Landauer's Principle

Landauer's principle bounds the heat transferred to an initially thermal reservoir when another system's von Neumann entropy decreases. In a finite-dimensional formulation, let the initial density matrix be , with , a Hermitian matrix , and . Here is Boltzmann's constant and is the temperature. A unitary matrix produces .

Using natural logarithms, write for von Neumann entropy, , and . Then

where is quantum relative entropy and is quantum mutual information. Both quantities are nonnegative, yielding the bound (Reeb and Wolf 2014).

The ideal reset of an unbiased bit to a definite value decreases entropy by , giving

In the finite-dimensional setting above, an unbiased bit cannot be reset exactly to a definite value (Reeb and Wolf 2014, pp. 12-13). This is not a heat cost for every logical operation. Initial independence, a thermal reservoir, and unitary joint evolution are essential hypotheses of this formulation. Finite reservoirs generally require a stronger bound (Reeb and Wolf 2014).