Past Selection Filtration

Past-Selection Filtrations (PSF): Unifying Quantum History with Operator-Algebraic Foundations and Macro-Record Insights Peter James Bavaro October 7, 2025 Abstract I formulate Past-Selection Filtrations (PSF): a filtration of von Neumann subalgebras ( At,Et ) with faithful, normal φ -preserving conditional expectations that capture “records of the past” in an operator-algebraic setting. Using root fidelity as operational visibility V ( ρ, σ ) = pF(ρ, σ) , I prove V ( t )is nonincreasing along the filtration and relate visibility to the quantum Chernoff exponent via sharp two-sided bounds. I construct an explicit dilation/instrument for Et , establish delayed-choice invariance under a φ -invariant normalizer hypothesis and give a two-qubit counterexample beyond it. For fragmented environments I derive a redundancy ⇒ erasure lower bound and, in the independentproduct case, obtain exponential visibility suppression with a dimension-independent constant c ( δ ) ≥ −1 2log (1 −δ2 ). I extend PSF to algebraic QFT; in this revision I restrict claims to settings where existence of Et is guaranteed (e.g., wedges in Minkowski vacuum via Bisognano– Wichmann or half-sided modular inclusions), and I provide a concrete formulation for such cases. I define nonnegative “past rates” from Bures–Fisher and Araki relative entropy and introduce a “past curvature” diagnostic. New in this revision: (i) complete proof of the L2 φ martingale convergence theorem for PSF with links to noncommutative martingale theory; (ii) a full proof of the approximate delayed-choice bound in L2 φ with explicit constants; (iii) a classical reduction showing PSF collapses to classical filtrations (Bhattacharyya/Chernoff); (iv) tightened AQFT hypotheses (wedges or half-sided modular inclusions); (v) expanded experimental protocols (swap tests vs. direct fidelity estimation; QCB estimation; redundancy/rate estimation) with finite-sample guidance; (vi) bibliographic additions (Tomiyama/Choi–Effros, Pisier–Xu/Junge, Wiesbrock/Borchers) and minor consistency fixes. Contents 1 Notation glossary (selected symbols) 3 2 Preliminaries and core identities 3 2.1 Faithful, phi-preserving conditional expectations and the predual ........... 4 3 Structural results for PSF 5 3.1 Approximate delayed choice (finite-dimensional L2 φbound) ............... 6 4 Macro-record limits: redundancy and erasure 7 5 Correlated environments: guaranteed bounds and a conjecture 8 1 6 Relation to Quantum Darwinism and Spectrum Broadcast Structures 9 7 Relativistic generalizations: PSF in algebraic QFT 9 8 Double-slit and delayed choice (equational) 10 9 Information geometry of the past 10 9.1 Past curvature ....................................... 10 9.2 L2 phi martingale convergence and literature links ................... 11 10 Consistent histories via PSF (with compatibility conditions) 11 11 Operational protocols and finite-sample guidance 12 12 Classical reduction and commutative limit 12 13 Counterexample without phi-invariant normalizer 13 Cat model A: fragmented qubit recorder (redundant pure records) 13 Cat model B: bosonic pointer via coherent states 13 Referee-Friendly Summary (assumptions, novelty, reproducibility) • Scope. Separable Hilbert spaces, with extensions to type III von Neumann algebras in AQFT. I assume von Neumann subalgebras At⊆ B ( HR )equipped with faithful, normal, φ -preserving conditional expectations Et : B ( HR ) →At forming a filtration ( t1≤t2⇒At1⊆At2 and Et1 = Et1◦Et2 ). Here “ φ -preserving” means φ◦Et = φ . For QFT, I restrict to regimes where Et is known to exist: (a) wedge algebras in Minkowski vacuum under Bisognano–Wichmann; or (b) increasing families obtained from half-sided modular inclusions (HSMI) so that σφ -invariance holds (Theorem 7.2). • Known ingredients. Uhlmann fidelity F[ 1 ], trace distance D, Fuchs–van de Graaf [ 2 ], quantum Chernoff bound/exponent [ 3 ], Stinespring/Naimark [ 4 , 5 ], SLD–QFI [ 6 ], Tomita–Takesaki [ 7 , 8 ], Tomiyama/Choi–Effros on conditional expectations [ 23 , 24 ], noncommutative martingales [ 25 , 26 ], modular inclusions in AQFT [28,29]. • Convention. Visibility is the root fidelity V( ρ, σ ) := pF(ρ, σ)∈ [0 , 1], optimized over reversible record operations on R (isometries/unitaries); fidelity invariance under isometries gives V max = V (Theorem 2.5). Complementarity is V 2 + D 2≤ 1(Theorem 2.6). Operationally: Fvia swap tests/overlap estimation or direct fidelity estimation (DFE); Dvia Helstrom discrimination; ξQCB via multi-copy error exponents; Vcoincides with interferometric fringe visibility when paths are balanced (Section 8). •New contributions. (a) PSF dilation/instrument with explicit expectation structure (Theorem 3.1). Heisenberg/ Schrödinger pictures are linked by the predual (Et)∗. (b) Delayed-choice invariance with φ -invariant normalizer (Theorem 3.4) and an explicit two-qubit counterexample outside the hypothesis (Section 13). 2 (c) Redundancy ⇒erasure (lower bound; equality in product case) (Theorems 4.2 and 4.3). (d) Product-case suppression with Chernoff–visibility bounds. I prove a universal upper bound V( ρL R, ρR R ) ≤exp−1 2PmξQCB ( ρL m, ρR m )  , and equality of Chernoff exponents (hence the two-sided bounds e−PmξQCB(ρL m,ρR m)≤ V ≤e−1 2PmξQCB(ρL m,ρR m) ) holds when all fragment pairs share a common Chernoff minimizer (e.g., i.i.d.). (Theorems 4.4 and 4.5.) (e) Correlated environments: safe bounds (Theorem 5.1) and a correlation-penalized subadditivity conjecture (Theorem 5.2) informed by sandwiched Rényi recoverability [20–22]. (f) Relativistic PSF restricted to cases with guaranteed expectations: wedges or HSMI-generated nests (Theorems 7.1 to 7.3). (g) Rates. Bures–Fisher past rate γ ( t )and Araki-entropy past rate γA ( t ) := 1 2 d dt S ( ωL t∥ωR t ) ≥ 0 (Theorem 9.2); past curvature diagnostics (Section 9.1). (h) L2 φ martingale convergence (Theorem 9.6) and approximate delayed-choice with full proof (Theorem 3.6). (i) Classical reduction (filtrations of σ-algebras; Bhattacharyya/Chernoff) (Section 12). (j) Experimental protocols with finite-sample guidance (Section 11). • What is testable. F (hence V = √F ) via two-copy overlap/DFE estimators; bounds on ξQCB from V via the sandwich (Eq. (3) ); multi-copy tests if one wants asymptotic ξQCB directly; Rδ via local tomography; Vvia reversible erasure operations; rates via time-resolved tomography. • Limitations. Correlation-sensitive upper bounds beyond the product case remain open; our subadditivity is conjectural. AQFT claims are limited to wedges or HSMI nests where σφ - invariance is guaranteed. 1 Notation glossary (selected symbols) Symbol Meaning F(ρ, σ)Uhlmann fidelity; V(ρ, σ) = pF(ρ, σ)(visibility). D(ρ, σ) = 1 2∥ρ−σ∥1Trace distance; V2+ D2≤1. Q(ρ, σ)Chernoff overlap; Q(ρ, σ) = mins∈[0,1] Tr(ρsσ1−s), ξQCB =−log Q. (At, Et)PSF algebra and φ-preserving conditional expectation. N(At)Normalizer of Atin U(HR). RδRedundancy: |{m:dm=1 2∥ρL m−ρR m∥1≥δ}|. dPSF Erasure cost: minimal number of fragments to trace to equalize marginals. 2 Preliminaries and core identities Let HS,HR be separable; B ( H )bounded operators; D ( H )density operators. CPTP maps Λ( X ) = PkKkXK† k,PkK† kKk=1. 3 2.1 Faithful, φ-preserving conditional expectations and the predual Theorem 2.1 (Takesaki).Let A⊆ M be von Neumann algebras and let φ be a faithful normal state on M . There exists a faithful normal conditional expectation E : M→A with φ◦E = φ iff A is invariant under the modular group σφ t[7,8]. Remark 2.2 (Uniqueness of φ -preserving expectations).If A⊂M is invariant under σφ t , then the φ -preserving conditional expectation E : M→A given by Takesaki is unique. I use this uniqueness in Theorem 3.4 when concluding AdUR◦Et◦Ad−1 UR=Et. Remark 2.3 (Tomiyama/Choi–Effros context).Norm-one projections onto von Neumann subalgebras are conditional expectations (Tomiyama) and unital completely positive idempotent maps onto operator systems carry a C∗-algebra structure (Choi–Effros); see [23,24]. Definition 2.4 (Predual map).For a normal CP map E : B ( HR ) → B ( HR )(Heisenberg picture), its predual E∗ : T1 ( HR ) → T1 ( HR )(Schrödinger picture) is the unique normal CP map satisfying Tr[E(X)ρ] = Tr[X E∗(ρ)] for all X∈ B(HR)and trace-class ρ. I write (Et)∗for the predual of Et. Record pair and visibility. Given a balanced two-path preparation, the records induce ( ρL R, ρR R ) ∈ D ( HR ) 2 . Write Uhlmann fidelity F ( ρ, σ ) := ∥√ρ√σ∥2 1 and trace distance D ( ρ, σ ) := 1 2∥ρ−σ∥1 . Define visibility V ( ρ, σ ) := pF(ρ, σ) and (operationally) optimize only over reversible record operations on R(isometries/unitaries); the optimum equals V(ρ, σ). Lemma 2.5 (Visibility as root fidelity).With the operational class restricted to isometries/unitaries on R, Vmax(ρL R, ρR R)=V(ρL R, ρR R) = qF(ρL R, ρR R).(1) Proof. Uhlmann’s theorem gives pF(ρ, σ) = maxU∈U(HR)Tr√ρ√σ U . In the interferometric setting, reversible operations on Ract as Uon one arm, so the achievable fringe amplitude equals the RHS; the maximizer is the partial isometry from the polar decomposition √σ√ρ = W|√σ√ρ| , i.e. U=W†. Lemma 2.6 (Complementarity). V(ρ, σ)2+D(ρ, σ)2≤1,1−D(ρ, σ)≤V(ρ, σ)≤q1−D(ρ, σ)2. (Fuchs–van de Graaf [2].) PSF filtration and tower property. A PSF is a family ( At,Et )with faithful normal φ and φ◦Et = φ such that t1≤t2⇒ At1⊆ At2 and Et1 = Et1◦Et2 = Et2◦Et1 . Taking preduals reverses the composition order. Proposition 2.7 (Monotonicity along the filtration).For t1≤t2, F(Et1)∗ρL R,(Et1)∗ρR R≥F(Et2)∗ρL R,(Et2)∗ρR R≥F(ρL R, ρR R),(2) hence V(t)is nonincreasing in t. Proof. Let t1≤t2 . The tower property implies Et1◦Et2 = Et1 = Et2◦Et1 . Passing to preduals and using (Et2◦Et1)∗= (Et1)∗gives (Et1)∗◦(Et2)∗= (Et1)∗. 4 . Apply fidelity monotonicity under the CPTP post - processing ( Et1 ) ∗ to the pair (( Et2 ) ∗ρL R, ( Et2 ) ∗ρR R ): F(Et1)∗(Et2)∗ρL R,(Et1)∗(Et2)∗ρR R≥F(Et2)∗ρL R,(Et2)∗ρR R. Since (Et1)∗◦(Et2)∗= (Et1)∗, the left-hand side equals F((Et1)∗ρL R,(Et1)∗ρR R), proving F(Et1)∗ρL R,(Et1)∗ρR R≥F(Et2)∗ρL R,(Et2)∗ρR R. The second inequality F (( Et2 ) ∗ρL R, ( Et2 ) ∗ρR R ) ≥F ( ρL R, ρR R )is the usual data - processing inequality under (Et2)∗. Hence the claim. Theorem 2.8 (Data processing for visibility).For any CPTP map Λon R ,V(Λ ρL R, Λ ρR R ) ≥ V(ρL R, ρR R). Proof. Immediate from monotonicity of Uhlmann fidelity F (Λ ρ, Λ σ ) ≥F ( ρ, σ ); taking square roots preserves the inequality. Equality condition. Equality holds for a given CPTP map Λiff Λis sufficient for the pair ( ρL R, ρR R )in the sense of Petz—i.e., there exists a CPTP recovery map R with R◦Λ(ρL/R R)=ρL/R R[13,14]. Theorem 2.9 (Chernoff–visibility sandwich).Let Q ( ρ, σ ) := min0≤s≤1Tr ρsσ1−s and ξQCB := −log Q. Then V(ρ, σ)2≤e−ξQCB(ρ,σ)≤V(ρ, σ).(3) Proof. By Audenaert et al. [3] ,F( ρ, σ ) ≤Q ( ρ, σ ) ≤pF(ρ, σ) . Taking −log and noting V = √F yields the stated inequalities. Theorem 2.10 (QFI–visibility law).For ρS ( θ ) = 1 21 Veiθ Ve−iθ 1 , the single-copy SLD quantum Fisher information is FQ= V2and for Ncopies F(N) Q=NV2[6]. Proof. Write ρ ( θ ) = 1 2I + V( cos θ σx + sin θ σy )  , i.e., a unitary encoding ρ ( θ ) = e−iθσz/2ρ (0) eiθσz/2 with generator H = σz/ 2. Let {λ±,|±⟩} be the spectral decomposition of ρ (0), where λ± = (1 ± V) / 2 and {|±⟩} are eigenstates of σx. The standard unitary-family formula gives FQ= 2 X i,j (λi−λj)2 λi+λj|⟨i|H|j⟩|2. Only the (+ ,− )and ( −, +) terms contribute; here ( λ+−λ− ) 2/ ( λ+ + λ− ) = V 2 and |⟨ + |H|−⟩|2 = 1 4 (since ⟨ + |σz|−⟩ = 1). Hence FQ = 2 · V 2·1 4 + 2 · V 2·1 4 = V 2 . Additivity over i.i.d. copies yields F(N) Q=NV2. 3 Structural results for PSF Theorem 3.1 (PSF dilation, instrument, and expectation structure).Let A⊆ B ( HR )admit a faithful normal φ -preserving conditional expectation E : B ( HR ) →A . Then there exist a Hilbert space K , an isometry V : HR→ HR⊗K , and a PVM {Pr} on K such that in the Heisenberg picture E(X)=V†(X⊗IK)V, X ∈ B(HR),(4) 5 and the associated instrument on states (Schrödinger picture) is Ir(ρ)=(I⊗Pr)V ρV †(I⊗Pr), E∗(ρ) = TrKV ρV †=X r Vrρ V † r, Vr:= (I⊗⟨r|)V. (5) Moreover, E is idempotent with range A and satisfies the A -bimodule property E ( aXb ) = a E ( X ) b for all a, b ∈A and X∈ B ( HR ); conversely, a normal unital CP idempotent map onto a von Neumann subalgebra with the bimodule property is a conditional expectation [13,14]. Remark 3.2 (Expectation structure; Naimark/Kraus in the commutant).Writing V = PrVr⊗|r⟩ (for any ONB {|r⟩}⊂K ) defines a POVM {Mr := V† rVr} with PrMr = I . By Naimark, there is a PVM {Pr}on Ksuch that Ir(ρ)=(I⊗Pr)V ρV †(I⊗Pr)implements {Mr}on R. Kraus operators in the commutant. When E is a conditional expectation onto a von Neumann subalgebra A⊂B ( HR )(Tomiyama), one can choose a Stinespring dilation with Kraus operators in the commutant A′ , i.e., E ( X ) = PjK† jXKj with each Kj∈A′ and PjK† jKj = 1. This makes the A -bimodule property immediate: E ( aXb ) = PjK† jaXbKj = a E ( X ) b for all a, b ∈A . See Tomiyama’s theorem and related structure results (e.g., Kawahigashi [ 27 ]) for expectations commuting with subalgebras. Definition 3.3 (Normalizer).For a von Neumann subalgebra At⊆ B ( HR ), its normalizer is N(At)={UR∈ U(HR) : URAtU† R=At}. Theorem 3.4 (Delayed-choice invariance: φ -invariant normalizer criterion).Let Et : B ( HR ) →At be the φ -preserving conditional expectation (Takesaki), and let U = US⊗UR with UR∈ N ( At )and φ◦AdUR=φ. Then the Heisenberg maps (id ⊗Et)and AdUcommute: (id ⊗Et)◦AdU= AdU◦(id ⊗Et).(6) Consequently, in the Schrödinger picture (id ⊗(Et)∗)◦AdU= AdU◦(id ⊗(Et)∗).(7) Proof sketch. Since URAtU† R = At and φ◦AdUR = φ , the map AdUR◦Et◦Ad−1 UR is a φ -preserving conditional expectation onto At . By Takesaki’s uniqueness, it equals Et . Tensoring with id on S yields the claim. Lemma 3.5 (State-dependent inner-product stability).Let φ ( X ) = Tr ( ρX )be faithful on a finite-dimensional B(HR)and set κ2 φ:= ∥ρ−1∥∞= 1/λmin(ρ). Then for all X, Y , ⟨U†X, U†Y⟩φ−⟨X, Y ⟩φ=Tr (ρ−UρU†)X†Y≤2εtκ2 φ∥X∥2,φ∥Y∥2,φ, where εt:= 1 2∥φ−φ◦AdUR∥1and ∥Z∥2 2,φ = Tr(ρZ†Z). 3.1 Approximate delayed choice (finite-dimensional L2 φbound) I quantify deviations when the normalizer/ φ -invariance hypotheses are only approximate. Let ⟨X, Y ⟩φ := φ ( X†Y )and ∥X∥2,φ := q⟨X, X⟩φ . Write Pt for the orthogonal projection on the φ -GNS Hilbert space onto the closure of At; for a φ-preserving expectation Et,Ptis implemented by Et. 6 Two deviation parameters. For UR∈ U(HR), define η+ t:= sup A∈At ∥A∥2,φ≤1 ∥AdUR(A)−A∥2,φ and η− t:= sup B∈At ⊥ ∥B∥2,φ≤1 ∥PtAdUR(B)∥2,φ .(8) Thus η+ t = ∥ ( I−Pt ) UPt∥ and η− t = ∥PtU ( I−Pt ) ∥ as operator norms on L2 φ , where U denotes the (bounded) operator on L2 φimplementing AdUR. Theorem 3.6 (Approximate delayed choice in L2 φ ).Assume finite-dimensional HR . For every Y∈ B(HR), Et(AdUR(Y)) −AdUR(Et(Y))  2,φ ≤max{η+ t, η− t} ∥Y∥2,φ.(9) Equivalently, at the operator level on L2 φone has the exact identity ∥PtU−UPt∥= max ∥PtU(I−Pt)∥,∥(I−Pt)UPt∥= max{η− t, η+ t}.(10) If moreover φ◦AdUR = φ (so that U acts unitarily on L2 φ ), the identity (10) still holds with the same constant. Work on L2 φ, where Etis the orthogonal projection Ptonto At. Then Et◦AdUR−AdUR◦Et↔PtU−UPt="0PtU(I−Pt) −(I−Pt)UPt0# in the orthogonal decomposition L2 φ=At⊕At⊥. Remark 3.7 (Connecting to state-drift parameters).The deviation pair ( η+ t, η− t )is intrinsic and requires no auxiliary assumptions. If one additionally quantifies departure from φ -invariance by εt := 1 2∥φ−φ◦AdUR∥1 and sets κ2 φ := ∥ρ−1∥∞ for φ ( X ) = Tr ( ρX )(finite B ( HR )), then Lemma 3.5 implies that U isa(1 + O ( κ2 φεt )) near - isometry on L2 φ . In symmetric near - normalizer regimes (empirically, η− t≈η+ t ), the bound (9) simplifies to  Et ( AdURY ) −AdUR ( EtY )  2,φ ≲η+ t∥Y∥2,φ up to an O(κ2 φεt)factor. 4 Macro-record limits: redundancy and erasure Assume HR=Nn m=1 Hm,ρL/R m= Tr=mρL/R R, and dm:= 1 2∥ρL m−ρR m∥1. Definition 4.1 (PSF erasure cost and redundancy).For U⊆ { 1 , . . . , n} write TrU := TrNm∈UHm . The erasure cost is dPSF(ρL R, ρR R) := min|U|: TrUρL R= TrUρR R,(11) and, for δ∈[0,1], the redundancy is Rδ:= {m:dm≥δ}. Theorem 4.2 (Redundancy lower-bounds erasure).Any erasure set U that equalizes the marginals must include all indices mwith dm≥δ. Consequently, dPSF(ρL, ρR)≥Rδ.(12) This bound is tight in the product/independence case (see Theorem 4.3). 7 Proof. If some m with dm≥δ is not traced, then ρL m = ρR m persists in the marginal, contradicting equality of TrUρLand TrUρR. Corollary 4.3 (Equality condition in the product case).If ρL/R R = NmρL/R m and S := {m : dm> 0 } , then dPSF =|S|. Hence dPSF =Rδfor any 0< δ ≤minm∈Sdm. Theorem 4.4 (Product-case exponential suppression).If ρL/R R = NmρL/R m , then fidelity factorizes and V(ρL R, ρR R) = Y m V(ρL m, ρR m).(13) For each fragment the Chernoff–visibility sandwich (3)gives e−ξQCB(ρL m,ρR m)≤ V( ρL m, ρR m ) ≤e−1 2ξQCB(ρL m,ρR m) . Multiplying over m yields the unconditional two-sided bounds e−PmξQCB(ρL m,ρR m)≤V(ρL R, ρR R)≤e−1 2PmξQCB(ρL m,ρR m).(14) -Moreover, the total Chernoff exponent satisfies +Moreover, the total Chernoff exponent satisfies −ξQCB(ρL R, ρR R)≥X m ξQCB(ρL m, ρR m),+ξQCB(ρL R, ρR R)≤X m ξQCB(ρL m, ρR m),(15) with equality iff the minimizing s⋆∈ [0 , 1] for Q ( ρL m, ρR m ; s )is common to all m (e.g., i.i.d.). In particular, if dm≥δfor at least Rδfragments, then V≤exp −1 2Rδc(δ), c(δ) := inf{ξQCB(ρ, σ) : 1 2∥ρ−σ∥1≥δ} ≥ −1 2log(1 −δ2).(16) Corollary 4.5 (Cat stability).For macroscopic redundancy, V(ρL R, ρR R)→0exponentially in Rδ. 5 Correlated environments: guaranteed bounds and a conjecture Theorem 5.1 (Monotonicity under fragment selection).For any subset S⊆ { 1 , . . . , n} and partial trace ΛS:= TrSc, ξQCB(ρL R, ρR R)≥ξQCB ρL S, ρR S,V(ρL R, ρR R)≤V ρL S, ρR S.(17) In particular, ξQCB(ρL R, ρR R)≥maxmξQCB(ρL m, ρR m). Conjecture 5.2 (Correlation-aware subadditivity).There exist correlation functionals C ( ρL R, ρR R )(e.g., based on sandwiched Rényi divergences) such that ξQCB(ρL R, ρR R)≥X m ξQCB(ρL m, ρR m)− C(ρL R, ρR R),(18) with C = 0 in the product case and C ≥ 0otherwise; cf. monotonicity/recoverability results in Frank and Lieb [20], Beigi [21], Müller-Lennert et al. [22]. 8 6 Relation to Quantum Darwinism and Spectrum Broadcast Structures Context. Quantum Darwinism (QD) explains objectivity via the proliferation of redundant records into many environment fragments [ 15 , 16 ]. Spectrum Broadcast Structures (SBS) capture objectivity by requiring fragment states conditioned on pointer outcomes to commute and be broadcast in many copies [18,19]. PSF’s additions. PSF packages redundancy and objectivity within a filtration of φ -preserving conditional expectations, tying: (i) redundancy to a concrete erasure cost lower bound dPSF ≥Rδ (Theorems 4.2 and 4.3); (ii) product-case scaling via a Chernoff–visibility upper bound and, when a common Chernoff minimizer exists (e.g., i.i.d.), the two-sided bounds (Theorem 4.4); and (iii) delayed-choice invariance to an explicit φ -invariant normalizer criterion (Theorem 3.4). The explicit c ( δ ) ≥−1 2log (1 −δ2 )bound links redundancy plateaus from QD to exponential visibility suppression; cf. Zwolak, Riedel, and Zurek [17] for QCB-based redundancy rates. 7 Relativistic generalizations: PSF in algebraic QFT I work with a Haag–Kastler net O7→ A ( O ) ⊆ B ( H )and restrict to regimes where modular/split hypotheses ensure the existence of φ-preserving conditional expectations. Standing modular-invariance assumption (made explicit). Fix a Rindler wedge W in Minkowski spacetime and let φ be the vacuum state. Let σφ s denote the modular group of ( A ( W ) , φ ) (Bisognano–Wichmann identifies σφ s with the one-parameter boost subgroup preserving W ). I consider any increasing family {At}t∈Iof von Neumann subalgebras with At⊂A(W)and σφ s(At)⊆Atfor all s∈Rand all t∈I. (19) By Takesaki, for each tthere is a unique φ-preserving conditional expectation Et:A(W)→At. Definition 7.1 (Relativistic PSF filtration: wedge case).Let {At}t∈I⊂A ( W )satisfy (19) and be increasing in t . Define the filtration by ( At, Et ) t with Et as above. For two normal states ωL/R on A ( W )set Vt := qF(ωL◦Et, ωR◦Et) . -Concrete example. If B0⊂A ( W )is any σφ -invariant subalgebra (e.g., a fixed-point algebra of a compact symmetry acting along the wedge edge), then - −At := vN σφ u ( B0 ) : |u|≤t− -is increasing in t and each At obeys (19) . +Concrete (trivial) example. If B0⊂A ( W )is fixed by σφ (e.g., the fixed-point +algebra of a compact symmetry along the wedge edge), then the constant family At≡B0+for all tsatisfies (19). ++++ Definition 7.2 (Relativistic PSF via HSMI).Alternatively, suppose ( At ) t≥0 arises from a half-sided modular inclusion A0⊂A with respect to ( A, φ )so that σφ s ( At ) ⊆At for all s≥ 0and t≥ 0. Then the φ-preserving conditional expectations Et:A→Atexist by [28,29] and form a PSF. Theorem 7.3 (No retrocausality in QFT (equivariance form)).If U = US⊗UR with UR∈ N ( At ) and φ◦AdUR=φ, then in the Schrödinger picture (id ⊗Et∗)◦AdU= AdU◦(id ⊗Et∗).(20) Hence for any normal state ρand Xon HS, DX⊗I, U (id ⊗Et∗)(ρ)U†ETr =DX⊗I, (id ⊗Et∗) UρU†ETr .(21) 9