Infinite Shannon entropy

Abstract

Even if a probability distribution is properly normalizable, its associated Shannon (or von Neumann) entropy can easily be infinite. We carefully analyze conditions under which this phenomenon can occur. Roughly speaking, this happens when arbitrarily small amounts of probability are dispersed into an infinite number of states; we shall quantify this observation and make it precise. We develop several particularly simple, elementary, and useful bounds, and also provide some asymptotic estimates, leading to necessary and sufficient conditions for the occurrence of infinite Shannon entropy. We go to some effort to keep technical computations as simple and conceptually clear as possible. In particular, we shall see that large entropies cannot be localized in state space; large entropies can only be supported on an exponentially large number of states. We are for the time being interested in single-channel Shannon entropy in the information theoretic sense, not entropy in a stochastic field theory or QFT defined over some configuration space, on the grounds that this simple problem is a necessary precursor to understanding infinite entropy in a field theoretic context.

13 pages; V2: 4 references added

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Cited by in corpus (14)

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• Entropy of a quantum channel
• Fundamental limits on quantum dynamics based on entropy change
• Entropy Maximization for Markov Decision Processes Under Temporal Logic Constraints
• Energy-constrained two-way assisted private and quantum capacities of quantum channels
• Clausius entropy for arbitrary bifurcate null surfaces
• Fermionic versus bosonic behavior of confined Wigner molecules
• Entanglement catalysis for quantum states and noisy channels
• Multi-partite analysis of average-subsystem entropies
• Asymptotic Normality for Plug-in Estimators of Generalized Shannon's Entropy
• Which number system is "best" for describing empirical reality?
• Carroll black holes in (A)dS and their higher-derivative modifications
• Spatial correlations and entanglement in a hybrid system of N fermion pairs with harmonic interaction