Quantum information as a non-Kolmogorovian generalization of Shannon's theory

In this article we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a new family of generalized entropic measures

Entropic measures of joint uncertainty: effects of lack of majorization

We compute R\'enyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two-dimensional quantum systems. The relative amount of uncertainty between two states depends on the uncertainty measure used. These results are not reproduced by a more standard duality relation. We show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics.Comment: 10 pages, 3 figure

On a generalized entropic uncertainty relation in the case of the qubit

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. R\'enyi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of R\'enyi entropies for any couple of (positive) entropic indices (\alpha,\beta). Thus, we have overcome the H\"older conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2] x [0 , 1/2] in the \alpha-\beta plane, and a semi-analytical expression on the line \beta = \alpha. It is seen that previous results are included as particular cases. Moreover, we present an analytical but suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.Comment: 15 pages, 6 figure

A family of generalized quantum entropies: definition and properties

We present a quantum version of the generalized $(h,\phi)$-entropies, introduced by Salicr\'u \textit{et al.} for the study of classical probability distributions. We establish their basic properties, and show that already known quantum entropies such as von Neumann, and quantum versions of R\'enyi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicr\'u form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum $(h,\phi)$-entropies under the action of quantum operations, and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement, and introduce a discussion on possible generalized conditional entropies as well.Comment: 26 pages, 1 figure. Close to published versio

Unified entropic measures of quantum correlations induced by local measurements

We introduce quantum correlations measures based on the minimal change in unified entropies induced by local rank-one projective measurements, divided by a factor that depends on the generalized purity of the system in the case of non-additive entropies. In this way, we overcome the issue of the artificial increasing of the value of quantum correlations measures based on non-additive entropies when an uncorrelated ancilla is appended to the system without changing the computability of our entropic correlations measures with respect to the previous ones. Moreover, we recover as limiting cases the quantum correlations measures based on von Neumann and R\'enyi entropies (i.e., additive entropies), for which the adjustment factor becomes trivial. In addition, we distinguish between total and semiquantum correlations and obtain some relations between them. Finally, we obtain analytical expressions of the entropic correlations measures for typical quantum bipartite systems.Comment: 10 pages, 1 figur

Approximate transformations of bipartite pure-state entanglement from the majorization lattice

We study the problem of deterministic transformations of an \textit{initial} pure entangled quantum state, $|\psi\rangle$, into a \textit{target} pure entangled quantum state, $|\phi\rangle$, by using \textit{local operations and classical communication} (LOCC). A celebrated result of Nielsen [Phys. Rev. Lett. \textbf{83}, 436 (1999)] gives the necessary and sufficient condition that makes this entanglement transformation process possible. Indeed, this process can be achieved if and only if the majorization relation $\psi \prec \phi$ holds, where $\psi$ and $\phi$ are probability vectors obtained by taking the squares of the Schmidt coefficients of the initial and target states, respectively. In general, this condition is not fulfilled. However, one can look for an \textit{approximate} entanglement transformation. Vidal \textit{et. al} [Phys. Rev. A \textbf{62}, 012304 (2000)] have proposed a deterministic transformation using LOCC in order to obtain a target state $|\chi^\mathrm{opt}\rangle$ most approximate to $|\phi\rangle$ in terms of maximal fidelity between them. Here, we show a strategy to deal with approximate entanglement transformations based on the properties of the \textit{majorization lattice}. More precisely, we propose as approximate target state one whose Schmidt coefficients are given by the supremum between $\psi$ and $\phi$. Our proposal is inspired on the observation that fidelity does not respect the majorization relation in general. Remarkably enough, we find that for some particular interesting cases, like two-qubit pure states or the entanglement concentration protocol, both proposals are coincident.Comment: Revised manuscript close to the accepted version in Physica A (10 pages, 1 figure

ТРАДИЦІЙНА КУЛЬТУРА ХХІ СТОЛІТТЯ У КОНТЕКСТІ ЖИТТЄЗДАТНОСТІ НЕМАТЕРІАЛЬНОЇ КУЛЬТУРНОЇ СПАДЩИНИ УКРАЇНИ (на прикладі фольклору)

This article provides a scientific analysis of the viability of the intangible cultural heritage in Ukraine as a component of traditional culture that promotes moral and aesthetic self-recovery of nation, preservation and development of customs, traditions, Ukrainian language, the revival of national consciousness, spirituality of people, the image of Ukraine in the European cultural space, and is the basis of the professional art development. On the example of folklore, the importance of "living culture" for humanity in the twentieth century is emphasized. The researches made by Dmitrenko M., L. Ivannikova, I. Kimakovych, I. Koval-Fuchylo, L. Kozar, A. Shalak, T. Shevchuk focus on topical issues of folklore study, the specific operation of traditional Ukrainian intangible culture, in particular the detection in the traditional culture of those phenomena that can adequately represent the global Ukrainian cultural and artistic integration process. However, the existing problems of the traditional culture of the XXI century in the context of sustainability of IKH in Ukraine as an example of folklore require more thorough study. It is noted that Ukraine is a multiethnic state, which has a huge variety of cultural traditions and ethnic and cultural features. The viability of the intangible cultural heritage of Ukraine reflects the traditions, rituals, customs, etc. for the operation of culture. The attention is focused on the ritual transmitted orally from generation to generation and which retains in modern edge all original elements and functions: socialization, communication, emotional, aesthetic, consuming – and is a prime example of respect for the festive and ritual tradition.Attention is focused also on traditional folk culture that is intertwined with folklore and is the root foundation of centuries-old culture of the peoples of Ukraine, retains its vitality, a system of spiritual values of the people directed to form patriotic education, comprehensive development of a nationally oriented person.It is noted adding to the Representative List of the Intangible Cultural Heritage of Humanity by UNESCO the element of Intangible Cultural Heritage of Ukraine – "Petrikivsky painting".The prospects of other elements of the cultural heritage of Ukraine that claim to international recognition were outlined.It is concluded that the traditional culture of the XXI century in the context of the viability of the intangible cultural heritage in Ukraine on the example of folklore as its spiritual component is a deep foundation of all variety of types, shapes, orientations of nation-type culture and reflects people's identity, based on the continuity of cultural traditions and preserves the most important elements of the spiritual side and historical experience of peoples. Developing for many centuries, cultural traditions function naturaly and perfectly in adaption the national culture to the society and a human to national statehood.Without knowledge of the depths of the indigenous culture and its values, one can not protect her. Intangible cultural heritage of the XXI century is in need of preservation, promotion on local and European levels.В статье проведено научный анализ жизнеспособности нематериального культурного наследия Украины в составляющей традиционной культуры, что способствует морально-эстетическому самоисцелению нации, сохранению и развитию обычаев, обрядов, традиций, украинского языка, возрождению национального сознания, духовности народа, имиджа Украины в европейском культурном пространстве и является основой развития профессионального искусства. На примере фольклора подчеркнуто значимость "живой культуры" для человечества в XXI веке с выявлением в культурном наследии тех феноменов, которые могут достойно представлять украинский народ в мировом культурно-художественном интеграционном процессе.У статті здійснено науковий аналіз життєздатності нематеріальної культурної спадщини України в складовій традиційної культури, що сприяє морально-естетичному самооздоровленню нації, збереженню та розвитку звичаїв, обрядів, традицій, української мови, відродженню національної свідомості, духовності народу, іміджу України в європейському культурному просторі та є підґрунтям розвитку професійного мистецтва. На прикладі фольклору підкреслено значущість "живої культури" для людства у ХХІ столітті з виявленням у культурній спадщині тих феноменів, що можуть гідно репрезентувати українців у світовому культурно-мистецькому інтеграційному процесі

Simultaneous quantum estimation of phase and indistinguishability in a two photon interferometer

With the rapid development of quantum technologies in recent years, the need for high sensitivity measuring techniques has become a key issue. In particular, optical sensors based on quantum states of light have proven to be optimal resources for high precision interferometry. Nevertheless, their performance may be severely affected by the presence of noise or imperfections. In this work we derive the quantum Fisher information matrix associated to the simultaneous estimation of an interferometric phase and the indistinguishability characterizing the probe state consisting of an even number of photons. We find the optimal measurement attaining the ultimate precision for both parameters in a single setup and perform an experiment based on a pair of photons with an unknown degree of indistinguishability entering a two-port interferometer.Comment: 10 pages, 4 figure

Collision entropy and optimal uncertainty

We propose an alternative measure of quantum uncertainty for pairs of arbitrary observables in the 2-dimensional case, in terms of collision entropies. We derive the optimal lower bound for this entropic uncertainty relation, which results in an analytic function of the overlap of the corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We compare our relation with other formulations of the uncertainty principle.Comment: The manuscript has been accepted for publication as a Regular Article in Physical Review

Polarization monotones of two-dimensional and three-dimensional random electromagnetic fields

We propose a formal resource-theoretic approach to quantify the degree of polarization of two- and three-dimensional random electromagnetic fields. This endows the space of spectral polarization matrices with the orders induced by majorization or convex mixing that naturally recover the best-known polarization measures