Finitely additive theory

1970 ◽  
pp. 15-21
Author(s):  
Flemming Topsøe
Keyword(s):  
Additive Theory

On a problem in the additive theory of numbers

1932 ◽  
Vol 34 (1) ◽  
pp. 637-644 ◽  
Author(s):  
C. J. A. Evelyn ◽  
E. H. Linfoot
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers

Note on a Result in the Additive Theory of Numbers

1938 ◽  
Vol s2-43 (1) ◽  
pp. 142-151 ◽  
Author(s):  
H. Davenport ◽  
H. Heilbronn
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers

Cyclic Spaces for Grassmann Derivatives and Additive Theory

1994 ◽  
Vol 26 (2) ◽  
pp. 140-146 ◽  
Author(s):  
J. A. Dias Da Silva ◽  
Y. O. Hamidoune
Keyword(s):  
Additive Theory

scholarly journals Effective Number Theory: Counting the Identities of a Quantum State

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1273
Author(s):  
Ivan Horváth ◽  
Robert Mendris
Keyword(s):  
Number Theory ◽  
Quantum Mechanics ◽  
Quantum Computation ◽  
Quantum State ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Quantum Algorithm ◽  
Effective Number ◽  
Additive Theory ◽  
Participation Number

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences.

Generalized Nonadditive Information Theory and Quantum Entanglement

2004 ◽  
Author(s):  
Sumiyoshi Abe
Keyword(s):  
Information Theory ◽  
Quantum Theory ◽  
Statistical Mechanics ◽  
Quantum Entanglement ◽  
Theory Of Measurement ◽  
Conditional Entropy ◽  
Tsallis Entropy ◽  
Von Neumann Entropy ◽  
Additive Theory ◽  
Von Neumann

Nonadditive classical information theory is developed in the axiomatic framework and then translated into quantum theory. The nonadditive conditional entropy associated with the Tsallis entropy indexed by q is given in accordance with the formalism of nonextensive statistical mechanics. The theory is applied to the problems of quantum entanglement and separability of the Werner-Popescu-type mixed state of a multipartite system, in order to examine if it has any points superior to the additive theory with the von Neumann entropy realized in the limit q → 1. It is shown that the nonadditive theory can lead to the necessary and sufficient condition for separability of the Werner-Popescu-type state, whereas the von Neumann theory can give only a much weaker condition…. Tsallis' nonextensive generalization of Boltzmann-Gibbs statistical mechanics [3, 15, 16] and its success in describing behaviors of a large class of complex systems naturally lead to the question of whether information theory can also admit an analogous generalization. If the answer is affirmative, then that will be of particular importance in connection with the problem of quantum entanglement and quantum theory of measurement [6, 8], in which necessities of a nonadditive information measure and an information content are suggested. One should also remember that there exists a conceptual similarity between a complex system and an entangled quantum system. In these systems, a "part" is indivisibly connected with the rest. An external operation on any part drastically influences the whole system, in general. Thus, the traditional reductionistic approach to an understanding of the nature of such a system may not work efficiently. In this chapter, we report a recent development in nonadditive quantum information theory based on the Tsallis entropy indexed by q [15] and its associated nonadditive conditional entropy [1]. This theory includes the ordinary additive theory with the von Neumann entropy in a special limiting case: q → To see if it has points superior to the additive theory, we apply it to the problems of separability and quantum entanglement.

ON A PROBLEM IN THE ADDITIVE THEORY OF NUMBERS (VI)

1933 ◽  
Vol os-4 (1) ◽  
pp. 309-314 ◽  
Author(s):  
C. J. A. EVELYN ◽  
E. H. LINFOOT
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers
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