mlx-qre 0.1.0

GPU-accelerated Quantum Relative Entropy on Apple Silicon via MLX

mlx-qre

GPU-accelerated Quantum Relative Entropy on Apple Silicon via MLX.

$$\Sigma = D(\rho | \sigma) = \mathrm{Tr}[\rho(\ln\rho - \ln\sigma)]$$

Installation

pip install -e .

Requires Python 3.10+ and Apple Silicon (M1/M2/M3/M4).

Quick Start

import mlx.core as mx
from mlx_qre import quantum_relative_entropy, random_density_matrix

# Two 100x100 density matrices on GPU
rho = random_density_matrix(100)
sigma = random_density_matrix(100)

# Compute D(rho || sigma) — eigendecomposition runs on Metal GPU
D = quantum_relative_entropy(rho, sigma)
mx.eval(D)
print(f"D(rho || sigma) = {D.item():.6f}")

# Batched: 500 pairs simultaneously
rho_batch = random_density_matrix(50, batch_size=500)
sigma_batch = random_density_matrix(50, batch_size=500)
D_batch = quantum_relative_entropy(rho_batch, sigma_batch)

Features

Module	Function	Description
qre	quantum_relative_entropy(rho, sigma)	D(rho || sigma) via GPU eigendecomposition
qre	von_neumann_entropy(rho)	S(rho) = -Tr[rho ln rho]
qre	relative_entropy_pure_state(psi, sigma)	Efficient D for pure states: -ln(psi|sigma|psi)
classical	kl_divergence(p, q)	Classical KL divergence
classical	jensen_shannon_divergence(p, q)	Symmetric JSD
classical	renyi_divergence(p, q, alpha)	Renyi divergence of order alpha
channels	thermal_attenuator(eta)	Gravitational channel eta = -g_00
channels	channel_entropy_production(K, rho, sigma)	Sigma through channel
channels	depolarizing_channel(p)	Depolarizing noise
channels	dephasing_channel(gamma)	Dephasing noise
petz	petz_recovery_map(K, sigma)	Construct Petz recovery R
petz	petz_recovery_fidelity(rho, sigma, K)	F(rho, R o N(rho))
petz	verify_petz_bound(rho, sigma, K)	Check F >= exp(-Sigma/2)
petz	retrodiction_quality(rho, sigma, K)	tau = 1 - F

Use Cases

• Gravitational entropy production: Sigma_grav = D(N_eta(rho) || N_eta(sigma)) with eta = 1/Q^2
• Quantum channel analysis: entropy production, data processing inequality
• Petz recovery bounds: F >= exp(-Sigma/2), retrodiction quality
• Quantum ML: kernel methods using QRE as a distance measure
• Neural entropy: EEG/neural signal entropy production analysis

Benchmark

python -m mlx_qre.benchmark

Compares MLX (Apple Silicon GPU) vs NumPy (CPU) across matrix sizes N = 10 to 1000.

Tests

pip install -e ".[dev]"
pytest tests/ -v

Theory

The quantum relative entropy D(rho || sigma) is the quantum generalization of KL divergence. In the retrocausality framework:

• Sigma = 2 ln Q: unified entropy production formula
• Petz bound: F >= exp(-Sigma/2) quantifies retrodiction cost
• tau = 1 - F: retrodiction deficit (0 = perfect, 1 = irreversible)
• Zero-entropy limit: Sigma -> 0 implies perfect retrodiction (no time arrow)

License

MIT License. Copyright (c) 2026 Sheng-Kai Huang.