1.0
Monotonicity Pass Rate
0.616
Max Entropy (6 modes)
Thermal Occupation Sweep: Sl+/-
Squeezing Parameter Sweep: Sl+/-
Multi-Mode Scaling: Mean Sl+
Participation Entropy vs Mode Number
TMSV: Monotones vs Squeezing
TMSV: Mode Weight Distribution
Physical Interpretation Test Cases
| Configuration | Modes | Sl+ | Sl- | Entropy |
|---|
| Thermal asymmetric | 2 | 0.0 | 0.0 | 0.0 |
| Squeezed mode 1 | 2 | 0.963 | 1.029 | 0.0 |
| Squeezed both | 2 | 0.930 | 0.917 | 0.489 |
| Entangled balanced | 2 | 0.362 | 0.427 | 0.693 |
| Three-mode chain | 3 | 0.863 | 0.955 | 0.201 |
| Four-mode star | 4 | 0.899 | 0.958 | 0.481 |
Key Findings
- Partial localization: The optimal mode participation entropy grows sub-logarithmically with mode number (0.372 at n=2 to 0.616 at n=6), well below the uniform bound ln(n).
- Squeezing dominance: Squeezing drives non-zero Sl values; purely thermal states yield Sl = 0. Maximum Sl+ of 0.987 at r = 2.0.
- Passive invariance: Sl+/- values are invariant under beam-splitter operations (variation less than 1e-15), while the optimal mode direction rotates.
- Monotonicity: Verified with 1.0 pass rate across 40 trials for both Sl+ and Sl-.
- Scaling: Mean Sl+ increases from 0.741 (2 modes) to 0.844 (6 modes) with decreasing variance.