23.10 Joint Manifold and Time-Information-Complexity Action

In previous articles, we separately established two geometric structures:

• Control Manifold $(\mathcal{M},G)$: Describes “complexity cost” of computation (Articles 23.8-9);
• Information Manifold $({\mathcal{S}}_Q,g_Q)$: Describes “information acquisition” of observation (Articles 23.6-7).

But real processes in computational universe involve both simultaneously:

• We need to control system evolution (spending time/energy);
• Simultaneously need to observe system state (acquiring information).

This raises a core question: How to optimize joint strategy of control and observation under given resources, maximizing information acquisition?

This article will construct joint manifold $\mathcal{N}=\mathcal{M}\times {\mathcal{S}}_Q$, define time-information-complexity action ${\mathcal{A}}_Q$ on it, and derive optimal computation trajectories (computation worldlines) through variational principle. This achieves complete unification of time, information, and complexity.

Core Questions:

• How to couple control manifold and information manifold into joint manifold?
• What is time-information-complexity action? Why is it “kinetic-potential” form?
• How do discrete computation paths converge to continuous computation worldlines?
• What dynamical equations do optimal computation worldlines satisfy?

This article is based on euler-gls-info/05-time-information-complexity-variational-principle.md.

1. Why Do We Need Joint Manifold? From Navigation to Optimal Strategy

1.1 Everyday Analogy: Dual Optimization of Driving Navigation

Imagine using navigation app to drive from home to airport:

Question 1: Which Route? (Control Optimization)

• Navigation plans a route, considering distance, time, congestion;
• This is finding shortest path on “map space” (control manifold);
• Cost is “travel time” (complexity cost).

Question 2: How to Know Road Conditions? (Information Acquisition)

• Navigation needs to continuously update road condition information (through GPS, sensors);
• This is moving on “information space” (information manifold);
• Cost is “data traffic/battery consumption” (information cost).

Core Insight: Optimal strategy needs to simultaneously optimize both problems:

• When choosing route, consider information quality (avoid information blind spots);
• When acquiring information, consider route constraints (cannot detour just to see road conditions).

This requires optimization on joint space (route × information), not separately!

1.2 Analogy in Computational Universe

In computational universe:

Control Manifold $\mathcal{M}$:

• Parameters are physical controls (e.g., quantum gate angles, circuit voltages);
• Metric $G_{ab}$ measures “computational cost” of control changes (time, energy);
• Geodesics are “fastest computation paths”.

Information Manifold ${\mathcal{S}}_Q$:

• Parameters are observation strategies (e.g., measurement basis selection, sampling schemes);
• Metric $g_{ij}^{(Q)}$ measures “information distance” of observation changes (Fisher information);
• Geodesics are “paths with gentlest information changes”.

Computation Worldline: Real computation process is a curve on joint space $\mathcal{M}\times {\mathcal{S}}_Q$: $$z(t)=(\theta (t),\varphi (t)),$$ where $\theta (t)$ is control parameter trajectory, $\varphi (t)$ is observation parameter trajectory.

graph TD
    A["Computation Process"] --> B["Control Part<br/>theta(t) in M"]
    A --> C["Observation Part<br/>phi(t) in S_Q"]

    B --> D["Control Cost<br/>Complexity Metric G"]
    C --> E["Information Cost<br/>Fisher Metric g_Q"]

    D --> F["Problem:<br/>Only Optimize Control,<br/>Ignore Information Acquisition"]
    E --> G["Problem:<br/>Only Optimize Information,<br/>Ignore Control Cost"]

    F --> H["Need:<br/>Joint Optimization!"]
    G --> H

    H --> I["Joint Manifold<br/>N = M × S_Q"]
    I --> J["Joint Action<br/>A_Q = ∫(Control Cost + Information Cost - Information Gain) dt"]

    J --> K["Optimal Computation Worldline<br/>z*(t) = (theta*(t), phi*(t))"]

    style A fill:#e1f5ff
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2. Construction of Joint Manifold: Product Space

Source Theory: euler-gls-info/05-time-information-complexity-variational-principle.md Section 3

2.1 Definition of Joint Manifold

Definition 2.1 (Joint Time-Information-Complexity Manifold, from euler-gls-info/05-time-information-complexity-variational-principle.md Definition 3.1)

For given task $Q$, define joint manifold

$${\mathcal{N}}_Q=\mathcal{M}\times {\mathcal{S}}_Q.$$

Its point $z=(\theta ,\varphi )$ simultaneously represents:

• Control State $\theta \in \mathcal{M}$ (e.g., quantum gate parameters);
• Task Information State $\varphi \in {\mathcal{S}}_Q$ (e.g., measurement basis parameters).

Everyday Interpretation:

• $\mathcal{M}$ is space of “physical knobs” (how to control system evolution);
• ${\mathcal{S}}_Q$ is space of “measurement knobs” (how to choose observation);
• ${\mathcal{N}}_Q$ is space of “adjusting both sets of knobs simultaneously”.

Example: Quantum Measurement

• Control parameter $\theta$: Parameters of Hamiltonian (e.g., magnetic field strength);
• Observation parameter $\varphi$: Parameters of measurement operator (e.g., rotation angle of measurement basis);
• Joint state $(\theta ,\varphi )$: “Simultaneously set evolution and measurement”.

2.2 Joint Metric: Weighted Product

Definition 2.2 (Joint Metric, from euler-gls-info/05-time-information-complexity-variational-principle.md Section 3.2)

On joint manifold ${\mathcal{N}}_Q$, define product-type metric

$$\mathbb{G}={\alpha }^{2}G\oplus {\beta }^2{g}_Q,$$

i.e., for tangent vector $v=(v^{\mathcal{M}},v^{{\mathcal{S}}_Q})\in T_{\theta }\mathcal{M}\oplus T_{\varphi }{\mathcal{S}}_Q$, define

$${\mathbb{G}}_z(v,v)={\alpha }^2{G}_{\theta }(v^{\mathcal{M}},v^{\mathcal{M}})+{\beta }^2{g}_{Q,\varphi }(v^{{\mathcal{S}}_Q},v^{{\mathcal{S}}_Q}),$$

where $\alpha ,\beta >0$ are coupling constants, used to balance weights of control cost and information cost.

Everyday Interpretation:

• $\alpha$ is “weight of control”: Larger $\alpha$, higher cost of control changes;
• $\beta$ is “weight of information”: Larger $\beta$, higher cost of information changes;
• Joint metric $\mathbb{G}$ is metric of “total cost”.

Mathematical Structure:

• This is standard Riemann product metric;
• In local coordinates: $\mathbb{G}=\mathrm{diag}({\alpha }^{2}G,{\beta }^2{g}_Q)$ (block diagonal);
• No coupling terms between different blocks: Control and information are “orthogonal”.

2.3 Velocity and Length of Joint Trajectory

Given a joint trajectory

$$z(t)=(\theta (t),\varphi (t)),t\in [0,T],$$

its instantaneous velocity squared is

$$|\dot{z}(t){|}_{\mathbb{G}}^2={\alpha }^2{G}_{ab}(\theta (t)){\dot{\theta }}^a(t){\dot{\theta }}^b(t)+{\beta }^2{g}_{ij}(\varphi (t)){\dot{\varphi }}^i(t){\dot{\varphi }}^j(t).$$

Geometric length of trajectory is

$$L_{\mathbb{G}}[z]={\int }_0^T\sqrt{|\dot{z}(t){|}_{\mathbb{G}}^2}\mathrm{d}t.$$

Everyday Interpretation:

• $|\dot{z}(t){|}_{\mathbb{G}}^2$ is “square of joint velocity” (similar to $v_x^2+v_y^2$);
• Contains two contributions: control velocity ${\alpha }^{2}G{\dot{\theta }}^2$ and information velocity ${\beta }^2{g}_Q{\dot{\varphi }}^2$;
• Geometric length $L_{\mathbb{G}}$ is “total distance” (time integral).

graph TD
    A["Joint Manifold<br/>N_Q = M × S_Q"] --> B["Point: z = (theta, phi)"]
    B --> C["theta in M<br/>Control State"]
    B --> D["phi in S_Q<br/>Information State"]

    A --> E["Joint Metric<br/>G = alpha²G ⊕ beta²g_Q"]
    E --> F["Control Part:<br/>alpha²G_ab dtheta^a dtheta^b"]
    E --> G["Information Part:<br/>beta²g_ij^(Q) dphi^i dphi^j"]

    H["Joint Trajectory<br/>z(t) = (theta(t), phi(t))"] --> I["Velocity Squared<br/>|z_dot|² = alpha²G·theta_dot² + beta²g_Q·phi_dot²"]

    I --> J["Geometric Length<br/>L = ∫√|z_dot|² dt"]

    J --> K["Problem:<br/>Only Length,<br/>No 'Information Gain'"]

    K --> L["Need to Introduce:<br/>Information Potential Function U_Q(phi)"]

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3. Information Potential Function: Quantifying Information Quality

Source Theory: euler-gls-info/05-time-information-complexity-variational-principle.md Section 3.3

3.1 Why Do We Need Potential Function?

Joint metric $\mathbb{G}$ alone is not enough!

Problem: Geodesics (shortest paths) on joint manifold only consider “minimum cost”, don’t consider “information quality”.

Everyday Analogy:

• Imagine mountain climbing treasure hunting:
  • Geometric Length: Distance from foot to summit (cost);
  • Treasure Location: Summit has treasure, mountainside doesn’t (gain);
  • Optimal Path: Not shortest path, but path balancing “cost and gain”!

In computational universe:

• Control-Information Length: Total resources spent on computation;
• Information Quality: Quality of information obtained at endpoint;
• Optimal Worldline: Trajectory with highest information quality under given resources.

Therefore, we need to introduce information potential function to encode “information quality”.

3.2 Definition of Information Potential Function

Definition 3.1 (Information Potential Function, from euler-gls-info/05-time-information-complexity-variational-principle.md Section 3.3)

Let information quality function of task $Q$ on information manifold be written as $I_Q:{\mathcal{S}}_Q\to \mathbb{R}$. Define information potential function

$$U_Q(\varphi )=I_Q(\varphi ),$$

or more generally, $U_Q(\varphi )=V(I_Q(\varphi ))$, where $V:\mathbb{R}\to \mathbb{R}$ is monotonic function.

Physical Meaning:

• $U_Q(\varphi )$ measures “information quality at information state $\varphi$”;
• Larger $U_Q$, higher information quality;
• In action, $U_Q$ appears with negative sign (like potential energy), making high information quality reduce action.

Everyday Interpretation:

• Imagine $U_Q(\varphi )$ is height of “information landscape”:
  • High places: Good information quality (e.g., measurement basis highly correlated with task);
  • Low places: Poor information quality (e.g., measurement basis uncorrelated with task).
• Optimal trajectory will “climb to high places” (regions with good information quality).

3.3 Example: Information Potential of Quantum Measurement

Consider measurement of quantum state $\rho$ under different measurement bases $\{{\hat{M}}_{\varphi }\}$:

• Information quality can be measured by fidelity: $I_Q(\varphi )=\mathrm{Tr}(\rho {\hat{M}}_{\varphi })$;
• Or measured by mutual information: $I_Q(\varphi )=I(\rho :M_{\varphi })$;
• Potential function $U_Q(\varphi )$ reaches maximum near “optimal measurement basis”.

Physical Picture:

• If choose measurement basis “orthogonal to state $\rho$”, $U_Q(\varphi )$ small (poor information quality);
• If choose measurement basis “aligned with state $\rho$”, $U_Q(\varphi )$ large (good information quality);
• Optimal strategy will dynamically adjust $\varphi (t)$ to track maximum of $U_Q$.

graph LR
    A["Information Manifold S_Q"] --> B["Parameter: phi<br/>(e.g., Measurement Basis Angle)"]
    B --> C["Information Quality Function<br/>I_Q(phi)"]

    C --> D["Example 1:<br/>Fidelity Tr(rho M_phi)"]
    C --> E["Example 2:<br/>Mutual Information I(rho:M_phi)"]
    C --> F["Example 3:<br/>Task-Related Entropy"]

    D --> G["Information Potential Function<br/>U_Q(phi) = I_Q(phi)"]
    E --> G
    F --> G

    G --> H["Physical Meaning:<br/>Height of 'Information Landscape'"]
    H --> I["High Places:<br/>Good Information Quality"]
    H --> J["Low Places:<br/>Poor Information Quality"]

    I --> K["Optimal Trajectory:<br/>Climb to High Places<br/>(Maximize Information)"]

    style A fill:#e1f5ff
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4. Time-Information-Complexity Action

Source Theory: euler-gls-info/05-time-information-complexity-variational-principle.md Section 4

4.1 Definition of Continuous Action

Definition 4.1 (Time-Information-Complexity Action, from euler-gls-info/05-time-information-complexity-variational-principle.md Definition 4.2)

For joint trajectory $z(t)=(\theta (t),\varphi (t))$, define continuous joint action

$${\mathcal{A}}_Q[\theta (\cdot ),\varphi (\cdot )]={\int }_0^T(\frac{1}{2}{\alpha }^2{G}_{ab}(\theta ){\dot{\theta }}^a{\dot{\theta }}^b+\frac{1}{2}{\beta }^2{g}_{ij}(\varphi ){\dot{\varphi }}^i{\dot{\varphi }}^j-\gamma U_Q(\varphi ))\mathrm{d}t,$$

where:

• First term: $\frac{1}{2}{\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b$ is control kinetic energy;
• Second term: $\frac{1}{2}{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j$ is information kinetic energy;
• Third term: $-\gamma U_Q(\varphi )$ is information potential energy (negative sign!);
• $\alpha ,\beta ,\gamma >0$ are coupling constants.

Everyday Interpretation:

• This is “kinetic-potential” form in classical mechanics:
  • $T=\frac{1}{2}m{v}^2$ (kinetic energy);
  • $V=U(x)$ (potential energy);
  • Action $S=\int (T-V)\mathrm{d}t$ (Hamilton action).
• In our framework:
  • “Kinetic energy” has two parts: control kinetic energy $+$ information kinetic energy;
  • “Potential energy” is information potential $-\gamma U_Q(\varphi )$ (negative sign makes high information quality reduce action).

4.2 Why “Kinetic-Potential” Form?

Principle: This is standard form of principle of least action.

In classical mechanics, real motion trajectories make action $S=\int (T-V)\mathrm{d}t$ extremal (usually minimum). This derives Euler-Lagrange equations (i.e., Newton’s second law).

In computational universe:

• Kinetic Terms: Penalize “rapid changes” (both control and information cannot change too fast);
• Potential Term: Reward “high information quality” (encourage trajectories to enter high $U_Q$ regions);
• Minimum Trajectory: Balances between “slow changes” and “high information quality”.

Physical Intuition:

• Imagine a ball rolling in valley:
  • Kinetic energy drives it to roll along inertial direction;
  • Potential energy drives it to roll toward valley bottom;
  • Real trajectory is balance of both (geodesic + potential gradient).
• Computation worldline similar:
  • Control/information kinetic energy drives trajectory to “maintain current direction”;
  • Information potential drives trajectory to “turn toward high information regions”;
  • Optimal trajectory is balance of both.

4.3 Discrete Action: From Paths to Continuous

At discrete level, computation path is a sequence of states $\gamma =(x_0,x_1,\dots ,x_n)$. Corresponding discrete action is

$${\mathcal{A}}_Q^{\mathrm{disc}}(\gamma )=\sum _{k=0}^{n-1}(\alpha \mathsf{C}(x_k,x_{k+1})+\beta d_{\mathrm{info},Q}(x_k,x_{k+1})-\gamma \Delta I_Q(x_k,x_{k+1})),$$

where:

• $\mathsf{C}(x_k,x_{k+1})$ is complexity cost (single-step time);
• $d_{\mathrm{info},Q}(x_k,x_{k+1})$ is information distance (single-step information change);
• $\Delta I_Q(x_k,x_{k+1})=I_Q({\varphi }_{k+1})-I_Q({\varphi }_k)$ is information quality increment.

Convergence Theorem 4.2 (Gamma-Convergence from Discrete to Continuous, from euler-gls-info/05-time-information-complexity-variational-principle.md Theorem 6.1)

Under appropriate regularity assumptions, when discrete step size $h\to 0$, discrete action ${\mathcal{A}}_Q^{\mathrm{disc}}$ converges to continuous action ${\mathcal{A}}_Q$ in $\Gamma$-convergence sense:

$$\Gamma \text{-}\underset{h\to 0}{\lim }{\mathcal{A}}_Q^{(h)}={\mathcal{A}}_Q.$$

In particular, limit of discrete optimal paths is continuous optimal worldline.

Everyday Interpretation:

• Discrete paths are “step-by-step jumps”;
• Continuous worldlines are “smooth curves”;
• $\Gamma$-convergence guarantees: Discrete optimal paths become continuous optimal worldlines in limit!

graph TD
    A["Discrete Computation Path<br/>gamma = (x_0,x_1,...,x_n)"] --> B["Single-Step Complexity<br/>C(x_k,x_k+1)"]
    A --> C["Single-Step Information Distance<br/>d_info(x_k,x_k+1)"]
    A --> D["Information Quality Increment<br/>Delta I_Q(x_k,x_k+1)"]

    B --> E["Discrete Action<br/>A_Q^disc = Sigma(alpha C + beta d - gamma Delta I)"]
    C --> E
    D --> E

    F["Refinement Limit h→0"] --> G["Gamma-Convergence"]
    E --> G

    G --> H["Continuous Action<br/>A_Q = ∫(½alpha²G theta_dot² + ½beta²g_Q phi_dot² - gamma U_Q) dt"]

    H --> I["First Term:<br/>Control Kinetic Energy<br/>½alpha²G·theta_dot²"]
    H --> J["Second Term:<br/>Information Kinetic Energy<br/>½beta²g_Q·phi_dot²"]
    H --> K["Third Term:<br/>Information Potential Energy<br/>-gamma U_Q(phi)"]

    I --> L["Minimization<br/>→ Optimal Worldline"]
    J --> L
    K --> L

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5. Euler-Lagrange Equations: Dynamics of Optimal Worldlines

Source Theory: euler-gls-info/05-time-information-complexity-variational-principle.md Section 5

5.1 Variational Principle: Why Minimize Action?

Principle of Least Action: Real physical trajectories make action $S=\int L\mathrm{d}t$ extremal (usually minimum).

In our framework:

• Lagrangian: $L(\theta ,\dot{\theta };\varphi ,\dot{\varphi })=\frac{1}{2}{\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+\frac{1}{2}{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j-\gamma U_Q(\varphi )$;
• Action: ${\mathcal{A}}_Q={\int }_0^{T}L\mathrm{d}t$;
• Minimum Trajectory: Trajectory $(thet{a}^{\ast }(t),{\varphi }^{\ast }(t))$ satisfying $\delta {\mathcal{A}}_Q=0$.

Varying ${\theta }^a$ and ${\varphi }^i$ separately, get Euler-Lagrange equations.

5.2 Euler-Lagrange Equation for Control Part

Varying control variable ${\theta }^a$, Euler-Lagrange equation is

$$\frac{\mathrm{d}}{\mathrm{d}t}({\alpha }^2{G}_{ab}(\theta ){\dot{\theta }}^b)-\frac{1}{2}{\alpha }^2({\partial }_a{G}_{bc})(\theta ){\dot{\theta }}^b{\dot{\theta }}^c=0.$$

Rewrite as Geodesic Equation:

Define Christoffel symbols

$${\Gamma }_{bc}^a(\theta )=\frac{1}{2}{G}^{ad}({\partial }_b{G}_{dc}+{\partial }_c{G}_{db}-{\partial }_d{G}_{bc}),$$

then Euler-Lagrange equation equivalent to

$${\ddot{\theta }}^a+{\Gamma }_{bc}^a(\theta ){\dot{\theta }}^b{\dot{\theta }}^c=0.$$

Physical Meaning:

• This is geodesic equation on control manifold $(\mathcal{M},G)$;
• Control trajectory $\theta (t)$ evolves along geodesic (inertial motion, no external force);
• Because control part in Lagrangian has no potential term!

5.3 Euler-Lagrange Equation for Information Part

Varying information variable ${\varphi }^i$, Euler-Lagrange equation is

$$\frac{\mathrm{d}}{\mathrm{d}t}({\beta }^2{g}_{ij}(\varphi ){\dot{\varphi }}^j)-\frac{1}{2}{\beta }^2({\partial }_i{g}_{jk})(\varphi ){\dot{\varphi }}^j{\dot{\varphi }}^k+\gamma {\partial }_i{U}_Q(\varphi )=0.$$

Rewrite as Geodesic Equation with Potential:

Define Christoffel symbols of information manifold ${\Gamma }_{jk}^i(\varphi )$, and raise potential gradient:

$${\ddot{\varphi }}^i+{\Gamma }_{jk}^i(\varphi ){\dot{\varphi }}^j{\dot{\varphi }}^k=-\frac{\gamma }{{\beta }^2}{g}^{ij}(\varphi ){\partial }_j{U}_Q(\varphi ).$$

Physical Meaning:

• Left side: Geodesic acceleration on information manifold $({\mathcal{S}}_Q,g_Q)$;
• Right side: Gradient of information potential $U_Q$ (covariantly raised), $-\frac{\gamma }{{\beta }^2}\nabla U_Q$;
• Information trajectory $\varphi (t)$ is not geodesic, but “geodesic with potential driven by potential gradient”!

Everyday Analogy:

• Imagine rolling ball on hillside:
  • If no gravity (flat ground), ball rolls along straight line (geodesic);
  • If gravity (hillside), ball is “pulled toward valley” (potential gradient);
  • Real trajectory is superposition of both (geodesic with potential).
• Information trajectory similar:
  • Geodesic part: Moves along “information inertia”;
  • Potential gradient part: “Pulled toward high information quality regions”.

5.4 Coupling Structure of Joint System

Summary: Optimal computation worldline $(thet{a}^{\ast }(t),{\varphi }^{\ast }(t))$ satisfies coupled system:

$$\{\begin{array}{ll}{\ddot{\theta }}^a+{\Gamma }_{bc}^a(\theta ){\dot{\theta }}^b{\dot{\theta }}^c=0& \text{(Control: Geodesic)}\\ {\ddot{\varphi }}^i+{\Gamma }_{jk}^i(\varphi ){\dot{\varphi }}^j{\dot{\varphi }}^k=-\frac{\gamma }{{\beta }^2}{g}^{ij}(\varphi ){\partial }_j{U}_Q(\varphi )& \text{(Information: Geodesic with Potential)}\end{array}$$

Core Insight:

• Control and information decoupled: $\theta$ equation doesn’t depend on $\varphi$, $\varphi$ equation doesn’t depend on $\theta$;
• But coupled through initial conditions and boundary conditions: Both need simultaneous optimization;
• This is because we chose product metric $\mathbb{G}={\alpha }^{2}G\oplus {\beta }^2{g}_Q$ (no cross terms).

graph TD
    A["Principle of Least Action<br/>delta A_Q = 0"] --> B["Calculus of Variations"]
    B --> C["Vary theta^a"]
    B --> D["Vary phi^i"]

    C --> E["Control Part<br/>Euler-Lagrange Equation"]
    D --> F["Information Part<br/>Euler-Lagrange Equation"]

    E --> G["Rewrite as Geodesic Equation<br/>theta_ddot^a + Gamma^a_bc theta_dot^b theta_dot^c = 0"]
    F --> H["Rewrite as Geodesic with Potential<br/>phi_ddot^i + Gamma^i_jk phi_dot^j phi_dot^k = -(gamma/beta²)nabla U_Q"]

    G --> I["Physical Meaning:<br/>Control Along Geodesic<br/>(Inertial Motion)"]
    H --> J["Physical Meaning:<br/>Information Driven by Potential<br/>(Climb to High Information Regions)"]

    I --> K["Coupling Structure:<br/>theta and phi Coupled Through<br/>Initial/Boundary Conditions"]
    J --> K

    K --> L["Optimal Computation Worldline<br/>z*(t) = (theta*(t), phi*(t))"]

    style A fill:#e1f5ff
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6. Optimization Under Resource Constraints: Lagrange Multiplier Method

Source Theory: euler-gls-info/05-time-information-complexity-variational-principle.md Section 7

6.1 Practical Problem: Constrained Optimization

In practical applications, we usually face constrained optimization problems:

Problem 1: Given time budget $T$, maximize endpoint information quality $I_Q(\varphi (T))$.

Problem 2: Given complexity budget $C_{\max }$, maximize endpoint information quality.

Problem 3: Given information quality target $I_{\mathrm{target}}$, minimize required time/complexity.

These are all constrained variational problems, can be converted to unconstrained problems using Lagrange multiplier method.

6.2 Example: Fixed Time, Maximize Endpoint Information

Problem:

$$\underset{\theta (\cdot ),\varphi (\cdot )}{\max }{I}_Q(\varphi (T)),\text{subject to }{\int }_0^T(\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2+\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2)\mathrm{d}t\le E_{\max }.$$

Convert to Unconstrained Problem:

Introduce Lagrange multiplier $\lambda$, define modified action

$${\tilde{\mathcal{A}}}_Q={\int }_0^T(\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2+\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2)\mathrm{d}t-\gamma I_Q(\varphi (T)).$$

Minimizing ${\tilde{\mathcal{A}}}_Q$ equivalent to original constrained problem (when constraint saturated).

Euler-Lagrange Equations:

• Bulk equations (inside $t\in (0,T)$) same as before;
• Boundary Condition (at $t=T$):

$${\beta }^2{g}_{ij}(\varphi (T)){\dot{\varphi }}^j(T)=\gamma {\partial }_i{I}_Q(\varphi (T)).$$

Physical Meaning:

• Boundary condition is “endpoint reflection condition”: At endpoint, information velocity proportional to information quality gradient;
• $\gamma /{\beta }^2$ controls “preference strength for endpoint information quality”;
• If $\gamma$ large, trajectory will “brake hard” at endpoint, rush toward high information regions.

6.3 Everyday Analogy: Optimal Braking Strategy

Imagine driving to destination:

• Constraint: Total energy limited (fuel tank capacity);
• Goal: Speed as low as possible when arriving (safe parking);
• Optimal Strategy:
  • Most time constant speed (save energy);
  • Near endpoint brake hard (speed drops to zero).

In computational universe:

• Constraint: Total computation energy limited;
• Goal: Endpoint information quality as high as possible;
• Optimal Strategy:
  • Most time evolve along geodesic (save energy);
  • Near endpoint “rush toward high information regions” (boundary condition).

graph TD
    A["Practical Optimization Problem"] --> B["Constraint 1:<br/>Fixed Time T"]
    A --> C["Constraint 2:<br/>Fixed Complexity Budget C_max"]
    A --> D["Constraint 3:<br/>Fixed Information Target I_target"]

    B --> E["Goal:<br/>Maximize Endpoint Information I_Q(phi(T))"]
    C --> E
    D --> F["Goal:<br/>Minimize Time/Complexity"]

    E --> G["Lagrange Multiplier Method"]
    F --> G

    G --> H["Modified Action<br/>A_Q_tilde = ∫(Kinetic Energy) - gamma I_Q(phi(T))"]

    H --> I["Bulk Equations:<br/>Same as Before<br/>(Geodesic + Potential)"]
    H --> J["Boundary Condition:<br/>beta²g·phi_dot(T) = gamma nabla I_Q(phi(T))"]

    I --> K["Solution: Optimal Worldline<br/>+ Boundary Reflection"]
    J --> K

    K --> L["Physical Meaning:<br/>Hard Brake at Endpoint<br/>Rush to High Information Regions"]

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7. Physical Example: Optimal Strategy for Quantum Measurement

7.1 Problem Setting: Adaptive Quantum Measurement

Consider a quantum system:

• Initial State: ${\rho }_0$ (known);
• Evolution: Hamiltonian $H(\theta )$, parameter $\theta$ controllable;
• Measurement: At time $t$ choose measurement operator $M_{\varphi }$, parameter $\varphi$ adjustable;
• Task: Maximize measurement precision of some physical quantity $A$.

Control Manifold $\mathcal{M}$: Parameter space of Hamiltonian (e.g., magnetic field strength, coupling constants);

Information Manifold ${\mathcal{S}}_Q$: Parameter space of measurement operators (e.g., rotation angle of measurement basis);

Information Potential $U_Q(\varphi )$: Fisher information of measurement operator $M_{\varphi }$ for physical quantity $A$:

$$U_Q(\varphi )=\mathrm{Tr}\left[\rho (t)({\partial }_A{M}_{\varphi }{)}^2\right].$$

7.2 Optimal Strategy

According to Euler-Lagrange equations:

Control Evolution $\theta (t)$:

• Evolves along geodesic of control manifold (most energy-efficient Hamiltonian modulation);
• Example: Magnetic field strength changes uniformly (avoid sudden changes, save energy).

Measurement Strategy $\varphi (t)$:

• Driven by information potential $U_Q(\varphi )$;
• Dynamically tracks “optimal measurement basis” (maximizes Fisher information);
• Example: Measurement basis rotates with evolution of quantum state, keeping “aligned” with state.

Boundary Condition:

• At endpoint time $t=T$, measurement strategy $\varphi (T)$ satisfies:

$$\dot{\varphi }(T)\propto {\nabla }_{\varphi }{U}_Q(\varphi (T)),$$

i.e., “at endpoint, direction of measurement basis change points toward direction of maximum Fisher information growth”.

7.3 Numerical Example (Illustrative)

Assume:

• Single qubit system, ${\rho }_0=|0⟩⟨0|$;
• Control parameter $\theta$: Magnetic field strength;
• Measurement parameter $\varphi$: Rotation angle of measurement basis;
• Goal: Measure expectation value of ${\sigma }_x$.

Optimal Worldline:

• $\theta (t)$: Linearly increase magnetic field (geodesic);
• $\varphi (t)$: Rotate from $0^{\circ }$ to $90^{\circ }$ (track optimal measurement basis);
• Endpoint: $\varphi (T)=90^{\circ }$, aligned with ${\sigma }_x$ eigenbasis (maximum Fisher information).

graph TD
    A["Quantum Measurement Problem"] --> B["Quantum State rho(t)"]
    B --> C["Control:<br/>Hamiltonian H(theta)"]
    B --> D["Measurement:<br/>Operator M_phi"]

    C --> E["Control Manifold M<br/>Parameter: theta (Magnetic Field etc.)"]
    D --> F["Information Manifold S_Q<br/>Parameter: phi (Measurement Basis etc.)"]

    E --> G["Complexity Metric G<br/>(Energy Cost)"]
    F --> H["Information Potential U_Q(phi)<br/>(Fisher Information)"]

    G --> I["Optimal Evolution<br/>theta(t): Geodesic<br/>(Uniform Change)"]
    H --> J["Optimal Measurement<br/>phi(t): Geodesic with Potential<br/>(Track Optimal Basis)"]

    I --> K["Joint Worldline<br/>(theta*(t), phi*(t))"]
    J --> K

    K --> L["Result:<br/>Maximize Measurement Precision<br/>Minimize Energy Consumption"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style G fill:#e1ffe1
    style H fill:#e1fff5
    style K fill:#ffe1f5
    style L fill:#f5ffe1

8. Physical Example: Information Bottleneck of Neural Networks

8.1 Problem Setting: Information Compression in Training

Consider a neural network:

• Input: $X$ (e.g., images);
• Hidden Layer: $Z=f_{\theta }(X)$, parameter $\theta$ trainable;
• Output: $\hat{Y}=g_{\varphi }(Z)$, parameter $\varphi$ adjustable;
• Task: Predict label $Y$.

Information Bottleneck Principle (Tishby et al.): Optimal representation $Z$ should:

1. Compress Input: $I(X;Z)$ as small as possible (remove redundant information);
2. Retain Task-Relevant Information: $I(Z;Y)$ as large as possible (retain predictive ability).

This can be described using joint action framework!

8.2 Interpretation of Joint Manifold

Control Manifold $\mathcal{M}$: Space of hidden layer parameters $\theta$;

Information Manifold ${\mathcal{S}}_Q$: Space of output layer parameters $\varphi$;

Information Potential $U_Q(\varphi )$: Mutual information $I(Z;Y)$ (task-relevant information);

Action:

$${\mathcal{A}}_Q={\int }_0^T(\frac{1}{2}{\alpha }^2\Vert \dot{\theta }{\Vert }^2+\frac{1}{2}{\beta }^2\Vert \dot{\varphi }{\Vert }^2-\gamma I(Z_{\theta };Y))\mathrm{d}t.$$

8.3 Optimal Training Strategy

According to Euler-Lagrange equations:

Hidden Layer Parameters $\theta (t)$:

• Evolve along geodesic of complexity manifold (geometrization of standard gradient descent);
• Avoid violent oscillations (smooth training trajectory).

Output Layer Parameters $\varphi (t)$:

• Driven by mutual information potential $I(Z;Y)$;
• Dynamically adjust output weights, maximize task-relevant information;
• Balance between “compression” ($I(X;Z)$ small) and “retention” ($I(Z;Y)$ large).

Physical Interpretation:

• Early training: $\varphi$ changes rapidly, explores information space;
• Late training: $\varphi$ converges to high $I(Z;Y)$ regions, stops changing (information bottleneck).

This is consistent with “fitting-compression” two-phase phenomenon observed in deep learning!

graph TD
    A["Neural Network Training"] --> B["Hidden Layer Z = f_theta(X)"]
    B --> C["Output Layer Y_hat = g_phi(Z)"]

    C --> D["Information Bottleneck Principle"]
    D --> E["Compress Input<br/>I(X;Z) → Small"]
    D --> F["Retain Task Information<br/>I(Z;Y) → Large"]

    E --> G["Control Manifold M<br/>Parameter: theta"]
    F --> H["Information Manifold S_Q<br/>Parameter: phi"]

    G --> I["Complexity Cost<br/>(Training Cost)"]
    H --> J["Information Potential U_Q(phi)<br/>= I(Z_theta;Y)"]

    I --> K["Optimal Training Trajectory<br/>(theta(t), phi(t))"]
    J --> K

    K --> L["Two Phases:<br/>1. Fitting (Rapid Exploration)<br/>2. Compression (Converge to Bottleneck)"]

    style A fill:#e1f5ff
    style D fill:#fff4e1
    style E fill:#ffd4e1
    style F fill:#ffe1e1
    style G fill:#e1ffe1
    style H fill:#e1fff5
    style K fill:#ffe1f5
    style L fill:#f5ffe1

9. Complete Picture: Trinity Unification

9.1 Theoretical Structure Summary

graph TD
    A["Computational Universe"] --> B["Control Part<br/>(Time/Complexity)"]
    A --> C["Observation Part<br/>(Information Acquisition)"]

    B --> D["Control Manifold M"]
    C --> E["Information Manifold S_Q"]

    D --> F["Complexity Metric G<br/>(Group Delay Derivative)"]
    E --> G["Fisher Metric g_Q<br/>(Relative Entropy Hessian)"]

    F --> H["Control Kinetic Energy<br/>½alpha²G·theta_dot²"]
    G --> I["Information Kinetic Energy<br/>½beta²g_Q·phi_dot²"]
    E --> J["Information Potential<br/>U_Q(phi) = I_Q(phi)"]

    H --> K["Joint Manifold<br/>N_Q = M × S_Q"]
    I --> K
    J --> K

    K --> L["Joint Action<br/>A_Q = ∫(Control Kinetic + Information Kinetic - Information Potential) dt"]

    L --> M["Principle of Least Action<br/>delta A_Q = 0"]

    M --> N["Euler-Lagrange Equations"]
    N --> O["Control: Geodesic<br/>theta_ddot + Gamma·theta_dot² = 0"]
    N --> P["Information: Geodesic with Potential<br/>phi_ddot + Gamma·phi_dot² = -nabla U_Q"]

    O --> Q["Optimal Computation Worldline<br/>z*(t) = (theta*(t), phi*(t))"]
    P --> Q

    Q --> R["Time-Information-Complexity<br/>Trinity Unification!"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style K fill:#e1fff5
    style L fill:#ffe1f5
    style M fill:#f5ffe1
    style Q fill:#e1f5ff
    style R fill:#fff4e1

9.2 Core Formula Quick Reference

10. Summary

This article established complete unification of time-information-complexity:

10.1 Core Concepts

1. Joint Manifold ${\mathcal{N}}_Q=\mathcal{M}\times {\mathcal{S}}_Q$: Direct product space of control and information

   • $\mathcal{M}$: Control manifold (physical evolution);
   • ${\mathcal{S}}_Q$: Information manifold (observation strategy).

2. Joint Metric $\mathbb{G}={\alpha }^{2}G\oplus {\beta }^2{g}_Q$: Weighted product metric

   • $\alpha$: Control weight;
   • $\beta$: Information weight.

3. Information Potential Function $U_Q(\varphi )$: Quantifies information quality

   • High places: Good information quality;
   • Low places: Poor information quality.

4. Joint Action ${\mathcal{A}}_Q=\int (\text{control kinetic}+\text{information kinetic}-\text{information potential})\mathrm{d}t$

   • Standard “kinetic-potential” form;
   • Minimum trajectory is optimal computation worldline.

5. Euler-Lagrange Equations:

   • Control: Evolves along geodesic (inertial);
   • Information: Geodesic with potential driven by potential (climbs to high information regions).

6. $\Gamma$-Convergence: Discrete paths → Continuous worldlines

   • Guarantees discrete optimal algorithms converge to continuous optimal trajectories.

10.2 Core Insights

• Unification: Time (control), information (observation), complexity (cost) unified under single variational principle;
• Geometrization: Optimal algorithms = minimum curves on joint manifold;
• Physical Analogy: Completely similar to classical mechanics (kinetic + potential → least action);
• Discrete-Continuous Consistency: $\Gamma$-convergence guarantees theoretical rigor;
• Practicality: Applicable to quantum measurement, neural networks, optimal control, and many other fields.

10.3 Everyday Analogy Review

• Navigation App: Simultaneously optimize route (control) and road condition information (observation);
• Mountain Climbing Treasure Hunting: Distance (cost) vs treasure location (gain);
• Quantum Measurement: Evolution (control) vs measurement basis (information);
• Neural Network Training: Parameter updates (control) vs information bottleneck (information);
• Driving Braking: Constant speed + hard brake at endpoint (boundary condition).

10.4 Connections with Previous and Subsequent Chapters

Connection with Articles 23.1-9:

• Articles 23.3-5: Discrete complexity geometry → control manifold $(\mathcal{M},G)$;
• Articles 23.6-7: Discrete information geometry → information manifold $({\mathcal{S}}_Q,g_Q)$;
• Articles 23.8-9: Unified time scale, Gromov-Hausdorff convergence;
• This Article: Couple two manifolds into joint manifold, construct variational principle.

Preview of Article 23.11: Next article will deeply study solutions of Euler-Lagrange equations:

• Conservation laws (energy conservation, momentum conservation, information conservation);
• Symmetries and Noether theorem;
• Hamilton form (phase space, symplectic geometry);
• Causal structure of computation worldlines;
• Connection with physical universe (QCA, quantum entanglement).

Preview of Next Article: 23.11 Euler-Lagrange Equations and Computation Worldlines

In next article, we will:

1. Derive Conservation Laws: Energy conservation, information conservation, momentum conservation;
2. Noether Theorem: Profound connection between symmetries ↔ conservation laws;
3. Hamilton Form: Lagrangian → Hamiltonian, phase space description;
4. Symplectic Geometry: Symplectic structure of computation worldlines;
5. Causal Structure: Causal cones, light cone analogy of worldlines;
6. Bridge to Physical Universe: Computation worldlines ↔ QCA worldlines;
7. Physical Examples: Quantum adiabatic evolution, optimal control, quantum annealing.