is monotone under any localized channel Φ : M→N:FBKM(Φ∗ωθ)≤ FBKM(ωθ). Proof sketch. Information–geometric derivation via monotonicity and convexity of Araki relative entropy [78, 79, 88]. Physical message. The best possible precision to estimate a local parameter in QFT is bounded by a metric computable from modular data and CDMs; coarse–graining (detector inefficiencies, losses) can only reduce this precision. F. Recovery maps, reversibility, and approximate Markov structure Given a channel (n.u.c.p. map) Φ : M→N and a faithful reference ϕ∈M∗ , the Petz recovery map RΦ ϕ : N→M is the unique (normal CP) map that perfectly reverses Φon the sufficiency subalgebra when equality holds in (21) [ 77 ]. In the W∗ setting, RΦ ϕ can be expressed using modular objects and the dual map Φ # in standard forms. A “rotated” family {RΦ ϕ,t}t∈R built with Connes cocycles yields quantitative stability: Theorem V.4 (Stability of data processing).For faithful ψ, ϕ ∈M∗and a channel Φ : M→N, S(ψkϕ)−S(Φ∗ψkΦ∗ϕ)≥ −log Fψ, RΦ ϕ◦Φ∗ψ, where F is the Uhlmann fidelity on M∗ , and RΦ ϕ is a suitable rotated Petz recovery (averaged over t ). Equality in (21) holds iff there exists a (possibly unrotated) Petz recovery achieving perfect reversal [ 66 – 68 ].
18 Conditional mutual information and Markov recovery. If Iω ( A : C|B )is small, then ωABC is close (in fidelity or relative entropy) to the output of a recovery channel acting on B (localized Petz map), i.e. A – B – C form an approximate quantum Markov chain [ 66 ]. This gives a rigorous operational criterion for near–Markovianity of local algebras in QFT. Physical message. The CDM framework upgrades data–processing from an inequality to actionable recovery maps on local algebras; small CMI means you can reconstruct missing regions from their neighbors, with quantitative error guarantees. G. Worked computations in QFT: wedges and gauge edges We now display two explicit computations that combine modular input and CDMs to yield physical predictions inside type–III algebras. (1) Rindler wedge: relative entropy and local discrimination. Let M = A ( W )be the right Rindler wedge algebra in a free scalar QFT, and let ω be the vacuum. By Bisognano–Wichmann, the modular flow is the boost and KW= 2πZx>0 x T00(t=0, x, y)dd−1y(+constant). For a coherent excitation ψ=ω◦Ad[W(f)] with W(f)∈M(Weyl unitary localized in W), one finds S(ψkω)W=hKWiψ−hKWiω= 2πZx>0 xhT00iψ−hT00iωdd−1y,(28) derived purely from modular theory and the quasi–free structure; the CDM–cone realization of ψ and ω yields the necessary modular objects without density matrices. For binary local discrimination between ψ and ω (Sec. V C), one computes the base norm kψ−ωkM∗ via characteristic functions of Weyl operators, obtaining a closed form for PW succ in terms of the one–particle covariance restricted to W. (2) Gauge subregion with an electric center: edge information. Let ACelec ( A )be the region algebra including the electric central subalgebra generated by gauge–invariant functions of the normal electric flux through ∂A . Using the reduction arrow rCelec (Sec. III B), the reduced CDM h(Celec) ω,A has a central direct–integral decomposition h(Celec) ω,A ∼ =Z⊕ hω,q dµω(q), with q labeling edge sectors (boundary flux values) and µω the induced classical measure. A localized instrument that reads the edge (e.g. coupling a probe to the boundary electric flux) is an abelian POVM on the center; the accessible information about q is upper bounded by the Shannon entropy H ( µω ), and the total locally accessible information about the preparation ensemble {pi, ωi} obeys the relative Holevo bound (26) with the decomposition splitting into a classical edge part and a bulk part (conditional on q ). Switching to a magnetic center corresponds to a CP arrow θE→B (Sec. III B), changing µω and hence the edge information content in a controlled, explicit way. Physical message. In wedges we obtain explicit relative entropies and optimal local tests for excitations; in gauge subregions we compute edge–resolved information and its dependence on center choice. Both are intrinsically type–III computations enabled by the CDM. H. Summary of Section V • Relative entropy S ( ·k· ), computed with CDMs, is the central quantitative tool on type–III algebras; it yields linear response (first law) via modular Hamiltonians and concrete energy relations in QFT geometries. • Mutual information and CMI admit split–based definitions via relative entropy; they obey data– processing and support local recovery maps with quantitative stability. • Optimal local discrimination follows the Helstrom formula with the predual norm; error exponents are controlled by relative entropy.
19 • Locally accessible classical information is bounded by a relative Holevo quantity χrel , requiring no von Neumann entropies. • Quantum Fisher information (BKM metric) provides metrological limits inside regions and degrades under localized channels. • Worked examples (wedges and gauge edges) illustrate fully operational predictions obtained without density matrices. VI. WORKED COMPUTATIONS IN TYPE–III QFT WITH THE CDM In this section we carry out detailed computations inside local, typically type–III, von Neumann algebras using the categorical density matrix (CDM). Our aim is twofold: (i) to show how the CDM makes physically relevant quantities calculable without ever invoking trace–class density operators, and (ii) to demonstrate that, whenever a traceful reduction exists (e.g. through a split inclusion), our formulas reduce exactly to the standard matrix expressions. We treat in detail two families of examples: (A) free scalar fields on a Rindler wedge (modular thermality and discrimination of coherent excitations), and (B) Abelian gauge theories on subregions with nontrivial centers (edge information and metrology). Along the way we emphasize the operational steps: instruments, probabilities, selective updates, and recovery. A. CCR framework, quasi–free vacuum, and wedge localization The Weyl CCR algebra. Let (K , σ )be a real symplectic space (one–particle test–function space with symplectic form σ(f, g)). The Weyl algebra W(K, σ)is generated by unitaries W(f),f∈K, satisfying W(f)W(g) = e−i 2σ(f,g)W(f+g), W(f)∗=W(−f).(29) Aquasi–free state ω is specified by a positive, symmetric bilinear form µ ( f, g )on Kobeying the CCR compatibility µ(f, f)µ(g, g)≥1 4σ(f, g)2,and ωW(f)= exp−1 2µ(f, f). For a free scalar field in the Minkowski vacuum, µ derives from the two–point function restricted to Cauchy data; we refer to [96, 97] for the precise construction. Local net and wedge algebra. Let W = {x∈R1+d : x1>|x0|} be the right Rindler wedge. The Haag–Kastler net O 7→ A ( O )is obtained by representing W in the GNS triple of ω and taking von Neumann closures of local subalgebras [ 57 , 96 ]. The wedge algebra M := A ( W )is (hyperfinite) type III 1 in standard models. Modular data for the wedge. By Bisognano–Wichmann [ 58 ], if Ωis the GNS vector of the vacuum state ω then Ωis cyclic and separating for M and the modular flow σω t coincides with the group of Lorentz boosts preserving W. Writing ∆ω=e−KW, the modular Hamiltonian is KW= 2πZx1>0 x1T00(t=0, x)ddx+ const.(30) The constant drops out of relative quantities and will be ignored. CDM for the wedge vacuum and excitations. In the CDM–cone picture, the wedge–restricted vacuum is the vector ξω,W ∈ P\ M representing ω|M , and any normal state ψ restricted to M has a unique representative ξψ,W . In the CDM– L1 picture, the Haagerup density hω,W ∈L1 ( M ) + satisfies hhω,W , Ai = ω(A)for A∈M. B. Relative entropy of coherent wedge excitations via CDM Coherent excitations. Let f∈ Khave support in W ; the Weyl unitary W ( f ) ∈M generates the coherent state ψ(A) := ωW(f)∗A W(f)(∀A∈M). Physically, this is a classical displacement of the free field.
20 Theorem VI.1 (Relative entropy of a unitary excitation) . Let M be a von Neumann algebra in standard form with cyclic separating vector Ω, ω ( A ) = h Ω , A Ω i . For any unitary U∈M and state ψ = ω◦Ad U , the Araki relative entropy satisfies S(ψkω) = hΩ, U∗KωUΩi−hΩ, KωΩi,(31) provided UΩlies in the form domain of K1/2 ω. Proof sketch. This is a standard consequence of Araki’s relative modular theory [ 76 ] and the unitary implementation of the automorphism; see also [ 59 ] for a QFT context. Using ∆ ψ|ω = U∗ ∆ ωU (relative modular operator intertwines under inner automorphisms) one has S(ψkω) = −hΩ, U∗log ∆ωUΩi+hΩ,log ∆ωΩi, which equals (31) since Kω=−log ∆ω. Corollary VI.2 (Coherent wedge excitation).For ψ=ω◦Ad W(f)localized in W, S(ψkω)W=hKWiψ−hKWiω= 2πZx1>0 x1hT00iψ−hT00iωddx, (32) where the last equality follows from (30). Physical interpretation. Equation (32) is a concrete, local prediction: for a coherent packet injected in the wedge, the relative entropy to the vacuum equals the modular energy injected, which here is the boost weight of the energy density. No density matrices are needed; the entire calculation occurs in M = A ( W ) using the CDM and modular theory. C. Localized measurement of a smeared field; Gaussian instrument Spectral measurement of φ ( g )(formal). Formally, the smeared field φ ( g )(with g supported in W ) is essentially selfadjoint and has a spectral measure Eφ(g)(dx). The (idealized) instrument is IB(A) := Zx∈B Eφ(g)(dx)A Eφ(g)(dx). Rigorous versions use bounded functional calculus on Weyl operators W ( λg )and limiting procedures [ 96 ]. For quasi–free states, φ ( g )is Gaussian with mean mψ ( g ) = ψ ( φ ( g )) and variance Varψ ( g ) = ψ(φ(g)2)−ψ(φ(g))2. Outcome law and posterior in CDM form. For a (quasi–free) state ψon Mwith CDM hψ(or ξψ), pψ(dx) = hhψ, Eφ(g)(dx)i=1 p2πVarψ(g)exp−(x−mψ(g))2 2 Varψ(g)dx, (33) hψ|x=I{x},∗(hψ) pψ(dx)(formal Dirac outcome; coarse–grained versions integrate over bins).(34) For coherent ψ and vacuum ω , mω ( g )=0while mψ ( g )is a linear functional of f ; Varψ ( g ) = Varω ( g ) since displacements do not change covariance. In practice one uses Gaussian pointer models (Ozawa dilation) to represent finite–resolution measurements [ 98 ]; the CDM formulas (11) – (13) then yield the same Gaussian statistics and posterior updates. Sequential measurements and conditional modular energy. Let I(1) measure φ ( g1 )and I(2) measure φ ( g2 ), both localized in W . Proposition IV.7 ensures order–independence if the smearings are spacelike separated; otherwise, the joint distribution is bivariate Gaussian with covariance inherited from the two–point function restricted to W . Conditioning on x1 produces a shifted posterior mean for the second measurement, and the CDM allows one to compute the conditional modular energy hKWiψ|x1 via the cone vector ξψ|x1. By the quantum–classical chain rule (Proposition IV.8), Zpψ(x1)hKWiψ|x1−hKWiωdx1≤S(ψkω)W, aone–shot bound on the average boost–weighted energy of postselected states.
21 D. Optimal local discrimination of coherent excitations Consider two coherent states ψj = ω◦Ad W ( fj )supported in W , with priors pj . The optimal success probability of a localized decision test in M=A(W)is PW succ =1 21 + kp0ψ0−p1ψ1kM∗. For quasi–free ω , the fidelity between ψ0, ψ1 is (CCR analogue of the Gaussian quantum fidelity) [ 64 , 97 ] F(ψ0, ψ1) = exp−1 4kf0−f1k2 C−1, where C is the covariance operator associated with µ restricted to W (the one–particle structure). By Uhlmann–Fuchs–van de Graaf inequalities, 1−pF(ψ0, ψ1)≤1 2kψ0−ψ1kM∗≤p1−F(ψ0, ψ1). Thus PW succ is sandwiched between explicit functions of the displacement norm kf0−f1kC−1 (computable from two–point functions). A tighter upper bound follows from the Pinsker inequality (Proposition V.1) with S(ψ0kψ1)computed via (32) when one state is a unitary excitation of the other. Operational upshot. Even though M is type III and no density matrices exist, the CDM enables fully quantum–optimal local discrimination bounds and (in Gaussian cases) essentially closed expressions for the best achievable performance by wedge–confined detectors. E. Edge information in Abelian gauge theory: electric center Region algebra with electric center. For a spatial region A in an Abelian gauge theory (e.g. Maxwell), the gauge–invariant local algebra Aphys ( A )generated by interior Wilson loops and electric fields strictly supported in A has a nontrivial center generated by gauge–invariant functions of the normal electric flux through ∂A . Let Celec be the abelian von Neumann algebra generated by these boundary observables; the region algebra is MA:= ACelec (A) = vNAphys(A)∪Celec. Fix the reduction arrow rCelec (Sec. III B) and consider the reduced CDM h(E) ω,A := rC,∗hω. Central decomposition and relative entropy split. The center Celec yields a direct–integral decomposition MA∼ =Z⊕ Q MA,q dµ(q), h(E) ω,A ∼ =Z⊕ hω,q dµω(q), where Qis the spectrum of the center (space of boundary flux sectors), µω is the edge sector distribution induced by ω , and MA,q are the fiber algebras (bulk conditionals). For two global states ω, ϕ , the Araki relative entropy on MAsplits as Sω|MAkϕ|MA=Hµωkµϕ+ZQ Sωqkϕqdµω(q),(35) with H the classical Kullback–Leibler divergence. This is a general fact for direct–integral von Neumann algebras (block–diagonal structure) [88]. Vacuum edge distribution and Shannon term. In free Maxwell theory, the vacuum–induced µω is Gaussian in the boundary flux q with covariance determined by the vacuum two–point function of the normal electric field smeared on ∂A (a positive quadratic form q7→ 1 2hq, G ∂A qi ) [ 69 , 70 ]. The Shannon edge term is then H(µω) = 1 2logdet2πe G∂A(finite after a choice of UV collar/measure), and contributes additively to S ( ω|MAkϕ|MA )for states with the same bulk conditionals (e.g. coherent interior excitations).
22 Holevo bound and sector readout. Consider an ensemble {pi, ωi} of global states; the locally accessible information in Aobeys (Sec. V D) I(i:y)≤X i piSωi|MAk¯ω|MA. Using (35) , this decomposes into an edge contribution H ( µ¯ω ) −PipiH ( µωi )(information in the sector label q ) plus a bulk conditional piece Rdµ¯ω ( q ) PipiS ( ωi,qk¯ωq ). Thus the CDM makes explicit how much information is encoded in the center (classical register) vs. the bulk, and how this depends on the chosen center via the arrow rCelec (switching to magnetic center replaces µω by another measure via a CP map θE→B). Edge metrology (local QFI). Couple a boundary source θ to the electric flux: Hedge ( θ ) = θ Q∂A , so that ωθ has the same bulk conditionals but a shifted edge distribution µωθ ( q ) = µω ( q−θ ). Then the local BKM quantum Fisher information (Sec. V E) for estimating θ equals the classical Fisher information of µω: FBKM(ωθ|MA) = ZQ (∂θlog µωθ(q))2µωθ(q)dq = VarµωQ∂A−1. For Gaussian µω this is the inverse variance of the edge flux, a boundary quadratic form computable from the vacuum two–point function. F. Split inclusion check: recovery of matrix formulas Interpose a type–I factor. Let Abe A and suppose the split property holds: there exists a type–I factor N such that A ( A ) ⊂N⊂ A ( e A )[ 53 ]. Pick an isomorphism θ : N→B ( HA ) ¯ ⊗1 and define the split reduction arrow r(N,ψR) as in (10) . For a global state ω , the reduced CDM h(N) A = r(N,ψR) ∗ ( hω )lies in L1(B(HA)) = S1(HA)and is the usual density matrix ρA. Consistency of computations. All CDM formulas now reduce to matrix formulas: •Relative entropy S(ψkω)on Nequals S(ρψkρω) = Tr(ρψ(log ρψ−log ρω)). •Instrument updates are ρ7→ PxKxρK† x/p(x)with p(x) = Tr(ρEx). •Holevo bound equals the standard χ=PipiTr(ρilog ρi)−Tr(¯ρlog ¯ρ). Therefore, the CDM is a genuine extension of the matrix formalism: it collapses to it when a traceful window is inserted, while remaining valid without the split. G. Implementation notes: numerical approximations For practical computations (e.g. plotting PW succ vs. displacement or evaluating edge Fisher information), one may approximate: 1. Discretize the one–particle space on a lattice adapted to W or ∂A , compute the discrete covariance Cand symplectic form σ. 2. Compute modular kernels (for wedges these follow from boosts) to evaluate hKWiψ using stress– tensor discretizations. 3. Construct Gaussian instruments (Ozawa dilation) with finite resolution ∆and evaluate pψ ( x )and posteriors; verify the one–shot bound from Proposition IV.8. 4. Edge sector covariance G ∂A from boundary correlators of normal electric fields; then H ( µω )and FBKM follow analytically (Gaussian) or numerically. All steps are regulator–transparent in the CDM language: the dependence on the split factor N or on the center choice Cappears as an explicit arrow label.
23 H. Takeaway of Section VI • For wedges, the CDM plus modular theory yields exact, local formulas for relative entropy, detector statistics, and optimal discrimination—all inside a type–III factor, no density matrices needed. • For gauge subregions, the CDM makes edge/center information manifest, splits relative entropy into a Shannon edge term plus bulk conditionals, and provides metrological and discrimination figures tied to the boundary. • Via the split property, every CDM computation reduces to familiar matrix formulas whenever a type–I window is interposed, confirming consistency. VII. BEYOND MODULAR THEORY: WHAT THE CDM ADDS, WITH DETAILED EXAMPLES Modular theory equips each faithful normal state on a von Neumann algebra with deep canonical structure (Tomita operator, modular flow, KMS, relative entropy). However, it is not an operational calculus: it does not by itself prescribe state updates upon measurement, it does not provide localized instruments, it does not separate classical registers induced by centers, and it does not supply state– independent reduction maps except under restrictive hypotheses. This section makes those limits precise and shows—with detailed, physically motivated examples—how the categorical density matrix (CDM) closes the gap. Each computation lives inside type–III local algebras and, when a traceful window is inserted (split property), collapses exactly to matrix formulas. A. What modular theory gives—and what it doesn’t What modular theory gives. Given a von Neumann algebra M and a faithful cyclic separating vector Ω(e.g. vacuum restricted to a local algebra), Tomita–Takesaki provides: •the modular conjugation Jand modular operator ∆with S=J∆1/2; •the modular flow σΩ t(A)=∆itA∆−it, under which ωΩ(A) = hΩ, AΩiis KMS; • relative modular operators ∆ ψ|ϕ and Araki relative entropy S ( ψkϕ )with data processing under all normal CP maps. These are state– and algebra–intrinsic. In QFT they produce geometric flows (e.g. boosts on wedges), passivity, and relative–entropy inequalities that underlie energy and information constraints [ 57 – 59 , 76 ]. What modular theory does not give by itself. 1. No selective update calculus. Modular objects quantify distinguishability and KMS structure but do not provide a rule of the form “after outcome x , the state is ω7→ ωx .” There is no intrinsic object in modular theory that implements outcome–conditioned state transitions; physically, this is the realm of instruments and CP maps. 2. No canonical reduction to subalgebras in type III. There is no partial trace on type–III algebras. A state–independent conditional expectation E : M→N onto a subalgebra N exists only if N is invariant under the modular flow of some faithful normal weight (Takesaki’s theorem); in general inclusions (e.g. sharp double cones, generic center choices) this fails, hence modular theory alone does not pick a reduction map [87]. 3. No separation of classical edge registers. In gauge theories, local algebras possess centers generated by boundary observables. Modular theory does not label the center choice nor split classical (edge) from quantum (bulk) contributions operationally. 4. No localized discrimination/communication instruments. Relative entropy yields bounds, but the instruments that achieve optimal discrimination or that implement channel coding live in the CP–map world (Davies–Lewis–Ozawa) [61, 98]. Modular theory alone doesn’t provide them.
24 How the CDM completes the picture. The CDM supplies (i) a state carrier ( hω∈L1 ( M ) + or ξω∈ P\ M ); (ii) arrows (normal CP maps) for restriction, coarse–graining, and measurement; (iii) explicit reduction arrows rC that encode center/edge choices; and (iv) a quantum–classical decomposition for instruments. Thus, everything modular theory knows remains available (via ∆, J and relative entropy), while the missing operational layer is now rigorous and computable in type III. B. No canonical partial trace: a precise obstruction Let M be a type–III factor and N⊂M a von Neumann subalgebra. One might seek a state–independent, normal conditional expectation E:M→Nto play the role of a “partial trace.” In general: Proposition VII.1 (Conditional expectations are exceptional) . Let M be a von Neumann algebra and N⊂M a von Neumann subalgebra. A faithful normal conditional expectation E : M→N exists if and only if there is a faithful normal semifinite weight φ on N such that N is globally invariant under σφ◦ι t , the modular automorphism group of the lifted weight on M (Takesaki’s theorem). In particular, for generic inclusions in type–III local QFT, no such Eexists. Thus, there is no canonical Heisenberg reduction to a subalgebra. The CDM, however, always affords the Schrödinger reduction ω7→ ω|N (pre-adjoint of the inclusion) and, when desired, reduction arrows rC (state–preserving Eϕ if available; split–based arrows otherwise). This makes reduction explicitly choice–dependent rather than ambiguous. C. Example I: selective measurement chains in a wedge Task. A detector in the right Rindler wedge W first measures φ ( g1 ), then—conditioned on a threshold event x1∈B⊂R —applies a localized channel Φ B (e.g. a feedback unitary), and finally measures φ ( g2 ). We want: (i) p(x1), (ii) p(x2|x1), and (iii) the conditional modular energy hKWiω|x1. Modular–only limitation. Modular theory provides KW and relative entropies, but gives no notion of “ωupdated by outcome x1”; hence (ii) and (iii) are undefined as state–conditioned quantities. CDM solution. Let I(1) and I(2) be the localized instruments of Sec. IV. Using the CDM: p(x1) = hhω,I(1) dx1(frm[o]−−)i, hω|x1=I(1) dx1,∗(hω) p(x1). Apply the feedback channel: hω|x17→ h0 ω|x1 = Φ B,∗ ( hω|x1 ). Then the second outcome law is p ( x2|x1 ) = hh0 ω|x1,I(2) dx2(frm[o]−−)i. Finally, the conditional modular energy is hKWiω|x1=hξω|x1, KWξω|x1i. By the qc chain rule (Prop. IV.8), Zp(x1)S(ωx1kϕ)dx1≤S(ωkϕ), which in turn bounds the average postselected modular energy cost via the first law (Sec. V). This yields operational limits for heralded state preparation by wedge–localized detectors—not accessible with modular theory alone. D. Example II: optimal local discrimination (Helstrom) in a region Task. Inside M=A(O), optimally discriminate between two normal states ω0, ω1with priors p0, p1. Modular–only limitation. Relative entropy S ( ω0kω1 )bounds exponents, but modular theory does not furnish the measurement (instrument) nor the one–shot optimal success probability.
25 CDM solution. The Helstrom optimum is [63, 64] PM succ =1 21 + kp0ω0−p1ω1kM∗, achieved by a two–outcome instrument with effects given by the positive/negative parts of the signed functional p0ω0−p1ω1 (a well–defined object in the predual). Existence and localization follow from general instrument dilations [ 61 , 98 ]. In Gaussian wedge examples (Sec. VI), Psucc is expressed in terms of one–particle covariances restricted to O ; CDM provides the update states and permits multi–step, adaptive tests, within M. E. Example III: accessible information without S(ρ) Task. A sender encodes label i into ωi supported in A ( O ); a receiver performs a localized instrument. What is the maximal I(i:y)? Modular–only limitation. The textbook Holevo χ = Pipi [ S ( ρi ) −S ( ¯ρ )] uses von Neumann entropies, which do not exist in type III. CDM solution. The relative Holevo quantity χrel =X i piS(ωik¯ω),¯ω=X i piωi, yields the bound I ( i : y ) ≤χrel for any localized instrument (Prop. V.2). In Gaussian families (small coherent shifts) this reduces to a quadratic functional of local means and covariance, giving a computable bits–per–photon bound for a detector confined to O. F. Example IV: recovery from coarse–graining and approximate Markov chains Task. A localized channel Φ : M→N discards microscopic details (resolution loss, detector inefficiency). Given ωand a reference ϕ, can we construct a recovery map and quantify reversibility? Modular–only limitation. Monotonicity (21) holds abstractly but does not produce a reversing map. CDM solution. The (rotated) Petz map RΦ ϕ : N→M is a normal CP arrow built from modular objects and Connes cocycles; it satisfies the stability inequality (Theorem V.4) [ 66 – 68 , 77 ]. In a tripartite split setting, small conditional mutual information I ( A : C|B )implies the existence of a localized recovery on B that approximately reconstructs the state on ABC . This is operational in CDM: one writes and evaluates the recovery channel on CDMs, entirely within type III local algebras. G. Example V: center/edge resolution in gauge theories Task. Quantify how much classical information a subregion A ’s edge stores about the preparation, and how bulk information depends on the center choice (electric vs. magnetic). Modular–only limitation. Modular theory does not separate the classical register induced by the center, nor does it label the choice of center. CDM solution. Choosing a center C defines a reduction arrow rC ; the reduced CDM decomposes as h(C) ω,A ∼ =R⊕hω,x dµω(x)over Spec(C). Then (Sec. VI E) Sω|MAkϕ|MA=H(µωkµϕ) + ZS(ωxkϕx)dµω(x), splits edge (classical) and bulk (quantum) contributions. The accessible information splits similarly; switching centers is a CP arrow θC→C0 , changing edge statistics explicitly [ 69 , 70 ]. Concrete predictions: for Maxwell theory the edge distribution is Gaussian in boundary flux, so edge Shannon terms and edge QFI are boundary quadratic forms computable from two–point functions. H. Example VI: resolution coarse–graining as a quasi–free channel Task. Degrade spatial resolution in a bosonic field theory and quantify information loss under coarse–graining.
32 Proposition IX.1 (Characteristic function and Gaussian law) . Let ω be quasi–free with covariance µ , and let ψ=ω◦Ad W(f). The characteristic function of Φ(g)in ψis χψ(t) := ψeitΦ(g)= expi t σ(f, g)−1 2t2µ(g, g),(41) and the outcome distribution pψhas the Gaussian density pψ(x) = 1 p2π µ(g, g)exp−(x−σ(f, g))2 2µ(g, g).(42) Hence the mean and variance are Eψ[Φ(g)] = σ(f, g),Varψ(Φ(g)) = µ(g, g), i.e. coherent displacements shift the mean but do not change the variance. Proof. Using (38) and eitΦ(g)=W(tg), χψ(t) = ωW(−f)W(tg)W(f). Apply the Weyl relations twice: W(−f)W(tg)W(f) = ei σ(f,tg)W(tg) = eitσ(f,g)W(tg), so by (39) we get (41). The density (42) is the Fourier inversion of a Gaussian characteristic function: pψ(x) = 1 2πZR dt e−itx χψ(t) = 1 2πZR dt exp−1 2µ(g, g)t2+it(σ(f, g)−x), and the standard Gaussian integral yields (42). Physical reading. A linear, homodyne–type detector measuring Φ( g )in a coherent packet observes aGaussian histogram whose mean equals the classical overlap σ ( f, g )and whose variance equals the vacuum noise µ ( g, g )fixed by the two–point function. No density matrices are required; CDMs furnish the expectation functional that pairs with the spectral projections of Φ(g). C. Finite–resolution Gaussian instrument and selective posteriors Projective measurements of unbounded fields are unrealistic. A standard Ozawa dilation yields a Gaussian instrument with finite resolution s > 0having density of effects Ex:= 1 √2πs2exp−(x−Φ(g))2 2s2,ZR Exdx =frm[o]−−,(43) and instrument maps IB ( A ) = RBMxAMxdx with Mx = (2 πs2 ) −1/4exp−(x−Φ(g))2 4s2 (all functions of Φ(g), hence localized and commuting with eitΦ(g)) [98]. Proposition IX.2 (Outcome law with resolution s).For any normal state ψ, pψ,s(x) := hCDM(ψ), Exi=1 p2π(µ(g, g) + s2)exp−(x−σ(f, g))2 2 (µ(g, g) + s2).(44) Proof. Because Exis a Borel function of Φ(g), its characteristic function factorizes: Zeitxpψ,s(x)dx =ψeitΦ(g)e−1 2s2t2. Using (41) , the product is Gaussian with variance µ ( g, g ) + s2 and mean σ ( f, g ); Fourier inversion yields (44).
33 Selective posterior on the abelian subalgebra. Because Mx is a function of Φ( g ), the instrument is nondemolition for Φ( g ); the posterior restricted to the abelian algebra C∗ (Φ( g )) is classical Bayes with Gaussian prior N(m, V )and Gaussian likelihood N(x|y, s2). Writing m:= σ(f, g), V := µ(g, g), the posterior distribution of Φ(g)given outcome xis Gaussian N(mpost, Vpost)with mpost(x) = V V+s2x+s2 V+s2m, Vpost =V s2 V+s2.(45) Equivalently, the posterior characteristic function of eitΦ(g)in the CDM is χψ|x(t) = expit mpost(x)−1 2t2Vpost. In the full noncommutative algebra, (13) provides the selective CDM update hψ|x = Idx,∗ ( hψ ) /pψ,s ( x ); the formulas above describe its restriction to C∗(Φ(g)). Physical reading. Finite detector resolution adds variance in quadrature ( V7→ V + s2 ) and shrinks the posterior variance by the usual Bayesian factor V s2/ ( V + s2 ). The CDM calculus supplies both the outcome law (nonselective) and the selective posterior (state update), even in a type–III local algebra. D. Toy gauge edge: discrete Gaussian and edge Shannon entropy Consider a simple abelian edge degree of freedom: the (dimensionless) normal electric flux q∈Z through a single boundary link of region A in a U (1) lattice gauge vacuum. The reduced state on the center is purely classical with discrete Gaussian distribution pσ(q) = 1 Z(σ)exp−q2 2σ2, Z(σ) := X n∈Z exp−n2 2σ2,(46) where σ2> 0encodes the (dimensionless) edge variance induced by vacuum correlations on ∂A . This is the lattice analogue of the Gaussian edge measure discussed in Sec. VI E. The edge Shannon entropy is Hedge(σ) := −X q∈Z pσ(q) log pσ(q) = log Z(σ) + 1 2σ2Eσ[q2],(47) with Eσthe expectation under pσ. Closed forms via Jacobi theta. Introduce α := 1 2σ2 and the Jacobi theta function ϑ3 (0 , q ) := Pn∈Zqn2 . Then Z(σ) = ϑ3 0, e−α,Eσ[q2] = −d dα log ϑ3 0, e−α,(48) so that Hedge(σ) = log ϑ3 0, e−α−αd dα log ϑ3 0, e−α, α =1 2σ2.(49) For numerics, use the modular identity ϑ3 0, e−α=rπ αϑ30, e−π2 α,(50) which is exponentially convergent for small α(large σ) [99, §20]. Continuum (large– σ ) limit and first correction. Using (50) with α 1, ϑ3 (0 , e−π2/α ) = 1+ O ( e−π2/α ), we obtain Hedge(σ) = 1 2log2πe σ2+Oe−π2 α=1 2log2πe σ2+Oe−2π2σ2.(51) Thus the discrete edge entropy approaches the continuum Gaussian formula with exponentially small corrections in σ.
34 Multiple independent edge links. For d independent boundary links with variances σ2 k , the edge distribution factors and the entropy adds: H(d) edge = d X k=1 Hedge(σk) = d X k=1 "log ϑ3 0, e−1/(2σ2 k)−1 2σ2 k d dα log ϑ3(0, e−α)α=1 2σ2 k#,(52) with continuum limit H(d) edge ∼1 2log(2πe)dQkσ2 k. Physical reading. Hedge quantifies the classical information available by reading the edge center (e.g. electric flux) alone. Equations (49) – (51) give closed and numerically stable formulas that (i) capture lattice discreteness exactly, and (ii) recover the continuum edge Shannon term of Sec. VI E as σ2grows. E. Remarks on implementation and CDM provenance • In the field example, expectations ψ ( · )are implemented by pairing with the CDM hψ (or ξψ ). The characteristic function χψ ( t ) = hhψ, W ( tg ) i evaluates using Weyl relations and quasi–free data; Fourier inversion gives the histogram observed by a detector. Selective updates are hψ|x = Idx,∗(hψ)/pψ,s(x); in C∗(Φ(g)) they reduce to (45). • In the edge example, the CDM restricted to the center is a probability measure; equations (47) – (49) compute its Shannon entropy. Switching to a different center (e.g. magnetic) composes with the CP arrow θE→B , changing the distribution and hence Hedge in a way that is explicit in the CDM framework. X. EXTENSIONS, DYNAMICS, AND OUTLOOK In this section we extend the categorical density matrix (CDM) beyond the static, flat–spacetime, Abelian–edge examples treated so far. Mathematically, we: (i) formulate the CDM in the locally covariant setting where spacetimes and regions form a category and quantum operations are functorial; (ii) develop dynamics via quantum Markov semigroups (QMS) on von Neumann algebras, including modular detailed balance and entropy production; (iii) introduce Rényi–type divergences (Petz quasi–entropies) applicable in type III, which control one–shot information tasks; and (iv) treat non–Abelian gauge edges, where the center becomes a block algebra labeled by irreducible representations. Each subsection closes with worked examples stressing physical predictions that modular theory alone does not supply, but which the CDM renders explicit. A. Local covariance and naturality of the CDM Let Loc be the category whose objects are oriented, time–oriented globally hyperbolic spacetimes ( M, g )and whose morphisms are isometric embeddings ψ : ( M, g ) ,→ ( M0, g0 )with causally convex images. A locally covariant QFT is a covariant functor A : Loc →W∗ Alg assigning to ( M, g )a von Neumann algebra A ( M )and to ψ a normal ∗ –monomorphism A ( ψ ) : A ( M ) → A ( M0 )(time–slice axiom, Einstein causality, etc.) [80]. Definition X.1 (CDM as a natural state assignment) . Fix a class of normal states S ( M ) ⊂ A ( M ) ∗ for each object. A locally covariant CDM is a choice, for every (M, g)and ω∈ S(M), of a representative CDMM(ω)∈L1(A(M))+∩{h:khk1= 1}or P\ A(M), such that for every morphism ψand ω0∈ S(M0)one has CDMMω0◦A(ψ)=A(ψ)∗CDMM0(ω0).(53) Proposition X.2 (Naturality and composition) . If A is a locally covariant QFT and S is stable under pullback by A ( ψ ), then the assignments ω7→ hω (Haagerup L1 ) and ω7→ ξω (natural cone) define a locally covariant CDM in the sense of (53). In particular, for nested embeddings ψ1, ψ2one has CDMMω◦A(ψ2◦ψ1)=A(ψ1)∗A(ψ2)∗CDMM00 (ω). Proof. Functoriality is exactly Proposition III.1(2) composed with functoriality of A ; the cone case is identical via Riesz representation in the standard form.
35 Physical reading. Equation (53) says: change the spacetime, then reduce, or reduce, then change the spacetime—the CDM gives the same state in the smaller world. This is crucial for curved backgrounds (cosmology, black holes): predictions made in a small laboratory region agree whether we first embed the lab in a larger spacetime or not. Example (curved spacetime detectors). Let ( M, g )be stationary with a timelike Killing field; let ωβ be a KMS state at inverse temperature β for the dynamics αt induced by the Killing flow on A ( M ). Embedding an accelerated lab worldline ψ : R,→ M , a two–level Unruh–DeWitt detector is modeled by an instrument I localized in a tube around ψ ( R ). The outcome law pωβ and sequential post–selections are computed by CDMs via (11) – (13) ,independently of whether we first embed the lab (via A ( ψ )) or first write the instrument—this is (53) at work. Modular theory alone supplies the KMS property but not the instrument calculus. B. Quantum Markov semigroups, detailed balance, and entropy production Let M be a von Neumann algebra. A quantum Markov semigroup (QMS) is a family { Φ t}t≥0 of normal, unital, CP maps on M with Φ 0 = id and Φ t+s = Φ t◦ Φ s , strongly continuous on M (or σ –weakly continuous on bounded sets). The (Heisenberg) generator L is defined on a suitable core by d dt Φt(A)|t=0 =L(A). Stationary states and modular detailed balance. Let ϕ be a faithful normal state with ϕ◦ Φ t = ϕ for all t (stationarity). The semigroup satisfies (modular) detailed balance with respect to ϕ if, for all A, B ∈M , hA, L(B)iϕ=hL(A), Biϕ,hX, Y iϕ:= Z1 0 ds ϕ σϕ −is(X)∗Y, the Kubo–Mori inner product induced by the modular flow σϕ [ 94 , 95 ]. This is the correct noncommutative W∗analogue of “microscopic reversibility.” Theorem X.3 (Spohn inequality in W∗ form) . Let { Φ t} be a QMS on M with faithful stationary state ϕ and modular detailed balance. For any faithful state ψ the function t7→ S (Φ t,∗ψkϕ )is nonincreasing, and d dtt=0+ S(Φt,∗ψkϕ) = −σ(ψ)≤0, where σ ( ψ )is the entropy production given by a quadratic form determined by L and the Kubo–Mori inner product. Equality holds iff ψis a fixed point of the adjoint semigroup Φt,∗. Proof sketch. Differentiate S (Φ t,∗ψkϕ )at t = 0 + using the Gâteaux derivative of Araki relative entropy in the standard form and the adjoint generator L∗ acting on the CDM. Detailed balance identifies the derivative with a negative Kubo–Mori quadratic form (Spohn identity) [ 88 , 94 ]. Monotonicity follows from Grönwall’s inequality. Physical reading. The CDM allows us to propagate states under Φ t,∗ (Schrödinger picture) and compute how distinguishability from equilibrium decays. The result is a genuine second law of thermodynamics for local, possibly type III, dynamics—beyond modular theory, which does not provide such open evolutions. Example (Davies generator in a wedge). Let M = A ( W )and take ϕ the wedge vacuum (KMS for boosts). Weakly couple M to a quasi–free Gaussian environment stationary under boosts; Davies theory yields a QMS { Φ t} satisfying modular detailed balance. Then S (Φ t,∗ψkϕ )decreases and the modular energy hKWiΦt,∗ψ relaxes towards hKWiϕ , with a decay rate fixed by the spectral density. The CDM supplies the trajectory ψt := Φ t,∗ψ and selective updates when measurements are interleaved—impossible in modular theory alone. C. Petz quasi–entropies and Rényi–type divergences For α∈ (0 , 1) ∪ (1 ,∞ )and faithful normal states ψ, ϕ ∈M∗ the Petz Rényi divergence is defined by [84, 88] DP α(ψkϕ) := 1 α−1log Dξψ,∆α−1 ψ|ϕξψE.(54) This reduces to Umegaki/Araki in the limit α→ 1and satisfies data processing under all normal CP maps.
36 Proposition X.4 (Data processing for Petz Rényi) . For any n.u.c.p. Φ : M→N and α∈ (0 , 1) ∪ (1 , 2], DP α(ψkϕ)≥DP α(Φ∗ψkΦ∗ϕ). Proof sketch. Monotonicity of Petz quasi–entropies follows from operator convexity/monotonicity and the Kadison–Schwarz inequality in the standard form; see [84, 88]. One–shot tasks. DP α controls Hoeffding/Chernoff error exponents and strong converse bounds in hypothesis testing; in type III this gives operational statements entirely in M using CDMs. For ensembles, Rényi–Holevo quantities χP α = PipiDP α ( ωik¯ω )bound one–shot accessible information with finite sample sizes. Example (finite–sample discrimination in a wedge). For two coherent states in M = A ( W ), DP α ( ψ0kψ1 ) is computable from quasi–free modular data; this yields tight nonasymptotic upper/lower bounds on the error probability under n localized uses of a detector instrument. Modular theory alone provides only the α→1asymptotics. D. Non–Abelian gauge edges and block centers Consider a compact gauge group G and a spatial region A . The gauge–invariant region algebra with an electric center has a block structure labeled by conjugacy–invariant functions of the boundary holonomy. In lattice regularizations this appears as a direct sum over representation labels assigned to the links that intersect ∂A. Proposition X.5 (Block decomposition for non–Abelian electric centers) . Let C be the von Neumann algebra generated by class functions of the boundary holonomy on ∂A. Then MA:= AC(A)∼ =Z⊕ b G MA,R dµ(R), where b G is the unitary dual (irreducible representations), µ a Plancherel–type measure induced by the vacuum, and MA,R fiber algebras (bulk conditionals). For normal states ω, ϕ the restriction to MA decomposes as ω|MA∼ =Z⊕ ωRdµω(R), ϕ|MA∼ =Z⊕ ϕRdµϕ(R), with probability measures µω, µϕon b G. Consequently, Sω|MAkϕ|MA=Hµωkµϕ+Zb G S(ωRkϕR)dµω(R).(55) Proof sketch. C is abelian, generated by central functions of boundary holonomy; the Gelfand spectrum is (a quotient of) b G . Direct–integral decomposition then yields a block–diagonal representation of MA and states. Relative entropy on a direct integral equals the sum of the classical part plus the average of fiber entropies [88]. Physical reading. Equation (55) generalizes the Abelian “edge Shannon +bulk” split: the edge stores classical information about the representation content on the boundary, while the bulk carries quantum information conditional on the edge sector. Example (SU(2) boundary charge sensing). Let G = SU (2) and suppose the vacuum–induced edge measure µω is supported on low spins j with weights pj . An instrument that couples a probe to a boundary Wilson loop reads a (noisy) function of j ; the accessible information about j is upper bounded by H ( {pj} ). Conditional on j , interior excitations can be discriminated or estimated with performance quantified by Pjpjχ(j) rel and PjpjF(j) BKM , respectively. Switching to a magnetic center changes the block labels from representations to conjugacy classes (characters), implemented by a CP arrow θE→B.
37 E. Further examples: selective control with dissipation; tomography; coding Selective control with dissipation. Combine a localized QMS { Φ t} with a sequence of instruments {Ik}(feedback conditioned on outcomes). The CDM update is h7→ Ixn,∗◦Φtn,∗◦···◦Ix1,∗◦Φt1,∗(h) p(x1, . . . , xn). Spohn’s inequality bounds the average entropy production; the qc chain rule (Prop. IV.8) bounds the average postselection cost. Modular theory alone has no notion of these controlled trajectories. Local tomography under split windows. Inside split factors Nα one can perform standard tomographically complete measurements; Corollary VIII.11 ensures convergence of reconstructed CDMs to the intrinsic type–III state on A ( ˜ A ). This yields practical recipes for tomography of local states using only matrix tools in Nαwhile guaranteeing the correct continuum limit. Coding with recovery. Encoding a logical state in A ( A )and subjecting it to a noise channel Φ : A ( A ) → A ( B ), the Petz recovery RΦ ϕ on A ( B )gives a local decoder with performance controlled by Theorem V.4. This is a QFT version of approximate error correction, fully inside type III algebras. F. Outlook We close with a nonexhaustive list of directions that our framework opens: •Interacting models. Extending worked computations (Sec. VI) to interacting integrable models and CFTs, where modular Hamiltonians are known in limited geometries, and assessing performance of CDM–based metrology and discrimination. •Curved backgrounds. Exploiting local covariance (Sec. X A) to study detectors, coding, and recovery in cosmological spacetimes and near black holes; modular KMS structures abound but instruments and selective updates demand the CDM. •Beyond Petz. Systematizing rotated and measured recovery maps on W∗ algebras, sharpening equality conditions and finite–blocklength bounds. •Numerics. Developing algorithms that combine split windows with Haagerup L1 discretizations to evaluate CDMs, relative entropies, and QFI in lattice approximations, with guaranteed convergence (Corollary VIII.11). •Non–Abelian and gravity edges. Quantifying edge registers for non–Abelian gauge groups and gravitational constraints (ADM charges) within the same CDM formalism, including sector–resolved metrology and discrimination. The overarching message remains: modular theory gives indispensable structure, but the operational layer of quantum physics in QFT—measurements, updates, one–shot information, control and recovery—requires precisely the sort of functorial state object and arrow calculus that the CDM provides. XI. CONCLUSIONS What we set out to solve The central obstruction in local quantum field theory (QFT) is that reduced density matrices do not exist for generic local algebras: they are type III factors. This makes the textbook operational calculus—partial traces, trace–class density operators, von Neumann entropies, Kraus updates—ill-defined in the very regime where the local physics is most fundamental. Building on modular theory, we introduced the categorical density matrix (CDM): a functorial state carrier CDM(ω)∈L1(M)+∩{h:khk1= 1}or P\ M, for any von Neumann algebra M (especially type III), together with arrows given by normal unital completely positive (CP) maps. This single construct replaces every operation one performs with density matrices by a categorical counterpart that is available in all types and reduces to matrices in type I/II.
38 Main conceptual contributions 1. State representation without traces. States are carried either by Haagerup densities hω∈ L1 ( M ) + (core trace pairing) or by standard–form cone vectors ξω∈ P\ M . Both are canonical and equivalent, and both exist for type III algebras. 2. Arrows, not partial traces. Restriction, coarse–graining, and measurement are arrows in W∗ AlgnCP . There is no need for a partial trace. Where a normal conditional expectation exists (Takesaki’s theorem), it is one of our arrows; where it does not (generic type III inclusions), the preadjoint of the inclusion provides Schrödinger reduction, and split inclusions provide traceful windows. 3. Measurement calculus in type III. Instruments (Davies–Lewis–Ozawa) are CP–valued measures; in CDM form they yield Born probabilities, selective and nonselective updates, and closed composition laws inside local algebras. Selective, adaptive protocols are now rigorously defined in type III. 4. Information theory with relative entropy. All operational quantities—discrimination, accessible information, metrology, recovery—are written with Araki relative entropy and modular objects, hence are valid in type III. The “Holevo bound” becomes χrel = PipiS ( ωik¯ω ), and the quantum Fisher information is the BKM metric (the Hessian of symmetrized relative entropy). 5. Centers and edge modes as explicit classical registers. A center choice C is an explicit arrow rC : A ( O ) → AC ( O ). The reduced CDM then decomposes over Spec ( C )into a classical edge measure and bulk conditionals, yielding an exact Shannon+bulk split of relative entropy and accessible information. Changing centers (electric ↔ magnetic) is a CP arrow that changes the edge register explicitly. 6. Consistency and reduction. In type I, CDM( ω )is the density matrix and cone vectors are ρ1/2 . All formulas collapse to matrix formulas (Born rule, Kraus updates, Umegaki entropy, Holevo χ , Petz recovery). With the split property, type–I windows provide regulators whose removal recovers intrinsic type III values (monotone convergence of relative entropies and related quantities). 7. Local dynamics and second law. Quantum Markov semigroups with modular detailed balance have a Spohn inequality in W∗ form; CDMs propagate under the preadjoint and quantify entropy production and relaxation towards KMS states in regions. Operational dictionary (replace density matrices by CDMs) Matrix world −→ CDM world Tr(ρA)7→ hhω, Aior hξω, A ξωi Partial trace Tr ¯ A7→ Preadjoint of inclusion MA,→M(restriction arrow) Kraus channel ρ7→ PKiρK† i7→ h7→ Φ∗(h)for n.u.c.p. Φ POVM {Ex}:p(x) = Tr(ρEx)7→ p(x) = hhω, Exi Selective update ρ7→ PKxρK† x p(x)7→ h7→ h|x=Ix,∗(h)/p(x) Umegaki S(ρkσ)7→ Araki S(ψkϕ)(cone/relative modular operator) Holevo χ7→ χrel =PipiS(ωik¯ω) SLD QFI (special case) 7→ BKM QFI (monotone metric; reduces to SLD for commuting tangents) Petz recovery (matrix) 7→ Petz/rotated Petz via modular objects and Connes cocycles Physics: what becomes computable Beyond modular theory, which is structural but not operational, the CDM makes the following local tasks computable in type III factors:
39 •Detector statistics and updates: Outcome laws for realistic finite–resolution instruments, selective posteriors, and sequential/adaptive protocols localized in spacetime regions. •Local discrimination: One–shot Helstrom success probabilities using the predual norm (and tight bounds via fidelity and relative entropy), and asymptotic exponents via Araki/Petz Rényi divergences. •Accessible information: Bounds by χrel without any von Neumann entropies; decomposition into edge (Shannon) and bulk (conditional) parts in gauge theories. •Metrology: BKM quantum Fisher information in regions, with Cramér–Rao bounds and monotonicity under localized noise channels. •Recovery and error correction: Constructive (rotated) Petz maps that reverse localized channels approximately whenever conditional mutual information is small. •Thermalization and second law in regions: Entropy production and relaxation towards KMS via Spohn inequalities for quantum Markov semigroups, fully within local algebras. Worked example highlights •Gaussian measurement of Φ( g ) : In a coherent state ω◦Ad W ( f ), the histogram of Φ( g )is exactly Gaussian with mean σ(f, g)and variance µ(g, g). A finite–resolution Gaussian instrument adds variance in quadrature and yields Bayesian posteriors within the abelian subalgebra C∗ (Φ( g )). All steps are carried by CDMs with no density matrices. •Wedge modular energy and relative entropy: For unitary excitations U∈ A ( W ), S ( ω◦Ad Ukω ) = hKWiUΩ−hKWiΩ , giving a concrete boost–weighted energy formula and tight discrimination bounds. •Gauge edges: A center choice yields a direct–integral decomposition; the edge Shannon entropy is computable (e.g. discrete Gaussian in a toy model with closed forms via theta functions), and accessible information cleanly splits into edge + bulk contributions. Changing centers is a CP arrow that changes the edge register explicitly. Conceptual resolutions 1. Penrose’s critique revisited. The density matrix is not the state in QFT regions; the CDM is. It is functorial, canonical, and strictly more general. Where density matrices exist, CDMs reproduce them exactly. 2. No partial trace in type III—no problem. All reductions are arrows; selective and nonselective updates are CP maps on CDMs. There is no ambiguity once the arrow (e.g. center choice, split window) is named; dependence becomes explicit and composable. 3. Edge modes are classical registers. Centers encode classical data; the CDM makes this manifest and quantitatively separates classical (edge) from quantum (bulk) information in operational tasks. 4. Modular theory + CDM = structure + operations. Modular theory supplies ∆, J , KMS, and relative entropy; CDM supplies instruments, updates, and channels. Together they form a complete local quantum calculus. Limitations and scope •Faithfulness and domains. Some formulas (e.g. Araki relative entropy, BKM metric) presume faithful normal states or require domain control for modular generators. This is standard in operator algebraic QFT but demands care in exotic states.
40 •Localization of instruments. Realistic instruments are modeled via Ozawa dilations with probes in spacetime collars; ideal projective measurements of unbounded fields are limits. All locality statements assume standard Haag–Kastler axioms. •Edge regularization. Edge entropies depend on geometric/UV data (e.g. collar thickness) and center choice; in CDM this dependence is explicit (arrow labels), but physical interpretation must respect gauge and geometric constraints. •Interacting models. We presented detailed computations in quasi–free settings and gauge toy models; extending closed forms to strongly interacting theories (beyond free or integrable) remains challenging, though the formalism itself already applies. Where this leaves us We have provided a full, operational framework for doing quantum information and measurement theory inside type III von Neumann algebras. The categorical density matrix is both the conceptual and technical device that makes this possible: it turns choices (algebras, centers, splits, coarse–grainings) into arrows, it replaces traces by canonical pairings with L1 ( M )or the natural cone, and it renders modular +instrument physics tractable. In this sense, CDM elevates local QFT from a structural theory to a computational one, without sacrificing the deep modular underpinnings. Immediate opportunities •Experiment–adjacent predictions: Local discrimination and metrology bounds for Unruh–like detectors and analogue gravity platforms; sector–resolved edge sensing in tabletop U (1) and SU (2) lattice implementations. •Coding and recovery in AQFT: Quantitative, localized error correction with Petz recovery on nets of algebras; conditional mutual information as a design target for approximate quantum Markov structures. •Numerics with guarantees: Split–window matrix calculations that converge monotonically to intrinsic type III quantities (relative entropy, χrel , QFI), enabling practical finite–volume simulations with certified limits. •Curved backgrounds: Locally covariant CDMs give detector predictions on black hole and cosmological spacetimes; KMS structures exist, but only CDMs make selective measurements and one–shot tasks precise. Take–home formulas For rapid use and comparison with matrix quantum mechanics: Born rule: p(x) = hhω, Exi(or p(x) = hξω, Exξωi). Selective update: hω|x=Ix,∗(hω) p(x). Relative entropy: S(ψkϕ) = −hξψ,log ∆ψ|ϕξψi. Relative Holevo: χrel =X i piS(ωik¯ω). BKM QFI: FBKM =d2 dθ2θ=0 S(ωθkω0) + S(ω0kωθ). Recovery (type I): RΦ ϕ(X) = ρ1/2 ϕΦ†Φ(ρϕ)−1/2XΦ(ρϕ)−1/2ρ1/2 ϕ. Edge split: S(ω|MAkϕ|MA) = H(µωkµϕ) + ZS(ωxkϕx)dµω(x).
41 Final remark. Penrose’s observation that the density matrix is not the quantum state finds a precise resolution here: in local QFT the categorical density matrix is the correct, functorial state object. It reproduces the density matrix in traceful cases and extends the operational calculus to the type III world where physics actually takes place. Appendix A: Notation, Conventions, and Operator–Algebraic Background 1. General conventions • H denotes a (separable) complex Hilbert space, B ( H )the bounded operators, Sp ( H )the Schatten classes. • Avon Neumann algebra M⊆B ( H )is a unital ∗ –subalgebra closed in the weak operator topology; equivalently M=M00 (bicommutant theorem). •The commutant is M0={X∈B(H):[X, A]=0∀A∈M}; the center is Z(M) = M∩M0. • The predual M∗ is the Banach space of normal linear functionals on M ; every normal state ω∈M∗ is positive with ω(frm[o]−−)=1. • A linear map Φ : M→N between von Neumann algebras is normal, unital, completely positive (n.u.c.p.) if it is σ –weakly continuous on bounded sets, preserves the unit, and is CP in the matrix sense. Its preadjoint Φ∗:N∗→M∗satisfies (Φ∗ω)(A) = ω(Φ(A)). • Our ambient category is W∗ AlgnCP with objects W∗ –algebras and arrows n.u.c.p. maps; the monoidal product is the spatial tensor ¯ ⊗. 2. Tomita–Takesaki and standard form (quick reference) Let M⊆B ( H )and Ω ∈ H be cyclic and separating for M . The Tomita operator S0 ( A Ω) = A∗ Ω extends to a closed, antilinear S with polar decomposition S = J ∆ 1/2 , where J is antiunitary (modular conjugation) and ∆positive selfadjoint (modular operator). The modular flow is σΩ t ( A ) = ∆ itA ∆ −it ; ωΩ ( A ) = h Ω , A Ω i is KMS at inverse temperature 1with respect to σΩ . The standard form of M is a quadruple ( M, HM, J, P\ M ), unique up to unitary equivalence, where JMJ = M0 and P\ M⊂ HM is a closed self–dual cone such that each normal state ω has a unique representative ξω∈ P\ M with ω(A) = hξω, Aξωi. Relative modular objects. For faithful normal states ψ, ϕ with cone vectors ξψ, ξϕ , the relative Tomita operator Sψ|ϕ is defined on Mξϕ by Sψ|ϕ ( Aξϕ ) = A∗ξψ , with polar decomposition Sψ|ϕ = Jψ|ϕ ∆ 1/2 ψ|ϕ . Araki’s relative entropy is S(ψkϕ) = −hξψ,log ∆ψ|ϕξψi ∈ [0,∞]. Appendix B: Haagerup Lp(M), the Core Trace, and CDM–L1 1. Crossed product and dual action Let M be a von Neumann algebra and φ a faithful normal semifinite weight. The modular flow σφ:R y Madmits the crossed product N:= MoσφR, which carries: •the canonical faithful normal semifinite trace TrN; •the dual action bσ:R y Nwith bσt(λ(s)) = eitsλ(s)and bσt(π(A)) = π(A). Here πand λare the standard covariant representations of Mand R(see [86, 87]).
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