To use all functions of this page, please activate cookies in your browser.
my.chemeurope.com
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
- My watch list
- My saved searches
- My saved topics
- My newsletter
Joint quantum entropy
In this article, we will use S(ρ,σ) for the joint quantum entropy. Additional recommended knowledge
BackgroundIn information theory, for any classical random variable X, the classical Shannon entropy H(X) is a measure of how uncertain we are about the outcome of X. For example, if X is a probability distribution concentrated at one point, the outcome of X is certain and therefore its entropy H(X) = 0. At the other extreme, if X is the uniform probability distribution with n possible values, intuitively one would expect X is associated with the most uncertainty. Indeed such uniform probability distributions have maximum possible entropy H(X) = log2(n). In quantum information theory, the notion of entropy is extended from probability distributions to quantum states, or density matrices. For a state ρ, 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 S(ρ) (or sometimes H(ρ). DefinitionGiven 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 ρAB, the joint quantum entropy is then Each subsystem has it own entropy. The state of the subsystems are given by the partial trace operation. PropertiesThe 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 ρAB exhibits quantum entanglement, then the entropy of each subsystem may be larger than the joint entropy. This is equivalent to the fact that the conditional quantum entropy may be negative, while the classical conditional entropy may never be. Consider a maximally entangled state such as a Bell state. If ρAB is a Bell state, say, 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 log2 = 1. Thus the joint entropy of the combined system is less than that of subsystems. This is because for entangled states, definite states cannot be assigned to subsystems, resulting in positive 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 MeasuresThe joint quantum entropy S(ρAB) can be used to define of the conditional 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. See alsoReferences
|
|
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Joint_quantum_entropy". A list of authors is available in Wikipedia. |