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Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero.This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crystal referred to the crystal state, whose entropy is zero according to the third law of thermodynamics.
The residual entropy of a fluid has some special significance. In 1976, Yasha Rosenfeld published a landmark paper, showing that the transport coefficients of pure liquids, when expressed as functions of the residual entropy, can be treated as monovariate functions, rather than as functions of two variables (i.e. temperature and pressure, or ...
The reduced von Neumann entropy of ρ AB with respect to system A is (). S(ρ B) is defined in the same way. It can now be seen that the definition of quantum mutual information, corresponding to the classical definition, should be as follows.
The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
As with many other objects in quantum information theory, quantum relative entropy is defined by extending the classical definition from probability distributions to density matrices. Let ρ be a density matrix. The von Neumann entropy of ρ, which is the quantum mechanical analog of the Shannon entropy, is given by
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons , nats , or hartleys .
Despite the foregoing, there is a difference between the two quantities. The information entropy Η can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.
Equivalently, the min-entropy () is the largest real number b such that all events occur with probability at most . The name min-entropy stems from the fact that it is the smallest entropy measure in the family of Rényi entropies. In this sense, it is the strongest way to measure the information content of a discrete random variable.