![]() This post was a warm-up for the next post: Rényi differential privacy. Information theory clarifies data discussions.Why Kullback-Leibler divergence is not a distance.Then for α not equal to 1,Īs with Lebesgue norms, you use varying values of the parameter to emphasize various features. Let p without a subscript be the vector of all the p i. In the limit as α goes to ∞, the Rényi entropy of X converges to the negative log of the probability of the most probable outcome. This case is also called collision entropy and is used in quantum information theory. That is, when people speak of Rényi entropy without qualification, they often have in mind the case α = 2. When the order α is not specified, it’s implicit default value is 2. We introduce a novel measure for the quantum property of nonstabilizerness-commonly known as magic-by considering the Renyi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. When α = 1 we get the more familiar Shannon entropy: It is simply log 2 n, the log of the number of values X takes on with positive probability. Then the H 0 is known as the max-entropy, or Hartley entropy. Max-entropy: α = 0Īssume all the probabilities p i are positive. Much of the analysis about the Shannon and Rnyi entropies is. And for each discrete random variable X, H α is a monotone non-decreasing function of α. p(x) logp(x)dx with entropy power N1 N e2h (provided that Nr(X) > 0 for some r > 1). All are additive for independent random variables. For example, let X be the random variable defined on [1, ∞) with densityĮach value of α gives a possible entropy measure. Corrections to universal Rnyi entropy in quasiparticle excited states of quantum chains Center for Joint Quantum Studies and Department of Physics, School of. The definition of Rényi entropy can be extended to continuous random variables byīut unlike the discrete case, Rényi entropy can be negative for continuous random variables, and so Rényi entropy is typically only used for discrete variables. Rényi entropy of continuous random variable In the case α = 1 or ∞ this expression means the limit as α approaches 1 or ∞ respectively. He started with Cauchy’s functional equation: If p and q are indepen- dent than I(pq)I(p)+I(q). If a discrete random variable X has n possible values, where the ith outcome has probability p i, then the Rényi entropy of order α is defined to beįor 0 ≤ α ≤ ∞. Renyi’s entropy History: Alfred Renyi was looking for the most general definition of information measures that would preserve the additivity for indepen- dent events and was compatible with the axioms of probability. The most common way of measuring information is Shannon entropy, but there are others. Rényi entropy, developed by Hungarian mathematician Alfréd Rényi, generalizes Shannon entropy and includes other entropy measures as special cases.
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