Landauer's Principle and the balance equation
February 29, 2016
I’ve been working on my thesis over reading week, and I think I’ve finished my introduction to Landauer’s Principle. I ended up writing a pretty detailed derivation of the balance equation, and thus Landauer’s bound, so I thought it might be useful to post here.
Landauer's principle states that there is a minimal energetic cost for a state transformation on a system via the action of a thermal reservoir at temperature .
Joules per Kelvin is Boltzmann's constant. In particular, if is the change of entropy of the system , and is the change in energy of the reservoir , then
This principle has generated interest since its inception in 1961; see [Section 1, RW14] An improved Landauer Principle with finite-size corrections, D. Reeb and M. Wolf, 2014 (v3). for a recent summary. First, the bound has allusions to practicality: perhaps the energy efficiency of our computers will be limited. For changing the state of a classical or quantum bit however, the bound is at most
which is extremely small for reasonable temperatures ; yet, modern processors are within several orders of magnitude of this limit, as shown to the right.
Energy cost of changing state for modern silicon transistors, as compared to a theoretical minimum for classical bits encoded in electron charge at room temperature. Figure reproduced from Figure 1(a) of Energy dissipation and transport in nanoscale devices by Eric Pop (2010).
Moreover, in 1973 Bennett showed that any Turing machine program may be implemented in a reversible manner, so that . Reversible computing is an area of considerable practical interest and continuing theoretical work.
More fundamentally, Landauer's bound is a direct relationship between energy and information (entropy). From now on, we will use natural units so that . In fact, Landauer's principle follows from the entropy balance equation
where is the entropy production.
We will define and prove , in a finite dimensional quantum unitary setup, following [Section 3, RW14] and [JP14] A note on the Landauer principle in quantum statistical mechanics, V. Jaksic and C. Pillet, 2014 (v1).. We assume the system is described by a finite dimensional Hilbert space , with self-adjoint Hamiltonian . The initial state on the system is given by a density matrix non-negative trace-one operator on . Likewise, we assume the environment is described by a finite dimensional Hilbert space with self-adjoint Hamiltonian , and initial state
the Gibbs state Gibbs states on are invariant under the free dynamics ; in this finite dimensional context, they are uniquely so. They thus have the interpretation of thermal equilibrium states. at temperature . The system and environment start uncoupled, so the joint initial state is . The evolution of the joint system is given by a unitary operator , leading to the final joint state . We decouple the systems, yielding
as the final state on the system, environment, respectively.
We identify two quantities of interest during this process: , the change of entropy of the system of interest, and , the change of energy of the environment, defined as Note the sign convention.
where is the von Neumann entropy. Recall the relative entropy of two faithful states and has with equality if and only if . See sections 2.5-2.6 of Entropic Fluctuations in Quantum Statistical Mechanics. An Introduction by Jaksic et al (2011) for a review of entropy functions in finite dimensional quantum mechanics.
With this function, we define the entropy production
By definition of relative entropy, the entropy production may be written
We then recognize the first term as an entropy, and expand the second term using the following claim.
Claim. If and are positive (and thus self-adjoint) on a finite dimensional Hilbert space, then
Proof. If have spectral decompositions and , then . With this,
Since entropy is invariant under a unitary transformation As may immediately be seen by the spectral theorem: if has spectral decomposition , then , since eigenvalues are invariant under unitary transformations (change of basis)., we have . Furthermore, by definition of the partial trace,
which is simply . Using this argument for the third term as well, we are left with
using that . Then,
Additionally, using ,
using that .
We thus have the balance equation (equation ). Landauer’s Principle (equation ) follows by simply noting as it is a relative entropy.
We may interpret as a microscopic Clausius formulation of the Second Law of Thermodynamics [BHN+14] The second laws of quantum thermodynamics F. Brandao et al, 2013 (v4). . More specifically, we may interpret as the Clausius entropy change of the environment. Note: We’re making an analogy to the Clausius formulation of the Second Law of Thermodynamics, but not a formal relationship. The balance equation presented here and the 2nd law are part of two different frameworks. Then, with a minus sign to account for our sign convention, we will interpret , and the Second Law is
In this language then, serves as the entropy production, which gives it its name. The classical Second Law, however, is a statement about macroscopic quantities obtained from the behavior of particles. Within the theory of quantum mechanics and our assumptions, however, the balance equation is exact on a microscopic level.