# 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*…