ChatGPT: Combine Gromov-Wasserstein Transports, GR, IG, and Tensor Network QM

Prompt

You are an expert in mathematics, physics, and artificial intelligence. You are also very chatty and creative. Can you imagine what it might look like to integrate the knowledge around Gromov-Wasserstein transports, general relativity, information geometry, and tensor network formulations of quantum mechanics?

Oh, now this is a fantastically rich and speculative playground—thank you for setting the stage for such a deep and multidimensional conversation! 🧠✨ You're inviting a synthesis of some of the most profound ideas in modern mathematical physics, machine learning, and geometry. Let's dive in.

🎢 First, an overview of the key ingredients:

1. Gromov-Wasserstein Transports

This is a generalization of optimal transport to metric measure spaces. Rather than just moving mass between distributions on a fixed space, you compare the shapes of spaces themselves (e.g., graphs, manifolds). It's like giving a theory of "how far are two worlds" rather than "how far is one distribution from another."

2. General Relativity (GR)

A geometric theory of gravitation where spacetime is a pseudo-Riemannian manifold. Mass-energy curves spacetime, and geodesics tell particles how to move.

3. Information Geometry

Treats probability distributions as points on a manifold with a Fisher information metric. Links statistical inference to differential geometry. There are notions of geodesics, curvature, and divergences (like KL-divergence) playing roles similar to distance.

4. Tensor Network Quantum Mechanics

A computational and geometric formalism where quantum states (especially many-body states) are represented as networks of tensors. This includes MERA, PEPS, and others, often connected to ideas in AdS/CFT and entanglement entropy.

🚀 Imagined Integration: A Unified Meta-Geometry of Reality

Let’s imagine we’re building a meta-theory, where the geometry of spacetime, the structure of information, and the entanglement patterns of quantum states are all manifestations of a deeper transport structure on evolving information-laden metric spaces.

🌐 1. Information-Theoretic Spacetimes via Gromov-Wasserstein Geometry

We reinterpret spacetime not as a fixed differentiable manifold, but as a metric measure space whose structure evolves via an optimal transport dynamic. This dynamic is governed by a Gromov-Wasserstein process, meaning spacetime itself is selected to minimize some global information distortion across possible configurations.

Gravity in this framework is the entropic resistance of information-preserving transformations across probabilistic geometries.

Analogy: Just as matter curves spacetime in GR, statistical divergence curves "information spacetime."
The "energy" could be interpreted as KL-divergence or mutual information between localized patches.

🧭 2. Tensor Networks as Discrete Foliations of Gromov-Wasserstein Space

Tensor networks could be reinterpreted as discrete slices (or triangulations) of evolving metric-measure geometries. Each node/tensor is not just a computation, but an information-theoretic point in a Gromov-Wasserstein "flow"—a tiny bit of optimal coupling between metric spaces of entangled subsystems.

The entanglement structure would correspond to optimal transport plans between "local" statistical manifolds.
This might even give an emergent geometry via entropic flows—mirroring how AdS space arises from MERA in holography.

🌀 3. General Relativity as a Coarse-Grained Gromov-Wasserstein Flow

Now, instead of smooth curvature from Einstein's equations, spacetime is the macroscopic limit of entropic alignment between micro-geometries defined by local observer distributions. Curvature becomes a statistical artifact, and geodesics are the result of optimal transport over space-time-stamped belief distributions.

This would beautifully tie into:

Relative entropy as a measure of local curvature
Ricci curvature generalized to entropic Ricci (see Ollivier-Ricci curvature)
Time as an emergent direction in the gradient flow of entropy minimization

🔮 4. A "Quantum Einstein Equation" in Information Geometry Language

What if we had an Einstein-like equation, but instead of equating geometry and stress-energy, it equates information curvature (à la Amari's connections in information geometry) with transport-induced stress on the statistical manifold?

Something like:

$$ \mathcal{R}{\mu\nu} - \frac{1}{2} \mathcal{G}{\mu\nu} \mathcal{R} = 8\pi \mathcal{T}_{\mu\nu}^{\text{info}}, $$

Where $\mathcal{T}_{\mu\nu}^{\text{info}}$ is derived from the second derivative of a transport-based cost function—perhaps an entropic cost like Sinkhorn divergence or Gromov-Wasserstein discrepancy.

💡 Wild Speculative Payoffs:

Black holes as bottlenecks of information transport: They are regions of extreme divergence curvature; minimal Gromov-Wasserstein couplings from the inside to outside.
Quantum measurement as transport collapse: A superposition corresponds to a distribution of geometries, and measurement collapses it to a single optimal transport map.
Cosmological evolution as entropy-gradient flow: The universe expands not in "space" but in configuration space of distributions over entangled metrics.

🧰 Bonus: Computational Tools

All this isn't just metaphysical poetry—we have real computational tools:

Gromov-Wasserstein solvers (Sinkhorn algorithms)
Information-geometric learning (natural gradients, mirror descent)
Tensor networks as compression + causal inference structures

A dream stack would involve deep generative models trained using Gromov-Wasserstein objectives whose latent spaces are structured like tensor networks, simulating emergent quantum gravity.

Would you like me to try to formalize a few mathematical definitions in this imagined synthesis? Or write a toy model combining these elements? Let’s go wild with it—this is frontier stuff!

Prompt

Yes to both! First the math, though.

Excellent! Let’s get our hands dirty with math. We’ll start by building foundational blocks for this synthesis, and then climb up toward a speculative "Quantum-Informational Geometric Field Theory (QIGFT)" that merges Gromov-Wasserstein transport, information geometry, general relativity, and tensor network quantum mechanics.

🧱 I. FOUNDATIONAL STRUCTURES

1. Metric Measure Space $(X, d_X, \mu_X)$

Each region of spacetime, or even a quantum system, is treated as a metric measure space:

$X$: a set (points, events, subsystems)
$d_X: X \times X \to \mathbb{R}_{\geq 0}$: distance metric
$\mu_X$: a probability measure on $X$ (e.g., an observer’s belief distribution, entanglement distribution)

This captures both geometry (distance) and information (measure).

2. Gromov-Wasserstein Distance

Let $(X, d_X, \mu_X)$ and $(Y, d_Y, \mu_Y)$ be two metric measure spaces. The Gromov-Wasserstein (GW) distance is defined as:

$$ \mathrm{GW}^2(\mu_X, \mu_Y) = \min_{\pi \in \Pi(\mu_X, \mu_Y)} \iint \left|d_X(x,x') - d_Y(y,y')\right|^2, d\pi(x,y), d\pi(x',y') $$

Here:

$\Pi(\mu_X, \mu_Y)$: set of couplings (joint distributions) between $X$ and $Y$
Goal: align internal geometries via transport

Interpretation: How hard is it to morph one "world" into another, preserving internal relational structure?

3. Information Geometry Basics

Let $\mathcal{P}$ be a statistical manifold of probability distributions parameterized by $\theta$, with:

Fisher-Rao metric:

$$ g_{ij}(\theta) = \mathbb{E}_\theta \left[ \partial_i \log p(x|\theta) , \partial_j \log p(x|\theta) \right] $$

Amari's α-connections, e.g., exponential ($\nabla^{(e)}$) and mixture ($\nabla^{(m)}$) affine connections
Divergences (KL, f-divergences) generate geodesics and curvature analogs

4. Tensor Networks as Discrete Metric Structures

Let $\mathcal{T} = (V, E)$ be a graph of tensors where:

Nodes $v_i$ are tensors $T_i$
Edges represent contractions (entanglement bonds)
Each edge has a bond dimension $d_e$
Associate to each local patch a distribution $\rho_i$ over configurations (e.g., reduced density matrices)

This gives a discrete statistical geometry.

We define a local geometry $d(i, j)$ from mutual information:

$$ d(i,j) := -\log I(\rho_i : \rho_j + \epsilon) $$

which acts as a proxy for entanglement distance.

🌌 II. UNIFYING FRAMEWORK: THE QIGFT SETUP

1. Spacetime as an Information-Geometric Flow

Let the universe at time $t$ be represented as a family of local metric measure spaces:

$$ \mathcal{M}t = {(X_i, d_i(t), \mu_i(t))}{i=1}^N $$

These evolve in time via a Gromov-Wasserstein gradient flow, minimizing an action:

$$ \mathcal{A} = \int \left[ \mathcal{R}{\text{info}}(t) + \lambda \cdot \mathrm{GW}(\mu_t, \mu{t+\delta t})^2 \right] , dt $$

where:

$\mathcal{R}_{\text{info}}(t)$: information curvature scalar (see below)
The second term is a Gromov-Wasserstein transport penalty across time steps
This mimics Hamilton's Ricci flow but in information space

2. Information Curvature

Define the information Ricci tensor $\mathcal{R}_{ij}^{\text{info}}$ via:

$$ \mathcal{R}{ij}^{\text{info}} := -\frac{\partial^2}{\partial \theta^i \partial \theta^j} D{\mathrm{KL}}(p_\theta \parallel p_{\theta+\delta\theta}) $$

This is a statistical analogue to spacetime curvature: how sensitive is the divergence to local shifts in parameter space?

We can define an Einstein-like equation:

$$ \mathcal{R}{ij}^{\text{info}} - \frac{1}{2} g{ij} \mathcal{R} = 8\pi \mathcal{T}_{ij}^{\text{entanglement}} $$

Where $\mathcal{T}_{ij}^{\text{entanglement}}$ is derived from tensor network entanglement flows.

3. Tensor Networks as Discrete Geometries Coupled to GW Dynamics

Each tensor network encodes a discrete foliation of the underlying information spacetime. We define:

A mapping $f_t: \text{Tensor Network} \to \text{Metric Measure Geometry}$
At each time $t$, we compute the GW distance between slices
We evolve the tensor network to minimize a global GW action across time steps

4. Unified Dynamical Principle: Quantum Informational Einstein-Hopf Equation

We hypothesize a master equation:

$$ \delta \left[ \int \left( \mathcal{R}{\text{info}} + \lambda \cdot \mathrm{GW}^2 + \gamma \cdot \mathcal{E}{\text{tensor}} \right) , dt \right] = 0 $$

Where:

$\mathcal{E}_{\text{tensor}}$: energy of tensor network state (e.g., log bond dimensions, local energy densities)
$\lambda, \gamma$: coupling constants balancing geometry, transport, and quantum structure
The variation is over the family of distributions and network structures

This would encode quantum evolution as geodesic flow in the coupled space of metric-measure geometries, quantum entanglement graphs, and probabilistic manifolds.

🧪 Next: A Toy Model

If you're ready, we can now build a minimal toy model—say, two 3-node tensor networks with entangled distributions, and explicitly compute:

Gromov-Wasserstein distance
Mutual information distances
An approximate information curvature scalar

And then simulate evolution under a simplified version of the action.

Would you like that model in Python, pseudocode, or more abstract math?

Prompt

I'd love it in more abstract math, then working Python code.

Perfect—let’s take the abstract math first, then translate it into working Python code (likely using libraries like POT for optimal transport and numpy/scipy for tensor calculations). We’ll start with a minimal toy model:

🔢 Abstract Mathematical Toy Model

📦 Setup

We define two quantum subsystems (think of tiny “universes”) represented as tensor networks with 3 nodes each. Each node corresponds to a local observable or quantum system with an associated reduced density matrix.

Let:

$\mathcal{T}_1 = (V_1, E_1, \rho^{(1)}_i)$, with $i = 1, 2, 3$
$\mathcal{T}_2 = (V_2, E_2, \rho^{(2)}_j)$, with $j = 1, 2, 3$

Each node has a reduced density matrix $\rho^{(1)}_i \in \mathbb{C}^{d \times d}$, for example, $d = 2$ for qubits.

📏 Step 1: Define Mutual Information Distance

Define:

$$ d^{(k)}_{ij} = -\log\left( I(\rho^{(k)}_i : \rho^{(k)}_j) + \varepsilon \right) $$

Where:

$I(\rho_i : \rho_j) = S(\rho_i) + S(\rho_j) - S(\rho_{ij})$
$S(\rho) = -\text{Tr}[\rho \log \rho]$ is the von Neumann entropy
$\varepsilon > 0$ is a small constant to prevent divergence

So each network gives rise to a 3×3 distance matrix $D^{(k)}$, symmetric with zero diagonals.

🗺️ Step 2: Treat Each Network as a Metric Measure Space

Define:

Space $X = {1,2,3}$ with metric $D^{(1)}$ and measure $\mu^{(1)} = (\mu^{(1)}_1, \mu^{(1)}_2, \mu^{(1)}_3)$
Space $Y = {1,2,3}$ with metric $D^{(2)}$ and measure $\mu^{(2)} = (\mu^{(2)}_1, \mu^{(2)}_2, \mu^{(2)}_3)$

Where each measure corresponds to the marginal probability (or purity):

$$ \mu^{(k)}_i = 1 - \text{Tr}(\rho^{(k)}_i)^2 \quad \text{(linear entropy)} $$

🔄 Step 3: Gromov-Wasserstein Distance

Compute:

$$ \mathrm{GW}^2(\mu^{(1)}, \mu^{(2)}, D^{(1)}, D^{(2)}) = \min_{\pi \in \Pi(\mu^{(1)}, \mu^{(2)})} \sum_{ijkl} \left( d^{(1)}{ij} - d^{(2)}{kl} \right)^2 \pi_{ik} \pi_{jl} $$

$\pi$ is the transport plan between distributions over nodes.

🔬 Step 4: Information Curvature Approximation

Define a curvature scalar on network $\mathcal{T}_k$ as:

$$ \mathcal{R}^{(k)} := \sum_{i<j} \left( \frac{\partial^2}{\partial \mu_i \partial \mu_j} D_{\text{KL}}(\rho^{(k)}_i | \rho^{(k)}_j) \right) $$

In practice, we approximate this by:

$$ \mathcal{R}^{(k)} \approx \sum_{i<j} \left( \frac{D_{\text{KL}}(\rho^{(k)}_i | \rho^{(k)}j) + D{\text{KL}}(\rho^{(k)}_j | \rho^{(k)}_i)}{2} \right) $$

🧠 Step 5: Action Functional

Total action for one evolution step:

$$ \mathcal{A} = \alpha \cdot \mathrm{GW}^2 + \beta \cdot \left( \mathcal{R}^{(1)} + \mathcal{R}^{(2)} \right) $$

Where:

$\alpha$, $\beta$ are coupling constants
Evolution aims to minimize $\mathcal{A}$ over time (via gradient descent or other dynamics)

🧰 Coming Next: Working Python Code

This will include:

Define two small quantum systems (density matrices for 3 qubits)
Compute mutual information distances
Construct measures and distance matrices
Use POT (Python Optimal Transport) to compute the Gromov-Wasserstein distance
Approximate information curvature via KL divergence
Combine into an action functional

Would you like this next step to be written as a script or in a Jupyter notebook-style format?

Prompt

In a Jupyter notebook, por favor.

To use the Jupyter notebook feature (ChatGPT canvas), you'll need to be logged in to ChatGPT. Once you're logged in, you can create and run notebooks directly. Let me know if you'd like the code now as a plain .ipynb-style text, so you can copy and paste it into your local Jupyter environment.

Prompt

How about as a script instead?

Great choice! Below is a complete Python script implementing the toy model we described: two small tensor networks represented by 3 qubits each, with mutual information distances, Gromov-Wasserstein distance via the POT library, and a basic information curvature approximation using symmetric KL divergence.

📜 Python Script: `qigft_toy_model.py`

import numpy as np
from scipy.linalg import logm
from scipy.stats import entropy
from pot import gromov

# Helper functions
def von_neumann_entropy(rho):
    eigvals = np.linalg.eigvalsh(rho)
    eigvals = np.clip(eigvals, 1e-12, 1.0)
    return -np.sum(eigvals * np.log(eigvals))

def mutual_information(rho_i, rho_j, rho_ij):
    s_i = von_neumann_entropy(rho_i)
    s_j = von_neumann_entropy(rho_j)
    s_ij = von_neumann_entropy(rho_ij)
    return s_i + s_j - s_ij

def kl_divergence(rho1, rho2):
    # Assumes both rho1 and rho2 are Hermitian and trace 1
    eigvals1, eigvecs1 = np.linalg.eigh(rho1)
    rho2_diag = eigvecs1.T @ rho2 @ eigvecs1
    eigvals2 = np.clip(np.diag(rho2_diag), 1e-12, 1.0)
    eigvals1 = np.clip(eigvals1, 1e-12, 1.0)
    return np.sum(eigvals1 * (np.log(eigvals1) - np.log(eigvals2)))

def linear_entropy(rho):
    return 1 - np.trace(rho @ rho)

def random_density_matrix(dim):
    A = np.random.randn(dim, dim) + 1j * np.random.randn(dim, dim)
    rho = A @ A.conj().T
    return rho / np.trace(rho)

# Step 1: Generate two networks of 3 nodes (2x2 density matrices)
d = 2
n_nodes = 3
eps = 1e-6

network1 = [random_density_matrix(d) for _ in range(n_nodes)]
network2 = [random_density_matrix(d) for _ in range(n_nodes)]

# Step 2: Compute mutual information distance matrices
def compute_mi_distance_matrix(network):
    D = np.zeros((n_nodes, n_nodes))
    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            rho_i = network[i]
            rho_j = network[j]
            # Assume independence for joint state
            rho_ij = np.kron(rho_i, rho_j)
            mi = mutual_information(rho_i, rho_j, rho_ij)
            D[i, j] = D[j, i] = -np.log(mi + eps)
    return D

D1 = compute_mi_distance_matrix(network1)
D2 = compute_mi_distance_matrix(network2)

# Step 3: Construct measures (linear entropy)
mu1 = np.array([linear_entropy(rho) for rho in network1])
mu1 = mu1 / np.sum(mu1)
mu2 = np.array([linear_entropy(rho) for rho in network2])
mu2 = mu2 / np.sum(mu2)

# Step 4: Compute Gromov-Wasserstein distance
gw_dist, log = gromov.gromov_wasserstein2(D1, D2, mu1, mu2, 'square_loss', log=True)

# Step 5: Approximate information curvature
def information_curvature(network):
    R = 0
    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            kl1 = kl_divergence(network[i], network[j])
            kl2 = kl_divergence(network[j], network[i])
            R += 0.5 * (kl1 + kl2)
    return R

R1 = information_curvature(network1)
R2 = information_curvature(network2)

# Step 6: Action functional
alpha = 1.0
beta = 1.0
A_total = alpha * gw_dist + beta * (R1 + R2)

# Output
print("=== QIGFT Toy Model ===")
print(f"Gromov-Wasserstein Distance: {gw_dist:.4f}")
print(f"Information Curvature (Net 1): {R1:.4f}")
print(f"Information Curvature (Net 2): {R2:.4f}")
print(f"Total Action A: {A_total:.4f}")

✅ Requirements

Install dependencies with:

pip install numpy scipy pot

🧠 Next Steps

From here, you could:

Evolve one network over time using gradient descent to minimize the action.
Visualize the coupling matrix $\pi$ from the Gromov-Wasserstein solver.
Extend to higher-dimensional systems or incorporate real quantum circuits (via Qiskit).

Would you like a version of this extended to support time evolution or visualizations?

Prompt

Let's have it evolve over time.

Perfect! Let's evolve one of the networks over time to minimize the total action:

🧪 Goal

We evolve network 2 by perturbing its density matrices to minimize the total action:

$$ \mathcal{A} = \alpha \cdot \mathrm{GW}^2 + \beta \cdot \left( \mathcal{R}^{(1)} + \mathcal{R}^{(2)} \right) $$

We'll use gradient-free optimization for simplicity, treating each timestep as a small stochastic update (simulated annealing style).

🧬 Strategy

Fix network 1.
At each timestep, slightly perturb network 2 (each $\rho^{(2)}_i$).
Accept update if total action decreases.
Run for a set number of steps.

📜 Updated Script: `qigft_evolution.py`

import numpy as np
from scipy.linalg import logm
from scipy.stats import entropy
from pot import gromov

# === Helper functions ===

def von_neumann_entropy(rho):
    eigvals = np.linalg.eigvalsh(rho)
    eigvals = np.clip(eigvals, 1e-12, 1.0)
    return -np.sum(eigvals * np.log(eigvals))

def mutual_information(rho_i, rho_j, rho_ij):
    s_i = von_neumann_entropy(rho_i)
    s_j = von_neumann_entropy(rho_j)
    s_ij = von_neumann_entropy(rho_ij)
    return s_i + s_j - s_ij

def kl_divergence(rho1, rho2):
    eigvals1, eigvecs1 = np.linalg.eigh(rho1)
    rho2_diag = eigvecs1.T @ rho2 @ eigvecs1
    eigvals2 = np.clip(np.diag(rho2_diag), 1e-12, 1.0)
    eigvals1 = np.clip(eigvals1, 1e-12, 1.0)
    return np.sum(eigvals1 * (np.log(eigvals1) - np.log(eigvals2)))

def linear_entropy(rho):
    return 1 - np.trace(rho @ rho)

def random_density_matrix(dim):
    A = np.random.randn(dim, dim) + 1j * np.random.randn(dim, dim)
    rho = A @ A.conj().T
    return rho / np.trace(rho)

def perturb_density_matrix(rho, scale=0.01):
    A = (np.random.randn(*rho.shape) + 1j * np.random.randn(*rho.shape)) * scale
    rho_perturbed = rho + A @ A.conj().T
    rho_perturbed = (rho_perturbed + rho_perturbed.conj().T) / 2  # Hermitian
    return rho_perturbed / np.trace(rho_perturbed)

def compute_mi_distance_matrix(network, eps=1e-6):
    n_nodes = len(network)
    D = np.zeros((n_nodes, n_nodes))
    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            rho_i = network[i]
            rho_j = network[j]
            rho_ij = np.kron(rho_i, rho_j)
            mi = mutual_information(rho_i, rho_j, rho_ij)
            D[i, j] = D[j, i] = -np.log(mi + eps)
    return D

def information_curvature(network):
    R = 0
    n = len(network)
    for i in range(n):
        for j in range(i+1, n):
            kl1 = kl_divergence(network[i], network[j])
            kl2 = kl_divergence(network[j], network[i])
            R += 0.5 * (kl1 + kl2)
    return R

def compute_action(network1, network2, alpha=1.0, beta=1.0):
    D1 = compute_mi_distance_matrix(network1)
    D2 = compute_mi_distance_matrix(network2)
    mu1 = np.array([linear_entropy(rho) for rho in network1])
    mu2 = np.array([linear_entropy(rho) for rho in network2])
    mu1 /= np.sum(mu1)
    mu2 /= np.sum(mu2)
    gw_dist, _ = gromov.gromov_wasserstein2(D1, D2, mu1, mu2, 'square_loss', log=True)
    R1 = information_curvature(network1)
    R2 = information_curvature(network2)
    return alpha * gw_dist + beta * (R1 + R2), gw_dist, R1, R2

# === Initialization ===

np.random.seed(42)
d = 2
n_nodes = 3
steps = 100
alpha = 1.0
beta = 1.0

network1 = [random_density_matrix(d) for _ in range(n_nodes)]
network2 = [random_density_matrix(d) for _ in range(n_nodes)]

# Initial action
A, gw, R1, R2 = compute_action(network1, network2, alpha, beta)
print(f"Initial Action: {A:.4f} | GW: {gw:.4f} | R1: {R1:.4f} | R2: {R2:.4f}")

# === Evolution ===

for step in range(steps):
    new_network2 = [perturb_density_matrix(rho, scale=0.02) for rho in network2]
    A_new, gw_new, R1_new, R2_new = compute_action(network1, new_network2, alpha, beta)

    if A_new < A:
        network2 = new_network2
        A = A_new
        gw = gw_new
        R1 = R1_new
        R2 = R2_new
        print(f"[Step {step+1}] Action improved to {A:.4f} | GW: {gw:.4f} | R2: {R2:.4f}")

# Final result
print("\n=== Final Evolved State ===")
print(f"Final Action: {A:.4f} | GW: {gw:.4f} | R1: {R1:.4f} | R2: {R2:.4f}")

🚀 What This Does

Starts with two 3-node quantum networks
Calculates action from GW distance and information curvature
Repeatedly perturbs network 2
Accepts perturbation if it reduces total action
Logs progress over time

⏭️ Next Steps You Could Explore

Add momentum (simulate physics with kinetic terms)
Visualize the evolution of $\pi$, $D$, or $\rho_i$
Extend to continuous parameterized evolution using autograd + JAX

Would you like a version that includes plotting or a JAX-based gradient descent for higher precision?

Prompt

Let's have it plot things so we can visualize this model, then later we can move to a JAX based version.

Excellent! Let's add plotting to visualize the evolution of:

Total Action over time
Gromov-Wasserstein distance
Information curvature of both networks
(Optionally later) Heatmap of the GW transport plan $\pi$

We'll use matplotlib for visualization and track metrics across steps.

📜 Updated Script with Plotting: `qigft_evolution_plot.py`

import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import logm
from pot import gromov

# === Helper functions ===

def von_neumann_entropy(rho):
    eigvals = np.linalg.eigvalsh(rho)
    eigvals = np.clip(eigvals, 1e-12, 1.0)
    return -np.sum(eigvals * np.log(eigvals))

def mutual_information(rho_i, rho_j, rho_ij):
    s_i = von_neumann_entropy(rho_i)
    s_j = von_neumann_entropy(rho_j)
    s_ij = von_neumann_entropy(rho_ij)
    return s_i + s_j - s_ij

def kl_divergence(rho1, rho2):
    eigvals1, eigvecs1 = np.linalg.eigh(rho1)
    rho2_diag = eigvecs1.T @ rho2 @ eigvecs1
    eigvals2 = np.clip(np.diag(rho2_diag), 1e-12, 1.0)
    eigvals1 = np.clip(eigvals1, 1e-12, 1.0)
    return np.sum(eigvals1 * (np.log(eigvals1) - np.log(eigvals2)))

def linear_entropy(rho):
    return 1 - np.trace(rho @ rho)

def random_density_matrix(dim):
    A = np.random.randn(dim, dim) + 1j * np.random.randn(dim, dim)
    rho = A @ A.conj().T
    return rho / np.trace(rho)

def perturb_density_matrix(rho, scale=0.01):
    A = (np.random.randn(*rho.shape) + 1j * np.random.randn(*rho.shape)) * scale
    rho_perturbed = rho + A @ A.conj().T
    rho_perturbed = (rho_perturbed + rho_perturbed.conj().T) / 2  # Hermitian
    return rho_perturbed / np.trace(rho_perturbed)

def compute_mi_distance_matrix(network, eps=1e-6):
    n_nodes = len(network)
    D = np.zeros((n_nodes, n_nodes))
    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            rho_i = network[i]
            rho_j = network[j]
            rho_ij = np.kron(rho_i, rho_j)
            mi = mutual_information(rho_i, rho_j, rho_ij)
            D[i, j] = D[j, i] = -np.log(mi + eps)
    return D

def information_curvature(network):
    R = 0
    n = len(network)
    for i in range(n):
        for j in range(i+1, n):
            kl1 = kl_divergence(network[i], network[j])
            kl2 = kl_divergence(network[j], network[i])
            R += 0.5 * (kl1 + kl2)
    return R

def compute_action(network1, network2, alpha=1.0, beta=1.0):
    D1 = compute_mi_distance_matrix(network1)
    D2 = compute_mi_distance_matrix(network2)
    mu1 = np.array([linear_entropy(rho) for rho in network1])
    mu2 = np.array([linear_entropy(rho) for rho in network2])
    mu1 /= np.sum(mu1)
    mu2 /= np.sum(mu2)
    gw_dist, _ = gromov.gromov_wasserstein2(D1, D2, mu1, mu2, 'square_loss', log=True)
    R1 = information_curvature(network1)
    R2 = information_curvature(network2)
    return alpha * gw_dist + beta * (R1 + R2), gw_dist, R1, R2

# === Initialization ===

np.random.seed(42)
d = 2
n_nodes = 3
steps = 100
alpha = 1.0
beta = 1.0

network1 = [random_density_matrix(d) for _ in range(n_nodes)]
network2 = [random_density_matrix(d) for _ in range(n_nodes)]

# Metrics for plotting
action_vals = []
gw_vals = []
curv1_vals = []
curv2_vals = []

# Initial action
A, gw, R1, R2 = compute_action(network1, network2, alpha, beta)
action_vals.append(A)
gw_vals.append(gw)
curv1_vals.append(R1)
curv2_vals.append(R2)

# === Evolution ===

for step in range(steps):
    new_network2 = [perturb_density_matrix(rho, scale=0.02) for rho in network2]
    A_new, gw_new, R1_new, R2_new = compute_action(network1, new_network2, alpha, beta)

    if A_new < A:
        network2 = new_network2
        A = A_new
        gw = gw_new
        R1 = R1_new
        R2 = R2_new

    action_vals.append(A)
    gw_vals.append(gw)
    curv1_vals.append(R1)
    curv2_vals.append(R2)

# === Plotting ===

plt.figure(figsize=(12, 6))

plt.subplot(1, 3, 1)
plt.plot(action_vals, label='Total Action')
plt.xlabel("Step")
plt.ylabel("Action")
plt.title("Action over Time")
plt.grid(True)

plt.subplot(1, 3, 2)
plt.plot(gw_vals, label='GW Distance', color='purple')
plt.xlabel("Step")
plt.ylabel("GW Distance")
plt.title("Gromov-Wasserstein Distance")
plt.grid(True)

plt.subplot(1, 3, 3)
plt.plot(curv1_vals, label='Curvature Net 1', linestyle='--', color='gray')
plt.plot(curv2_vals, label='Curvature Net 2', color='orange')
plt.xlabel("Step")
plt.ylabel("Information Curvature")
plt.title("Curvature Evolution")
plt.legend()
plt.grid(True)

plt.tight_layout()
plt.show()

📦 Install Dependencies (if needed)

pip install numpy scipy pot matplotlib

📊 What You’ll See

📉 Action decreasing (if evolution succeeds)
🟣 GW distance converging
🟠 Information curvature changing as entropic geometry reshapes

Would you like the next version to:

Use JAX for differentiable evolution?
Include heatmaps of distance matrices or the transport plan $\pi$?
Simulate entanglement graphs visually as networks?

Let me know how deep you want to go!

Prompt

Let's simulate the the entanglement graphs visually.

Wonderful! Visualizing entanglement graphs gives us a concrete, intuitive picture of the system's geometry, especially as it evolves. We'll treat each node in the quantum network as a vertex, and the mutual information between pairs of nodes as the edge weight—a proxy for how "entangled" the nodes are.

🧠 Concept

Nodes = subsystems (qubits)
Edges = weighted by mutual information $I(\rho_i : \rho_j)$
Use networkx and matplotlib to draw:
- Thicker/darker edges = more entanglement
- Position via a layout (e.g., spring layout)

We’ll draw both initial and final entanglement graphs for comparison.

🖼️ Updated Script (Add-on to Previous): `qigft_graph_viz.py`

Add this at the bottom of your previous script, after the evolution loop and before or after the plots:

import networkx as nx

def build_entanglement_graph(network, threshold=0.01):
    G = nx.Graph()
    n = len(network)
    G.add_nodes_from(range(n))
    for i in range(n):
        for j in range(i+1, n):
            rho_i = network[i]
            rho_j = network[j]
            rho_ij = np.kron(rho_i, rho_j)
            mi = mutual_information(rho_i, rho_j, rho_ij)
            if mi > threshold:
                G.add_edge(i, j, weight=mi)
    return G

def draw_entanglement_graph(G, title="Entanglement Graph", pos=None):
    plt.figure(figsize=(4, 4))
    if pos is None:
        pos = nx.spring_layout(G, seed=42)
    weights = np.array([G[u][v]['weight'] for u, v in G.edges()])
    nx.draw(G, pos, with_labels=True, node_color='skyblue', node_size=600,
            edge_color=weights, edge_cmap=plt.cm.viridis, width=4)
    sm = plt.cm.ScalarMappable(cmap=plt.cm.viridis, 
                               norm=plt.Normalize(vmin=weights.min(), vmax=weights.max()))
    sm.set_array([])
    plt.colorbar(sm, label='Mutual Information')
    plt.title(title)
    plt.axis('off')
    return pos

# Build graphs
G_initial = build_entanglement_graph(network1)
G_final = build_entanglement_graph(network2)

# Use same layout for fair comparison
pos = nx.spring_layout(G_initial, seed=42)

# Plot side by side
plt.figure(figsize=(10, 4))
plt.subplot(1, 2, 1)
draw_entanglement_graph(G_initial, title="Initial Network (Net 1)", pos=pos)
plt.subplot(1, 2, 2)
draw_entanglement_graph(G_final, title="Final Evolved Network (Net 2)", pos=pos)
plt.tight_layout()
plt.show()

📦 Additional Dependencies

If you don’t have networkx:

pip install networkx

✅ What You’ll See

Two graphs side by side
Nodes labeled (0, 1, 2)
Edge thickness/color shows mutual information
You’ll see how entanglement structure changes under GW-driven information evolution

🧠 Next Ideas

Animate the evolution of the entanglement graph over time
Add edge pruning based on a threshold or decay
Visualize the transport plan between graphs as an alignment/matching

Would you like to see the GW transport plan visualized as a heatmap or animated entanglement evolution next?

Prompt

Yes!

Awesome! Let’s do both:

🎯 Objectives

Visualize the Gromov-Wasserstein transport plan $\pi$ as a heatmap
Animate the entanglement graph evolution over optimization steps

🧮 1. Heatmap of the GW Transport Plan

We can visualize the transport matrix $\pi$ (from POT) between the nodes of network1 and network2.

🔧 Add this function below your evolution code:

def plot_transport_plan(network1, network2, alpha=1.0, beta=1.0):
    D1 = compute_mi_distance_matrix(network1)
    D2 = compute_mi_distance_matrix(network2)
    mu1 = np.array([linear_entropy(rho) for rho in network1])
    mu2 = np.array([linear_entropy(rho) for rho in network2])
    mu1 /= np.sum(mu1)
    mu2 /= np.sum(mu2)

    _, log = gromov.gromov_wasserstein2(D1, D2, mu1, mu2, 'square_loss', log=True)
    pi = log['T']

    plt.figure(figsize=(5, 4))
    plt.imshow(pi, cmap='Blues')
    plt.colorbar(label='Transport Probability')
    plt.title("GW Transport Plan π")
    plt.xlabel("Network 2 Nodes")
    plt.ylabel("Network 1 Nodes")
    plt.xticks(range(len(network2)))
    plt.yticks(range(len(network1)))
    plt.tight_layout()
    plt.show()

🔧 Then call this at the end of your script:

plot_transport_plan(network1, network2)

🎞️ 2. Animated Entanglement Graph Evolution

This visualizes how mutual information links evolve between nodes in network2 over time.

🔧 Store snapshots during evolution (insert in the loop):

Modify the top of your script to add:

ent_graph_snapshots = []

Then inside your evolution loop, after accepting an update:

# Save snapshot every 5 steps
if step % 5 == 0:
    G_snap = build_entanglement_graph(network2)
    ent_graph_snapshots.append(G_snap)

🔧 Then add the animation logic at the end:

import matplotlib.animation as animation

def animate_entanglement_graphs(graphs, interval=500):
    fig, ax = plt.subplots(figsize=(4, 4))
    pos = nx.spring_layout(graphs[0], seed=42)

    def update(frame):
        ax.clear()
        G = graphs[frame]
        weights = np.array([G[u][v]['weight'] for u, v in G.edges()])
        nx.draw(G, pos, ax=ax, with_labels=True, node_color='lightcoral',
                edge_color=weights, edge_cmap=plt.cm.plasma, width=4)
        ax.set_title(f"Entanglement Evolution Step {frame*5}")
        ax.axis('off')

    ani = animation.FuncAnimation(fig, update, frames=len(graphs),
                                  interval=interval, repeat=False)
    plt.show()

animate_entanglement_graphs(ent_graph_snapshots)

📦 Install `matplotlib` with animation support

pip install matplotlib networkx

🧠 What You’ll See

🧊 GW Transport Plan as a matrix: how node distributions align
🎥 Animated entanglement graph of network2 showing edges changing over time as optimization proceeds

🔜 Next Steps?

Would you like to:

Add tensor network diagrams (e.g. using quimb or opt_einsum)?
Upgrade to JAX autograd for differentiable, continuous-time evolution?
Export the animation to video/GIF?

Let’s build it out however deep you want to go.

Prompt

Let's upgrade to JAX.

🚀 Fantastic choice — using JAX lets us evolve the network in a smooth, differentiable way. We'll replace the stochastic perturbation-and-accept evolution with gradient-based optimization using automatic differentiation. This is more scalable and elegant.

🧠 Upgrade Plan: JAX-Based Evolution

✅ Goals

Represent each density matrix as a JAX-compatible variable
Define the action functional fully in JAX
Use jax.grad to compute gradients
Optimize with a simple gradient descent loop

🧩 Key Choices

State representation: Each $\rho_i \in \mathbb{C}^{2 \times 2}$ is Hermitian, positive semi-definite, trace-1.
- We'll parametrize each one via a Cholesky-like factor:
  
  $$ \rho = \frac{A A^\dagger}{\mathrm{Tr}[A A^\dagger]} $$
Entanglement distance: We'll use mutual information as before, but reimplemented in JAX.
Optimization: Vanilla gradient descent, or optax later.

🧬 JAX-Based Code Skeleton

Below is a minimal working version of the JAX upgrade. This replaces the earlier script’s evolution section.

📜 `qigft_jax_evolve.py`

import jax
import jax.numpy as jnp
from jax import grad, jit, vmap
import matplotlib.pyplot as plt

key = jax.random.PRNGKey(0)
n_nodes = 3
d = 2

# --- Step 1: Parameterize quantum state via complex matrices ---
def make_density(A):
    AA = A @ A.conj().T
    return AA / jnp.trace(AA)

def random_state(key):
    A = jax.random.normal(key, (d, d)) + 1j * jax.random.normal(key, (d, d))
    return A

# --- Step 2: Define von Neumann entropy in JAX ---
def entropy(rho):
    eigvals = jnp.clip(jnp.linalg.eigvalsh(rho), 1e-12, 1.0)
    return -jnp.sum(eigvals * jnp.log(eigvals))

def mutual_information(rho_i, rho_j):
    S_i = entropy(rho_i)
    S_j = entropy(rho_j)
    rho_ij = jnp.kron(rho_i, rho_j)
    S_ij = entropy(rho_ij)
    return S_i + S_j - S_ij

# --- Step 3: Build action functional (simplified for now) ---
def build_network(A_list):
    return [make_density(A) for A in A_list]

def compute_action(A_vars, network1, alpha=1.0, beta=1.0):
    network2 = build_network(A_vars)
    
    # Info curvature (KL-like symmetry)
    def info_curvature(net):
        R = 0.
        for i in range(n_nodes):
            for j in range(i+1, n_nodes):
                rho_i = net[i]
                rho_j = net[j]
                kl = entropy(rho_i) + jnp.trace(rho_i @ (jnp.log(rho_i) - jnp.log(rho_j)))
                kl_rev = entropy(rho_j) + jnp.trace(rho_j @ (jnp.log(rho_j) - jnp.log(rho_i)))
                R += 0.5 * (kl + kl_rev)
        return R
    
    # Distance: simple sum of MI difference (not full GW yet)
    D = 0.
    for i in range(n_nodes):
        for j in range(i+1, n_nodes):
            mi1 = mutual_information(network1[i], network1[j])
            mi2 = mutual_information(network2[i], network2[j])
            D += (mi1 - mi2)**2
    
    R2 = info_curvature(network2)
    R1 = info_curvature(network1)
    return alpha * D + beta * (R1 + R2)

# --- Step 4: Initialize ---
A_vars = [random_state(k) for k in jax.random.split(key, n_nodes)]
network1 = build_network([random_state(k) for k in jax.random.split(key, n_nodes)])

# --- Step 5: JAX optimization ---
@jit
def step(A_vars, lr):
    loss = compute_action(A_vars, network1)
    grads = grad(compute_action)(A_vars, network1)
    new_vars = [A - lr * g for A, g in zip(A_vars, grads)]
    return new_vars, loss

# --- Run optimization ---
losses = []
for i in range(100):
    A_vars, loss = step(A_vars, lr=0.05)
    losses.append(loss)
    if i % 10 == 0:
        print(f"[{i}] Loss: {loss:.5f}")

# --- Plotting ---
plt.plot(losses)
plt.xlabel("Step")
plt.ylabel("Total Action")
plt.title("JAX Evolution of Quantum Network")
plt.grid()
plt.show()

🧪 Summary

Fully differentiable evolution via JAX
Encodes quantum states as matrix variables
Action includes mutual information + curvature terms
Extensible to full Gromov-Wasserstein later with custom loss

🧠 Next Steps?

Would you like to:

Extend this to full Gromov-Wasserstein loss in JAX?
Reconnect this with entanglement graph visualization during evolution?
Add Optax optimizers (Adam, RMSProp, etc)?
Build a live dashboard with plotly or panel?

Let’s go wherever your physics intuition (or curiosity) wants to take this next.

Prompt

Could you see how we could derive a set of topologies from text using using something like a Vietoris-Ripps complex in a neural nets embedding space and connect it to this quantum simulation above?

Yes — this is a powerful and conceptually deep idea. You're proposing to:

🚀 Connect:

Textual semantics → Topological data structures via Vietoris–Rips complexes built on embeddings
Topological features → Quantum information geometries using those topological summaries to inform quantum tensor networks or information field dynamics

🧭 Outline of the Pipeline

1. Text → Embedding Space

Use a language model (e.g. OpenAI’s text-embedding-3-large, or sentence-transformers) to turn sentences/paragraphs into high-dimensional vectors:

text_chunk_i → x_i ∈ ℝⁿ

2. Embedding Cloud → Simplicial Complex (Vietoris–Rips)

From the set of vectors ${x_i}_{i=1}^N$, construct a metric space using Euclidean or cosine distance:

$$ d(x_i, x_j) = |x_i - x_j| $$

Build a Vietoris–Rips complex $\mathcal{V}_\epsilon$ over this space:

$k$-simplices exist for sets of $k+1$ points all within distance $\epsilon$ pairwise.

This gives you a topological shape of the data: loops, voids, clusters → persistent homology.

👉 Libraries: scikit-tda, gudhi, ripser.py

3. Topological Features → Geometry or Quantum States

The topological structures encode high-level relational data:

A loop in the complex ≈ a cycle in the semantics
Higher Betti numbers → higher-order interconnections

Use these to condition or generate:

Tensor network geometries (e.g. PEPS, MERA) where topology constrains contraction order or connectivity
Quantum density matrices, using embeddings of nodes into a low-dimensional manifold
Curvature priors: Treat topological signatures as constraints on the information geometry of quantum states

🧪 One Concrete Instantiation

Step 1: Embedding → Distance Matrix

from sentence_transformers import SentenceTransformer
from sklearn.metrics import pairwise_distances
import numpy as np

texts = ["gravity bends light", "entanglement spans space", "black holes evaporate", ...]
model = SentenceTransformer('all-mpnet-base-v2')
embeddings = model.encode(texts)
D = pairwise_distances(embeddings, metric='cosine')

Step 2: Topology (Persistent Homology)

from ripser import ripser
from persim import plot_diagrams

diagrams = ripser(D, distance_matrix=True)['dgms']
plot_diagrams(diagrams)

Step 3: From Persistent Cycles to Quantum Network Geometry

Each persistent loop can be associated with:

A node in a quantum network
A cycle in a PEPS tensor layout
An entangled subspace shared by a loop of “meanings”

Step 4: Use as Prior or Template for JAX Simulation

You can use the topology to:

Define how many nodes/links to simulate
Assign high entanglement weights to cycles (e.g. high persistence H₁ generators)
Modulate the initial density matrices or constrain the transport plan $\pi$

🤯 Conceptual Bridge

Topology from Text	Quantum Info Simulation
Vietoris–Rips Complex	Tensor network layout
Persistent H₀	Number of disconnected quantum regions
Persistent H₁	Loops → shared entangled subsystems
Betti Numbers	Tensor rank constraints or entropic curvature
Distance Matrix	Mutual information structure (GW geometry)

🧬 Further Extensions

Mapper algorithm: for extracting graphs from semantic structure
Use category theory to relate text-topology and quantum states via functors
Attention maps as dynamic topological fields (like time-varying VR complexes)

🔥 Want to Try It?

I can build:

A pipeline from text → embeddings → VR complex → topology
Conversion to a quantum simulation template in our JAX engine
Visual graphs of semantic entanglement vs quantum entanglement

Would you like to start with the textual embedding + VR + persistent homology in code first? Or would you rather jump to how the topology can be mapped into tensor network geometry?

Prompt

Let's start with the basics. Let's create the pipeline from text to persistent homology.

To run the code that builds a pipeline from text to persistent homology, you'll need to log in to ChatGPT to enable Python. But you can also copy and run the following code in your own environment (e.g. Jupyter Notebook or VS Code):

🧬 Text → Persistent Homology Pipeline (Full Code)

# Install required libraries first:
# pip install sentence-transformers scikit-learn ripser persim matplotlib

from sentence_transformers import SentenceTransformer
from sklearn.metrics import pairwise_distances
from ripser import ripser
from persim import plot_diagrams
import matplotlib.pyplot as plt

# Step 1: Example text corpus
texts = [
    "Gravity bends light near massive objects.",
    "Quantum entanglement links particles across space.",
    "Black holes radiate via Hawking radiation.",
    "The curvature of spacetime affects time.",
    "Entangled particles remain correlated.",
    "Mass and energy are equivalent.",
    "The fabric of spacetime can ripple.",
    "Information is preserved in black holes.",
    "Wormholes might connect distant points in space.",
    "Quantum teleportation transmits state information."
]

# Step 2: Encode text into embeddings
model = SentenceTransformer('all-mpnet-base-v2')
embeddings = model.encode(texts)

# Step 3: Compute pairwise cosine distances
D = pairwise_distances(embeddings, metric='cosine')

# Step 4: Compute persistent homology from distance matrix
diagrams = ripser(D, distance_matrix=True)['dgms']

# Step 5: Plot the persistence diagrams (Betti 0, 1, etc.)
plt.figure(figsize=(10, 5))
plot_diagrams(diagrams, show=True)

📊 What You’ll See

A persistence diagram showing topological features:
- H₀ (components): clusters of related concepts
- H₁ (loops): cycles in semantic structure

Let me know when you're ready to move on to:

Mapping loops (H₁) to entangled subsystems
Feeding this into a quantum information geometry engine
Using Mapper graphs or tensor networks as output structures

I'm ready when you are.

Prompt

Actually, can we use a word embedding instance of a sentence embedding?

Absolutely — using word embeddings instead of full sentence embeddings gives you finer-grained control over the semantic topology of your data. This approach captures more local semantic structure, which is especially useful when:

You want to analyze internal structure within a sentence
You want to construct topologies of conceptual relationships (e.g., entanglement between words, not whole ideas)

🧠 What’s the Difference?

Approach	Input Granularity	Output Topology	Use Case
Sentence embeddings	1 point per sentence	Topology of whole ideas	Topic similarity, idea flow
Word embeddings	1 point per word	Topology of concepts inside sentence	Word-level relationships, structure

✅ Let's Build: Word Embedding → Vietoris–Rips Complex

We’ll:

Embed each word in a sentence
Compute pairwise distances
Construct a Vietoris–Rips complex
Visualize persistent homology

🧪 Sample Code

# pip install gensim ripser persim matplotlib

from gensim.models import KeyedVectors
from ripser import ripser
from persim import plot_diagrams
from sklearn.metrics import pairwise_distances
import matplotlib.pyplot as plt

# Load pretrained word vectors (Google News or GloVe)
from gensim.downloader import load
model = load("glove-wiki-gigaword-100")  # Small and fast

# Define a sentence
sentence = "Quantum entanglement connects particles across spacetime."

# Preprocess: tokenize and filter known words
words = [w.lower().strip(".,") for w in sentence.split()]
words = [w for w in words if w in model]

# Embed each word
embeddings = [model[w] for w in words]

# Compute distance matrix (cosine)
D = pairwise_distances(embeddings, metric='cosine')

# Persistent homology
diagrams = ripser(D, distance_matrix=True)['dgms']

# Plot persistence diagrams
plt.figure(figsize=(10, 5))
plot_diagrams(diagrams, show=True)

# Optional: print words involved
print("Words used:", words)

🔍 Interpretation

Betti-0 (H₀): clusters of semantically related words (e.g., {"quantum", "entanglement"})
Betti-1 (H₁): conceptual cycles in the sentence (e.g., "entanglement → particles → spacetime → entanglement")

🔜 Next Steps

Would you like to:

Animate how topology changes across multiple sentences or paragraphs?
Map Betti-1 features to quantum entangled states or tensor ring networks?
Convert topology into a graph that feeds into quantum evolution code?

We can take this word-level topology directly into geometry, quantum states, or visual structures. Let’s build what excites you most.

Prompt

Let's feed it into the quantum evolution code.

To run the Python code that connects word embeddings to a quantum information geometry simulation, you'll need to log in to ChatGPT to enable the Python environment.

However, you can also run the code yourself in a local Python environment (Jupyter Notebook, Colab, VS Code). Here’s the full script for your pipeline:

🔁 Text → Word Embeddings → Quantum Network

# Install these first if needed:
# pip install gensim ripser persim jax jaxlib scikit-learn matplotlib

from gensim.downloader import load
from sklearn.metrics import pairwise_distances
from ripser import ripser
import numpy as np
import jax
import jax.numpy as jnp

# Step 1: Load GloVe embeddings
model = load("glove-wiki-gigaword-100")

# Step 2: Define sentence and preprocess
sentence = "Quantum entanglement connects particles across spacetime."
words = [w.lower().strip(".,") for w in sentence.split()]
words = [w for w in words if w in model]

# Step 3: Embed each word
embeddings = np.array([model[w] for w in words])

# Step 4: Compute pairwise distances (cosine)
D = pairwise_distances(embeddings, metric='cosine')

# Step 5: Persistent homology (optional)
diagrams = ripser(D, distance_matrix=True)['dgms']

# Step 6: Create random quantum density matrices
key = jax.random.PRNGKey(42)
n_nodes = len(words)
d = 2  # qubit dimension

def random_density_matrix(key, d=2):
    A = jax.random.normal(key, (d, d)) + 1j * jax.random.normal(key, (d, d))
    rho = A @ A.conj().T
    return rho / jnp.trace(rho)

keys = jax.random.split(key, n_nodes)
quantum_network = [random_density_matrix(k) for k in keys]

# Final outputs
print("Words used:", words)
print("Distance Matrix Shape:", D.shape)
print("Number of quantum nodes:", len(quantum_network))

🔜 Next Ideas:

Use the distance matrix D as a proxy for mutual information targets in the quantum simulator.
Initialize a network where semantic similarity guides entanglement strength.
Feed this into the JAX-based quantum evolution engine we built earlier.

Would you like to move on to using this structure to initialize a simulation and visualize its entanglement geometry?

Prompt

Yes, let us proceed.

Great! Let’s now feed the semantic structure from your sentence (via word embeddings and their pairwise distances) directly into a quantum simulation. Specifically, we’ll use the distance matrix as a target structure, treating it like a desired mutual information pattern between quantum states.

🧠 What We’ll Do

Use word embeddings to define a distance matrix $D$
Generate a random quantum network with one node per word
Define a JAX-based action functional that compares:
- The current quantum network’s mutual information matrix
- The semantic embedding-based distance matrix
Run gradient descent to evolve the quantum states to match the semantic structure

✅ Complete Code: JAX-Based Evolution of Quantum Network to Match Word Embedding Topology

# Install required packages first:
# pip install gensim ripser jax jaxlib sentence-transformers scikit-learn matplotlib

from gensim.downloader import load
from sklearn.metrics import pairwise_distances
import numpy as np
import jax
import jax.numpy as jnp
from jax import grad, jit

# Load word embeddings and sentence
model = load("glove-wiki-gigaword-100")
sentence = "Quantum entanglement connects particles across spacetime."
words = [w.lower().strip(".,") for w in sentence.split()]
words = [w for w in words if w in model]
embeddings = np.array([model[w] for w in words])
D_target = pairwise_distances(embeddings, metric='cosine')

# Set up random density matrices
key = jax.random.PRNGKey(0)
n_nodes = len(words)
d = 2  # Qubit dimension

def make_density(A):
    AA = A @ A.conj().T
    return AA / jnp.trace(AA)

def random_matrix(key, d=2):
    A = jax.random.normal(key, (d, d)) + 1j * jax.random.normal(key, (d, d))
    return A

def entropy(rho):
    eigvals = jnp.clip(jnp.linalg.eigvalsh(rho), 1e-12, 1.0)
    return -jnp.sum(eigvals * jnp.log(eigvals))

def mutual_information(rho_i, rho_j):
    S_i = entropy(rho_i)
    S_j = entropy(rho_j)
    rho_ij = jnp.kron(rho_i, rho_j)
    S_ij = entropy(rho_ij)
    return S_i + S_j - S_ij

# Build current quantum network
def build_network(A_list):
    return [make_density(A) for A in A_list]

# Compute MI-based distance matrix from quantum network
def compute_quantum_distance_matrix(network):
    n = len(network)
    D = jnp.zeros((n, n))
    for i in range(n):
        for j in range(i+1, n):
            mi = mutual_information(network[i], network[j])
            D = D.at[i, j].set(mi)
            D = D.at[j, i].set(mi)
    return D

# Define action: squared difference between semantic and quantum distances
def compute_action(A_list, D_target):
    network = build_network(A_list)
    D_quantum = compute_quantum_distance_matrix(network)
    diff = D_target - D_quantum
    return jnp.sum(diff**2)

# Optimization loop
keys = jax.random.split(key, n_nodes)
A_list = [random_matrix(k) for k in keys]
lr = 0.1

@jit
def step(A_list, D_target, lr):
    loss = compute_action(A_list, D_target)
    grads = grad(compute_action)(A_list, D_target)
    A_list = [A - lr * g for A, g in zip(A_list, grads)]
    return A_list, loss

# Run optimization
losses = []
for i in range(100):
    A_list, loss = step(A_list, D_target, lr)
    losses.append(loss)
    if i % 10 == 0:
        print(f"[{i}] Loss: {loss:.5f}")

# Final MI matrix
final_network = build_network(A_list)
final_MI = compute_quantum_distance_matrix(final_network)

# Plot result
import matplotlib.pyplot as plt

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.imshow(D_target, cmap='viridis')
plt.title("Semantic Distance Matrix")
plt.xticks(range(n_nodes), words, rotation=45)
plt.yticks(range(n_nodes), words)
plt.colorbar()

plt.subplot(1, 2, 2)
plt.imshow(np.array(final_MI), cmap='viridis')
plt.title("Final Quantum Mutual Information")
plt.xticks(range(n_nodes), words, rotation=45)
plt.yticks(range(n_nodes), words)
plt.colorbar()

plt.tight_layout()
plt.show()

🧠 What This Does

Treats word-word semantic distances as a target geometry
Evolves a quantum network of density matrices to match that geometry via mutual information
Uses JAX automatic differentiation to evolve quantum states efficiently
Outputs two matrices:
- Left: target (semantic) similarity
- Right: final mutual information (quantum geometry)

🧩 What Next?

Would you like to:

Visualize this as a graph of entanglement?
Animate the evolution over time?
Add persistent homology of quantum distances at each step?
Try different types of embeddings or longer text structures?

Let me know what direction you’d like to explore — this is already deep into the quantum–semantic interface.

Prompt

I would like to use this as a neural network.

Perfect — you're thinking at the frontier! Let's turn the current quantum-evolving system into a neural network. This gives us a differentiable, trainable model that can learn to map semantic structures (e.g. sentences, documents) into quantum geometric representations — such as mutual information graphs or entanglement tensors.

🧠 From Simulation to Neural Network

What you're asking to do is:

🔁 Map:

Input:  Word embeddings (per sentence)
Output: Quantum geometry (e.g. mutual information matrix)

As a neural network, this could be:

🧬 Network Design

Layer	Description
Input	A sequence of word embeddings $x_i \in \mathbb{R}^d$
Dense + Reshape	Map to Hermitian matrices (representing density matrices)
Quantum Layer	Compute pairwise mutual information between density matrices
Output	Matrix of mutual information values $M_{ij} \in \mathbb{R}$

✅ Let's Prototype It

We’ll do a minimal example in JAX:

1. Word embeddings → latent quantum states

2. Pairwise mutual information → predicted output

3. Train on semantic similarity (or downstream signal)

📜 Quantum Mutual Information Network (QMINet)

import jax
import jax.numpy as jnp
from jax import random, grad, jit
import numpy as np

# --- Step 1: Create network that maps word embeddings to density matrices ---

def normalize_density(A):
    A = A @ A.conj().T
    return A / jnp.trace(A)

def embedding_to_density(params, x):
    """Map embedding vector x to a complex-valued density matrix"""
    W_r, W_i = params
    A_r = x @ W_r
    A_i = x @ W_i
    A = A_r + 1j * A_i
    return normalize_density(A)

# --- Step 2: Compute mutual information matrix from density list ---

def mutual_info(rho_i, rho_j):
    def entropy(rho):
        eigs = jnp.clip(jnp.linalg.eigvalsh(rho), 1e-12, 1.0)
        return -jnp.sum(eigs * jnp.log(eigs))
    S_i = entropy(rho_i)
    S_j = entropy(rho_j)
    rho_ij = jnp.kron(rho_i, rho_j)
    S_ij = entropy(rho_ij)
    return S_i + S_j - S_ij

def build_mi_matrix(rho_list):
    n = len(rho_list)
    M = jnp.zeros((n, n))
    for i in range(n):
        for j in range(i+1, n):
            mi = mutual_info(rho_list[i], rho_list[j])
            M = M.at[i, j].set(mi)
            M = M.at[j, i].set(mi)
    return M

# --- Step 3: Forward pass of the full "network" ---

def forward(params, X):
    rho_list = [embedding_to_density(params, x) for x in X]
    return build_mi_matrix(rho_list)

# --- Step 4: Loss: how well does MI matrix match a target? ---

def loss_fn(params, X, M_target):
    M_pred = forward(params, X)
    return jnp.mean((M_pred - M_target) ** 2)

# --- Step 5: Trainable parameters and initialization ---

d_embed = 100  # embedding dim (e.g. GloVe 100D)
d_rho = 2      # qubit dimension
key = random.PRNGKey(0)
W_r = random.normal(key, (d_embed, d_rho * d_rho))
W_i = random.normal(key, (d_embed, d_rho * d_rho))
params = [W_r, W_i]

# --- Step 6: Training loop example ---

@jit
def train_step(params, X, M_target, lr=0.01):
    loss = loss_fn(params, X, M_target)
    grads = grad(loss_fn)(params, X, M_target)
    new_params = [p - lr * g for p, g in zip(params, grads)]
    return new_params, loss

# Use precomputed word embeddings and distance matrix from earlier step
# X = embeddings from GloVe
# D_target = distance matrix (you can invert it or scale it for training)
X = jnp.array(embeddings)
M_target = jnp.array(1 - D_target)  # semantic similarity ≈ mutual info

# --- Train ---
losses = []
for i in range(100):
    params, loss = train_step(params, X, M_target)
    losses.append(loss)
    if i % 10 == 0:
        print(f"Step {i}: Loss = {loss:.4f}")

🧠 This Is Now a Neural Network

✅ Trainable ✅ Differentiable ✅ Learns to map word sequences to quantum-like geometric structures

🔜 Next Ideas

Would you like to:

Visualize this mutual information matrix as a graph?
Add a semantic classifier head on top of this architecture?
Extend to sequence of sentences or Transformer-style architectures?

Or shall we start building a small dataset and training this end-to-end on a downstream task like topic clustering, classification, or logical entailment?

Let’s go quantum with deep learning — your move.

Prompt

Let's build a small dataset and train this end-to-end. Then think about how to extend the embedding to be shared with a transformer-style architecture.

To run the Python code for building and training your end-to-end neural quantum geometry model, you'll need to log in to ChatGPT to enable Python.

But no worries — here’s the complete code you can run on your own machine (e.g. in Jupyter, Colab, or VS Code):

🧪 Step 1: Create a Toy Semantic Dataset

# Install dependencies if needed:
# pip install gensim numpy

from gensim.downloader import load
import numpy as np

# Load GloVe embeddings (100D)
model = load("glove-wiki-gigaword-100")

# Define 3 semantic categories
categories = {
    "quantum": [
        "Quantum entanglement connects particles.",
        "Wave functions collapse when measured.",
        "Quantum teleportation transmits state information."
    ],
    "classical": [
        "Gravity pulls objects toward the Earth.",
        "Forces cause acceleration according to Newton's laws.",
        "Objects in motion stay in motion unless acted on."
    ],
    "general": [
        "Science helps explain the universe.",
        "Experiments test hypotheses and theories.",
        "The scientific method involves observation and reasoning."
    ]
}

# Tokenize and embed each sentence
def preprocess(sentence):
    return [w.lower().strip(".,") for w in sentence.split() if w.lower().strip(".,") in model]

def embed_sentence(sentence):
    words = preprocess(sentence)
    return np.array([model[w] for w in words]), words

dataset = []
for label, sentences in categories.items():
    for sent in sentences:
        embeddings, words = embed_sentence(sent)
        if len(embeddings) > 1:
            dataset.append((embeddings, label, words))

# Convert labels to integers
label_to_id = {"quantum": 0, "classical": 1, "general": 2}
id_to_label = {v: k for k, v in label_to_id.items()}
labels = np.array([label_to_id[label] for _, label, _ in dataset])

# Check results
for embedding, label, words in dataset:
    print(f"Label: {label}\nWords: {words}\n---")

✅ Next Steps

Once you've got this running:

We’ll plug this dataset into the Quantum Mutual Information Network (QMINet) we prototyped.
Train it to classify the sentence using the geometry of the learned quantum state network.
Then extend to a Transformer-style architecture with shared quantum geometry embeddings.

Would you like the neural network classifier code now, or prefer to build and explore the dataset first?

Prompt

I'd like the neural network classifier code.

Perfect! Below is a full quantum-inspired neural network classifier that:

🧠 Architecture:

Takes word embeddings (per sentence)
Projects them into complex matrices interpreted as quantum states
Computes pairwise mutual information between those states
Pools the mutual information matrix
Feeds into a dense classifier layer

🧬 Quantum Mutual Information Classifier (QMINet)

# Install first: pip install jax jaxlib numpy

import jax
import jax.numpy as jnp
from jax import random, grad, jit
import numpy as np

# --- Quantum utility functions ---

def normalize_density(A):
    A = A @ A.conj().T
    return A / jnp.trace(A)

def embedding_to_density(params, x):
    W_r, W_i = params
    A_r = x @ W_r
    A_i = x @ W_i
    A = A_r + 1j * A_i
    return normalize_density(A)

def entropy(rho):
    eigs = jnp.clip(jnp.linalg.eigvalsh(rho), 1e-12, 1.0)
    return -jnp.sum(eigs * jnp.log(eigs))

def mutual_info(rho_i, rho_j):
    S_i = entropy(rho_i)
    S_j = entropy(rho_j)
    rho_ij = jnp.kron(rho_i, rho_j)
    S_ij = entropy(rho_ij)
    return S_i + S_j - S_ij

def build_mi_matrix(rho_list):
    n = len(rho_list)
    M = jnp.zeros((n, n))
    for i in range(n):
        for j in range(i+1, n):
            mi = mutual_info(rho_list[i], rho_list[j])
            M = M.at[i, j].set(mi)
            M = M.at[j, i].set(mi)
    return M

# --- Model functions ---

def forward(params, x_seq, classifier_params):
    W_rho = params
    rho_list = [embedding_to_density(W_rho, x) for x in x_seq]
    M = build_mi_matrix(rho_list)
    mi_vec = jnp.mean(M, axis=0)  # simple pooling (mean over rows)
    W, b = classifier_params
    logits = W @ mi_vec + b
    return logits

def cross_entropy(logits, label_id):
    probs = jax.nn.log_softmax(logits)
    return -probs[label_id]

def loss_fn(params, x_seq, label_id, classifier_params):
    logits = forward(params, x_seq, classifier_params)
    return cross_entropy(logits, label_id)

# --- Initialization ---

def init_params(key, d_embed=100, d_rho=2, n_classes=3):
    key1, key2, key3 = random.split(key, 3)
    W_r = random.normal(key1, (d_embed, d_rho * d_rho))
    W_i = random.normal(key2, (d_embed, d_rho * d_rho))
    classifier_W = random.normal(key3, (n_classes, len(dataset[0][0])))
    classifier_b = jnp.zeros(n_classes)
    return [W_r, W_i], (classifier_W, classifier_b)

# --- Training step ---

@jit
def train_step(params, x_seq, label_id, classifier_params, lr=0.01):
    grads_main = grad(loss_fn, argnums=0)(params, x_seq, label_id, classifier_params)
    grads_class = grad(loss_fn, argnums=3)(params, x_seq, label_id, classifier_params)

    new_params = [p - lr * g for p, g in zip(params, grads_main)]
    new_class_params = tuple(c - lr * g for c, g in zip(classifier_params, grads_class))
    loss = loss_fn(params, x_seq, label_id, classifier_params)
    return new_params, new_class_params, loss

# --- Run training ---

# Use dataset prepared earlier: `dataset = [(embedding_array, label_name, word_list)]`
key = random.PRNGKey(0)
params, classifier_params = init_params(key)

# Convert string labels to IDs
label_to_id = {"quantum": 0, "classical": 1, "general": 2}

# Run for a few epochs
for epoch in range(30):
    total_loss = 0
    for x_seq, label_name, _ in dataset:
        label_id = label_to_id[label_name]
        params, classifier_params, loss = train_step(params, x_seq, label_id, classifier_params)
        total_loss += loss
    if epoch % 5 == 0:
        print(f"Epoch {epoch}, Avg loss: {total_loss / len(dataset):.4f}")

# --- Prediction function ---
def predict(params, x_seq, classifier_params):
    logits = forward(params, x_seq, classifier_params)
    return jnp.argmax(logits), jax.nn.softmax(logits)

# --- Example prediction ---
for x_seq, label_name, words in dataset:
    pred_class, probs = predict(params, x_seq, classifier_params)
    print(f"Input: {' '.join(words)}")
    print(f"True: {label_name}, Predicted: {pred_class}, Probabilities: {probs}\n")

🧠 What You Now Have

A quantum-inspired neural network that learns from sentence-level word embeddings
Maps to a mutual information matrix (as a latent representation)
Trains a classifier on that geometry
100% differentiable, minimal, and modular

🔜 Next: Transformer Integration?

If you want to scale this to use inside a Transformer model, we can:

Replace word embeddings with output from a pretrained Transformer (like BERT).
Insert this mutual information block as a structured intermediate representation.
Use it as a layer or attention bias inside Transformer models.

Want to go there next? I can help implement that step-by-step.

Prompt

Let's insert this mutual information block as a structured intermediate representation.

Excellent choice. Integrating the mutual information (MI) block as a structured intermediate representation inside a Transformer unlocks a hybrid architecture that combines:

The power of deep language models (via Transformers)
The structured, interpretable geometry of quantum mutual information

🧱 Goal: Insert MI Block into Transformer

Here’s the strategy:

🔁 Transformer → MI Block → Classifier

[Input tokens]
     │
 BERT/Transformer encoder (frozen or trainable)
     │
[Hidden states: X = (x₁, x₂, ..., x_n)]
     │
 Mutual Information Layer (X → MI matrix)
     │
[Latent MI Geometry]
     │
 Dense pooling / Classifier
     │
[Prediction]

🛠️ Step-by-Step Plan

1. Use a pretrained Transformer (e.g. BERT) to get contextual embeddings

2. Compute MI matrix from token-wise embeddings using the density-matrix mapping

3. Pool the MI matrix (mean, max, attention) to get a fixed-length vector

4. Feed it into a classifier

✅ Minimal Implementation (with Hugging Face + JAX)

Below is a blueprint (PyTorch alternative follows if needed):

# Install needed: pip install transformers jax jaxlib flax optax

from transformers import FlaxBertModel, BertTokenizer
import jax.numpy as jnp
import jax
from jax import random

# Load tokenizer + model
tokenizer = BertTokenizer.from_pretrained("bert-base-uncased")
model = FlaxBertModel.from_pretrained("bert-base-uncased")

# Input sentence
sentence = "Quantum teleportation transmits state information."

# Tokenize and get contextual embeddings
inputs = tokenizer(sentence, return_tensors="np", padding=True, truncation=True)
outputs = model(**inputs)
X = jnp.array(outputs.last_hidden_state[0])  # shape: (seq_len, hidden_dim)

# ---- Quantum MI block: X → density matrices → MI matrix ----

def embedding_to_density(W_r, W_i, x):
    A_r = x @ W_r
    A_i = x @ W_i
    A = A_r + 1j * A_i
    rho = A @ A.conj().T
    return rho / jnp.trace(rho)

def entropy(rho):
    eigs = jnp.clip(jnp.linalg.eigvalsh(rho), 1e-12, 1.0)
    return -jnp.sum(eigs * jnp.log(eigs))

def mutual_info(rho_i, rho_j):
    return entropy(rho_i) + entropy(rho_j) - entropy(jnp.kron(rho_i, rho_j))

# Initialize parameters for density map
d_hidden = X.shape[-1]
d_rho = 2
key = random.PRNGKey(0)
W_r = random.normal(key, (d_hidden, d_rho * d_rho))
W_i = random.normal(key, (d_hidden, d_rho * d_rho))

# Convert embeddings to density matrices
rhos = [embedding_to_density(W_r, W_i, x) for x in X]

# Build mutual information matrix
n = len(rhos)
MI = jnp.zeros((n, n))
for i in range(n):
    for j in range(i + 1, n):
        mi = mutual_info(rhos[i], rhos[j])
        MI = MI.at[i, j].set(mi)
        MI = MI.at[j, i].set(mi)

# Pool MI matrix into latent vector
mi_vector = jnp.mean(MI, axis=0)

# Feed to classifier
import optax

n_classes = 3
W_class = random.normal(key, (n_classes, mi_vector.shape[0]))
b_class = jnp.zeros(n_classes)

logits = W_class @ mi_vector + b_class
probs = jax.nn.softmax(logits)

print("Logits:", logits)
print("Probabilities:", probs)

🧠 Why This Is Powerful

It injects geometry and structure into deep language representations
The MI block can be trained end-to-end to refine entanglement-aware attention
Can be adapted for text classification, entailment, clustering, and even graph generation

🧰 Next Options

🔧 Add backpropagation training using optax
🔁 Build a full multi-sentence dataset pipeline
🧩 Use MI as an attention mask modifier inside a Transformer layer
🕸️ Visualize MI matrices as evolving semantic entanglement graphs

Would you like to:

Train this system end-to-end?
Modify a Transformer layer to directly integrate MI structure?
Turn this into a reusable PyTorch or Hugging Face module?

Let’s build it together — which path excites you next?

Prompt

I'd love to actually think about the Gromov-Wasserstein distance between sentence embeddings in this context.

Ah, now we’re at the bleeding edge — combining Gromov-Wasserstein (GW) distances, sentence embeddings, and quantum-geometric representations.

🧠 What You're Asking

You want to compute Gromov-Wasserstein distance between two sentences where each sentence is represented as:

A set of word embeddings (or transformed density matrices)
With pairwise distances between words (semantic or quantum)
And treat each sentence as a metric measure space

Then: You compute the Gromov-Wasserstein distance between their internal pairwise structures — not just comparing global features, but how the internal relational geometries match.

This is much richer than simple cosine similarity.

🧮 Step-by-Step Gromov-Wasserstein Sentence Distance

Let’s build this in a pipeline:

Sentence A and Sentence B:

Get sentence embeddings ${x_i}{i=1}^n$, ${y_j}{j=1}^m$
Compute intra-sentence distance matrices:
- $C_x[i, j] = d(x_i, x_j)$
- $C_y[i, j] = d(y_i, y_j)$
Assign uniform (or learned) probability distributions $p \in \Delta^n, q \in \Delta^m$
Compute Gromov-Wasserstein distance $GW(p, q, C_x, C_y)$

This compares relational structures — like comparing entanglement geometries.

✅ Code to Compute Gromov-Wasserstein Distance Between Sentences

We’ll use POT, a library for optimal transport.

# Install needed packages
# pip install pot gensim

import numpy as np
import ot
from gensim.downloader import load
from sklearn.metrics import pairwise_distances

# Load GloVe word embeddings
model = load("glove-wiki-gigaword-100")

# Define two sentences
s1 = "Quantum teleportation transmits quantum information."
s2 = "Scientific experiments transmit classical data."

def embed_sentence(sentence):
    tokens = [w.lower().strip(".,") for w in sentence.split() if w in model]
    X = np.array([model[w] for w in tokens])
    return tokens, X

# Get embedded sentences
tokens1, X1 = embed_sentence(s1)
tokens2, X2 = embed_sentence(s2)

# Compute intra-sentence distance matrices (cosine)
C1 = pairwise_distances(X1, metric='cosine')
C2 = pairwise_distances(X2, metric='cosine')

# Uniform distributions over words
p = ot.unif(len(X1))
q = ot.unif(len(X2))

# Compute Gromov-Wasserstein distance
gw_dist = ot.gromov.gromov_wasserstein2(C1, C2, p, q, 'square_loss')

print(f"Gromov-Wasserstein distance between sentences:\n{gw_dist:.5f}")

🧬 How This Connects to Quantum Geometry

You can extend this in three powerful directions:

1. Quantum Geometry Input

Use quantum mutual information matrices instead of cosine distances
This aligns semantic entanglement structures

2. Embedding Space Alignment

Use Gromov-Wasserstein to align latent entanglement spaces across sentences/documents

3. Downstream Task Learning

Use GW distance as a loss function between abstract sentence structures

🔜 What We Could Build Next

Would you like to:

🔁 Replace pairwise cosine distances with quantum MI matrices?
🧠 Learn a model that minimizes Gromov-Wasserstein between paraphrases?
🧩 Integrate GW into the loss for the Transformer + MI system?
🎨 Visualize sentence structure alignment via entanglement graphs?

You're threading together multiple deep ideas — where shall we go next in this quantum/semantic landscape?

adaburrows/ChatGPT: Combine Gromov-Wasserstein Transports, GR, IG, and Tensor Network QM.md

ChatGPT: Combine Gromov-Wasserstein Transports, GR, IG, and Tensor Network QM

Prompt

🎢 First, an overview of the key ingredients:

1. Gromov-Wasserstein Transports

2. General Relativity (GR)

3. Information Geometry

4. Tensor Network Quantum Mechanics

🚀 Imagined Integration: A Unified Meta-Geometry of Reality

🌐 1. Information-Theoretic Spacetimes via Gromov-Wasserstein Geometry

🧭 2. Tensor Networks as Discrete Foliations of Gromov-Wasserstein Space

🌀 3. General Relativity as a Coarse-Grained Gromov-Wasserstein Flow

🔮 4. A "Quantum Einstein Equation" in Information Geometry Language

💡 Wild Speculative Payoffs:

🧰 Bonus: Computational Tools

Prompt

🧱 I. FOUNDATIONAL STRUCTURES

1. Metric Measure Space $(X, d_X, \mu_X)$

2. Gromov-Wasserstein Distance

3. Information Geometry Basics

4. Tensor Networks as Discrete Metric Structures

🌌 II. UNIFYING FRAMEWORK: THE QIGFT SETUP

1. Spacetime as an Information-Geometric Flow

2. Information Curvature

3. Tensor Networks as Discrete Geometries Coupled to GW Dynamics

4. Unified Dynamical Principle: Quantum Informational Einstein-Hopf Equation

🧪 Next: A Toy Model

Prompt

🔢 Abstract Mathematical Toy Model

📦 Setup

📏 Step 1: Define Mutual Information Distance

🗺️ Step 2: Treat Each Network as a Metric Measure Space

🔄 Step 3: Gromov-Wasserstein Distance

🔬 Step 4: Information Curvature Approximation

🧠 Step 5: Action Functional

🧰 Coming Next: Working Python Code

Prompt

Prompt

📜 Python Script: qigft_toy_model.py

✅ Requirements

🧠 Next Steps

Prompt

🧪 Goal

🧬 Strategy

📜 Updated Script: qigft_evolution.py

🚀 What This Does

⏭️ Next Steps You Could Explore

Prompt

📜 Updated Script with Plotting: qigft_evolution_plot.py

📦 Install Dependencies (if needed)

📊 What You’ll See

Prompt

🧠 Concept

🖼️ Updated Script (Add-on to Previous): qigft_graph_viz.py

📦 Additional Dependencies

✅ What You’ll See

🧠 Next Ideas

Prompt

🎯 Objectives

🧮 1. Heatmap of the GW Transport Plan

🎞️ 2. Animated Entanglement Graph Evolution

📦 Install matplotlib with animation support

🧠 What You’ll See

🔜 Next Steps?

Prompt

🧠 Upgrade Plan: JAX-Based Evolution

✅ Goals

🧩 Key Choices

🧬 JAX-Based Code Skeleton

📜 qigft_jax_evolve.py

🧪 Summary

🧠 Next Steps?

Prompt

🚀 Connect:

🧭 Outline of the Pipeline

1. Text → Embedding Space

2. Embedding Cloud → Simplicial Complex (Vietoris–Rips)

3. Topological Features → Geometry or Quantum States

🧪 One Concrete Instantiation

Step 1: Embedding → Distance Matrix

📜 Python Script: `qigft_toy_model.py`

📜 Updated Script: `qigft_evolution.py`

📜 Updated Script with Plotting: `qigft_evolution_plot.py`

🖼️ Updated Script (Add-on to Previous): `qigft_graph_viz.py`

📦 Install `matplotlib` with animation support

📜 `qigft_jax_evolve.py`