1,199 research outputs found
Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization
Kullback-Leibler relative-entropy has unique properties in cases involving
distributions resulting from relative-entropy minimization. Tsallis
relative-entropy is a one parameter generalization of Kullback-Leibler
relative-entropy in the nonextensive thermostatistics. In this paper, we
present the properties of Tsallis relative-entropy minimization and present
some differences with the classical case. In the representation of such a
minimum relative-entropy distribution, we highlight the use of the q-product,
an operator that has been recently introduced to derive the mathematical
structure behind the Tsallis statistics. One of our main results is
generalization of triangle equality of relative-entropy minimization to the
nonextensive case.Comment: 15 pages, change of title, revision of triangle equalit
Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon
logarithmic entropic functional () is inadequate for wide classes of
strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's
{\it Conceptual inadequacy of the Shannon information in quantum measurements},
among many other systems exhibiting various forms of complexity. On the other
hand, the Shannon and Khinchin axioms uniquely mandate the BG form
; the Shore and Johnson axioms follow the same
path. Many natural, artificial and social systems have been satisfactorily
approached with nonadditive entropies such as the one (), basis of nonextensive
statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953
uniqueness theorems have already been generalized in the literature, by Santos
1997 and Abe 2000 respectively, in order to uniquely mandate . We argue
here that the same remains to be done with the Shore and Johnson 1980 axioms.
We arrive to this conclusion by analyzing specific classes of strongly
correlated complex systems that await such generalization.Comment: This new version has been sensibly modified and updated. The title
and abstract have been modifie
Nonadditive conditional entropy and its significance for local realism
Based on the form invariance of the structures given by Khinchin's axiomatic
foundations of information theory and the pseudoadditivity of the Tsallis
entropy indexed by q, the concept of conditional entropy is generalized to the
case of nonadditive (nonextensive) composite systems. The proposed nonadditive
conditional entropy is classically nonnegative but can be negative in the
quantum context, indicating its utility for characterizing quantum
entanglement. A criterion deduced from it for separability of density matrices
for validity of local realism is examined in detail by employing a bipartite
spin-1/2 system. It is found that the strongest criterion is obtained in the
limit q going to infinity.Comment: 12 pages, 1 figur
Extensive nonadditive entropy in quantum spin chains
We present details on a physical realization, in a many-body Hamiltonian
system, of the abstract probabilistic structure recently exhibited by
Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy ( density matrix; ) can conform, for an anomalous value of q (i.e., q
not equal to 1), to the classical thermodynamical requirement for the entropy
to be extensive. Moreover, we find that the entropic index q provides a tool to
characterize both universal and nonuniversal aspects in quantum phase
transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its
T=0 critical transverse field, we obtain
). The present results
suggest a new and powerful approach to measure entanglement in quantum
many-body systems. At the light of these results, and similar ones for a d=2
Bosonic system discussed by us elsewhere, we conjecture that, for blocks of
linear size L of a large class of Fermionic and Bosonic d-dimensional many-body
Hamiltonians with short-range interaction at T=0, we have that the additive
entropy (i.e., for , and for d>1), hence it is not extensive, whereas, for anomalous values of
the index q, we have that the nonadditive entropy (), i.e., it is extensive. The present discussion neatly illustrates that
entropic additivity and entropic extensivity are quite different properties,
even if they essentially coincide in the presence of short-range correlations.Comment: 9 pages, 4 figures, Invited Paper presented at the international
conference CTNEXT07, satellite of STATPHYS23, 1-5 July 2007, Catania, Ital
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