Introduction to Quantum Physics and Information Processing

226 Introduction to Quantum Physics and Information Processing

5. Entropy and measurement: When an observable

ˆ

A is measured in a

state ρ, the outcomes a have a probability distribution

P(a) = ha|ρ|ai, (11.33)

where |ai is the eigenstate of

ˆ

A. We then find that the Shannon entropy of

the measurement outcomes is greater than the von Neumann entropy of the

state:

H(A) ≥ S(ρ), (11.34)

with equality when

ˆ

A commutes with ρ. This means that measurement in-

creases the randomness in the system unless we measure a commuting observ-

able.

11.2.2 Entropy of composite systems

Some of the properties of the von Neumann entropy for composite systems

are similar to those ...