arXiv subjects

Alexey E. Rastegin

Publications and source records attributed to Alexey E. Rastegin.

At least 37 records · Page 2Linked to original sources

arXiv2017

Rényi and Tsallis formulations of separability conditions in finite dimensions

Separability conditions for a bipartite quantum system of finite-dimensional subsystems are formulated in terms of Rényi and Tsallis entropies. Entropic uncertainty relations often lead to entanglement criteria. We propose new approach based on the convolution of discrete probability distributions. Measurements on a total system are constructed of local ones according to the convolution scheme. Separability conditions are derived on the base of uncertainty relations of the Maassen-Uffink type as well as majorization relations. On each of subsystems, we use a pair of sets of subnormalized vectors that form rank-one POVMs. We also obtain entropic separability conditions for local measurements with a special structure, such as mutually unbiased bases and symmetric informationally complete measurements. The relevance of the derived separability conditions is demonstrated with several examples.

quant-ph↗

arXiv2017

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.

quant-ph↗

arXiv2017

On entropic uncertainty relations in the presence of a minimal length

Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. As the corresponding entropies differ, known entropic uncertainty relations are changed. Using differential Shannon entropies, we give a state-dependent formulation with correction term. State-independent uncertainty relations are obtained in terms of the Rényi entropies and the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution.

quant-ph↗

arXiv2017

Rényi formulation of entanglement criteria for continuous variables

Entanglement criteria for an $n$-partite quantum system with continuous variables are formulated in terms of Rényi entropies. Rényi entropies are widely used as a good information measure due to many nice properties. Derived entanglement criteria are based on several mathematical results such as the Hausdorff-Young inequality, Young's inequality for convolution and its converse. From the historical viewpoint, the formulations of these results with sharp constants were obtained comparatively recently. Using the position and momentum observables of subsystems, one can build two total-system measurements with the following property. For product states, the final density in each global measurement appears as a convolution of $n$ local densities. Hence, restrictions in terms of two Rényi entropies with constrained entropic indices are formulated for $n$-separable states of an $n$-partite quantum system with continuous variables. Experimental results are typically sampled into bins between prescribed discrete points. For these aims, we give appropriate reformulations of the derived entanglement criteria.

quant-ph↗

arXiv2017

Rényi and Tsallis entropies related to eigenfunctions of quantum graphs

For certain families of finite quantum graphs, we study the question of how eigenfunctions are distributed over the graph. To characterize properties of the distribution, generalized entropies of the Rényi and Tsallis types are considered. The presented approach is similar to entropic uncertainty relations of the Maassen-Uffink type. Using the Riesz theorem, we derive lower bounds on symmetrized generalized entropies of eigenfunctions. A quality of such estimates will depend on boundary conditions used at vertices of the given graph. Rényi and Tsallis entropies of eigenfunctions of star graphs are separately examined. Relations between generalized entropies and variances of eigenfunctions are considered as well. When such relations remain valid on average, they may be used in studies of quantum ergodicity.

math-ph↗

arXiv2016

Entropic uncertainty relations for successive measurements of canonically conjugate observables

Uncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg-Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. The Rényi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive measurements of canonically conjugate operators. Entropic uncertainty relations for the case of position and momentum are separately discussed.

quant-ph↗

arXiv2016

Notes on use of generalized entropies in counting

We address an idea of applying generalized entropies in counting problems. First, we consider some entropic properties that are essential for such purposes. Using the $α$-entropies of Tsallis-Havrda-Charvát type, we derive several results connected with Shearer's lemma. In particular, we derive upper bounds on the maximum possible cardinality of a family of $k$-subsets, when no pairwise intersections of these subsets may coincide. Further, we revisit the Minc conjecture. Our approach leads to a family of one-parameter extensions of Brégman's theorem. A utility of the obtained bounds is explicitly exemplified.

math.CO↗

arXiv2016

Majorization entropic uncertainty relations for quantum operations

Majorization uncertainty relations are derived for arbitrary quantum operations acting on a finite-dimensional space. The basic idea is to consider submatrices of block matrices comprised of the corresponding Kraus operators. This is an extension of the previous formulation, which deals with submatrices of a unitary matrix relating orthogonal bases in which measurements are performed. Two classes of majorization relations are considered: one related to tensor product of probability vectors and another one related to their direct sum. We explicitly discuss an example of a pair of one-qubit operations, each of them represented by two Kraus operators. In the particular case of quantum maps describing orthogonal measurements the presented formulation reduces to earlier results derived for measurements in orthogonal bases. The presented approach allows us also to bound the entropy characterizing results of a single generalized measurement.

quant-ph↗

arXiv2016

Separability conditions based on local fine-grained uncertainty relations

Many protocols of quantum information processing use entangled states. Hence, separability criteria are of great importance. We propose new separability conditions for a bipartite finite-dimensional system. They are derived by using fine-grained uncertainty relations. Fine-grained uncertainty relations can be obtained by consideration of the spectral norms of certain positive matrices. One of possible approaches to separability conditions is connected with upper bounds on the sum of maximal probabilities. Separability conditions are often formulated for measurements that have a special structure. For instance, mutually unbiased bases and mutually unbiased measurements can be utilized for such purposes. Using resolution of the identity for each subsystem of a bipartite system, we construct some resolution of the identity in the product of Hilbert spaces. Separability conditions are then formulated in terms of maximal probabilities for a collection of specific outcomes. The presented conditions are compared with some previous formulations. Our results are exemplified with entangled states of a two-qutrit system.

quant-ph↗

arXiv2016

On conclusive eavesdropping and measures of mutual information in quantum key distribution

We address the question of quantifying eavesdropper's information gain in an individual attack on systems of quantum key distribution. It is connected with the concept of conclusive eavesdropping introduced by Brandt. Using the BB84 protocol, we examine the problem of estimating a performance of conclusive entangling probe. The question of interest depends on the choice of a quantitative measure of eavesdropper's information about the error-free sifted bits. The Fuchs--Peres--Brandt probe realizes a very powerful individual attack on the BB84 scheme. In the usual formulation, Eve utilizes the Helstrom scheme in distinguishing between the two output probe states. In conclusive eavesdropping, the unambiguous discrimination is used. Comparing these two versions allows to demonstrate serious distinctions between widely used quantifiers of mutual information. In particular, the so-called Rényi mutual information does not seem to be a completely legitimate measure of an amount of mutual information. It is brightly emphasized with the example of conclusive eavesdropping.

quant-ph↗

arXiv2016

Quantum-coherence quantifiers based on the Tsallis relative $α$-entropies

The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum level. Various measures could be proposed to quantify coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. Basic properties of the considered coherence quantifiers are derived. Trade-off relations between coherence and mixedness are examined. The property of monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.

quant-ph↗

arXiv2016

Rényi and Tsallis formulations of noise-disturbance trade-off relations

We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the Rényi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional Rényi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.

quant-ph↗

arXiv2015

On the Brukner-Zeilinger approach to information in quantum measurements

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 information can be used for estimating the map norm of quantum operations.

quant-ph↗

arXiv2015

Uncertainty and certainty relations for successive projective measurements of a qubit in terms of Tsallis' entropies

We study uncertainty and certainty relations for two successive measurements of two-dimensional observables. Uncertainties in successive measurement are considered within the following two scenarios. In the first scenario, the second measurement is performed on the quantum state generated after the first measurement with completely erased information. In the second scenario, the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome. Induced quantum uncertainties are characterized by means of the Tsallis entropies. For two successive projective measurement of a qubit, we obtain minimal and maximal values of related entropic measures of induced uncertainties. Some conclusions found in the second scenario are extended to arbitrary finite dimensionality. In particular, a connection with mutual unbiasedness is emphasized.

quant-ph↗

arXiv2015

On uncertainty relations and entanglement detection with mutually unbiased measurements

We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.

quant-ph↗

arXiv2015

On generalized entropies and information-theoretic Bell inequalities under decoherence

We consider information-theoretic inequalities of the Bell type in the presence of decoherence. It is natural that too strong coupling with the environment can prevent an observation of quantum correlations. In this regard, the use of various entropic functions may give additional capabilities to reveal desired correlations. It was already shown that the Bell and Leggett-Garg inequalities in terms conditional Tsallis entropies are more sensitive in the cases of detection inefficiencies. In this paper, we study capabilities of generalized conditional entropies of the Tsallis type in analyzing the Bell theorem in decoherence scenarios. Two forms of the conditional Tsallis $q$-entropy are known in the literature. We show that each of them can be used for defining a metric in the probability space of interest. Such metrics can be used in realizing the so-called triangle principle. The triangle principle has recently been proposed as a unifying approach to questions of local realism and non-contextuality. Applying the triangle principle leads to the two families of $q$-metric inequalities of the Bell type. Information-theoretic formulations in terms of the $q$-entropic metrics are first discussed for the CHSH scenario in dephasing environment. Then we also revisit $q$-entropic inequalities of the Leggett-Garg type. An environmental influence is modeled by the phase damping channel and by the depolarizing channel.

quant-ph↗

arXiv2014

Fine-grained uncertainty relations for several quantum measurements

We study fine-grained uncertainty relations for several quantum measurements in a finite-dimensional Hilbert space. The proposed approach is based on exact calculation or estimation of the spectral norms of corresponding positive matrices. Fine-grained uncertainty relations of the state-independent form are derived for an arbitrary set of mutually unbiased bases. Such relations are extended with a recent notion of mutually unbiased measurements. The case of so-called mutually biased bases is considered in a similar manner. We also discuss a formulation of fine-grained uncertainty relations in the case of generalized measurements. The general approach is then applied to two measurements related to state discrimination. The case of three rank-one projective measurements is further examined in details. In particular, we consider fine-grained uncertainty relations for mutually unbiased bases in three-dimensional Hilbert space.

quant-ph↗

arXiv2014

Tests for quantum contextuality in terms of $q$-entropies

The information-theoretic approach to Bell's theorem is developed with use of the conditional $q$-entropies. The $q$-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional $q$-entropies for all $q\geq1$. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of $n$-cycle scenario is briefly mentioned. The new inequalities with conditional $q$-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The $q$-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the $q$-entropic formulation.

quant-ph↗