Measure and probability theory
The notion of relative entropy of states is a generalization of the notion of entropy to a situation where the entropy of one state is measured “relative to” another state.
is also called
For states on finite probability spaces
For two finite probability distributions and , their relative entropy is
Alternatively, for two density matrices, their relative entropy is
For states on classical probability spaces
For states on quantum probability spaces (von Neumann algebras)
Let be a von Neumann algebra and let , be two states on it (faithful, positive linear functionals).
The relative entropy of relative to is
where is the relative modular operator? of any cyclic and separating vector representatives and of and .
This is due to (Araki).
This definition is independent of the choice of these representatives.
In the case that is finite dimensional and and are density matrices of and , respectively, this reduces to the above definition.
Relative entropy of states on von Neumann algebras was introduced in
A characterization of relative entropy on finite-dimensional C-star algebras is given in
- D. Petz, Characterization of the relative entropy of states of matrix algebras (pdf)
A survey of entropy in operator algebras is in
- Erling Størmer, Entropy in operator algebras (pdf)
Revised on July 15, 2011 22:12:00
by Urs Schreiber