Landauer Erasure with Side Information
Quantify how retained information lowers an ideal bit-erasure work bound.
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Landauer Erasure with Side Information
Quantify how retained information lowers an ideal bit-erasure work bound.
Worked Example
Consider erasing a bit X that is a Bernoulli random variable with p = 0.5. We have side information Y, which is related to X through a symmetric error channel with error probability e = 0.1, meaning Y = X XOR E. The temperature T is 300 K.
- H(X) = h(0.5) = 0.6931471805599453 nats
- P(Y=1) = p(1-e) + (1-p)e = 0.5(0.9) + 0.5(0.1) = 0.5
- H(Y) = h(0.5) = 0.6931471805599453 nats
- H(E) = h(0.1) = 0.3250829733914481 nats
- H(X|Y) = H(X) + H(E) - H(Y) = 0.6931471805599453 + 0.3250829733914481 - 0.6931471805599453 = 0.3250829733914481 nats
- Minimum work with side info = kB * T * H(X|Y) = (1.380649e-23) * 300 * 0.3250829733914481 = 1.3464764463897883e-21 J
Experiment
In this experiment, you can observe how retaining side information can theoretically lower the thermodynamic cost of erasure.
- Open the Landauer Erasure with Side Information interactive module.
- Set Bit probability to 0.5 and Temperature kelvin to 300.
- Set Side-channel error to 0.1.
- Set Erasure to With side information. You should observe the work is around 1.346e-21 J.
- Change Erasure to Without side information. Notice that the charged entropy increases to H(X), and the minimum work bound roughly doubles to 2.871e-21 J.
- Change Side-channel error to 0.5 to make Y completely independent of X. The conditional entropy H(X|Y) becomes equal to H(X), and side information offers no thermodynamic advantage.
Limitations
This model calculates the formal ideal lower bounds for erasure work (Wmin). The calculations are isothermal and reversible bounds and do not apply to finite-rate, real-world erasure devices where additional thermodynamic losses invariably occur.
The device is assumed to operate classically. We do not consider any quantum formulations that might yield negative conditional entropies. At T = 0 K, the ideal bound trivially evaluates to exactly 0 J, which is a theoretical abstraction and not a claim about any physically realizable zero-temperature process.
The side memory Y is retained throughout the process. Resetting or erasing this side memory later is outside the scope of this particular erasure calculation.
References
The formal basis for conditional entropy and its relation to Landauer's principle can be found in the following resources.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Landauer Erasure with Side Information module.
Try Landauer Erasure with Side Information on Riano →