Relative entropy (divergence) of Bregman type recently proposed by T. D. Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism. It has been observed that two different expressions arise due to two different forms of quantum divergences. It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas one of Csiszàr type is related to the third-choice thermostatistics. The superiority of the third-choice thermostatistics to the first-choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q ∈ (0, 1) and fidelity F smaller than 1 whereas the one obtained by using Csiszàr type is free from such anomalies. Moreover, it has been noted that these two measures show different qualitative behavior with respect to F. Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore–Johnson axioms. The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further. The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.
Relative entropy (divergence) of Bregman type recently proposed by T. D. Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism. It has been observed that two different expressions arise due to two different forms of quantum divergences. It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas one of Csiszàr type is related to the third-choice thermostatistics. The superiority of the third-choice thermostatistics to the first-choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q ∈ (0, 1) and fidelity F smaller than 1 whereas the one obtained by using Csiszàr type is free from such anomalies. Moreover, it has been noted that these two measures show different qualitative behavior with respect to F. Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore–Johnson axioms. The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further. The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.
The purity of Werner state in non-extensive formalism associated with two different constraints has been calculated in a previous paper by Bağci et al.17 Two different results have been obtained corresponding to ordinary probability and escort probability. The former has been shown to result in negative values thereby leading authors to deduce the advantage of escort probabilities over ordinary probabilities. However, these results have only been for a limited interval of q values which lie between 0 and 1. In this paper, we solve the same problem for all values of non-extensive index q by using a perturbative approach and show that the simultaneous use of both types of constraint is necessary in order to obtain the solution for a whole spectrum of non-extensive index. In this sense, the existence of these different constraints in non-extensive formalism must not be seen as a deficiency in the formalism but rather must be welcomed as a means of providing solution for all values of parameter q.
A general Werner-type state is studied from two viewpoints: (i) an application of the dynamical interaction of the objective system with its environment, represented by a unital positive operator-valued measure (POVM), which ensures an increase of entropy, making the system evolve from an initial pure state to a mixed state of the Werner type, and (ii) the space of the objective system is constrained to have a given value of the fidelity. Then, the maximum entropy principle is shown to yield the Werner-type state as a canonical ensemble with a projector Hamiltonian. This novel observation is illustrated by examples of bipartite systems, the separability criteria of which are given in terms of the values of temperature. The present viewpoint may cast light on the relevance of thermostatistics to the physics of quantum entanglement. In addition, the POVM scheme presented here offers a way of experimentally generating the Werner-type thermal states.
Density matrices are a central tool in quantum physics, but it is also used in machine learning. A positive definite matrix called kernel matrix is used to represent the similarities between examples. Positive definiteness assures that the examples are embedded in an Euclidean space. When a positive definite matrix is learned from data, one has to design an update rule that maintains the positive definiteness. Our update rule, called matrix exponentiated gradient update, is motivated by the quantum relative entropy. Notably, the relative entropy is an instance of Bregman divergences, which are asymmetric distance measures specifying theoretical properties of machine learning algorithms. Using the calculus commonly used in quantum physics, we prove an upperbound of the generalization error of online learning.
Density matrices are a central tool in quantum physics, but it is also used in machine learning. A positive definite matrix called kernel matrix is used to represent the similarities between examples. Positive definiteness assures that the examples are embedded in an Euclidean space. When a positive definite matrix is learned from data, one has to design an update rule that maintains the positive definiteness. Our update rule, called matrix exponentiated gradient update, is motivated by the quantum relative entropy. Notably, the relative entropy is an instance of Bregman divergences, which are asymmetric distance measures specifying theoretical properties of machine learning algorithms. Using the calculus commonly used in quantum physics, we prove an upperbound of the generalization error of online learning.
We present some basic elements of the theory of generalised Brègman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two former approaches to Brègman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Brègman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.
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.