Background
In information theory, for any classical random variable, the classical Shannon entropy is a measure of how uncertain we are about the outcome of . For example, if is a probability distribution concentrated at one point, the outcome of is certain and therefore its entropy . At the other extreme, if is the uniform probability distribution with possible values, intuitively one would expect is associated with the most uncertainty. Indeed such uniform probability distributions have maximum possible entropy .
In quantum information theory, the notion of entropy is extended from probability distributions to quantum states, or density matrices. For a state, the von Neumann entropy is defined by
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 (or sometimes .
Read more about this topic: Joint Quantum Entropy
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