# Von Neumann entropy in the algebraic approach

In the algebraic approach to QM a quantum system has associated to it one $\ast$ algebra $\mathfrak{A}$ generated by its observables. A state is a positive, normalized linear functional $\omega : \mathfrak{A}\to \mathbb{C}$. Furthermore, we can define at this level pure and mixed states.

More than that, we can even deal with situations on which the usual density operator wouldn't exist. This is what motivates for instance the KMS condition characterizing thermal states.

Now on the traditional approach, where we have one fixed representation of $\mathfrak{A}$ on a Hilbert space of states $\mathfrak{H}$, if the system is described by a density operator $\rho \in \mathfrak{A}(\mathfrak{H})$ we can define its Von-Neumann entropy to be

$$S(\rho)=-\operatorname{tr}(\rho\ln \rho),$$

this is rather important not just to understand thermal behavior, but also to understand entanglement and the flow of information.

My question here is: at the level of the algebra $\mathfrak{A}$ and of the algebraic states $\omega$, can we define the Von-Neumann entropy, or is it a "representation-dependent" thing?

I find it strange, because I got the impression from some readings that "all physics should be independent of a particular representation".

• You may take a look at this review paper. It would be too long and too complicated (at least for me) to provide a full answer. Maybe someone else may do that. By the way, relative entropy seems easier to define than entropy itself. – yuggib Jun 7 '18 at 10:13