On the Conservation of Information and the Second Law of Thermodynamics

How can these fundamental laws be consistent?

The conservation of quantum information, quantified in term of von Neumann entropy, is a fundamental consequence of the linearity and unitarity of quantum mechanics. As the terms information and entropy are often used interchangeably in several branches of sciences, this may sound very strange to anyone familiar with the second law of thermodynamics, which says that entropy generally increases with time. How can these claims be consistent?

First, there is nothing quantum with this apparent contradiction. The same question could be asked in classical physics. For a Hamiltonian system, the dynamics are always reversible, so information is conserved by Liouville’s theorem. One could then wonder how entropy can increase for a classical system if entropy is a measure of information.

This article is the second part of my series about entropy and the second law of thermodynamics. You should definitly check out the first part before that one.

Short answer

The entropy (or information) constrained by Liouville’s theorem is not the same as the entropy that the Second Law is talking about. The latter is talking about the amount of hidden information (information that is inaccessible to macroscopic measurements), while the former is the total information content of a system. Or in other words, the irreversibility of thermodynamics is a statistical effect and does not conflict the reversibility of classical /quantum mechanics.

Still, there is much more depth to this question than it looks, so let’s dive in.

What are entropy and information ?

The term entropy is used in different contexts, which often results in confusion. We find entropy in information theory (Shanon entropy), statistical thermodynamics (Boltzman and Gibbs entropy) and quantum mechanics (von Neuman theory) just to name a few. While they are all defined with the same formula (up to a constant), they describe different concepts.