arXiv subjects

Alexey E. Rastegin

Publications and source records attributed to Alexey E. Rastegin.

At least 55 records · Page 3Linked to original sources

arXiv2014

Notes on general SIC-POVMs

An unavoidable task in quantum information processing is how to obtain data about the state of an individual system by suitable measurements. Informationally complete measurements are relevant in quantum state tomography, quantum cryptography, quantum cloning, and other questions. Symmetric informationally complete measurements (SIC-POVMs) form an especially important class of such measurements. We formulate some novel properties and relations for general SIC-POVMs in a finite-dimensional Hilbert space. For a given density matrix and any general SIC-POVM, the so-called index of coincidence of generated probability distribution is exactly calculated. Using this result, we obtain state-dependent entropic bounds for a single general SIC-POVM. Lower entropic bounds are derived in terms of the Rényi $α$-entropies for $α\in[2;\infty)$ and the Tsallis $α$-entropies for $α\in(0;2]$. A lower bound on the min-entropy of a SIC-POVM is separately examined. For a pair of general SIC-POVMs, entropic uncertainty relations of the Maassen-Uffink type are considered.

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arXiv2014

Formulation of Leggett-Garg inequalities in terms of $q$-entropies

As is well known, the macroscopic realism and the noninvasive measurability together lead to Leggett-Garg inequalities violated by quantum mechanics. We consider tests of the Leggett-Garg type with use of the $q$-entropies. For all $q\geq1$, quantum mechanics predicts violations of an entire family of $q$-entropic inequalities of the Leggett-Garg type. Violations are exemplified with a quantum spin-$s$ system. In general, entropic Leggett-Garg inequalities give only necessary conditions that some probabilistic model is compatible with the macrorealism in the broader sense. The presented $q$-entropic inequalities allow to widen a class of situations, in which an incompatibility with the macrorealism can be tested. In the considered example, both the strength and range of violations are somehow improved by varying $q$. We also examine $q$-entropic inequalities of the Leggett-Garg type in the case of detection inefficiencies, when the no-click event may occur in each observation. With the use of the $q$-entropic inequalities, the required amount of efficiency may be reduced.

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arXiv2014

On quantum conditional entropies defined in terms of the $f$-divergences

We consider a family of quantum conditional entropies based on the concept of quantum $f$-divergences. First, we explicitly formulate conditions under which the notion of quantum conditional entropy is well defined in this way. In particular, we demand that the value of conditional entropy be independent of any extension of the principal Hilbert space. Using fundamental properties of the quantum $f$-divergences, several interesting relations for such conditional entropies are formulated. We separately examine an especially important case of quantum conditional entropies related to the Tsallis divergence.

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arXiv2014

Uncertainty and certainty relations for the Pauli observables in terms of the Rényi entropies of order $α\in(0;1]$

We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order $α\in(0;1]$. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the conditions for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi $α$-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.

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arXiv2014

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the Rényi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis $α$-entropic bounds for $α\in(0;2]$. Relations in a model of detection inefficiences are obtained. In terms of Rényi's entropies, lower bounds are given for $α\in[2;\infty)$. State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered. We also obtain lower bounds in term of the so-called symmetrized entropies. The presented results for mutually unbiased bases are extensions of some bounds previously derived in the literature. We further formulate new properties of symmetric informationally complete measurements in a finite-dimensional Hilbert space. For a given state and any SIC-POVM, the index of coincidence of generated probability distribution is exactly calculated. Short notes are made on potential use of this result in entanglement detection. Further, we obtain state-dependent entropic uncertainty relations for a single SIC-POVM. Entropic bounds are derived in terms of the Rényi $α$-entropies for $α\in[2;\infty)$ and the Tsallis $α$-entropies for $α\in(0;2]$. In the Tsallis formulation, a case of detection inefficiences is briefly mentioned. For a pair of symmetric informationally complete measurements, we also obtain an entropic bound of Maassen-Uffink type.

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arXiv2014

Jarzynski equality for quantum stochastic maps

Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are applied for exemplary systems subjected to quantum channels acting on a finite-dimensional Hilbert space.

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arXiv2013

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Pinsker's and Fannes' type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum $f$-divergence is used for its estimating from below. For order $α\in(0,1)$, a family of lower bounds of Pinsker type is obtained. For $α>1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi's entropies. The Fano inequality is extended to Tsallis' entropies for all $α>0$. The deduced bounds on the Tsallis conditional entropy are used for obtaining inequalities of Fannes' type.

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arXiv2013

Unified-entropy trade-off relations for a single quantum channel

Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamiołkowski-Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix obtained as reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver $(q,s)$-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.

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arXiv2013

Non-equilibrium equalities with unital quantum channels

A general tool for description of open quantum systems is given by the formalism of quantum operations. Most important of them are trace-preserving maps also known as quantum channels. We discuss those conditions on quantum channels under which the Jarzynski equality and related fluctuation theorems hold. It is essential that the representing quantum channel be unital. Under the mentioned condition, we first derive the corresponding Jarzynski equality. For bistochastic map and its adjoint, we further formulate a theorem of Tasaki-Crooks type. In the context of unital channels, some notes on heat transfer between two quantum systems are given. We also consider the case of a finite system operated by an external agent with a feedback control. When unital channels are applied at the first stage and, for a mutual-information form, at the further ones, we obtain quantum Jarzynski-Sagawa-Ueda relations. These are extension of the previously given results to unital quantum operations.

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arXiv2013

Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis' entropies

Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $α$-entropies. For all real $α\in(0;1]$ and integer $α\geq2$, lower bounds on the sum of three $α$-entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate of one of the Pauli observables. Using concavity with respect to the parameter $α$, we derive approximate lower bounds for non-integer $α\in(1;+\infty)$. In the case of pure states, the developed method also allows to obtain upper bounds on the entropic sum for real $α\in(0;1]$ and integer $α\geq2$. For applied purposes, entropic bounds are often used with averaging over the individual entropies. Combining the obtained bounds leads to a band, in which the rescaled average $α$-entropy ranges in the pure-state case. A width of this band is essentially dependent on $α$. It can be interpreted as an evidence for sensitivity in quantifying the complementarity.

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arXiv2012

Notes on entropic characteristics of quantum channels

One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange are derived. We also give upper bounds on the map $(q,s)$-entropies in terms of the output entropy, corresponding to the completely mixed input.

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arXiv2012

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.

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arXiv2012

Number-phase uncertainty relations in terms of generalized entropies

Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.

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arXiv2012

Entropic formulation of the uncertainty principle for the number and annihilation operators

An entropic approach to formulating uncertainty relations for the number-annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz-Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in this inequality and further derive uncertainty relations in terms of both the Renyi and Tsallis entropies. Entropic uncertainty relations for continuous distribution as well as relations for discretized one are presented.

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arXiv2011

On unified-entropy characterization of quantum channels

We consider properties of quantum channels with use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map $(q,s)$-entropy is naturally defined as the unified $(q,s)$-entropy of rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For tensor product of a pair of quantum channels, we derive two-sided estimating of the output entropy of a maximally entangled input state.

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arXiv2011

Fano type quantum inequalities in terms of $q$-entropies

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 $q\geq0$, 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 $0\leq{q}\leq2$, Fano type quantum inequalities with freely variable parameters are obtained.

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arXiv2011

Upper continuity bounds on the relative $q$-entropy for $q>1$

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.

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arXiv2011

Some general properties of unified entropies

Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes' sense and stability in Lesche's sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum Rényi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurement is proved.

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