Fundamental properties of Tsallis relative entropy

Quantum Information Processing, 2011

Generalizations of the quantum Fano inequality are considered. The notion of q-entropy exchange is introduced. This quantity is concave in each of its two arguments. For q ≥ 0, the inequality of Fano type with q-entropic functionals is established. The notion of coherent information and the perfect reversibility of a quantum operation are discussed in the context of q-entropies. By the monotonicity property, the lower bound of Pinsker type in terms of the trace norm distance is obtained for the Tsallis relative q-entropy of order q = 1/2. For 0 ≤ q ≤ 2, Fano type quantum inequalities with freely variable parameters are obtained.

Journal of Physics A: Mathematical and Theoretical, 2018

Measurements can be considered as a genuine example of processes that crush quantum coherence. In the case of an observable with degeneracy, the formulations of Lüders and von Neumann are known. These pictures postulate the two different states of a system immediately following the act of measurement. Hence, they are associated with divers variants of coherence losses during the measurement. Recent studies have focused on several ways to characterize quantum coherence appropriately. One of the existing types of quantifier is based on quantum α-divergences of the Tsallis type. In this paper, we introduce coherence quantifiers associated with the Lüders picture of quantum measurements. The are shown to satisfy the same properties as coherence α-quantifiers related to some orthonormal basis. Further, we consider losses of quantum coherence during a generalized measurement. The proposed approach is exemplified with unambiguous state discrimination; extreme properties of the states to be discriminated are clearly shown.

Physical Review A, 2020

The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched Rényi relative entropy of coherence (SRREOC) and unified (α, β)-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a ddimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a d-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a d-dimensional quantum system.

Quantum Information Processing, 2016

In this paper, we study the ordering states with Tsallis relative α-entropies of coherence and l1 norm of coherence for single-qubit states. We show that any Tsallis relative α-entropies of coherence and l1 norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative α-entropies of coherence, such as C1 ,C2 and C 1 2 , and show any one of these three measures and C l 1 will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.

Science China Physics, Mechanics & Astronomy, 2018

In this paper, we consider the quantum uncertainty relations of two generalized relative entropies of coherence based on two measurement bases. First, we give quantum uncertainty relations for pure states in a d-dimensional quantum system by making use of the majorization technique; these uncertainty relations are then generalized to mixed states. We find that the lower bounds are always nonnegative for pure states but may be negative for some mixed states. Second, the quantum uncertainty relations for single qubit states are obtained by the analytical method. We show that the lower bounds obtained by this technique are always positive for single qubit states. Third, the lower bounds obtained by the two methods described above are compared for single qubit states.

Methodology and Computing in Applied Probability, 2016

The behavior of generalized relative complexity measures is studied for assessment of structural dependence in a random vector. A related optimality criterion to sampling network design, which provides a flexible extension of mutual information based methods previously introduced, is formulated. Aspects related to practical implementation and conceptual issues regarding the meaning and potential use of this new approach are discussed. Numerical examples are used for illustration.

Reports on Progress in Physics, 2014

We present a comprehensive and up to date review on the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.

Quantum Information Processing, 2013

For an arbitrary strictly convex function f defined on the non-negative real line we determine the structure of all transformations on the set of density operators which preserve the quantum f-divergence.

Entropy, 2011

E.T. Jaynes, originator of the maximum entropy interpretation of statistical mechanics, emphasized that there is an inevitable trade-off between the conflicting requirements of robustness and accuracy for any inferencing algorithm. This is because robustness requires discarding of information in order to reduce the sensitivity to outliers. The principal of nonlinear statistical coupling, which is an interpretation of the Tsallis entropy generalization, can be used to quantify this trade-off. The coupled-surprisal,

Mathematics

The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are converge...

Journal of Mathematical Physics, 2011

Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative q-entropy is generally unbounded for q > 1. Upper bounds on the quantum relative q-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015

We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total inf...