Markov Property: Independence of Causal Diamonds

“Causal diamond chains satisfy Markov property; modular Hamiltonian obeys inclusion-exclusion formula.”

🎯 Core of This Article

In previous article, we learned modular Hamiltonian is localized on null boundaries. Now the key question is:

How do multiple causal diamonds combine? How do their modular Hamiltonians add?

Answer: Through Markov property!

$$\boxed{I(A:C|B)=0}$$

This seemingly simple formula reveals profound nature of causal structure:

Theorem (Inclusion-Exclusion Formula): For causal diamond family $\{D_j\}$, modular Hamiltonian satisfies:

$$K_{{\bigcup }_j{D}_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

This is perfect union of quantum information and causal geometry!

🧊 Analogy: Independence of Puzzle Pieces

Imagine you are assembling a huge jigsaw puzzle:

graph TB
    subgraph "Three Puzzle Pieces"
        A["Piece A"]
        B["Piece B"]
        C["Piece C"]
    end

    subgraph "Puzzle Relations"
        A ---|"Contact"| B
        B ---|"Contact"| C
        A -.-|"Isolated"| C
    end

    A -.->|"Given B"| INDEP["A and C Independent"]
    C -.->|"Given B"| INDEP

    B -.->|"Screens"| SCREEN["B Screens A and C"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffe1e1
    style INDEP fill:#e1ffe1

Puzzle Analogy:

• Piece A: Causal diamond $D_A$
• Piece B: Causal diamond $D_B$ (between A and C)
• Piece C: Causal diamond $D_C$
• Contact: Causal diamonds have overlapping regions
• Independence: Given B, information of A and C is independent

Key Insight:

• If B is between A and C, and B “screens” all paths from A to C
• Then given B’s information, A and C are conditionally independent
• This is intuitive meaning of Markov property!

📐 Conditional Mutual Information and Markov Property

Review: Mutual Information

For quantum state ${\rho }_{ABC}$ (defined on region $A\cup B\cup C$), define mutual information:

$$I(A:C):=S(A)+S(C)-S(AC)$$

where $S(A)=-\mathrm{tr}({\rho }_A\log {\rho }_A)$ is von Neumann entropy.

Physical Meaning: $I(A:C)$ measures quantum correlation (entanglement + classical correlation) between $A$ and $C$.

Conditional Mutual Information

Conditional mutual information is defined as:

$$I(A:C|B):=S(AB)+S(BC)-S(B)-S(ABC)$$

Physical Meaning: Remaining correlation between $A$ and $C$ given information of $B$.

Markov Property

If:

$$I(A:C|B)=0$$

then $A$ and $C$ are conditionally independent given $B$, or $(A,B,C)$ satisfies Markov property.

graph LR
    A["Region A"] -.->|"Entanglement"| AC["A-C Correlation"]
    C["Region C"] -.->|"Entanglement"| AC

    B["Region B<br/>(Screening Layer)"] -->|"Blocks"| AC

    AC -.->|"Markov Property"| MARKOV["I(A:C|B) = 0"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffe1e1
    style MARKOV fill:#e1ffe1

Strong Subadditivity (SSA)

Markov property is equivalent to strong subadditivity (SSA) of entropy:

$$S(ABC)+S(B)\le S(AB)+S(BC)$$

SSA is cornerstone of quantum information theory, proved by Lieb-Ruskai in 1973.

Rearranging SSA gives Markov property:

$$I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)\ge 0$$

Equality holds if and only if Markov property holds.

🔗 Markov Property of Causal Diamonds

Null Plane Regions

Consider null plane regions:

In Minkowski spacetime, null plane is defined by equation:

$${\mathcal{N}}^{+}=\{x\mid u:=t-x=\text{const}\}$$

This is a null hypersurface ($u=$ constant defines light front cone).

graph TB
    subgraph "Minkowski Spacetime (t,x)"
        PAST["u₁ = t - x<br/>(Past Null Plane)"]
        MID["u₂ = t - x<br/>(Middle Null Plane)"]
        FUTURE["u₃ = t - x<br/>(Future Null Plane)"]
    end

    PAST -->|"Causal Future"| MID
    MID -->|"Causal Future"| FUTURE

    MID -.->|"Screens"| SCREEN["u₂ Screens u₁ and u₃"]

    style PAST fill:#e1f5ff
    style MID fill:#fff4e1
    style FUTURE fill:#ffe1e1
    style SCREEN fill:#e1ffe1

Casini-Teste-Torroba Theorem

Theorem (Casini-Teste-Torroba, 2017): For null plane region families in quantum field theory, Markov property holds.

Specifically, let:

• $A$: Region behind null plane ${\mathcal{N}}_1$
• $B$: Region between null planes ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$
• $C$: Region ahead of null plane ${\mathcal{N}}_2$

where ${\mathcal{N}}_1\prec {\mathcal{N}}_2$ (causal order).

Then:

$$I(A:C|B)=0$$

Proof Idea:

1. Use Null-Modular double cover (previous article)
2. Modular Hamiltonian completely localized on null plane boundaries
3. No direct “connection” between null planes (except through middle region)
4. Therefore conditional mutual information is zero

Physical Meaning

Physical Meaning of Markov Property:

• Causal Screening: Middle null plane $B$ completely screens causal connection between $A$ and $C$
• Information Flow: Information can only flow from $A$ to $C$ through $B$, no “shortcuts”
• Independence: Given complete information of $B$, $A$ and $C$ have no additional correlation

graph LR
    A["Region A<br/>(Past)"] -->|"Information Flow"| B["Region B<br/>(Present)"]
    B -->|"Information Flow"| C["Region C<br/>(Future)"]

    A -.-|"No Direct Path"| C

    B -.->|"Screens"| SCREEN["Complete Screening<br/>I(A:C|B) = 0"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffe1e1
    style SCREEN fill:#e1ffe1

🧮 Inclusion-Exclusion Formula

Additivity of Modular Hamiltonian

For two disjoint regions $A$ and $B$ ($A\cap B=\varnothing$):

$$K_{A\cup B}=K_A+K_B$$

This is additivity of modular Hamiltonian.

But what if $A$ and $B$ overlap?

Inclusion-Exclusion Principle

Theorem (Inclusion-Exclusion): For arbitrary finite family of causal diamonds $\{D_1,D_2,\dots ,D_n\}$:

$$K_{{\bigcup }_{j=1}^n{D}_j}=\sum _{k=1}^n(-1{)}^{k-1}\sum _{1\le j_1<\cdots <j_k\le n}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

Expanded Form (case $n=3$):

$$K_{D_1\cup D_2\cup D_3}=K_{D_1}+K_{D_2}+K_{D_3}-K_{D_1\cap D_2}-K_{D_1\cap D_3}-K_{D_2\cap D_3}+K_{D_1\cap D_2\cap D_3}$$

graph TB
    UNION["K_{D₁ ∪ D₂ ∪ D₃}"] --> SINGLE["Single Terms<br/>+ K_D₁ + K_D₂ + K_D₃"]
    UNION --> DOUBLE["Double Intersections<br/>- K_{D₁∩D₂} - K_{D₁∩D₃} - K_{D₂∩D₃}"]
    UNION --> TRIPLE["Triple Intersection<br/>+ K_{D₁∩D₂∩D₃}"]

    SINGLE -.->|"(-1)⁰"| SIGN1["Positive Sign"]
    DOUBLE -.->|"(-1)¹"| SIGN2["Negative Sign"]
    TRIPLE -.->|"(-1)²"| SIGN3["Positive Sign"]

    style UNION fill:#fff4e1
    style SINGLE fill:#e1ffe1
    style DOUBLE fill:#ffe1e1
    style TRIPLE fill:#e1f5ff

Proof Idea

Key Tool: Markov property + algebraic properties of modular flow

Steps:

1. Use relationship between modular Hamiltonian and relative entropy: $$K_D\sim S_{\mathrm{rel}}({\rho }_D\Vert {\omega }_D)$$
2. Relative entropy satisfies inclusion-exclusion formula (fundamental theorem of information theory)
3. Markov property ensures cross terms cancel correctly
4. Induction generalizes to arbitrary $n$

Geometric Intuition

Geometric meaning of inclusion-exclusion formula:

Question: How to calculate $K_{D_1\cup D_2}$?

Naive Idea: $K_{D_1}+K_{D_2}$

Problem: Overlapping region $D_1\cap D_2$ is counted twice!

Correction: $K_{D_1}+K_{D_2}-K_{D_1\cap D_2}$

This is exactly $n=2$ case of inclusion-exclusion formula:

$$K_{D_1\cup D_2}=K_{D_1}+K_{D_2}-K_{D_1\cap D_2}$$

graph TB
    subgraph "Two Causal Diamonds"
        D1["D₁"] -.->|"Overlap"| OVERLAP["D₁ ∩ D₂"]
        D2["D₂"] -.->|"Overlap"| OVERLAP
    end

    subgraph "Counting Correction"
        COUNT["K_D₁ + K_D₂"] -->|"Double Count"| OVERLAP
        COUNT -->|"Subtract"| CORRECT["- K_{D₁ ∩ D₂}"]
    end

    CORRECT -.-> RESULT["K_{D₁ ∪ D₂}"]

    style D1 fill:#e1f5ff
    style D2 fill:#fff4e1
    style OVERLAP fill:#ffe1e1
    style RESULT fill:#e1ffe1

📊 Example: Causal Diamond Chain

Scenario: Three Causal Diamonds in Series

Consider three causal diamonds arranged in temporal order:

$$D_1\prec D_2\prec D_3$$

where $D_i\prec D_j$ means $D_i$ is in causal past of $D_j$.

graph LR
    D1["D₁<br/>(Past)"] -->|"Causal Order"| D2["D₂<br/>(Present)"]
    D2 -->|"Causal Order"| D3["D₃<br/>(Future)"]

    D1 -.->|"No Intersection"| NOTE1["D₁ ∩ D₃ = ∅"]
    D1 -.->|"Intersect"| OVERLAP12["D₁ ∩ D₂ ≠ ∅"]
    D2 -.->|"Intersect"| OVERLAP23["D₂ ∩ D₃ ≠ ∅"]

    style D1 fill:#e1f5ff
    style D2 fill:#fff4e1
    style D3 fill:#ffe1e1

Modular Hamiltonian Calculation

Union

$$K_{D_1\cup D_2\cup D_3}=?$$

Apply Inclusion-Exclusion Formula

$$\begin{array}{rl}{K}_{D_1\cup D_2\cup D_3}& =K_{D_1}+K_{D_2}+K_{D_3}\\ & -K_{D_1\cap D_2}-K_{D_2\cap D_3}-K_{D_1\cap D_3}\\ & +K_{D_1\cap D_2\cap D_3}\end{array}$$

Simplification

Since $D_1\cap D_3=\varnothing$ (causal order ensures no intersection):

$$K_{D_1\cap D_3}=0,K_{D_1\cap D_2\cap D_3}=0$$

Therefore:

$$K_{D_1\cup D_2\cup D_3}=K_{D_1}+K_{D_2}+K_{D_3}-K_{D_1\cap D_2}-K_{D_2\cap D_3}$$

Markov Verification

Markov Property requires:

$$I(D_1:D_3|D_2)=0$$

That is: Given $D_2$, $D_1$ and $D_3$ are conditionally independent.

Verification:

1. $D_2$ is between $D_1$ and $D_3$
2. Null boundaries of $D_2$ screen all causal paths from $D_1$ to $D_3$
3. Therefore Markov property holds

Conclusion: Inclusion-exclusion formula is completely consistent with Markov property!

🔬 Relative Entropy and Modular Hamiltonian

Relative Entropy Definition

For region $D$ and state ${\rho }_D$ (relative to reference state ${\omega }_D$), define relative entropy:

$$S({\rho }_D\Vert {\omega }_D):=\mathrm{tr}({\rho }_D\log {\rho }_D)-\mathrm{tr}({\rho }_D\log {\omega }_D)$$

Relationship with Modular Hamiltonian

For vacuum state ${\omega }_D=|0⟩⟨0|$:

$$S({\rho }_D\Vert {\omega }_D)=⟨K_D{⟩}_{\rho }-S({\rho }_D)$$

where:

• $⟨K_D{⟩}_{\rho }=\mathrm{tr}({\rho }_D{K}_D)$: Expectation value of modular Hamiltonian
• $S({\rho }_D)=-\mathrm{tr}({\rho }_D\log {\rho }_D)$: von Neumann entropy

Physical Meaning:

• Relative entropy measures “distinguishability” between ${\rho }_D$ and ${\omega }_D$
• Modular Hamiltonian is generator of relative entropy

Information-Theoretic Origin of Inclusion-Exclusion

Theorem (Information Theory): Relative entropy satisfies:

$$S({\rho }_{A\cup B}\Vert {\omega }_{A\cup B})=S({\rho }_A\Vert {\omega }_A)+S({\rho }_B\Vert {\omega }_B)-S({\rho }_{A\cap B}\Vert {\omega }_{A\cap B})+\Delta$$

where $\Delta \ge 0$ is conditional mutual information term, and $\Delta =0$ when Markov property holds.

Causal Diamond Case: Due to Markov property, $\Delta =0$, therefore:

$$S({\rho }_D\Vert {\omega }_D)\text{satisfies inclusion-exclusion formula}$$

From $K_D\sim S({\rho }_D\Vert {\omega }_D)$, modular Hamiltonian also satisfies inclusion-exclusion!

graph TB
    REL["Relative Entropy<br/>S(ρ‖ω)"] --> IE["Inclusion-Exclusion Formula"]
    MOD["Modular Hamiltonian<br/>K_D"] --> REL

    IE --> MARKOV["Markov Property<br/>I(A:C|B) = 0"]
    MOD -.->|"Equivalent"| REL

    MARKOV -.->|"Ensures"| IE

    style REL fill:#fff4e1
    style MOD fill:#e1f5ff
    style MARKOV fill:#e1ffe1

🌐 Generalization: General Regions

Beyond Causal Diamonds

Inclusion-exclusion formula holds not only for causal diamonds, but for any region family satisfying Markov property.

General Theorem: For region family $\{A_j\}$ in quantum field theory, if it satisfies:

• Locality: Spacelike separated regions commute
• Markov Property: Appropriate conditional independence

then modular Hamiltonian satisfies inclusion-exclusion formula.

Spherical Regions

Example: In CFT (conformal field theory), consider spherical regions $B_r$ (sphere of radius $r$).

For two concentric spheres $B_{r_1}$ and $B_{r_2}$ ($r_1<r_2$):

$$K_{B_{r_2}}\ne K_{B_{r_1}}+K_{B_{r_2}\setminus B_{r_1}}$$

Reason: Spherical regions do not satisfy Markov property!

Modular Hamiltonian of spherical regions requires additional geometric terms, cannot be simply decomposed into boundary contributions.

Null Planes vs Spheres

Comparison:

Conclusion: Null plane regions are natural choice of causal structure, spherical regions are artificial!

💡 Key Points Summary

1. Markov Property

$$I(A:C|B)=0$$

Given middle region $B$, $A$ and $C$ are conditionally independent.

2. Causal Screening

Null plane regions naturally satisfy Markov property because:

• Middle null plane screens all causal paths
• Information flow must pass through middle region

3. Inclusion-Exclusion Formula

$$K_{{\bigcup }_j{D}_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

Modular Hamiltonian satisfies inclusion-exclusion, correcting double counting of overlapping regions.

4. Casini-Teste-Torroba Theorem

Null plane region families satisfy Markov property (in quantum field theory).

5. Connection with Relative Entropy

$$K_D\sim S({\rho }_D\Vert {\omega }_D)$$

Inclusion-exclusion of modular Hamiltonian originates from information-theoretic properties of relative entropy.

🤔 Thought Questions

Question 1: Why Don’t Spherical Regions Satisfy Markov Property?

Hint: Consider “annular” region between two concentric spheres.

Answer: Boundaries of spherical regions are spacelike hypersurfaces, not null hypersurfaces. Spacelike hypersurfaces cannot completely “screen” causal connections because:

• Quantum fluctuations can create correlations between spacelike separated points (though cannot transmit signals)
• Entanglement can span spacelike separated regions
• Therefore Markov property of spherical regions does not hold

Question 2: If Causal Diamonds Don’t Satisfy $D_i\prec D_j$ (Causal Order), Does Inclusion-Exclusion Still Hold?

Hint: Consider case where two causal diamonds are “side by side”.

Answer: Still holds! Inclusion-exclusion formula holds for arbitrary causal diamond families, does not require causal order. Key points:

• Modular Hamiltonian of each causal diamond is localized on its null boundaries
• Modular Hamiltonian of overlapping regions is correctly calculated through inclusion-exclusion
• Markov property ensures no “hidden” cross correlation terms

Question 3: How Is Markov Property Manifested in AdS/CFT?

Hint: Consider subregions of boundary CFT.

Answer: In AdS/CFT correspondence:

• Bulk AdS: Causal diamonds → Entanglement wedges
• Boundary CFT: Subregions → Boundary regions
• Markov Property: Manifested as Markov chain structure of boundary subregions

Specifically, if boundary regions $A$, $B$, $C$ satisfy causal order, then their entanglement wedges satisfy Markov property: $$I_{\mathrm{bulk}}(A:C|B)=0$$

This is realized through RT formula (Ryu-Takayanagi) and quantum extremal surfaces!

Question 4: What Is Physical Meaning of Sign $(-1{)}^{k-1}$ in Inclusion-Exclusion Formula?

Hint: Consider correction for “double counting”.

Answer: Sign $(-1{)}^{k-1}$ reflects parity of counting:

• $k=1$ (single term): Positive sign, direct contribution
• $k=2$ (double intersection): Negative sign, subtract double counted part
• $k=3$ (triple intersection): Positive sign, compensate for over-subtraction
• $k=4$: Negative sign, correct again…

This is quantum version of inclusion-exclusion principle, originating from fundamental identity of set theory!

📖 Source Theory References

Content of this article mainly from following source theories:

Core Source Theory

Document: docs/euler-gls-causal/unified-theory-causal-structure-time-scale-partial-order-generalized-entropy.md

Key Content:

• Definition and causal meaning of Markov property
• Complete statement of inclusion-exclusion formula
• Markov property of null plane regions
• Connection with relative entropy

Important Theorem (original text):

“For null plane region families, Markov property holds, modular Hamiltonian satisfies inclusion-exclusion formula:

$$K_{\cup D_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$ “

Classical Literature

Casini-Teste-Torroba (2017):

• Proof of Markov property for null plane regions
• Quantum field theory realization of inclusion-exclusion formula
• Connection with QNEC

Lieb-Ruskai (1973):

• Strong subadditivity (SSA) theorem
• Foundation of quantum entropy inequalities

Petz (1986):

• Properties of relative entropy
• Modular theory and quantum information

🎯 Next Steps

We’ve understood how Markov property ensures “independence” of causal diamonds. Next article will explore how multiple observers reach consensus through causal structure.

Next Article: 06-observer-consensus_en.md - Consensus Geometry and Causal Networks

There, we will see:

• Complete formalization of observers (nine-tuple)
• Three levels of consensus: causal, state, model
• Convergence of relative entropy Lyapunov function
• Communication graph and information propagation
• From local observers to global spacetime

Back: Causal Structure Chapter Overview

Previous: 04-null-modular-cover_en.md