The joint quantum entropy generalizes the classical joint entropy to the context of quantum information theory. Intuitively, given two quantum states
Contents
In this article, we will use
Background
In information theory, for any classical random variable
In quantum information theory, the notion of entropy is extended from probability distributions to quantum states, or density matrices. For a state
Applying the spectral theorem, or Borel functional calculus for infinite dimensional systems, we see that it generalizes the classical entropy. The physical meaning remains the same. A maximally mixed state, the quantum analog of the uniform probability distribution, has maximum von Neumann entropy. On the other hand, a pure state, or a rank one projection, will have zero von Neumann entropy. We write the von Neumann entropy
Definition
Given a quantum system with two subsystems A and B, the term joint quantum entropy simply refers to the von Neumann entropy of the combined system. This is to distinguish from the entropy of the subsystems. In symbols, if the combined system is in state
the joint quantum entropy is then
Each subsystem has its own entropy. The state of the subsystems are given by the partial trace operation.
Properties
The classical joint entropy is always at least equal to the entropy of each individual system. This is not the case for the joint quantum entropy. If the quantum state
Consider a maximally entangled state such as a Bell state. If
then the total system is a pure state, with entropy 0, while each individual subsystem is a maximally mixed state, with maximum von Neumann entropy
Notice that the above phenomenon cannot occur if a state is a separable pure state. In that case, the reduced states of the subsystems are also pure. Therefore all entropies are zero.
Relations to other entropy measures
The joint quantum entropy
and the quantum mutual information:
These definitions parallel the use of the classical joint entropy to define the conditional entropy and mutual information.