Gap Between Thermal and Enhanced Thermal Operations

Key Findings

• Classical gap is zero: For diagonal state transformations, TO and EnTO give identical achievable sets across all dimensions, temperatures, and energy structures.
• Coherence opens the gap: The gap emerges at coherence level ~0.30 and grows to 0.200 at coherence 0.5, demonstrating its quantum-coherent origin.
• Gaussian closure confirmed: 0 out of 50 Gaussian trials show any gap, confirming Hu et al.'s theorem.
• TO fraction scales down: From 0.565 at d=2 to 0.345 at d=5; from 0.475 at beta=0.1 to 0.285 at beta=5.0.
• Mean energy change: TO-achievable pairs have mean energy change of -0.453, consistent with thermalization.