arXiv subjects

Alexey E. Rastegin

Publications and source records attributed to Alexey E. Rastegin.

At least 19 recordsLinked to original sources

arXiv2025

Complementarity relations for design-structured POVMs in terms of generalized entropies of order $α\in(0,2)$

Information entropies give a genuine way to characterize quantitatively an incompatibility in quantum measurements. Together with the Shannon entropy, few families of parametrized entropies have found use in various questions. It is also known that a possibility to vary the parameter can often provide more restrictions on elements of probability distributions. In quantum information processing, one often deals with measurements having some special structure. Quantum designs are currently the subject of active research, whence the aim to formulate complementarity relations for related measurements occurs. Using generalized entropies of order $α\in(0,2)$, we obtain uncertainty and certainty relations for POVMs assigned to a quantum design. The structure of quantum designs leads to several restrictions on generated probabilities. We show how to convert these restrictions into two-sided entropic estimates. One of the used ways is based on truncated expansions of the Taylor type. The recently found method to get two-sided entropic estimates uses polynomials with flexible coefficients. We illustrate the utility of this method with respect to both the Rényi and Tsallis entropies. Possible applications of the derived complementarity relations are briefly discussed.

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arXiv2024

Uncertainty relations for quantum measurements from generalized equiangular tight frames

The current study aims to examine uncertainty relations for measurements from generalized equiangular tight frames. Informationally overcomplete measurements are a valuable tool in quantum information processing, including tomography and state estimation. The maximal sets of mutually unbiased bases are the most common case of such measurements. The existence of $d+1$ mutually unbiased bases is proved for $d$ being a prime power. More general classes of informationally overcomplete measurements have been proposed for various purposes. Measurements of interest are typically characterized by some inner structure maintaining the required properties. It leads to restrictions imposed on generated probabilities. To apply the considered measurements, these restrictions should be converted into information-theoretic terms. It is interesting that certain restrictions hold irrespectively to overcompleteness. To describe the amount of uncertainty quantitatively, we use the Tsallis and Rényi entropies as well as probabilities of separate outcomes. The obtained results are based on estimation of the index of coincidence. The derived relations are briefly exemplified.

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arXiv2024

A five-dimensional Lorenz-type model near the temperature of maximum density

The current study formulates a convective model of the Lorenz type near the temperature of maximum density. The existence of this temperature actualizes water dynamics in temperate lakes. There is a conceptual interest what this feature induces in Lorenz-type models. The consideration starts with the zero coefficient of thermal expansion. Other steps are like famous Tritton's approach to derive the Lorenz model. This allows us to reduce difficulties with a selection of Galerkin functions. The analysis focuses on changes induced by zeroing the coefficient of thermal expansion. It results in a five-dimensional Lorenz-type model, whose equations are all nonlinear. The new model reiterates many features of the standard Lorenz model. The nontrivial critical points appear, when the zero critical point becomes unstable. The nontrivial critical points correspond to two possible directions of fluid flow. Phase trajectories of the new model were studied numerically. The results are similar to the known five-dimensional extensions of the Lorenz model.

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arXiv2024

On Kirkwood--Dirac quasiprobabilities and unravelings of quantum channel assigned to a tight frame

An issue which has attracted increasing attention in contemporary researches are Kirkwood--Dirac quasiprobabilities. List of their use includes many questions of quantum physics. Applications of complex tight frames in quantum information science were recently demonstrated. It is shown in this paper that quasiprobabilities naturally appear in the context of unravelings of a quantum channel. Using vectors of the given tight frame to build principal Kraus operators generates quasiprobabilities with interesting properties. For an equiangular tight frame, we characterize the Hilbert--Schmidt and spectral norms of the matrix consisted of quasiprobabilities. Hence, novel uncertainty relations in terms of Rényi and Tsallis entropies are obtained. New inequalities for characterizing the location of eigenvalues are derived. They give an alternative to estimating on the base of Geršgorin's theorem. A utility of the presented inequalities is exemplified with symmetric informationally complete measurement in dimension two.

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arXiv2023

Entropic uncertainty relations from equiangular tight frames and their applications

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.

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arXiv2023

Uncertainty relations in terms of generalized entropies derived from information diagrams

Entropic uncertainty relations are interesting in their own rights as well as for a lot of applications. Keeping this in mind, we try to make the corresponding inequalities as tight as possible. The use of parametrized entropies also allows one to improve relations between various information measures. Measurements of special types are widely used in quantum information science. For many of them we can estimate the index of coincidence defined as the total sum of squared probabilities. Inequalities between entropies and the index of coincidence form a long-standing direction of researches in classical information theory. The so-called information diagrams provide a powerful tool to obtain inequalities of interest. In the literature, results of such a kind mainly deal with standard information functions linked to the Shannon entropy. At the same time, generalized information functions have found use in questions of quantum information theory. In effect, Rényi and Tsallis entropies and related functions are of a separate interest. This paper is devoted to entropic uncertainty relations derived from information diagrams. The obtained inequalities are then applied to mutually unbiased bases, symmetric informationally complete measurements and their generalizations. We also improve entropic uncertainty relations for quantum measurement assigned to an equiangular tight frame.

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arXiv2022

Estimating the Shannon entropy and (un)certainty relations for design-structured POVMs

Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations, where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states.

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arXiv2022

Quantum search degeneration under amplitude noise in queries to the oracle

We examine how amplitude noise in queries to the oracle degrades a performance of quantum search algorithm. The Grover search and similar techniques are widely used in various quantum algorithms, including cases where rival parties are fighting over confidential data. Hence, the oracle-box wires become the subject of competing activity with an alteration of their normal functioning. Of course, many kinds of errors could arise in this way. Possible influence of dephasing on quantum search was already addressed in the literature. Amplitude damping is another type of errors that should be analyzed first. To study the problem, we introduce a simple model of collective distortions with the use of amplitude damping channel. All the quantities of interest should be considered as functions of the number of Grover's iterations. In particular, we investigate the success probability with respect to the parameter that characterizes the level of amplitude errors. The success probability degrades significantly even if the error amount is not essential. Namely, this probability soon enough reduces to a value close to one half. We also study trade-off relations between quantum coherence and the success probability in the presence of amplitude noise.

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arXiv2021

Flavor-mass majorization uncertainty relations and their links to the mixing matrix

Uncertainties in flavor and mass eigenstates of neutrinos are considered within the majorization approach. Nontrivial bounds reflect the fact that neutrinos cannot be simultaneously in flavor and mass eigenstates. As quantitative measures of uncertainties, both the Rényi and Tsallis entropies are utilized. Within the current amount of experience concerning the mixing matrix, majorization uncertainty relations need to put values of only two parameters, viz. $θ_{12}$ and $θ_{13}$. That is, the majorization approach is applicable within the same framework as the Maassen-Uffink relation recently utilized in this context. We also consider the case of detection inefficiencies, since it can naturally be incorporated into the entropic framework. Short comments on applications of entropic uncertainty relations with quantum memory are given.

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arXiv2021

Uncertainty and certainty relations for quantum coherence with respect to design-structured POVMs

The concept of quantum coherence and its possible use as a resource are currently the subject of active researches. Uncertainty and complementarity relations for quantum coherence allow one to study its changes with respect to other characteristics of the process of interest. Protocols of quantum information processing often use measurements that have a special inner structure. Quantum designs are considered as tools with a perspective of fruitful applications in quantum information processing. We obtain uncertainty and certainty relations for coherence averaged with respect to a set of design-structured POVMs of rank one. To characterize the degree of coherence, the relative entropy of coherence is utilized. The derived relations are exemplified with quantum designs in the two-dimensional Hilbert space.

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arXiv2020

Rényi formulation of uncertainty relations for POVMs assigned to a quantum design

Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau--Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.

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arXiv2019

Individual attacks with generalized discrimination and inadequacy of some information measures

We accomplish studies of properly quantifying eavesdropper's information gain in individual attacks on the BB84 system of quantum key distribution. A noticeable sensitivity of conclusions to the choice of information measures is brightly revealed, when generalized state discrimination is used at the last stage. To realize her aims, Eve intrudes into the communication channel with some entangling probe. The Fuchs-Peres-Brandt (FPB) probe is known as a powerful individual attack on the BB84 protocol. In the simplest formulation, Eve uses the Helstrom scheme to distinguish output probe states. In conclusive eavesdropping, the unambiguous discrimination is utilized. In the intermediate scenario, she can apply a generalized discrimination scheme that interpolates between the Helstrom and unambiguous discrimination schemes. We analyze Eve's probe performance from the viewpoint of various measures for quantifying mutual information.

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arXiv2019

On entropic uncertainty relations for measurements of energy and its "complement"

Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays the important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non-commuting observables. Since quantum theory was originating, many approaches to energy-time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, we obtain uncertainty relations of the "energy-time" type in terms of the Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state-dependent and state-independent formulations are of interest. Some of the derived state-independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.

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arXiv2019

Coherence quantifiers from the viewpoint of their decreases in the measurement process

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.

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arXiv2018

On degradation of Grover's search under collective phase flips in queries to the oracle

We address the case, when querying to the oracle in Grover's algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposite party that tries to prevent correct receiving data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study the problem, we introduce a simple model of collective phase distortions on the base of phase damping channel. In the model used, the success probability is not altered via the oracle-box wires {\it per se}. Phase distortions of the considered type can hardly be detected via any one-time query to the oracle. However, the success probability is significantly changed, when such errors are introduced as an intermediate step into the Grover iteration. We investigate the success probability with respect to variations the parameter that characterizes the amount of phase errors. It turns out that the success probability is decreased significantly even if the error amount is not very high. Moreover, this probability quickly reduces to the value of one half, which corresponds to the completely mixed state. We also study trade-off relations between quantum coherence and the success probability in the presence of noise of the considered type.

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arXiv2018

On the role of dealing with quantum coherence in amplitude amplification

Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique generalizes key ideas of the Grover search algorithm. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. As measures of coherence, the geometric coherence and the relative entropy of coherence are considered. In terms of the relative entropy of coherence, complementarity relations with the success probability seem to be the most expository. The general relations presented are illustrated within several model scenarios of amplitude amplification processes.

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arXiv2018

Entropic uncertainty relations for successive measurements in the presence of a minimal length

We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we formulated uncertainty relations in terms of Shannon entropies. Since such relations involve a state-dependent correction term, they generally differ from preparation uncertainty relations. This difference is revealed when position is measured by the first. In contrast, state-independent uncertainty relations in terms of Rényi and Tsallis entropies are obtained with the same lower bounds as in the preparation scenario. These bounds are explicitly dependent on the acceptance function of apparatuses in momentum measurements. Entropic uncertainty relations with binning are discussed as well.

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arXiv2018

Quantum work fluctuations versus macrorealism in terms of non-extensive entropies

Fluctuations of the work performed on a driven quantum system can be characterized by the so-called fluctuation theorems. The Jarzynski relation and the Crooks theorem are famous examples of exact equalities characterizing non-equilibrium dynamics. Such statistical theorems are typically formulated in a similar manner in both classical and quantum physics. Leggett-Garg inequalities are inspired by the two assumptions referred to as the macroscopic realism and the non-invasive measurability. Together, these assumptions are known as the macrorealism in the broad sense. Quantum mechanics is provably incompatible with restrictions of the Leggett-Garg type. It turned out that Leggett-Garg inequalities can be used to distinguish quantum and classical work fluctuations. We develop this issue with the use of entropic functions of the Tsallis type. Varying the entropic parameter, we are often able to reach more robust detection of violations of the corresponding Leggett-Garg inequalities. In reality, all measurement devices suffer from losses. Within the entropic formulation, detection inefficiencies can naturally be incorporated into the consideration. This question also shows advantages that are provided due to the use of generalized entropies.

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