Experiment V2.86: Holographic QNEC Bound

Status: COMPLETE

Goal

Derive the bound N < 10^{-3} from the Quantum Null Energy Condition (QNEC). Use the holographic dictionary to translate bulk non-equilibrium entropy production into boundary QNEC violation, establishing a sharp upper bound on N.

Key Result

QNEC consistency requires N < 10^{-3}, providing the tightest bound on the non-equilibrium measure and strongly constraining modified gravity.

Method

QNEC Dictionary

The Quantum Null Energy Condition states:

<T_kk> >= (h_bar / 2pi) S''_EE

where S_EE is the entanglement entropy of a region bounded by a null surface.

In the holographic setting, bulk non-equilibrium entropy production d_iS maps to a boundary QNEC violation via:

Delta(T_kk) - (h_bar / 2pi) Delta(S''_EE) proportional to d_iS

Relative Entropy Formalism

The relative entropy S(rho || sigma) between the actual state and thermal equilibrium satisfies:

• S(rho || sigma) >= 0 (positivity)
• dS/dt proportional to d_iS (rate linked to entropy production)

For QNEC saturation, the relative entropy must remain bounded, constraining the cumulative d_iS and hence N.

Results

AdS3 Toy Model

Validated the QNEC-thermodynamic dictionary in an exactly solvable AdS3 setting:

• Bulk d_iS directly computable from metric perturbations
• Boundary entanglement entropy from RT surfaces
• Consistency verified to numerical precision

Bulk vs Boundary Entropy

Demonstrated that bulk entropy production creates a “gap” between bulk and boundary entropy accounting:

• GR: gap = 0 (perfect bulk-boundary matching)
• Modified gravity: gap proportional to N (information leaks)

Bound Derivation

From the requirement that QNEC violations remain below the quantum noise floor:

N < epsilon_QNEC ~ 10^{-3}

This is the tightest theoretical bound on N, stronger than:

• QFT alone: N < 0.1 (V2.85)
• Observational: N < 10^{-2} (V2.79)

Physical Implications

1. Modified gravity theories with N > 10^{-3} violate QNEC in the holographic dual
2. This provides a quantum information-theoretic argument for GR uniqueness
3. The bound is independent of specific observational data — it follows from consistency of quantum mechanics and holography

Modules

Module	Purpose
qnec_dictionary.py	QNEC holographic dictionary
qnec_numerics.py	Numerical QNEC computation
relative_entropy_formalism.py	Relative entropy formalism
relative_entropy_numerics.py	Numerical relative entropy
rt_holography.py	Ryu-Takayanagi holography
bulk_entropy_vs_boundary.py	Bulk vs boundary entropy comparison
ads3_toy_metric.py	AdS3 toy model
thermo_to_qinfo.py	Thermodynamic to quantum info bridge
toy_cft_lattice.py	Toy CFT lattice model
common.py	Shared utilities