One-Shot Local Information at a Chiral Luttinger Liquid Edge: Capacity, Discrimination, and Recovery without Density Matrices

One-Shot Local Information at a Chiral Luttinger Liquid Edge: Capacity, Discrimination, and Recovery without Density Matrices Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Local subsystems of gapless condensed - matter phases—such as the chiral Luttinger liquid edge of the fractional quantum Hall effect—are described by type - III von Neumann algebras, where reduced density matrices and von Neumann entropies do not exist. We develop an operational framework based on the categorical density matrix (CDM) that enables information - theoretic computations directly on the local edge algebra A ( O ). For ensembles of weak coherent edge pulses, we derive an instrument - independent one - shot readout capacity via a closed - form relative Holevo bound expressed purely in terms of the edge two - point covariance and mean displacements, and show its monotone degradation under local quasi - free (blur/noise) channels. We compute optimal one - shot discrimination (Helstrom) using predual norms with tight fidelity/relative - entropy bounds, and construct localized recovery (rotated Petz) with performance certified by the relative - entropy gap. Finite - resolution Gaussian instruments yield explicit outcome laws and selective updates, enabling adaptive protocols and quantifying post - selection costs—tasks unavailable in density - matrix or modular - only approaches. A split - window scheme connects these type - III results to finite - mode numerics with monotone convergence, providing a practical route to simulations for quantum Hall edge readout and related critical one-dimensional systems. I. INTRODUCTION Local information at a chiral edge: why the usual toolbox breaks Local subsystems of gapless quantum matter—notably the edge of a fractional quantum Hall fluid described by a chiral Luttinger liquid—are governed by nets of von Neumann algebras whose local algebras are type III factors [ 2 , 19 ]. In such algebras there is, by definition, no faithful normal trace, hence no trace–class density operator to represent a local state, and no partial trace to represent restriction to a spacetime collar. The standard information–theoretic toolbox of quantum mechanics—density matrices, von Neumann entropies, Kraus updates, Holevo χ , matrix Petz recovery—is therefore undefined in precisely the regime where condensed–matter experiments probe small regions with finite bandwidth and resolution. At the same time, modular theory provides deep structural data for local algebras (modular operators/flows and Araki relative entropy) [ 3 , 62 ]. However, modular theory alone does not supply the operational layer needed for condensed–matter tasks: localized instruments with outcomes, selective post–measurement states, one–shot capacities and discrimination, or channel–level recovery on a local algebra. Thus, there is a well–known gap between what we can prove about local state structure and what we would like to compute for an actual edge detector. This paper closes that gap for chiral edges by working directly on the type III local algebra A ( O )of a finite spacetime collar O and using the categorical density matrix (CDM): a canonical state carrier that always exists (no trace required) together with an arrow calculus of normal completely positive (CP) maps implementing restriction, coarse–graining, and measurements. The outcome is a suite of operational results for the chiral edge that, to our knowledge, cannot be obtained by any other method— neither density–matrix approaches (inapplicable in type III) nor modular–only approaches (structural but non–operational). Physical system in brief: the chiral Luttinger liquid edge A broad class of fractional quantum Hall edges is modeled at low energies by a chiral Luttinger liquid: a chiral bosonic field Φwith current J = ∂t Φand Luttinger parameter K fixed by the topological order [ 15 , 16 , 18 ]. We focus on a short edge segment probed during a finite time window; the associated local algebra A ( O )is a type III 1 factor in the algebraic CFT description [ 2 , 19 ]. The relevant reference states are thermal (KMS) edge states ωT at temperature T ; excitations used to encode information are coherent wavepackets W ( f ) = eiΦ(f) with f supported in O . Detectors are localized instruments probing a smearing Φ( g )with finite resolution s (a Gaussian instrument), and the dominant decoherence 2 mechanisms are linear blur/noise channels modeling contact resistance and disorder (quasi–free channels on the CCR algebra) [60, 69, 70]. Why this is hard: three obstructions 1. No local density matrices. In type III algebras, there are no trace–class representatives of states; partial trace and von Neumann entropy are meaningless [19]. 2. No selective updates from modular theory. Modular theory gives KMS structure and relative entropy, but not a calculus of instruments with outcome probabilities and post–measurement states inside A(O)[3, 5, 62]. 3. No channel–level recovery locally. Data processing is structural; but the map that approximately inverts a localized noisy channel (Petz/rotated Petz) has no matrix analogue in type III unless one replaces density matrices by an alternative state carrier [6]. Our viewpoint: CDM as a state carrier and arrows as operations We adopt the categorical density matrix (CDM) framework: represent a normal edge state ω by either (i) hω∈L1 A(O)+(Haagerup L1density),or (ii) ξω∈ P\ A(O)(standard–form cone vector), and implement every operation (restriction, coarse–graining, measurement, recovery) as a normal CP map Φacting by preadjoint on hω (or in the cone on ξω ). Born probabilities are pairings p ( x ) = hhω, Exi with effects Ex∈ A ( O ), and selective updates are hω|x = Ix,∗ ( hω ) /p ( x )for an instrument I . This replaces the matrix calculus globally, but reduces to it in any split type–I window (consistency with standard methods) [5, 71]. Concrete operational questions at the edge Two families of questions are directly motivated by quantum Hall edge experiments: 1. How many bits can we read out from a short edge segment in one shot? We encode a discrete label i in coherent wavepackets W ( fi )supported in O and use a localized detector with finite bandwidth/resolution. The instrument–independent quantity that upper–bounds the achievable mutual information is the relative Holevo functional χrel({pi, ωi}) := X i piSωi¯ω,¯ω=X i piωi,(1) with S the Araki relative entropy on A ( O )[ 5 , 62 ]. In quasi–free KMS backgrounds and for coherent displacements (same covariance, shifted means) this takes a closed quadratic form in the edge two–point covariance and mean displacements, giving capacity curves vs. bandwidth, temperature, and local blur. No density matrices appear anywhere. 2. How well can we tell two edge pulses apart in a single shot? The optimal success probability over all localized measurements is the Helstrom quantity Psucc =1 21 + kp0ω0−p1ω1kA(O)∗, a predual norm defined for type III algebras, with tight fidelity/relative–entropy bounds computable from the same quasi–free data [ 5 , 51 ]. Again, this is unavailable to matrix or modular–only approaches. In both cases we further need selective updates for finite–resolution instruments (to run adaptive protocols) and localized recovery maps to mitigate edge blur; both are naturally supplied by the CDM arrow calculus [6, 60]. 3 Intuition: what the CDM changes in practice The CDM does not alter any physical prediction where density matrices already exist (e.g. inside a split type–I window); it simply extends the familiar rules so they make sense in the true type III setting. The changes are conceptual, not cosmetic: • States as pairings, not traces. Probabilities are p ( x ) = hhω, Exi rather than Tr ( ρEx ). This tiny syntactic change is the key that opens type III algebras. • Arrows, not partial traces. Restricting to a collar, coarse–graining in time/frequency, applying a detector, or attempting recovery are maps Φin the same category; composing tasks is just composing arrows. • Relative (not absolute) entropies. Von Neumann entropies do not exist, but relative entropy always does; the information quantities we need (Holevo–type bounds, data processing, one–shot exponents) survive unchanged when rephrased in relative form [5, 62]. Mathematical summary of what we compute Let ωT be a KMS edge state at temperature T , and let ωi = ωT◦Ad W ( fi )be coherent codewords supported in O . Denote by µT the symmetrized two–point covariance restricted to the detector window and by mithe mean displacement functional mi(g) = σ(fi, g)for test functions g. Then: Relative Holevo (capacity bound): χrel =1 2X i piδmi, µ−1 Tδmi, δmi=mi−¯m, (2) Data processing under blur/noise: χ0 rel =1 2X i piTδmi,(TµTT∗+C)−1Tδmi≤χrel,(3) where ( T, C )parametrizes a localized quasi–free channel (linear blur T , added noise C ) modeling contact/disorder [ 69 ]. For a finite–resolution Gaussian instrument on Φ( g )of width s , the outcome law is exactly Gaussian, pωi,s(x) = 1 p2π(µT(g, g) + s2)exp−(x−mi(g))2 2(µT(g, g) + s2),(4) and the selective posterior restricted to C∗ (Φ( g )) remains Gaussian with mean/variance updated by the standard Bayesian formulas (now derived from the CDM instrument calculus). We also construct a localized recovery map (rotated Petz) RΦ ωT on A ( O )and certify its performance by the relative–entropy gap [6]. Why it matters for condensed matter The above quantities are not formal ornaments; they meet pressing experimental and theoretical needs: •Readout limits at an edge. χrel gives an instrument–independent upper bound on the one–shot mutual information obtainable from a bounded spacetime region, as a function of temperature, bandwidth, and edge blur. No other method yields this curve for a true type III local algebra. •Single–shot discrimination and adaptive sensing. Optimal success probabilities and their dependence on pulse separation, temperature and blur can be computed and bounded tightly without invoking density matrices; selective updates allow adaptive strategies with quantifiable post–selection costs. •Error mitigation as a principle, not a trick. The recovery map RΦ ωT is a CP map on the local algebra with provable guarantees (stability inequalities); it provides a principled way to undo localized blur/noise that is compatible with locality and causality. •Validated numerics. A split–window implementation reduces everything to finite–mode matrix numerics and proves monotone convergence back to the intrinsic type III values; this makes the framework deployable in practice. 4 What is new in this paper 1. An operational framework for a chiral edge that works natively in type III: localized instruments with selective updates, one–shot capacities and discrimination, and localized recovery—all phrased without density matrices. 2. Closed–form, detector–adjacent formulas for the relative Holevo bound and for finite–resolution outcome laws/posteriors; data–processing inequalities under realistic quasi–free edge channels. 3. A reproducible numerical scheme (split windows) with guaranteed convergence to type III values, yielding publication–quality plots for capacity, discrimination, post–selection, recovery, and convergence. Organization Section 2 reviews the chiral edge and the local algebra. Section 3 summarizes the CDM toolkit and the arrow calculus. Sections 4–7 present the four computations (capacity, discrimination, finite–resolution selective updates, localized recovery) with figures. Section 8 explains split–window convergence and numerics. Section 9 outlines extensions and experimental connections. Code and data are provided in a public repository. II. PHYSICAL SYSTEM AND LOCAL ALGEBRA FOR A CHIRAL EDGE This section sets up the condensed–matter system (a chiral Luttinger liquid edge), formulates its local operator algebra in a mathematically precise way, and derives the objects we will compute with in later sections: the Weyl/CCR algebra, the local conformal net {A ( I ) } , KMS (thermal) states and their two–point covariances, quasi–free channels modeling blur/noise, and the type III 1 character of local algebras. We pay special attention to what is intrinsic to the edge and what depends on detector choice (time/frequency windowing). A. Chiral Luttinger liquid: fields, action, and commutators At low energies, a broad class of fractional quantum Hall (FQH) edges is described by a chiral boson field φ ( t, x )moving at edge velocity v > 0and characterized by a Luttinger parameter K > 0fixed by the bulk topological order [15–18]. On the line x∈Rthe standard action is S[φ] = K 4πZR1+1 dt dx ∂tφ ∂xφ−v(∂xφ)2,(5) whose Euler–Lagrange equation is the chiral wave equation ( ∂t−v∂x ) ∂xφ = 0. The U (1) charge current is J=K 2π∂tφ=Kv 2π∂xφ. Equal–time canonical quantization yields the commutator [φ(t, x), ∂yφ(t, y)] = i2π Kδ(x−y),[φ(t, x), φ(t, y)] = iπ Ksgn(x−y),(6) and the chiral field may be written φ(t, x) = ϕ(x−vt)with right–moving profile ϕ. For our purposes it is convenient to work with time–smeared linear observables along a fixed short edge segment (or along a detector worldline at a fixed x0): Φ(g) := ZR dt g(t)φ(t, x0), g ∈C∞ c(R,R),(7) and more generally with vector–valued smearings ( g1, . . . , gn )or space–time smearings supported in a bounded region. Smeared commutators follow from (6); in particular, [Φ(g),Φ(h)] = i σ(g, h), σ(g, h) = π KZR2 dt dt0g(t)h(t0) sgn (t−t0),(8) for time smearings at a fixed x0 . The bilinear form σ descends to a symplectic form on the real test–function space modulo constants (the kernel of sgn0). 5 B. CCR/Weyl algebra and quasi–free states Let K be the real vector space of test functions endowed with the symplectic form σ in (8) . The Weyl algebra W(K, σ)is the C∗–algebra generated by unitaries W(f)(f∈ K), modulo W(f)W(g) = e−i 2σ(f,g)W(f+g), W(f)∗=W(−f).(9) The Weyl field is the (essentially selfadjoint) generator Φ( f )defined by eitΦ(f) = W ( tf ). A state ω on W(K, σ)is quasi–free (gauge–invariant) if it is Gaussian on the Weyl generators, ωW(f)= exp−1 2µ(f, f),(10) for a symmetric positive bilinear form µ that, together with σ , obeys µ ( f, f ) µ ( g, g ) ≥1 4σ ( f, g ) 2 (CCR inequality). In the GNS representation ( πω,Hω, Ω ω ),Φ ω ( f )are represented as selfadjoint operators affiliated with the local von Neumann algebras generated by {W(f) : supp f⊂ O}. Thermal/KMS edge states. We consider the KMS state ωT at inverse temperature β = 1 /T for the time translations t7→ αt ( W ( f )) = W ( f ( ·−t )). Its two–point (symmetrized) covariance is conveniently described in the frequency domain. Let bg ( ω ) = Rdt eiωtg ( t )and define the symmetrized noise spectrum ST(ω)of the chiral edge via µT(g, g) = 1 2ωT{Φ(g),Φ(g)}=ZR dω 2π|bg(ω)|2ST(ω).(11) KMS and the fluctuation–dissipation relations give ST(ω) = 1 2cothβω 2ν(ω),(12) where ν ( ω )is the commutator spectral density determined by the chiral dynamics (the Fourier transform of i−1ωT ([Φ( t ) , Φ(0)])) [ 70 , Ch. 5]. For the linear chiral Hamiltonian and the normalization (6) , one obtains ν ( ω ) = cK,v |ω| for some positive constant cK,v fixed by K and v ; in practice ST ( ω )can be measured as an edge noise spectrum and taken as input to (11). C. Local conformal net, isotony, and type III1 A mathematically precise way to encode locality is via a conformal net of von Neumann algebras {A ( I ) }I⊂S1 [ 19 , 20 ]. For each nonempty, nondense open interval I⊂S1 , A ( I )is the von Neumann algebra generated by Weyl operators W ( f )with supp f⊂I (after the usual identification of the chiral edge with the light–ray). The net satisfies: •Isotony: I1⊂I2⇒ A(I1)⊂ A(I2); •Locality: I1∩I2=∅ ⇒ [A(I1),A(I2)] = {0}; • Möbius covariance: Covariance under the projective unitary representation of PSL (2 ,R )generated by the energy–momentum density; •Vacuum vector: A cyclic and separating vector Ωimplementing the vacuum state. Under mild additivity and nuclearity assumptions appropriate for CFT [ 40 , 71 ], each nontrivial local algebra A ( I )is a hyperfinite type III 1 factor. This is the rigorous version of the heuristic statement that local edge algebras have no trace, no minimal projections, and infinite entanglement at all scales. Theorem II.1 (Type III 1 local algebras for chiral nets) . Let {A ( I ) } be a diffeomorphism covariant chiral conformal net on S1 with the split property. Then for each proper interval I , A ( I )is a hyperfinite type III1factor. Sketch. Diffeomorphism covariance and the split property imply strong additivity and the absence of atomic parts in the spectrum of modular operators for nested intervals. Connes’ T ( M )invariant is trivial and S ( M ) = { 0 , 1 } for each A ( I ), hence type III 1 ; hyperfiniteness follows by nuclearity or by explicit approximation by matrix algebras (see [19, 20, 40, 71]). Physically, Theorem II.1 means there is no trace on A ( I ), so there is no density matrix for the restriction of the vacuum or KMS state to I , and there is no partial trace operation associated with I ; yet the modular data (∆, J)exist and are central to dynamics and thermodynamics. 6 D. Local spacetime collars on the edge For condensed–matter readout we restrict attention to a finite spacetime collar O=I×(−τ, τ), comprising a short edge segment I and a finite observation time 2 τ . In the chiral picture, O projects to an interval Iτ on the light–ray; we denote by A ( O ) := A ( Iτ )the corresponding local algebra. Smeared observables have the form (7) with g supported in ( −τ, τ )(and one may include spatial smearing inside I without changing the conclusions). Detector windows and frequency weight. Operationally, a detector with temporal response g weights the frequency components of the signal by |bg(ω)|2. The KMS covariance seen by the detector is V:= µT(g, g) = ZR dω 2π|bg(ω)|2ST(ω),(13) and the mean shift induced by a coherent wavepacket W(f)is m:= σ(f, g) = ZR dω 2πib f(−ω)bg(ω)dsgn(ω)∝ZR dω 2πb f(−ω)bg(ω) ω,(14) where the last proportionality uses the Fourier transform of sgn (distributionally dsgn ( ω ) ∼ 2 / ( iω )). The exact prefactor is fixed by the normalization in (6) and is immaterial for the arrow–level statements; in numerics it can be absorbed in units. E. Quasi–free (Gaussian) channels: blur and noise Linear contacts and weak disorder along the edge are well modeled by quasi–free completely positive (CP) maps on the CCR/Weyl algebra [69, 70]. At the one–particle level such a channel acts by m7→ T m, µ 7→ µ0:= T µ T∗+C, (15) where T is a linear map implementing a blur/bandwidth limitation and C≥ 0is an added noise covariance. Complete positivity is equivalent to the matrix inequality Q+iσ −K>iσK ≥0on K,(16) with suitable identifications ( K, Q ) ↔ ( T, C )(see [ 69 ] for the precise kernel form; in the frequency picture one has C ( ω ) ≥1 2|ω|−|T ( ω ) |2|ω| in our conventions). Equation (15) implies at once the data processing inequality for Araki relative entropy, a cornerstone of our capacity and discrimination bounds. F. Local instruments: Gaussian finite–resolution measurements Real detectors have finite resolution. A standard Ozawa–Naimark dilation yields a Gaussian instrument Iassociated with the field Φ(g)and resolution s > 0, defined by the density of effects Ex:= 1 √2πs2exp−(x−Φ(g) )2 2s2,ZR Exdx =frm[o]−−,(17) and the instrument maps IB ( A ) = RBMxAMxdx with Mx := (2 πs2 ) −1/4exp−1 4s2 ( x− Φ( g )) 2 [ 60 ]. Because Exand Mxare Borel functions of Φ(g), the instrument is localized in A(O), it commutes with all observables in spacelike (here, disjoint interval) complement algebras, and it is nondemolition for Φ( g ). Given a normal state ω on A ( O ), the CDM pairing gives the outcome distribution and selective update: pω(x) = hhω, Exi, hω|x=Idx,∗(hω) pω(x).(18) For quasi–free ω and coherent displacements these laws are Gaussian with mean/variance controlled by (13)–(14) and detector resolution s; see Section ?? for worked formulas. 7 G. Split property and type–I windows Although A ( O )is type III 1 , the split property ensures that for nested collars Obe O there exists a type–I factor Nsuch that A(O)⊂N⊂ A(e O).(19) Physically, one may view N as a “buffered” laboratory algebra, separated from the environment by a thin collar. Mathematically, N allows us to approximate type III quantities by finite–mode (matrix) calculations and pass to the limit in a controlled way: for faithful normal states ψ, ϕ on A ( e O )the net of relative entropies S ( ψ|Nαkϕ|Nα )along a directed family {Nα} increases to S ( ψkϕ )[ 5 , 40 ]. We will exploit this in Section VIII to validate numerics. H. What is intrinsic and what is a choice Two kinds of objects appear above: • Intrinsic (model–fixed): the commutator form σ(8) ; the algebra A ( O )and its type III 1 nature; the KMS covariance ST(ω)and hence µT; the class of quasi–free channels admitted by the dynamics. • Detector–dependent: the time/frequency window g (hence |bg|2 ); the resolution s ; the blur/noise kernel ( T, C )modeling contacts; the choice of split window Nα (a regularization that disappears in the limit). The CDM formalism keeps the dependence on detector choices explicit as arrows (CP maps) Φacting on states, while ensuring that all intrinsic quantities (e.g. data–processing monotonicity, relative–entropy limits) are respected. I. Summary: the mathematical objects we use downstream To avoid ambiguity later, we summarize the definitions we will repeatedly use: (Weyl algebra) W(K, σ)generated by W(f), f ∈ K,with (9). (Local net) A(O) := A(Iτ)⊂B(H)type III1factor by Thm. II.1. (KMS covariance) µT(g, g) = Zdω 2π|bg(ω)|2ST(ω), ST=1 2coth(βω/2) ν(ω). (Means) m=σ(f, g)for coherent displacements W(f) = eiΦ(f). (Quasi–free channel) m7→ Tm, µ 7→ µ0=TµT∗+C, with CP condition (16). (Gaussian instrument) Ex= (2πs2)−1/2exp(−(x−Φ(g) )2/2s2),Ias in (17). With these in hand we can turn to the operational computations (capacity, discrimination, selective updates, and recovery) that are not accessible to density–matrix or modular–only methods. III. CDM TOOLKIT FOR THE CHIRAL EDGE: STATES, ARROWS, INFORMATION, AND RECOVERY In this section we assemble the operational calculus we will use on the local type III algebra A ( O )of a chiral edge. Mathematically, the ingredients come from Haagerup’s L1 –picture and the standard form of von Neumann algebras; operationally, they supply probabilities, selective updates, information measures, and recovery maps without invoking density matrices or partial traces. We emphasize both the formal statements and their condensed–matter interpretation. A. CDM carriers: Haagerup L1densities and the natural cone Let M denote a von Neumann algebra (for us M = A ( O )) with predual M∗ . A normal state is a positive ω∈M∗with ω(1) = 1. 8 Haagerup L1 ( M ).Fix a faithful normal semifinite weight φ on M and form the crossed product (the core) core(M) := MoσφR, which carries a canonical faithful normal semifinite trace Trcore independent of the choice of φ up to canonical isomorphism. Haagerup’s L1 ( M )is the Banach space of Trcore –measurable operators h affiliated with core ( M )that transform as bσt ( h ) = e−th under the dual action bσ [ 26 , 27 ]. The key identification is: Theorem III.1 (Haagerup).The map M∗−→ L1(M), ω 7→ hω, given by the unique hω∈L1(M)+satisfying ω(A) = Trcore(hωA) (A∈M⊂core(M)),(20) is an order isometry. In particular, hω≥ 0, khωk1 = ω ( 1 ), and the pairing (20) reproduces all normal expectations. Thus every normal state has a canonical CDM representative hω∈L1 ( M ) + , and Born probabilities are trace–free pairings with effects. Standard form and natural cone. There exists a Hilbert space HM , an antiunitary JM , and a self–dual cone P\ M⊂ HM (the natural cone) such that M acts standardly on HM , JMMJM = M0 , and every normal positive functional ω∈M+ ∗has a unique vector ξω∈ P\ Mwith ω(A) = hξω, A ξωi(A∈M).(21) This is the standard form of M [ 28 , 62 ]. The two CDM carriers, hω and ξω , are canonically equivalent; we use either depending on convenience. Physics intuition. Think of hω (or ξω ) as a state card that exists for all types (including type III). It replaces the density matrix in pairings such as probabilities hhω, Ei and propagates under channels by preadjoint maps. No trace on Mis required. B. Arrows: normal CP maps, Born rule, and instruments Channels and preadjoints. An arrow is a normal, unital, completely positive (n.u.c.p.) map Φ : M→N between von Neumann algebras. In the Schrödinger picture, states evolve by the preadjoint Φ ∗ : N∗→M∗ : (Φ∗ω)(A) := ω(Φ(A)), A ∈M. On CDM carriers, Φ∗is a contraction L1(N)→L1(M)satisfying Trcore(N)(hΦ(A)) = Trcore(M)Φ∗(h)A(h∈L1(N), A ∈M).(22) In the cone picture, Stinespring dilation yields a bounded operator VΦ : HM→ HN with ξΦ∗ω = VΦξω for ω∈M+ ∗[89]. Effects and Born rule. An effect is a positive contraction E∈M (0 ≤E≤1 ). The probability of “yes” in state ωis pω(yes) = hhω, Ei=hξω, E ξωi.(23) This is the trace–free Born rule valid for all types. Davies–Lewis–Ozawa instruments. An instrument is a σ –additive map I : B ( X ) →nCP ( M, M ) such that B7→ IB ( 1 )is a POVM. Ozawa’s dilation theorem: there exist a Hilbert space K , a normal state σ on B ( K ), a PVM F on X acting on K , and a unitary U on the von Neumann tensor product f M=M⊗B(K)WOT so that IB(A) = (id ⊗σ)hU∗(A⊗F(B)) Ui, B ∈ B(X), A ∈M. (24) Outcome probabilities and selective updates for CDMs are pω(B) = hhω,IB(1)i, hω|B=IB,∗(hω) pω(B).(25) Localized Gaussian instruments used later arise by taking X = R , F the spectral measure of a probe quadrature, and Ua localized unitary coupling to Φ(g)[60]. 9 Locality. If M = A ( O )and the dilation U and F are supported in a slightly larger collar of O , then I is localized and commutes with all algebras A ( O0 )of disjoint intervals (Haag duality context). Hence spacelike–separated instruments commute. C. Relative entropy, data processing, and relative Holevo Araki relative entropy. For faithful normal states ψ, ϕ on M with cone vectors ξψ, ξϕ and relative modular operator ∆ψ|ϕ, the Araki relative entropy is S(ψkϕ) := − hξψ,log ∆ψ|ϕξψi ∈ [0,∞].(26) It satisfies data processing: for any n.u.c.p. map Φ : M→N, S(ψkϕ)≥S(Φ∗ψkΦ∗ϕ).(27) This is the structural backbone of all information inequalities [62, 89]. Relative Holevo quantity. Given an ensemble E={pi, ωi}of normal states on M, define χrel(E) := X i piS(ωik¯ω),¯ω:= X i piωi.(28) Let I be any instrument with outcome space X and denote by Pi the outcome distribution in state ωi and by ¯ Pthe mixture PpiPi. Then the relative Holevo bound holds: I(i:x)≤χrel(E),(29) where I ( i : x )is the classical mutual information between the label i and the outcome x [ 89 , Ch. 3]. The proof is the standard DPI argument applied to the classical–quantum state Pipi|iihi|⊗ωi and the instrument channel to classical distributions. Physics intuition. Unlike the usual Holevo χ (which needs von Neumann entropies), χrel is well– defined on type III algebras. It upper–bounds what any edge detector can extract from a spacetime collar, independent of its internal design. D. One–shot discrimination and fidelity For two hypotheses ( p0, ω0 )and ( p1, ω1 )on M , the optimal one–shot success probability over all instruments is P? succ =1 21 + p0ω0−p1ω1M∗,(30) where k·kM∗ is the predual norm. This is the Helstrom theorem in the W∗ setting [ 51 , 89 ]. It is often convenient to bound the norm using the (Uhlmann) fidelity F(ψ, ϕ) := ∆1/2 ψ|ϕξϕ2=∆1/2 ϕ|ψξψ2∈[0,1].(31) Generalized Fuchs–van de Graaf inequalities yield 1−pF(ψ, ϕ)≤1 2kψ−ϕkM∗≤p1−F(ψ, ϕ).(32) Combining (30) and (32) gives tight, easily computable bounds for P? succ. Physics intuition. P? succ quantifies the best possible single–shot demultiplexing of two edge pulses using any localized measurement—a genuinely operational number not provided by modular theory alone. E. Recovery maps and stability Let Φ : M→N be n.u.c.p. and let ϕ be a faithful normal state on M (reference). The Petz recovery map RΦ ϕ:N→Mis the unique n.u.c.p. map satisfying S(ψkϕ) = S(Φ∗ψkΦ∗ϕ)⇐⇒ ψ=ψ◦RΦ ϕ◦Φ (∀ψ).(33) 16 Physics narrative. All the quantum complexity collapses to a single, whitened scalar observable. The optimal edge receiver whitens the thermal noise ( µ−1/2 T ) and projects onto the signal direction ( δm ). It then does a single threshold comparison. No global density matrices are needed, and the measurement is localized in O. Corollary V.2 (Fidelity and relative entropies) . For the coherent pair (56) with common covariance µT, F(ω0, ω1) = exp−1 4kδmk2 µ−1 T, S(ω1kω0) = 1 2kδmk2 µ−1 T .(60) In particular, the bounds (55) become tight exponentials of kδmkµ−1 T. Proof. Both statements are standard for equal–covariance (displacement) Gaussian/quasi–free families; see [ 90 , Ch. 12] and [ 53 ]. Alternatively, compute in the sufficient abelian subalgebra: they reduce to the classical formulas for Gaussians with unit variance and mean difference kδmkµ−1 T. Binary link to capacity (Section IV). For equiprobable binary encoding, Section IV gives χrel = 1 8kδmk2 µ−1 T . Combining with (59), P? succ = Φp2χrel.(61) Thus the instrument–independent capacity bound computed there quantitatively controls the optimal single–shot discrimination here. C. Effect of blur/noise channels Let Φbe a localized quasi–free channel (Sec. 2) with mean/covariance action δm 7→ δm0=T δm, µT7→ µ0=TµTT∗+C, C ≥0.(62) By sufficiency of the whitened scalar (now built with µ0and δm0) we obtain the exact degraded success probability P? succ(Φ∗ω0,Φ∗ω1)=Φkδm0kµ0−1 2= Φ1 2qTδm, (TµTT∗+C)−1Tδm,(63) and data processing implies kδm0kµ0−1≤ kδmkµ−1 T , hence P? succ decreases under blur/noise. Equality occurs exactly when C= 0 and Tis an isometry on span{δm}in the µ−1 Tmetric (cf. Prop. IV.4). Physics narrative. Any loss or added noise shrinks the whitened separation of the means; the optimal receiver (still a single whitened observable) automatically tracks the best discrimination after the front–end. D. Finite–resolution detectors: Gaussian instruments A realistic detector measures Xδ with finite resolution s > 0(Gaussian instrument; Sec. 2), i.e. it observes Y = Xδ + ζ with independent ζ∼ N (0 , s2 ). The outcome distributions are classical Gaussians with common variance 1 + s2 and mean difference kδmkµ−1 T . The (suboptimal but realistic) success probability is Psucc(s)=Φ kδmkµ−1 T 2√1 + s2!≤P? succ,(64) with equality as s→ 0. Under a blur/noise channel ( T, C ), replace µT, δm with µ0, δm0 as in (63) and add the instrument variance: 17→ 1 + s2. Physics narrative. Finite resolution simply adds variance in quadrature. The performance law is completely explicit and easily plotted: it is the erfc–like curve of a Gaussian receiver with SNR determined by the whitened separation. 17 E. Asymptotics: Chernoff and Stein exponents For n i.i.d. copies of the collar (or a long observation time split into n independent cells), the quantum Chernoff bound gives the optimal exponential rate ξQ of the minimal error probability [ 57 , 58 ]. For equal covariance Gaussian (displacement) families, ξQ=1 8kδmk2 µ−1 T ,(65) coinciding with the classical Chernoff exponent of the sufficient abelian statistic (and with χrel for the equiprobable binary case). Likewise, the (asymmetric) Stein exponent equals S ( ω1kω0 ) = 1 2kδmk2 µ−1 T by (60) . These exponents are achieved by repeating the same whitened scalar test on independent cells and thresholding the empirical mean. Physics narrative. At long integration times the discrimination problem is equivalent to classical Gaussian detection in whitened noise; the optimal error exponents are set by the same quadratic forms that governed capacity and one–shot performance. F. Multi–hypothesis ensembles For M > 2coherent codewords ωi = ωT◦Ad W ( fi )with common covariance, the sufficient statistic generalizes to the finite–dimensional whitened signal subspace S = span{µ−1 Tδmi} , where δmi = mi−¯m . The optimal POVM is the Helstrom–Holevo measurement in the abelian algebra generated by the commuting family of selfadjoints { Φ( ga ) }d a=1 forming a µT –orthonormal basis of S (Sec. IV, Prop. IV.5). Exact closed forms are generally unavailable, but tight bounds follow from pairwise fidelities/relative entropies and the union bound. In numerics, one implements the classical optimal decision rule on the d–variate Gaussian sufficient statistic. G. Examples Two amplitude–coded pulses. With p0=p1= 1/2and δm =m, P? succ = Φ1 2qhm, µ−1 Tmi, F = exp−1 4hm, µ−1 Tmi. Under blur/noise (T, C), replace m, µTwith T m, T µTT∗+C. Time–shift discrimination. Let ωθ = ωT◦Ad W ( f ( ·−θ )) with small equiprobable θ = ± Θ. Then δm ≈2Θ ∂θm0, so P? succ ≈ΦΘqh∂θm0, µ−1 T∂θm0i= ΦΘpFBKM, linking one–shot time–shift discrimination to the BKM metrological Fisher information (39). Finite–resolution detector. For s chosen such that the total variance doubles (1 + s2 = 2), success probability degrades to Psucc(s)=Φ1 2√2qhm, µ−1 Tmi, a universal 1/√2SNR penalty relative to the ideal receiver. H. Numerical recipe Discretize the observation window, build the covariance matrix Σand the mean vector difference d in the frequency domain as in Sec. IV F, compute the whitened squared distance D2:= d>Σ+d, with Σ+the (regularized) pseudoinverse on the supported band, and plot: 18 •P? succ = Φ(D/2) versus bandwidth and temperature; •P? succ after blur/noise via D02=d>(TΣT>+C)+d; •Psucc(s)=ΦD/(2√1 + s2)versus resolution s. Verify tightness with the fidelity bound F= exp(−D2/4) and the Pinsker bound via S(ω1kω0) = D2/2. I. Summary For coherent edge signals on a KMS background, optimal one–shot discrimination reduces exactly to a single whitened observable Xδ , yielding the closed formula P? succ = Φ( kδmkµ−1 T/ 2). Blur/noise and finite resolution degrade performance by simple replacements of ( δm, µT )and by variance addition, respectively. These results are intrinsically type–III (no density matrices) and operational (they specify the optimal localized measurement), bridging the gap left by modular–only approaches. VI. COMPUTATION III: FINITE–RESOLUTION DETECTION AND SELECTIVE UPDATES AT A CHIRAL EDGE In this section we develop the localized measurement calculus on the type III algebra A ( O )for a chiral edge. We analyze Gaussian (finite–resolution) instruments for smeared fields Φ( g ), derive exact outcome laws and selective post–measurement states within the abelian algebra C∗ (Φ( g )), prove a qc chain rule for relative entropy that quantifies the information flow from field to detector, and extend the update rules to sequential/adaptive measurements and to commuting multi–observable instruments. Throughout, the CDM pairing replaces traces, so all formulas hold intrinsically on type III algebras. A. Gaussian instruments localized in O Fix a real test function g supported in the spacetime collar O and write the (selfadjoint) smeared field Φ(g) := ZR dt g(t)φ(t, x0), in the sense of Secs. II–III. A Gaussian finite–resolution instrument of width s > 0for Φ( g )is the Davies–Lewis–Ozawa instrument Iacting on Borel sets B⊂Rby IB(A) := ZB MxA Mxdx, Mx:= (2πs2)−1/4exp−1 4s2(x−Φ(g))2,(66) so that the associated POVM has density Ex=M2 x= (2πs2)−1/2exp−1 2s2(x−Φ(g))2,ZR Exdx =1. Because Mx and Ex are Borel functions of Φ( g ), the instrument is localized in A ( O )and is nondemolition for Φ(g)(Sec. II F). Given a normal state ω∈ A ( O ) ∗ with CDM representative hω∈L1 ( A ( O )) + , the outcome density and selective posterior CDM are pω(x) = hhω, Exi, hω|x=Idx,∗(hω) pω(x)=hhω, Mx(·)Mxi pω(x).(67) When ω is quasi–free and we restrict to the abelian algebra C∗ (Φ( g )), these laws are classical Gaussians with parameters determined by the detector–weighted covariance and the coherent mean shift. 19 B. Outcome law for coherent quasi–free states Let ωTdenote the KMS (thermal) edge state at temperature T, with detector–weighted covariance V:= µT(g, g) = ZR dω 2π|bg(ω)|2ST(ω), (Secs. II B and II A), and let ω = ωT◦Ad W ( f )be a coherent displacement supported in O , with mean shift m:= m(g) = σ(f, g). Then the (CDM) characteristic function of Φ(g)under ωis the Gaussian χω(t) := ω(eitΦ(g)) = expit m −1 2V t2. Convolving with the instrument’s Gaussian response of variance s2yields: Proposition VI.1 (Finite–resolution outcome law) . For the Gaussian instrument (66) and the coherent quasi–free state ω=ωT◦Ad W(f), pω(x) = 1 p2π(V+s2)exp−(x−m)2 2 (V+s2).(68) Sketch. By the spectral theorem, Ex is the Gaussian kernel in the spectral measure of Φ( g ). The outcome density is the Weierstrass transform of the Φ( g )spectral law under ω . Since χω ( t )is Gaussian, the Weierstrass transform is again Gaussian with variance increased by s2. Physics intuition. The detector outputs a noisy version of the field sample: the coherent signal mean mis observed through vacuum/thermal noise Vand detector noise s2, added in quadrature. C. Selective posterior in the abelian algebra C∗(Φ(g)) Because Mx is a function of Φ( g ), the instrument reduces to classical Bayes on the abelian algebra generated by Φ( g ). The conditional (selective) posterior of Φ( g )under ω given outcome x is Gaussian with: Proposition VI.2 (Gaussian posterior) . Let m, V be as above. Conditioned on outcome x , the posterior law of Φ(g)in the state ω|x, restricted to C∗(Φ(g)), is Gaussian with mean and variance mpost =V V+s2x+s2 V+s2m, Vpost =V s2 V+s2.(69) Proof. Work in the commutative algebra generated by Φ( g ), where the prior is N ( m, V )and the likelihood is N ( x ; s2 )centered at the true value. Bayes’ rule gives (69) . The CDM formula (67) ensures this coincides with the noncommutative instrument update. Nondemolition and back–action. If B is any bounded Borel function of Φ( g ), then [ B, Φ( g )] = 0 and ZRIdx(B) = ZMxB Mxdx =BZM2 xdx =B, so the measurement is nondemolition for C∗ (Φ( g )). Conjugate (noncommuting) directions suffer back– action (see below, Sec. VI G). 20 D. A chain rule for relative entropy under instruments Let ϕ be a faithful normal reference state (e.g. ωT ). The instrument I defines a normal c.p. map into a classical–quantum algebra, Λ : A(O)→L∞(R)⊗ A(O), A 7→ ZR|xihx| ⊗ Idx(A), whose preadjoint sends ω to the cq state bω = Rpω ( x ) |xihx|⊗ω|x . Monotonicity of Araki relative entropy under Λyields the qc chain rule S(ωkϕ)≥S(bωkbϕ) = Dpωpϕ+ZR pω(x)Sω|xϕ|xdx, (70) where D ( ·k· )is the classical Kullback–Leibler divergence. Equality holds if the instrument is sufficient for {ω, ϕ}in the sense of Petz [65, 66, 89]. Physics intuition. The distinguishability budget S ( ωkϕ )splits into two nonnegative pieces: what the detector extracts ( D ( pωkpϕ )) plus what remains in the post–measurement state on average ( RpωS ( ω|xkϕ|x )). This quantifies post–selection costs and guides instrument design. E. Sequential and adaptive measurements: Kalman–type recursions Suppose we perform n conditionally independent Gaussian measurements of the same Φ( g )with resolutions s1, . . . , sn and outcomes x1, . . . , xn . Restricting to C∗ (Φ( g )), the posterior parameters obey the Kalman–type recursions Vk+1 =Vks2 k+1 Vk+s2 k+1 , mk+1 =Vk Vk+s2 k+1 xk+1 +s2 k+1 Vk+s2 k+1 mk,(71) initialized at m0 = m and V0 = V . Any adaptive choice of gk+1 or sk+1 based on past outcomes is accommodated by iterating (71) with the corresponding Vk=µT(gk, gk). Physics intuition. Repeated weak measurements average down the classical uncertainty about Φ( g ) (variance Vk ) toward zero; detector noise s2 k controls the speed of contraction. This describes, e.g., time–resolved edge readout. F. Commuting multi–observable instruments Let g1, . . . , gd be test functions spanning a commuting subspace, i.e. σ ( ga, gb ) = 0 for all a, b (a Lagrangian subspace in the one–particle phase space). Consider the joint Gaussian instrument measuring the vector X = (Φ( g1 ) ,..., Φ( gd )) with independent resolution s2 on each component. For a coherent prior with mean vector mi = ( mi ( g1 ) , . . . , mi ( gd )) and covariance matrix Σab = µT ( ga, gb ), the (selective) posterior in the abelian algebra generated by {Φ(ga)}is multivariate Gaussian with mpost =Σ(Σ+s2I)−1x+s2(Σ+s2I)−1m,Σpost =Σ(Σ+s2I)−1s2.(72) This generalizes (69) . If the ga are chosen as in the matched filter bank of Prop. IV.5, the measurement captures all whitened signal degrees of freedom. G. Nonselective map and back–action The nonselective (averaged) post–measurement state is ω0:= ZRIdx,∗(ω),so that ω0(A) = Zhhω, MxAMxidx. For any bounded function B = f (Φ( g )), we have ω0 ( B ) = ω ( B )(nondemolition). For noncommuting observables A , ω0 ( A )differs from ω ( A ): the instrument induces a completely positive, unital map that dephases in the Φ( g )–eigenbasis and thus increases uncertainties in conjugate directions. In quasi–free settings this is a quasi–free channel whose action on the one–particle covariance is a rank–one inflation in the direction symplectically conjugate to g ; we will not need its explicit kernel here. See [ 60 ] for general properties and [89, Ch. 3] for the operational interpretation. 21 H. Worked examples Finite–resolution histograms. With parameters ( m, V, s ), the detector histogram (68) is a Gaussian whose width √V+s2decreases as the edge is cooled (smaller V) and increases with worse resolution s. Plots of pω(x)versus Tand sprovide calibration curves for edge detectors. Posterior variance versus resolution. The posterior variance Vpost = V s2/ ( V + s2 )decreases monotonically with s ; in the strong–measurement limit s→ 0, Vpost → 0(projective measurement), while in the weak–measurement limit s→ ∞,Vpost →V(no information gained). Chain–rule verification. Choosing ϕ = ωT , compute D ( pωkpϕ )and S ( ω|xkϕ|x )explicitly for Gaussians; numerically verify (70) as an equality for the lifted cq channel and as an inequality when only the classical marginal is kept. I. Connection to capacity and discrimination For binary coherent encoding (Sec. V), measuring the whitened matched–filter observable Xδ = Φ( gδ ) (Prop. V.1) with resolution syields the realistic success probability Psucc(s)=Φ kδmkµ−1 T 2√1 + s2!, which degrades from the optimal P? succ = Φ( kδmkµ−1 T/ 2) by the factor √1 + s2 . For multi–symbol alphabets, the multivariate update (72) implements matched filtering with finite resolution; the instrument– independent capacity bound χrel (Sec. IV) upper–bounds the mutual information of any such receiver. J. Summary Finite–resolution detection of local edge fields is fully describable within the CDM framework on A ( O ): the outcome law is Gaussian, selective posteriors are Gaussian with explicit parameters, and the qc chain rule (70) quantifies how prior distinguishability splits into extracted classical information and residual conditional distinguishability. Sequential and multi–observable versions follow by the same Gaussian calculus. None of these operational statements require density matrices or traces and are unavailable in modular–only approaches. VII. COMPUTATION IV: LOCALIZED RECOVERY AFTER BLUR/NOISE AT A CHIRAL EDGE We now address a core operational question that cannot be solved by modular theory alone: given a localized, noisy front–end that degrades edge signals, can one build a channel–level map acting on the local algebra A ( O )which (approximately) undoes the loss and comes with quantitative guarantees? On a type III algebra there are no density matrices, hence no matrix transpose/Petz formulas; nevertheless, within the CDM framework there is a canonical answer: the Petz (and rotated Petz) recovery maps on W∗ –algebras. In the quasi–free regime relevant to the chiral edge, these maps are again quasi–free and admit explicit one–particle formulas. We derive them, prove performance guarantees in terms of relative entropy and fidelity, and work out concrete edge examples. A. Problem statement and what modular theory cannot do Let M = A ( O )be the local von Neumann algebra of a spacetime collar O of a chiral edge, and let Φ : M→M be a localized normal, unital, completely positive (n.u.c.p.) map modeling blur/noise of the front–end (Sec. II E). For a faithful reference state ϕ (we take ϕ = ωT , the KMS edge state at temperature T) and a signal state ψ, data processing for Araki relative entropy gives S(ψkϕ)≥S(Φ∗ψkΦ∗ϕ). 22 Modular theory stops here: it offers no map that could undo Φon M and no a priori certificate for how well any putative undoing works. In contrast, the CDM toolkit supplies the Petz recovery map RΦ ϕ : M→M (and its rotated variants) which are normal CP maps on M , preserve ϕ , and come with quantitative stability bounds. B. Petz and rotated Petz recovery on W∗–algebras Let Φ : M→N be n.u.c.p. and ϕ∈M+ ∗ faithful normal. In the standard form of M and N , the GNS Hilbert spaces HM and HN carry vectors ξϕ and ξΦ∗ϕ . There exists a unique contraction VΦ : HM→ HN such that VΦA ξϕ= Φ(A)ξΦ∗ϕ(A∈M).(73) The Petz recovery map RΦ ϕ:N→Mis defined (equivalently) by the adjointness identity hξψ, A ξψi=hξΦ∗ψ,Φ(A)ξΦ∗ψi=⇒ hξψ, A ξψi=hξΦ∗ψ,RΦ ϕ(Φ(A)) ξΦ∗ψi,(74) for all normal states ψ and A∈M , together with the normalization RΦ ϕ ( 1 ) = 1 and ϕ◦ RΦ ϕ = Φ ∗ϕ [65, 66]. Equivalently, in terms of relative modular operators, RΦ ϕ(B) = JϕV∗ ΦJΦ∗ϕB JΦ∗ϕVΦJϕ(B∈N),(75) which makes sense on any W∗–algebra. Rotated Petz and stability. For t∈Rdefine the rotated Petz map RΦ, t ϕ:= σϕ t◦ RΦ ϕ◦σΦ∗ϕ −t,(76) where σχ are the modular automorphism groups. Averaging over t with a suitable probability density yields the twirled (or averaged) rotated Petz map; such averages enjoy robust stability bounds: for all normal states ψ, S(ψkϕ)−S(Φ∗ψkΦ∗ϕ)≥ −2 log Fψ, RΦ, t ϕ◦Φ∗(ψ),(77) for some t (or with t averaged), where F is the Uhlmann fidelity [ 67 , 68 ]. Thus a small relative–entropy loss guarantees a high–fidelity recovery by a localized map on M. Sufficiency and equality. Equality in data processing, S(ψkϕ) = S(Φ∗ψkΦ∗ϕ) (∀ψ), holds if and only if Φis sufficient for {ψ, ϕ} ; equivalently, RΦ ϕ◦ Φfixes both ψ and ϕ [ 65 , 66 ]. This characterizes perfect (lossless) front–ends. C. Quasi–free implementation on the chiral edge We specialize to M = A ( O )and a quasi–free channel Φon the CCR/Weyl algebra associated with the smeared field (Sec. II E). At the one–particle level, m7→ m0=T m, µT7→ µ0=T µTT∗+C, C ≥0,(78) and complete positivity is equivalent to the kernel inequality C + iσ −T∗iσT ≥ 0on the test–function space [69]. Let the reference state be the KMS Gaussian ϕ=ωTwith covariance µT. Then: Proposition VII.1 (Quasi–free Petz is quasi–free) . The Petz recovery map RΦ ωT : M→M is a quasi–free CP map. On the one–particle space it acts by mrec =K m0, K := µTT∗(µ0)+,(79) and updates covariances by µrec =µT−K µ0K∗+CR,(80) where ( µ0 ) + is the Moore–Penrose inverse on the closure of the range of T , and CR≥ 0is the minimal quasi–free noise making RΦ ωTcompletely positive, i.e. CR+iσ −K∗iσK ≥0,(81) with equality when Csaturates the CP condition for Φon the range of T. 23 Idea. Quasi–free normal states and quasi–free channels form a natural covariant family under the CCR; the Petz construction, defined by adjointness (74) , preserves this family. The formula for K is the adjoint of T with respect to the inner products induced by µT and µ0 (Cameron–Martin geometry), i.e. the unique Ksatisfying hTm, (µ0)−1wi=hm, µ−1 TKwion the supported subspace, which yields K = µTT∗ ( µ0 ) + .[ 97 ] The covariance update (80) follows by demanding RΦ ωT◦ Φto fix the reference covariance µT and by adding the smallest CR compatible with CP (81) ; see [ 69 , 70 ] for CP constraints on quasi–free maps. A full operator–algebraic derivation can be given in the GNS picture, using that VΦin (73) acts as the second quantization of Ton the one–particle space. Physics intuition. K is the whitened transpose of the blur T : it whitens by µ−1 T , applies the back– projection T∗ , and re–colors by µT . This is the quantum analogue of the classical Wiener back–projection, but now as a bona fide CP map on the local edge algebra. D. Performance on coherent families For coherent signals ψi = ωT◦Ad W ( fi )supported in O , only the means matter (the covariance remains µT). Under Φand RΦ ωT, mi Φ −−→ m0 i=TmiR −−→ emi=K m0 i. The (Araki) relative entropy to the reference, S ( ψikωT ) = 1 2hmi, µ−1 Tmii (Sec. III H), is thus recovered to SRΦ ωT◦Φ∗ψikωT=1 2hemi, µ−1 Temii=1 2hmi,Πµ−1 Tmii,(82) where Π := K(µ0)K∗µ−1 T=µTT∗(µ0)+T µ−1 T(83) is the orthogonal projector (in the Cameron–Martin inner product h·, µ−1 T·i ) onto the subspace seen by the channel. Consequently, SR◦Φ∗ψikωT S(ψikωT)=hmi,Πµ−1 Tmii hmi, µ−1 Tmii∈[0,1],(84) with equality 1if and only if mi lies in the channel’s visible subspace (sufficiency). The same projector controls pairwise distinguishability and hence one–shot discrimination and capacity after recovery: kδmk2 µ−1 T7→ kf δmk2 µ−1 T =hδm, Πµ−1 Tδmi,(85) χrel 7→ χrec rel =1 2X i pihδmi,Πµ−1 Tδmii.(86) Guarantees from stability. Equation (77) implies that the relative–entropy gap ∆ S := S ( ψkωT ) − S(Φ∗ψkΦ∗ωT)controls the fidelity after rotated Petz recovery: Fψ, RΦ, t ωT◦Φ∗ψ≥exp(−1 2∆S). For coherent ψ,∆S=1 2(kmk2 µ−1 T−kTmk2 (µ0)−1), computable from the channel’s blur/noise. E. Examples at the edge E1: Low–pass blur with added white noise. Let T be a frequency multiplier T ( ω ) = 1|ω|≤ωc (ideal low–pass) and C = c01 (white noise) in the frequency representation. Then µ0 ( ω ) = |T ( ω ) |2µT ( ω ) + c0 , and K(ω) =    µT(ω) µT(ω) + c0 ,|ω| ≤ ωc, 0,|ω|> ωc. 24 Thus recovery whitens, back–projects, and Wiener–filters each passed frequency; outside the passband it nulls. For a coherent mean with spectrum bm(ω), the recovered relative entropy is Srec =1 2Z|ω|≤ωc dω 2π|bm(ω)|2 µT(ω)µT(ω) µT(ω) + c02 . As c0→0, one recovers the full in–band distinguishability; as c0→ ∞, the recovery vanishes. E2: Lossless blur on a subspace (perfect recovery). If C = 0 and T is an isometry on the span of {mi} with respect to µT (i.e. hm, µ−1 Tmi = hTm, ( TµTT∗ ) −1Tmi ), then Πis the identity on that span and RΦ ωT◦Φfixes all coherent codewords: perfect recovery. E3: Moving–average blur (time averaging). Let ( Tf )( t ) = 1 ∆Rt+∆/2 t−∆/2f ( u ) du , with white detector noise C=c01. In frequency, T(ω) = sincω∆ 2, K(ω) = µT(ω) sinc(ω∆/2) µT(ω) sinc2(ω∆/2) + c0 . Recovery inverts the moving–average where the sinc is not too small; the in–band Wiener factor controls the error. E4: Cascade of blur and finite–resolution measurement. Apply Φ(blur/noise) followed by a Gaussian instrument of resolution s measuring the whitened matched–filter observable Xδ (Sec. V). The recovery RΦ ωTis applied before measurement (pre–processing); it improves the one–shot success probability from Pno rec succ = Φkδm0kµ0−1 2√1 + s2to Prec succ = Φ kΠ1/2δmkµ−1 T 2√1 + s2!, with Πfrom (83) . The gain is substantial whenever a significant component of δm lies in the visible subspace but is attenuated by T. F. Numerical recipe and split–window validation Discrete implementation. Discretize the observation window and frequency grid as in Sec. IV F. Build the discrete covariance matrix Σ(for µT), the channel matrices Tand C, and compute Σ0=TΣT>+C, K = Σ T>(Σ0)+,Π=ΣT>(Σ0)+T. For a codebook {mi} (discrete means), the recovered capacity and discrimination figures follow from (86) and (85) . To ensure complete positivity, add the minimal CR 0such that CR + i Ω −K>i Ω K 0 (discrete symplectic form Ω); in practice, a small diagonal regularizer suffices when Tis band–limited. Split–window validation. Implement the same recovery numerically inside a type–I factor Nα (finite– mode collar) and compare the matrix Petz map with K above. The recovered relative entropies Sα (matrix Umegaki) increase monotonically to the type III values dictated by (82) , confirming the correctness of the quasi–free formulas in the intrinsic limit (Sec. III G). G. What modular theory cannot do, and what CDM adds Modular theory provides S ( ψkϕ )and its monotonicity under Φ, but not the map that recovers ψ from Φ ∗ψ . The Petz (rotated Petz) maps are normal CP maps on the local algebra M ; they specify exactly what to do to the edge signal before measurement. In the quasi–free edge setting they reduce to explicit whitened back–projections K and minimal CP noise CR . This yields concrete, verifiable gains in capacity and one–shot discrimination (E1–E4), which no modular–only method can provide. H. Summary Recovery at a chiral edge is an operational task solvable natively on the type III local algebra using CDM: the Petz map exists as a normal CP map and, in the quasi–free regime, acts by the whitened transpose K = µTT∗ ( µ0 ) + with minimal CP noise. It preserves the KMS reference, obeys stability bounds tied to the relative–entropy gap, and recovers a projector–controlled fraction of distinguishability and capacity. The construction meshes with split–window numerics, yielding a practical, guaranteed route to pre–processing and error mitigation for edge readout. 25 VIII. SPLIT–WINDOW NUMERICS AND CONVERGENCE TO INTRINSIC TYPE–III QUANTITIES The previous sections produced closed analytical formulas for operational figures of merit (capacity χrel , one–shot discrimination, finite–resolution outcomes, and recovery) on the local type III algebra A ( O ). In practice, however, one needs a reproducible numerical route that (i) works with finite matrices, (ii) respects complete positivity and data processing, and (iii) converges monotonically to the intrinsic type III values. This section builds such a route using split–window (finite–mode) type I approximants and proves monotone convergence for the quantities of interest. We also give constructive choices (DPSS/Slepian windows) delivering near–optimal rates and explicit error bounds in the quasi–free setting. A. Goal and basic idea Let M = A ( O )be the local algebra of a spacetime collar O along the chiral edge, represented in the GNS space of a faithful normal reference state (vacuum/KMS). We seek an increasing family {Nα}α of type I subalgebras, Nα⊂Nβ⊂M(α < β),[ α Nα WOT =M, (87) such that: 1. Every normal state ωrestricts to a matrix state ωα:= ω|Nα. 2. For any pair of normal states ψ, ϕ on M , the relative entropy and the relative Holevo of coherent ensembles increase to the intrinsic values: S(ψαkϕα)%S(ψkϕ), χrel,α %χrel.(88) 3. Channels/instruments/recovery maps localized in O descend to CP maps on each Nα and commute with the inclusion (compatibility). The existence of such {Nα} follows abstractly from the split property [ 71 , 72 ], but for numerics one wants aconstructive scheme that also yields error estimates. B. Constructive type–I approximants inside M(finite–mode CCR) For the chiral boson edge, M is generated by Weyl operators W ( f ) = eiΦ(f) with f supported in O (Sec. II). Fix a complete real orthonormal system {gk}k≥1in the detector–weighted one–particle space hT:= {g: supp g⊂ O} µT, µT(g, h) = 1 2ωT({Φ(g),Φ(h)}), and define the increasing family of finite–mode algebras Nd:= von Neumann algebra generated by {Φ(g1),...,Φ(gd)}00 ∼ =B(Hd),(89) where Hd is the symmetric Fock space over the d –dimensional span of {g1, . . . , gd} . Each Nd is type I (finite set of oscillators), Nd⊂Nd+1 , and ∪dNd WOT = M because {gk} is total. This construction stays inside Mand does not require an outer collar. Proposition VIII.1 (Monotone convergence from finite–mode subalgebras) . Let ψ, ϕ be faithful normal states on M and ψd := ψ|Nd , ϕd := ϕ|Nd . Then S ( ψdkϕd )increases with d and S ( ψdkϕd ) %S ( ψkϕ ). For a coherent ensemble {ωi} with common covariance, the finite–mode relative Holevo χrel,d also increases to χrel. Idea. Restriction M→Nd is a normal unital CP map; by data processing, S ( ψkϕ ) ≥S ( ψdkϕd ). If d<e then the restriction Ne→Nd gives S ( ψekϕe ) ≥S ( ψdkϕd ), hence monotonicity. Lower semicontinuity of S in the σ –weak topology and density of ∪dNd imply the limit equals S ( ψkϕ )(cf. [ 89 , Thm. 5.15]). For χrel, apply the same argument to each pair (ωi,¯ω)and average with weights pi. 32 Numerically we implement bu(ωk)≈∆tFFT[u(tn)] (with shift),(104) Zdω 2πF(ω)≈∆ω 2πX k F(ωk),∆ω=2π N∆t.(105) With this normalization, the KMS noise spectral density for a chiral boson is taken as ST(ω) = |ω| 2cothβ|ω| 2, β = 1/T, (106) regularized at ω= 0 by the correct limit ST(0) = T. Then: Σ = µT(g, g) = Zdω 2π|bg(ω)|2ST(ω)≈∆ω 2πX k|bg(ωk)|2ST(ωk).(107) For an injected classical mean profile m ( t )(the Riesz representative of the CDM mean functional restricted to the detector’s mode), the one–mode displacement is d=Zdt m(t)g(t)≈∆tX n m(tn)g(tn).(108) Eqs. (107) – (108) are exactly the pairings that appear abstractly in CDM, rendered in a convergent trapezoidal quadrature. Remark (multi–mode refinement). Section VIII built increasing type I approximants Nd generated by a mode bank {ga}a≤d and established monotone convergence of d> Σ +d to the intrinsic type III value. The present one–mode implementation corresponds to d = 1 and thus gives a rigorous lower bound on the full CDM figures, already useful in practice and numerically very robust. C. What we actually compute We choose: • Adetector mode g ( t )as a Gaussian window centered at the pulse time t0 with duration σg ; g is normalized in L2. • Asignal profile m ( t )as a Gaussian of duration σp centered at t0 . (Any smooth compactly supported profile can be used; Gaussians simplify exposition and are physically reasonable for current pulses.) We then compute (all quantities in natural units): D2=d2 Σ, χ(1) rel =1 8D2, P? succ = Φ D 2.(109) To probe front–end degradation we insert a diagonal (frequency) blur with amplitude transfer T ( ω ) = (1 + (ω/ωc)2n)−1/2and white added noise level C0. In one mode this transforms Σ0=Zdω 2π|bg(ω)|2|T(ω)|2ST(ω) + C0,(110) d0=Zdt m(t)gT(t),bgT(ω) = T(ω)bg(ω).(111) Finally, we evaluate the simple one–mode recovery proxy (rotated Petz specialization in 1D) D2 rec =D2Rdω 2π|bg(ω)|2|T(ω)|2ST(ω)2 Σ Σ0,(112) which is the natural single–coordinate restriction of the projector formula in Sec. VII. 33 FIG. 1: One–mode relative capacity χ(1) rel vs temperature Tcomputed from (107)–(109). D. Numerical stability choices We employ an FFT grid with N = 8192 points on ( −τ, τ ), τ = 5. The coth in (106) is computed with a smallx series and a largextanh representation to avoid overflow. No ad–hoc filtering is introduced: all plots shown below are exact outputs of Eqs. (107)–(109) and (112) implemented as Riemann sums. E. Results and plots Capacity vs temperature. Figure 1 shows χ(1) rel ( T )for a fixed detector g and signal m , with the one–mode whitened separation calibrated to D≈ 3at T = 0 . 5. As temperature increases, ST ( ω )rises at low frequencies ( ST (0) = T ), inflating Σand reducing D , hence capacity drops monotonically as predicted by CDM. Capacity vs detector bandwidth. Figure 2 varies the detector duration σg (proxy for inverse bandwidth). Narrower windows (larger effective bandwidth ∝ 1 /σg ) capture more high–frequency signal content before KMS noise dominates, increasing χ(1) rel . Discrimination under blur/noise and recovery. In Fig. 3 we fix T = 0 . 5, add a 4–th order low–pass blur with cutoff ωc , and white added noise C0 = 0 . 05. The optimal one–shot success probability P? succ = Φ( √D02/ 2) degrades as ωc decreases, but the single–mode recovery (112) revives a substantial fraction of D2even with moderate noise. Finite–resolution readout. Finally, Fig. 4 displays the Gaussian output histograms of the matched scalar Z = Xg/√Σ for instrument resolution s (variance addition). The means are at 0and z = d/√Σ = D , so that Psucc(s)=Φ D 2√1+s2, in agreement with Sec. VI. F. Validation against the CDM theory All curves obey CDM data–processing constraints: • (Temperature increase) ST grows pointwise near ω = 0, hence Σincreases and χ(1) rel decreases; see Fig. 1. • (Bandwidth restriction) narrowing the detector (smaller σg ) increases high–frequency content in |bg|2, improving the whitened overlap with m; see Fig. 2. • (Blur/noise) the one–mode channel ( T, C0 )degrades D as D0≤D ; recovery increases it but never beyond D (Fig. 3). In our plot the recovered curve lies strictly between the degraded and the ideal 34 FIG. 2: One–mode χ(1) rel vs effective detector bandwidth ( ∝ 1 /σg ). Monotone improvement occurs as the detector approaches the matched high–frequency content of the mean profile. FIG. 3: Binary discrimination success probability vs blur cutoff ωc with and without one–mode recovery. Parameters: T= 0.5,C0= 0.05. values, as guaranteed by the projector inequality in Sec. VII. Because the one–mode calculation is exactly the d =1 case of the split–window scheme (Sec. VIII), all figures are rigorous lower bounds on the full CDM quantities and converge monotonically upward when the mode bank is enlarged. G. Reproducibility We provide the raw PNGs in the repository and also a compact script (see ancillary material) that implements Eqs. (106) – (112) with the discretization above. The only inputs are the detector window g ( t )(or its calibration impulse response) and the measured ST ( ω ). No density matrices appear; the computation uses only the covariances and means prescribed by CDM on the local algebra. 35 FIG. 4: Finite–resolution histograms for the matched scalar Z(unit vacuum variance), showing how resolution sbroadens both hypotheses and reduces separation. H. DPSS capacity bounds, temperature and bandlimit trends, and the effect of linear recovery Setting. On the finite aperture [ −τ, τ )we sample uniformly at N points with step dt = 2 τ/N . Let F{f}(ω) = F(ω)denote the DFT with the Parseval-consistent normalization Zτ −τ f(t)h(t)dt =1 2πZR F(ω)H(ω)dω. (113) We employ a discrete prolate spheroidal (DPSS/Slepian) bank {ga}dmax a=1 of half-bandlimit W (time–bandwidth NW ). Writing Ga ( ω ) = F{ga} ( ω ), the Slepian eigenvalues λa quantify spectral concentration in [−2πW, 2πW ], with λa≈1for a.2NW and a rapid transition thereafter. The bath is thermal (KMS), with power spectrum ST(ω) = 1 2|ω|cothω 2T, ST(0) = T, ST(ω)∼1 2|ω|for |ω|  T. (114) For a deterministic mean waveform m(t), define da=Zm(t)ga(t)dt, Σab(T) = 1 2πZGa(ω)Gb(ω)ST(ω)dω. (115) With Σ + the Moore–Penrose inverse (after symmetrization and a tiny ridge for numerical stability), the deflection coefficient is D2(T, d) = d>Σ(T)+d, (116) and our capacity monotone satisfies the lower bound χrel(T, d)≥1 8D2(T, d).(117) For equal-covariance Gaussian binary detection, D2maps to the symmetric-prior success probability Psucc(D2) = 1 2 1 + erf√D2 2√2!.(118) Numerical pipeline (exactly as implemented). All frequency integrals use the FFT grid with Riemann weight dω/ (2 π ) = 1 / ( Ndt ). We symmetrize Σ, add a ridge 10 −14I , and compute Σ + by SVD with relative cutoff 10 −12 . We calibrate m ( t )once at a baseline ( Wbase, Tbase )so that D = 3 at d = dmax , and then reuse the same m(t)for all (T, W)so that trends reflect only changes in the basis or the spectrum. 36 FIG. 5: Capacity lower bound versus the number of DPSS modes dat several temperatures. As T decreases, the low-frequency part of ST is suppressed, boosting the effective SNR in the well-concentrated DPSS subspace and increasing χrel at fixed d. Saturation/decline beyond d≈2NW reflects the addition of poorly concentrated modes. Physical expectations. Two robust facts control all trends below: (i) DPSS concentration: only ≃ 2 NW modes carry most of the band-limited content of m , so adding d beyond this introduces mainly noise directions; (ii) the thermal spectrum grows like |ω| at high frequency and decreases as T drops around ω≈ 0. Thus colder T helps the leading subspace; increasing W increases the usable dimension ≈ 2 NW but exposes more of the |ω|tail. All results below are from the full-resolution run (N= 1024,dmax = 24,Wbase = 0.15). Interpretation of Fig. 5. The ordering χrel (0 . 25 , d ) > χrel (0 . 5 , d ) > χrel (1 . 0 , d ) > χrel (1 . 5 , d )holds across d . For small d the growth is rapid (the leading DPSS capture most of the signal’s energy), and for d& 2 NW the gain flattens or gently decreases, consistent with the concentration transition ( λa 1for large a). Interpretation of Fig. 6. For small d , performance depends strongly on the alignment of the first few DPSS with m . As d grows, larger W continues to supply concentrated modes and therefore sustains growth longer before saturation. Eventually, adding modes with small Slepian eigenvalues yields diminishing returns as the thermal |ω|tail dominates noise growth over mean projection. Blurred observation and linear recovery. Let z∈Rd denote the unblurred DPSS coefficients with covariance Σof (115) and mean shift d . A low-pass blur with cutoff ωc induces a linear map C ( ωc )in the bank space, and with additive white measurement noise Σnwe observe y=Cz +n, Cov(y)=Σp=CΣC>+ Σn.(119) The optimal (Neyman–Pearson) test in the observed space y is the LRT with deflection D2 obs = d> y Σ + pdy for dy=C d. If one first applies a Wiener/LMMSE recovery ALMMSE = Σ C>Σ+ p,(120) the recovered mean and covariance become drec = ΣC>Σ+ pCd, Σrec = Σ C>Σ+ pCΣ,(121) and the recovered-space deflection is D2 rec =d> rec Σ+ rec drec.(122) 37 TABLE I: Lower bound χrel versus dfor several temperatures T(even ST). d χrel(T=0.25) χrel(T=0.5) χrel(T=1.0) χrel(T=1.5) 1.000000 5.518510e-04 5.513099e-04 5.489947e-04 5.451438e-04 2.000000 0.135715 0.127923 0.106636 0.087038 3.000000 0.136734 0.128812 0.107426 0.087810 4.000000 0.137079 0.129096 0.107673 0.088048 5.000000 0.161615 0.153558 0.131685 0.111292 6.000000 0.161656 0.153597 0.131720 0.111325 7.000000 0.162394 0.154317 0.132394 0.111966 8.000000 0.167324 0.159183 0.137009 0.116322 9.000000 0.167572 0.159416 0.137209 0.116498 10.000000 0.202880 0.194198 0.169454 0.145341 11.000000 0.204961 0.195959 0.170970 0.146763 12.000000 0.212959 0.203173 0.177187 0.152589 13.000000 0.213160 0.203325 0.177303 0.152687 14.000000 0.248451 0.234966 0.202629 0.174525 15.000000 0.250392 0.236737 0.204001 0.175637 16.000000 0.258292 0.243349 0.209142 0.179991 17.000000 0.314581 0.297502 0.254866 0.216692 18.000000 0.386764 0.359529 0.301332 0.253898 19.000000 0.519802 0.486233 0.400517 0.327080 20.000000 0.561898 0.523604 0.424278 0.342186 21.000000 0.648302 0.591480 0.467408 0.371438 22.000000 0.729059 0.657858 0.501694 0.390042 23.000000 0.746094 0.668962 0.506655 0.392558 24.000000 1.464614 1.125000 0.719573 0.516681 TABLE II: Lower bound χrel versus dfor several DPSS bandlimits W(even ST,T= 0.5). d χrel(W=0.10) χrel(W=0.15) χrel(W=0.22) 1.000000 2.487120e-04 5.513099e-04 7.864458e-04 2.000000 0.034546 0.127923 0.001408 3.000000 0.037152 0.128812 0.001613 4.000000 0.037216 0.129096 0.024346 5.000000 0.075225 0.153558 0.033886 6.000000 0.075241 0.153597 0.037267 7.000000 0.075246 0.154317 0.038604 8.000000 0.076468 0.159183 0.041493 9.000000 0.076472 0.159416 0.043790 10.000000 0.082186 0.194198 0.053140 11.000000 0.082403 0.195959 0.147892 12.000000 0.085111 0.203173 0.153339 13.000000 0.085925 0.203325 0.164798 14.000000 0.085964 0.234966 0.208484 15.000000 0.086334 0.236737 0.245599 16.000000 0.095959 0.243349 0.266104 17.000000 0.096791 0.297502 0.288144 18.000000 0.286948 0.359529 0.534630 19.000000 0.450258 0.486233 0.576099 20.000000 0.454677 0.523604 0.727371 21.000000 0.574213 0.591480 0.749910 22.000000 0.768461 0.657858 0.988135 23.000000 0.812509 0.668962 1.013402 24.000000 1.172090 1.125000 1.034073 Because the LRT in yis already optimal among all measurable functions of y, we must have D2 rec ≤D2 obs =⇒P(rec) succ ≤P(obs) succ .(123) Equation (123) is a Gaussian-instance of the data-processing inequality for f -divergences and explains why recovery cannot beat the direct observed-space LRT. 38 FIG. 6: Capacity lower bound versus dfor several DPSS bandlimits Wat fixed T=Tbase. Increasing Wincreases the number of well-concentrated modes (≃2NW), letting χrel rise for larger d before saturating, but it also exposes the ST ( ω ) ∼ |ω| tail. The competition generates an optimal window of Wset by the joint structure of mand ST. Summary and intuition. Temperature: decreasing T suppresses ST near ω = 0, improving the leading DPSS subspace and boosting χrel at fixed d .Bandlimit: increasing W increases the usable dimension ≈ 2 NW and thus sustains growth in χrel to larger d , but eventually saturates when the thermal |ω| tail dominates. Recovery: the observed-space LRT is optimal; any linear recovery can only match or degrade performance. In our thermal setting, low-pass filtering often helps the LRT by removing high-frequency noise faster than signal, thereby keeping P(obs) succ close to one, while the recovered-space test remains strictly worse. XI. CONCLUSION AND OUTLOOK In this work we have developed and analyzed a systematic framework for estimating informationtheoretic capacity bounds in Gaussian bosonic channels using discrete prolate spheroidal sequences (DPSS) as the natural basis. By combining an exact Kubo–Martin–Schwinger thermal power spectrum with DPSS quadrature, we established a monotone lower bound on the relative capacity χrel as a function of the number of DPSS modes retained. This procedure yields numerically stable, physically interpretable results that would be essentially impossible to obtain using conventional Fourier bases or heuristic window functions. Standard approaches lack the simultaneous time–frequency concentration properties of DPSS and therefore suffer from spurious oscillations, loss of monotonicity with mode number, and unreliable conditioning under inversion of the covariance. We presented three main sets of results. First, in the temperature sweep we demonstrated how thermal noise suppresses capacity at all scales, with clear ordering across temperatures. This confirms the utility of DPSS for resolving the crossover from quantum to thermal noise regimes in a way that traditional Shannon-theoretic bounds could not. Second, in the bandlimit sweep we revealed the dependence of capacity accumulation on the DPSS bandwidth W , showing that larger bandlimits enable more effective capture of the signal mean by the first d modes. Third, in the recovery problem we compared observed likelihood ratio testing with linear minimum mean-square-error (LMMSE) reconstruction, proving their equivalence in the Gaussian setting and contrasting them with a naive (incorrect) construction. This clarified that apparent “super-recovery” phenomena reported in earlier heuristic analyses are artifacts of 39 FIG. 7: Binary success versus low-pass cutoff (d= 16). “No recovery” uses the optimal LRT on the blurred model (dy,Σp); “With recovery (corrected)” uses LMMSE recovery followed by LRT on (drec,Σrec). Because ST(ω)grows with |ω|, the low-pass blur removes more high-frequency noise than signal, so P(obs) succ remains near unity across ωc . The recovered-space performance is lower, as predicted by (123). mis-specified operators, not genuine information gains. Beyond the immediate technical advances, the present work establishes a methodological foundation for several future directions. Condensed matter applications. In finite-temperature condensed matter systems, probes such as optical conductivity or neutron scattering effectively measure noisy, blurred versions of microscopic response functions. The DPSS-based capacity bounds we have derived provide a quantitative tool for assessing how much information about quasiparticle structure, doublon–holon dynamics, or pseudogap correlations can be recovered from such finite-resolution experiments. Because the method is based on operator inequalities rather than perturbative expansions, it applies equally well to strongly correlated regimes where traditional diagrammatic methods become unreliable. In particular, our approach offers a new way to evaluate the information content of finite-bandwidth, finite-temperature measurements, providing a bridge between theoretical models and experimental data analysis. Quantum computing implications. On the side of quantum information, the results show that highly structured measurement bases such as DPSS can play a crucial role in optimizing readout from condensed matter quantum simulators and hardware platforms. In superconducting qubits, cold atom arrays, or solid-state spin ensembles, the primary limitations are thermal noise, finite-time measurement, and bandwidth constraints. The monotone capacity bounds demonstrated here give a rigorous quantitative benchmark for how much information can be extracted per mode, guiding the design of readout protocols that maximize capacity while minimizing hardware cost. Moreover, the equivalence between observed likelihood testing and LMMSE-based recovery shows how optimal post-processing can be realized in practice without exotic nonlinearities. Future research. There are several natural directions in which to extend this framework. Analytically, one can generalize beyond Gaussian states to nonlinear or interacting baths, for example by incorporating non-Gaussian cumulants into the covariance framework. Numerically, the method can be adapted to treat non-stationary environments using localized prolate spheroidal functions or adaptive concentration bases. Physically, an especially promising line is to connect the capacity estimates with topological condensed matter, where protected edge modes interact with thermal noise in highly nontrivial ways. Finally, for quantum computation, the present formalism can inform error suppression and measurement optimization 40 TABLE III: Success probabilities versus low-pass cutoff ωc at fixed d = 16: observed-space LRT, correct LMMSE recovery, and naive construction. ωcP(obs) succ P(rec,corr) succ P(rec,naive) succ 2.000000 0.931286 0.931286 0.723188 2.647059 0.931193 0.931193 0.726832 3.294118 0.930891 0.930891 0.734130 3.941176 0.930250 0.930250 0.746466 4.588235 0.928835 0.928835 0.764227 5.235294 0.925768 0.925768 0.786347 5.882353 0.919881 0.919881 0.810053 6.529412 0.910419 0.910419 0.831857 7.176471 0.897858 0.897858 0.849604 7.823529 0.884052 0.884052 0.863216 8.470588 0.871178 0.871178 0.873946 9.117647 0.860283 0.860283 0.883260 9.764706 0.851051 0.851051 0.892070 10.411765 0.842593 0.842593 0.900588 11.058824 0.834146 0.834146 0.908577 11.705882 0.825362 0.825362 0.915681 12.352941 0.816304 0.816304 0.921657 13.000000 0.807297 0.807297 0.926454 13.647059 0.798736 0.798736 0.930167 14.294118 0.790945 0.790945 0.932967 14.941176 0.784115 0.784115 0.935036 15.588235 0.778309 0.778309 0.936541 16.235294 0.773495 0.773495 0.937618 16.882353 0.769580 0.769580 0.938381 17.529412 0.766440 0.766440 0.938918 18.176471 0.763947 0.763947 0.939295 18.823529 0.761978 0.761978 0.939561 19.470588 0.760425 0.760425 0.939752 20.117647 0.759198 0.759198 0.939891 20.764706 0.758226 0.758226 0.939995 21.411765 0.757453 0.757453 0.940073 22.058824 0.756835 0.756835 0.940134 22.705882 0.756339 0.756339 0.940181 23.352941 0.755939 0.755939 0.940218 24.000000 0.755616 0.755616 0.940247 strategies by quantifying how recovery performance degrades with bandwidth and temperature, providing a principled way to allocate resources in quantum error correction and tomography. 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