Chapter 3: Markov Splicing and Information Recovery

Source Theory: euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md, §3.2-3.4

Introduction

In the previous chapter, we established the Null-Modular double cover structure of causal diamond chains. Now we face a key question: How to “splice” multiple adjacent causal diamonds into larger composite regions?

This is not simple geometric puzzle-solving, but involves profound information-theoretic principles. This chapter will demonstrate:

• Markov Splicing: Under specific conditions, modular Hamiltonians of adjacent regions can be losslessly spliced through inclusion-exclusion principle
• Inclusion-Exclusion Identity: Addition rules for modular Hamiltonians
• Information Recovery: Reconstructing complete states from partial information through Petz maps
• Non-Totally-Ordered Gap: Quantitative characterization of information loss when ideal conditions are violated

Everyday Analogy: Imagine splicing three audio segments: A→B→C. If segment B contains all correlation information between A and C, then you can losslessly splice A-B-C. But if there exists “hidden correlation” between A and C that segment B lacks, splicing will produce an information gap.

1. Inclusion-Exclusion Identity: Addition and Subtraction of Modular Hamiltonians

1.1 Inclusion-Exclusion of Monotonic Half-Spaces

Consider multiple half-space regions $\{R_{V_i}{\}}_{i=1}^N$ on the same null hyperplane (e.g., $E^{+}$), where $V_i(x_{\perp })$ are transversely dependent threshold functions.

Theorem B (Inclusion-Exclusion Identity): $$K_{{\cup }_i{R}_{V_i}}=\sum _{k=1}^N(-1{)}^{k-1}\sum _{1\le i_1<\cdots <i_k\le N}{K}_{R_{V_{i_1}}\cap \cdots \cap R_{V_{i_k}}}$$

Proof Core: Pointwise geometric identity $$(v-\underset{i}{\min }{V}_i{)}_{+}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{|I|=k}(v-\underset{i\in I}{\max }{V}_i{)}_{+}$$

Multiplying by second-order response kernel $2\pi T_{vv}$ and integrating gives the quadratic form inclusion-exclusion.

Mermaid Diagram: Inclusion-Exclusion Principle

graph TD
    A["Region Union<br/>${\cup }_i{R}_{V_i}$"] --> B["Single Region Contributions<br/>$+K_{V_1}+K_{V_2}+K_{V_3}$"]
    B --> C["Double Intersection Corrections<br/>$-K_{V_1\cap V_2}-K_{V_1\cap V_3}-K_{V_2\cap V_3}$"]
    C --> D["Triple Intersection Correction<br/>$+K_{V_1\cap V_2\cap V_3}$"]
    D --> E["Complete Modular Hamiltonian<br/>$K_{{\cup }_i{R}_{V_i}}$"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffe1f5
    style D fill:#f5e1ff
    style E fill:#e1ffe1

Everyday Analogy: Area of three circles: $S_{\text{total}}=S_1+S_2+S_3-S_{12}-S_{13}-S_{23}+S_{123}$. Modular Hamiltonian inclusion-exclusion follows exactly the same logic!

1.2 Distributed Regularization and Closure

Technical Point: Indicator function $1_{[a,\infty )}$ is not smooth, needs smoothing:

$$1_{[a,\infty )}^{\eta }:={\rho }_{\eta }\ast 1_{[a,\infty )}$$

where ${\rho }_{\eta }$ is a standard smoothing kernel. Define smoothed version of positive part function:

$$(x{)}_{+}^{\eta }:={\int }_{-\infty }^x{1}_{[0,\infty )}^{\eta }(t)dt$$

Strict Form of Inclusion-Exclusion Identity:

1. Smoothing: Construct $V_i^{\eta }$ for each $V_i$
2. Inclusion-Exclusion: Prove inclusion-exclusion identity for smoothed version
3. Limit: Let $\eta \to 0^{+}$, exchange limit and integral using dominated convergence theorem

Proposition B (Closure): Let $\mathfrak{k}:={\mathfrak{k}}_{{\cup }_i{R}_{V_i}}$ be the closed quadratic form of the union domain, with lower bound $a\in \mathbb{R}$. Take any $c>-a$, define shifted graph norm:

$$||\psi {|}_{\mathfrak{k},c}^2:=|\psi {|}^2+(\mathfrak{k}[\psi ]+c|\psi {|}^2)$$

If ${\psi }_n\to \psi$ converges in shifted graph norm, then quadratic form values on both sides of inclusion-exclusion identity converge simultaneously.

Everyday Analogy: Shifted graph norm is like “weighted distance”: not only considers the length of the vector itself, but also its “energy” (quadratic form value). This guarantees mathematical stability of the limit process.

2. Markov Splicing: Perfect Splicing of Totally-Ordered Cuts

2.1 Three-Segment Markovianity

Consider three adjacent causal diamonds $D_{j-1}$, $D_j$, $D_{j+1}$, with totally-ordered cuts on the same null hyperplane.

Theorem C (Markov Splicing):

Vacuum state satisfies the following two equivalent conditions:

(1) Conditional Mutual Information is Zero: $$I(D_{j-1}:D_{j+1}\mid D_j)=0$$

(2) Modular Hamiltonian Identity: $$K_{D_{j-1}\cup D_j}+K_{D_j\cup D_{j+1}}-K_{D_j}-K_{D_{j-1}\cup D_j\cup D_{j+1}}=0$$

Physical Meaning:

• Conditional mutual information zero means: Given information about middle region $D_j$, there is no additional correlation between left $D_{j-1}$ and right $D_{j+1}$
• This is Markovianity: $D_j$ “screens” the correlation between $D_{j-1}$ and $D_{j+1}$

Mermaid Diagram: Markov Splicing

graph LR
    A["$D_{j-1}$<br/>Left Region"] --> B["$D_j$<br/>Middle Region<br/>(Information Barrier)"]
    B --> C["$D_{j+1}$<br/>Right Region"]

    A -.->|"No Direct Correlation<br/>$I(j-1:j+1\Vert j)=0$"| C

    style A fill:#ffe1e1
    style B fill:#e1f5ff
    style C fill:#e1ffe1

Everyday Analogy: Three-person message relay game A→B→C:

• If B completely records A’s original words, information C obtains from B is exactly the same as directly from A
• At this point $I(A:C|B)=0$, B serves as “perfect intermediary”
• If B misses some content, then $I(A:C|B)>0$, an information gap appears

2.2 Combining Inclusion-Exclusion and Markovianity

Derivation: From inclusion-exclusion identity

$$\begin{array}{rl}{K}_{D_{j-1}\cup D_j\cup D_{j+1}}& =K_{D_{j-1}}+K_{D_j}+K_{D_{j+1}}\\ & -K_{D_{j-1}\cap D_j}-K_{D_j\cap D_{j+1}}-K_{D_{j-1}\cap D_{j+1}}\\ & +K_{D_{j-1}\cap D_j\cap D_{j+1}}\end{array}$$

Under totally-ordered cuts, intersections of adjacent regions degenerate:

• Boundary of $D_{j-1}\cap D_j$ shrinks to null set
• $D_{j-1}\cap D_{j+1}=\varnothing$ (not adjacent)

Combined with strong subadditivity (split property): boundary terms vanish.

Using relative entropy identity: $$S({\rho }_{ABC}\Vert {\rho }_A\otimes {\rho }_B\otimes {\rho }_C)=S_{AB}+S_{BC}-S_A-S_B-S_C-S_{ABC}$$

Connected to modular Hamiltonian through first-order variation ($\delta K=2\pi \delta S$), finally obtaining Markov splicing.

2.3 Lower Semicontinuity of Relative Entropy

Proposition C.2: Relative entropy $S(\rho \Vert \sigma )$ is lower semicontinuous in weak$\ast$ topology, and satisfies data processing inequality:

$$S(\rho \Vert \sigma )\ge S(\Phi \rho \Vert \Phi \sigma )$$

for any CPTP map $\Phi$.

Application: Given monotonic approximation $R_{V_{\alpha }}\uparrow R_V$, let ${\Phi }_{\alpha }$ be restriction channel to $R_{V_{\alpha }}$, then:

$$\underset{\alpha \to \infty }{\mathrm{liminf}}{I}_{\alpha }(A:C\mid B)\ge I(A:C\mid B)$$

This guarantees that Markovianity can be stably transmitted from discrete approximation to continuous limit.

3. Non-Totally-Ordered Gap: Stratification Degree and Information Loss

3.1 Definition of Stratification Degree

When cuts are not totally ordered (e.g., cut orders differ at different transverse points $x_{\perp }$), Markovianity fails.

Definition (Stratification Degree): Let $V_i^{\pm }(x_{\perp })$ be threshold functions on layers $E^{\pm }$ respectively, define:

$$\boxed{\kappa (x_{\perp }):=\#\{(a,b):a<b,(V_a^{+}-V_b^{+})(V_a^{-}-V_b^{-})<0\}}$$

Physical Meaning:

• $\kappa (x_{\perp })$ counts the number of inconsistent pairs of cut orders on layers $E^{+}$ and $E^{-}$ at transverse coordinate $x_{\perp }$
• Totally ordered: $\kappa \equiv 0$ (orders on two layers completely consistent)
• Non-totally ordered: $\kappa >0$ (appearance of “crossing”)

Mermaid Diagram: Stratification Degree

graph TD
    A["Totally-Ordered Cut<br/>κ=0"] --> A1["E+ Layer: V1 < V2 < V3<br/>E- Layer: V1 < V2 < V3"]
    A1 --> A2["Two Layers Order Consistent"]

    B["Non-Totally-Ordered Cut<br/>κ > 0"] --> B1["E+ Layer: V1 < V2 < V3<br/>E- Layer: V2 < V1 < V3"]
    B1 --> B2["(V1,V2) Forms Inconsistent Pair"]

    style A fill:#e1ffe1
    style A1 fill:#f0fff0
    style A2 fill:#e1f5e1
    style B fill:#ffe1e1
    style B1 fill:#fff0f0
    style B2 fill:#ffe1f0

Everyday Analogy: Traffic flow on two parallel lanes:

• Totally ordered: Vehicle orders on two lanes completely consistent ($\kappa =0$)
• Non-totally ordered: Vehicle orders on two lanes differ (overtaking), $\kappa$ counts number of “order-reversed” vehicle pairs

3.2 Markov Gap Line Density

Theorem C’ (Markov Gap for Non-Totally-Ordered):

Define Markov gap line density $\iota (v,x_{\perp })\ge 0$, satisfying:

$$I(D_{j-1}:D_{j+1}\mid D_j)=\iint \iota (v,x_{\perp })dv{d}^{d-2}{x}_{\perp }$$

and $\iota$ is monotone non-decreasing in $\kappa$.

Physical Meaning:

• $\iota (v,x_{\perp })$ characterizes local information leakage rate at spacetime point $(v,x_{\perp })$
• Integration gives total conditional mutual information $I(D_{j-1}:D_{j+1}\mid D_j)$
• Totally ordered: $\kappa =0$, then $\iota =0$, $I=0$ (perfect Markov)
• Non-totally ordered: $\kappa >0$, then $\iota >0$, $I>0$ (gap appears)

Lemma C.1 (Stratification Degree–Gap Comparison):

If $V_i^{\pm }$ are piecewise $C^1$ with finite crossing times, then there exists constant $c_{\ast }>0$ such that:

$$\iota (v,x_{\perp })\ge c_{\ast }\kappa (x_{\perp })1_{\{v\in [v_{-}(x_{\perp }),v_{+}(x_{\perp })]\}}$$

where $v_{-}(x_{\perp }):={\min }_i{V}_i^{+}(x_{\perp })$, $v_{+}(x_{\perp }):={\max }_i{V}_i^{+}(x_{\perp })$.

Quantitative Lower Bound: Combining with Fawzi-Renner inequality:

$$I(A:C\mid B)\ge -2\ln F({\rho }_{ABC},{\rho }_A\otimes {\rho }_{BC})$$

gives explicit lower bound estimate for the gap.

Everyday Analogy: Highway intersection (non-totally-ordered cut):

• More intersection points (larger $\kappa$)
• Higher traffic management difficulty (higher $\iota$)
• Total traffic delay ($I$) proportional to intersection complexity

4. Petz Recovery Map: Information Reconstruction

4.1 Problem Setup

Quantum Recovery Problem:

• Initial state: ${\rho }_{ABC}$ (three-region composite state)
• Operation: Forget subsystem $C$, obtain ${\rho }_{AB}={\mathrm{Tr}}_C[{\rho }_{ABC}]$
• Question: Can we recover original ${\rho }_{ABC}$ from ${\rho }_{AB}$?

Theorem D (Petz Recovery Map):

Denote $A=D_{j-1}$, $B=D_j$, $C=D_{j+1}$. Define forget channel:

$${\Phi }_{BC\to B}(X_{BC})={\mathrm{Tr}}_C[X_{BC}]$$

Its adjoint: $${\Phi }^{\ast }(Y_B)=Y_B\otimes {\mathbb{I}}_C$$

Take reference state ${\sigma }_{BC}={\rho }_{BC}$ (self-reference), Petz recovery map is defined as:

$$\boxed{{\mathcal{R}}_{B\to BC}(X_B)={\sigma }_{BC}^{1/2}({\sigma }_B^{-1/2}{X}_B{\sigma }_B^{-1/2}\otimes {\mathbb{I}}_C){\sigma }_{BC}^{1/2}}$$

where inverse is taken as pseudo-inverse on $\mathrm{supp}({\sigma }_B)$.

Perfect Recovery Condition:

$$({\mathrm{id}}_A\otimes {\mathcal{R}}_{B\to BC})({\rho }_{AB})={\rho }_{ABC}$$

If and only if $$I(A:C\mid B)=0$$

Mermaid Diagram: Petz Recovery

graph LR
    A["Initial State<br/>${\rho }_{ABC}$"] -->|"Forget $C$"| B["Partial State<br/>${\rho }_{AB}$"]
    B -->|"Petz Recovery<br/>${\mathcal{R}}_{B\to BC}$"| C["Recovered State<br/>${\tilde{\rho }}_{ABC}$"]

    A -.->|"Perfect Recovery<br/>$I(A:C\Vert B)=0$"| C

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#e1ffe1

Everyday Analogy:

• Initial: You have complete video file (${\rho }_{ABC}$)
• Loss: Deleted audio track (forgot $C$), only picture remains (${\rho }_{AB}$)
• Recovery: Petz map attempts to reconstruct audio from picture
  • If picture contains subtitles and $I(A:C|B)=0$, can perfectly recover dialogue
  • If picture lacks key information ($I(A:C|B)>0$), recovery is imperfect

4.2 Rotation Average and Stability

Technical Point: Unrotated Petz map may not satisfy fidelity inequality, needs rotation average.

Rotation-Averaged Petz Map ${\mathcal{R}}_{B\to BC}^{\mathrm{rot}}$ satisfies:

$$I(A:C\mid B)\ge -2\ln F\left({\rho }_{ABC},({\mathrm{id}}_A\otimes {\mathcal{R}}_{B\to BC}^{\mathrm{rot}})({\rho }_{AB})\right)$$

Equivalently, fidelity lower bound:

$$F\ge e^{-I(A:C\mid B)/2}$$

Physical Meaning:

• Smaller conditional mutual information $I(A:C\mid B)$, higher recovery fidelity $F$
• When $I=0$, $F=1$ (perfect recovery)
• When $I>0$, $F<1$ (lossy recovery)

Convention (Fidelity Definition): This text uses Uhlmann fidelity (unsquared):

$$F(\rho ,\sigma ):=|\sqrt{\rho }\sqrt{\sigma }{|}_1\in [0,1]$$

Fawzi-Renner Inequality:

$$I(A:C\mid B)\ge -2\ln F$$

provides an operational lower bound for conditional mutual information.

4.3 Geometric Picture of Recovery Error

Intuitive Understanding:

Let state space be high-dimensional Hilbert space, then:

• Markov State Manifold: States satisfying $I(A:C\mid B)=0$ form low-dimensional submanifold
• Non-Markov States: Deviate from this manifold, deviation measured by $I(A:C\mid B)$
• Petz Recovery: “Projects” ${\rho }_{AB}$ back to Markov manifold
• Fidelity $F$: Measures distance before and after projection

Everyday Analogy: GPS positioning error:

• Ideal GPS signals (Markov states): Three satellite signals $A,B,C$ satisfy $I(A:C|B)=0$
• Actual signals (with error): Signals have noise correlation, $I(A:C|B)>0$
• Positioning algorithm (Petz recovery): Attempts to recover $C$ information from $A,B$
• Positioning accuracy (fidelity $F$): Depends on signal noise level $I(A:C|B)$

5. Half-Sided Modular Inclusion: Skeleton of Algebraic Advancement

5.1 Definition of HSMI

Half-Sided Modular Inclusion (HSMI) is a core concept in algebraic quantum field theory.

Definition: Let $\mathcal{A}(D_j)\subset \mathcal{A}(D_{j+1})$ be inclusion relation of two von Neumann algebras, vacuum state $\Omega$ be cyclic separating vector. This inclusion is called right HSMI if there exists positive energy one-parameter semigroup $\{U(s){\}}_{s\ge 0}$ satisfying:

1. Covariance: $$U(s)\mathcal{A}(D_j)U(s{)}^{-1}\subset \mathcal{A}(D_j),\forall s\ge 0$$

2. Modular Flow Relation: $${\Delta }_{\mathcal{A}(D_{j+1})}^{\mathrm{i}t}=U(t){\Delta }_{\mathcal{A}(D_j)}^{\mathrm{i}t}U(-t)$$

where $\Delta$ is Tomita-Takesaki modular operator.

Theorem E (HSMI Advancement):

If $(\mathcal{A}(D_j)\subset \mathcal{A}(D_{j+1}),\Omega )$ is right HSMI, then there exists positive energy one-parameter semigroup covariant with ${\Delta }_{\mathcal{A}(D_{j+1})}^{\mathrm{i}t}$, intrinsically advancing $\mathcal{A}(D_j)$ to $\mathcal{A}(D_{j+1})$.

Physical Meaning:

• HSMI provides algebraic progressive structure of causal diamond chains
• Modular flow ${\Delta }^{\mathrm{i}t}$ geometrizes along chain as Lorentz boost (Bisognano-Wichmann property)
• Positive energy condition guarantees causal structure of quantum field theory

Mermaid Diagram: HSMI Advancement

graph LR
    A["$\mathcal{A}(D_j)$<br/>Small Algebra"] -->|"Inclusion"| B["$\mathcal{A}(D_{j+1})$<br/>Large Algebra"]

    A1["Modular Flow<br/>${\Delta }_j^{it}$"] -.->|"Covariant"| A
    B1["Modular Flow<br/>${\Delta }_{j+1}^{it}$"] -.->|"Covariant"| B

    A1 -->|"Advancement Semigroup<br/>$U(s)$"| B1

    style A fill:#e1f5ff
    style B fill:#e1ffe1
    style A1 fill:#fff4e1
    style B1 fill:#f5e1ff

Everyday Analogy: “Recursive computation” of quantum information:

• $\mathcal{A}(D_j)$: Currently known set of observables
• $\mathcal{A}(D_{j+1})$: Extended set of observables
• HSMI: Guarantees extension process preserves algebraic structure
• Modular flow ${\Delta }^{\mathrm{i}t}$: “Time evolution” rules for observables
• Advancement semigroup $U(s)$: “Evolution operator” from small set to large set

5.2 Wiesbrock-Borchers Structure Theorem

Wiesbrock-Borchers Theorem: HSMI is equivalent to existence of one-parameter unitary group $\{U(s)\}$ satisfying:

1. Borchers Commutation Relations: $$[U(s),{\Delta }_{\mathcal{A}(D_{j+1})}^{\mathrm{i}t}]=0,\forall s,t$$

2. Positive Energy Condition: $$⟨\Omega ,U(s)\Omega ⟩=e^{-Es},E\ge 0$$

3. Advancement Property: $$U(s)\mathcal{A}(D_j)U(s{)}^{-1}\subseteq \mathcal{A}(D_j)$$

Application: In causal diamond chains, HSMI guarantees algebraic consistency of chain advancement:

• Inclusion-exclusion identity (geometric level)
• Modular Hamiltonian splicing (physical level)
• Algebraic inclusion relations (operator level)

The three are fully compatible.

6. Applications: QNEC Chain Strengthening and Entanglement Wedge Splicing

6.1 QNEC Chain Strengthening

Quantum Null Energy Condition (QNEC):

$$⟨T_{vv}⟩\ge \frac{1}{2\pi }\frac{{\partial }^{2}S}{\partial v^2}$$

Near vacuum state, QNEC saturates: $$⟨T_{vv}{⟩}_{\mathrm{vac}}=\frac{1}{2\pi }\frac{{\partial }^2{S}_{\mathrm{vac}}}{\partial v^2}$$

Chain Strengthening: Combining inclusion-exclusion identity with QNEC, obtain inclusion-exclusion lower bound for joint region energy-entropy variation:

$$⟨T_{vv}{⟩}_{{\cup }_i{R}_{V_i}}\ge \frac{1}{2\pi }\frac{{\partial }^2{S}_{{\cup }_i{R}_{V_i}}}{\partial V_1\partial V_2\cdots \partial V_N}$$

When totally ordered, this lower bound takes equality, equivalent to Markov saturation.

6.2 Entanglement Wedge Splicing and Corner Charge

Holographic Duality (AdS/CFT): Boundary inclusion-exclusion/Markov corresponds in bulk to:

• Normal modular flow splicing of extremal surfaces
• Additivity of corner charges

JLMS Equality (Jafferis-Lewkowycz-Maldacena-Suh):

$$S_{\mathrm{boundary}}(R)=S_{\mathrm{bulk}}(EW[R])+\frac{A(\partial EW[R])}{4G_N}$$

where $EW[R]$ is the Entanglement Wedge.

Splicing Consistency: Under weak feedback and smooth corner conditions, boundary inclusion-exclusion–Markov lifts to bulk modular flow splicing, maintaining ledger consistency.

Everyday Analogy: Correspondence between map and terrain:

• Boundary inclusion-exclusion: Splicing of regions on map
• Bulk modular flow: “Evolution rules” of terrain
• JLMS equality: Map area $\leftrightarrow$ terrain entropy
• Splicing consistency: Map splicing rules compatible with terrain evolution rules

7. Numerical Verification and Experimental Schemes

7.1 Inclusion-Exclusion Verification

Two-Dimensional CFT Three-Block Chain: Take three causal diamonds $D_1,D_2,D_3$, numerically evaluate:

$${\epsilon }_{\mathrm{excl}}:=|K_{12}+K_{23}-K_2-K_{123}|$$

Expected Results:

• Totally-ordered cuts: ${\epsilon }_{\mathrm{excl}}\approx 0$ (within numerical error)
• Non-totally-ordered cuts: ${\epsilon }_{\mathrm{excl}}>0$, proportional to $\kappa$

Code Framework (conceptual):

Input: Boundary parameters V1, V2, V3 of three regions
Output: Inclusion-exclusion error ε_excl

1. Compute single-region modular Hamiltonians: K1, K2, K3
2. Compute union modular Hamiltonians: K12, K23, K123
3. Compute inclusion-exclusion error: ε_excl = |K12 + K23 - K2 - K123|
4. Plot error bars

7.2 Markov Splicing Verification

Conditional Mutual Information Measurement:

$$I(1:3|2)=S_{12}+S_{23}-S_2-S_{123}$$

Expected Results:

• Totally ordered: $I(1:3|2)\approx 0$
• Non-totally ordered: $I(1:3|2)>0$, satisfying $I\ge c_{\ast }\int \kappa d{x}_{\perp }$

Consistency with Inclusion-Exclusion: Through first-order variation relation $\delta K=2\pi \delta S$, verify:

$${\epsilon }_{\mathrm{excl}}\approx 2\pi I(1:3|2)$$

7.3 Petz Recovery Fidelity

Operational Steps:

1. Prepare initial state ${\rho }_{123}$ (three-region vacuum)
2. Forget $D_3$, obtain ${\rho }_{12}$
3. Apply Petz recovery ${\mathcal{R}}_{2\to 23}$, obtain ${\tilde{\rho }}_{123}$
4. Compute fidelity $F({\rho }_{123},{\tilde{\rho }}_{123})$

Expected Results:

• Totally ordered ($I(1:3|2)=0$): $F\approx 1$
• Non-totally ordered ($I(1:3|2)>0$): $F\approx e^{-I(1:3|2)/2}$

8. Chapter Summary

This chapter established Markov splicing theory for causal diamond chains, core results include:

8.1 Core Formulas

Inclusion-Exclusion Identity: $$K_{{\cup }_i{R}_{V_i}}=\sum _{k=1}^N(-1{)}^{k-1}\sum _{1\le i_1<\cdots <i_k\le N}{K}_{R_{V_{i_1}}\cap \cdots \cap R_{V_{i_k}}}$$

Markov Splicing: $$I(D_{j-1}:D_{j+1}\mid D_j)=0\iff K_{12}+K_{23}-K_2-K_{123}=0$$

Non-Totally-Ordered Gap: $$I(D_{j-1}:D_{j+1}\mid D_j)=\iint \iota (v,x_{\perp })dv{d}^{d-2}{x}_{\perp }\ge c_{\ast }\int \kappa (x_{\perp })d^{d-2}{x}_{\perp }$$

Petz Recovery Fidelity: $$F\ge e^{-I(A:C\mid B)/2}$$

8.2 Physical Picture

Mermaid Summary Diagram

graph TD
    A["Inclusion-Exclusion Identity<br/>Geometric Decomposition"] --> B["Totally-Ordered Cut<br/>κ=0"]
    B --> C["Markov Splicing<br/>I(j-1:j+1|j)=0"]
    C --> D["Perfect Recovery<br/>Petz Map F=1"]

    A --> E["Non-Totally-Ordered Cut<br/>κ > 0"]
    E --> F["Markov Gap<br/>I > 0"]
    F --> G["Lossy Recovery<br/>F < 1"]

    C --> H["HSMI Advancement<br/>Algebraic Structure"]
    H --> I["Chain Recursion<br/>Modular Flow Splicing"]

    style A fill:#e1f5ff
    style B fill:#e1ffe1
    style C fill:#ffe1e1
    style D fill:#f5e1ff
    style E fill:#fff4e1
    style F fill:#ffcccc
    style G fill:#ffaaaa
    style H fill:#e1e1ff
    style I fill:#f5f5ff

8.3 Key Insights

1. Universality of Inclusion-Exclusion Principle:

   • Starting from set-theoretic inclusion-exclusion formula
   • Generalizing to quadratic forms, modular Hamiltonians, relative entropy
   • Unified mathematical framework

2. Geometric Root of Markovianity:

   • Totally-ordered cuts $\iff$ stratification degree $\kappa =0$
   • Topological properties of null boundaries
   • Geometric realization of modular flow

3. Quantum Limit of Information Recovery:

   • Petz map achieves optimal recovery
   • Fidelity determined by conditional mutual information
   • Fawzi-Renner inequality provides operational lower bound

4. Dual Structure of Algebra and Geometry:

   • HSMI: Algebraic advancement
   • Modular flow: Geometric advancement
   • The two unified through Bisognano-Wichmann property

8.4 Preview of Next Chapter

Next chapter will discuss Scattering Scale and Windowed Readout:

• How to measure modular Hamiltonian through scattering phase $\arg \det S$?
• Birman-Krein formula and Wigner-Smith group delay
• Windowing techniques and parity threshold stability

End of Chapter

Source Theory: euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md, §3.2-3.4