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The Credit Ledger
Companion to State Credit and Protocol Efficiency
State Credit
Bias p = P(X=1)0.50
biased (low H)uniformbiased (low H)
Mutual info I(X;Y)0.00 bits
Capped at H(X) = 1.00 bits
Entropy H(X)1.000 bits
Negentropy (1 - H)0.000 bits
Correlation credit0.000 bits
Total state credit0.000 bits
Protocol Efficiency
Duration τ10 relaxation times
Thermodynamic length L1.5
Geodesic = 1.0, naive axis-aligned ≈ 5.0
Work Decomposition
Naive Landauer (kT ln 2)0.693 kT
W_rev (after credit)0.693 kT
W_irr (protocol)0.113 kT
Total Work0.806 kT
Savings vs naive Landauer: -16.2%
The Combined Bound
W ≥ kT ln 2 · [Hmax - Cstate] + kT L²/(2τ)
State credit (negentropy + correlations) reduces Wrev.
Protocol efficiency (thermodynamic length) reduces Wirr.
Try This
- Maximum credit: Set p = 0.1 (biased) and I = 0.47 (max for that H). Watch total work drop to ~17% of naive.
- Fast vs slow: With p = 0.5, compare τ = 2 vs τ = 50. Short protocols waste energy on irreversible dissipation.
- Geodesic advantage: With L = 1 (optimal) vs L = 5 (naive), see how poor protocols cost ~25× more Wirr.