nonadditive conditional entropy and its significance for local realism based on the form invariance of the structures given by khinchin s axiomatic foundations of information theory and the pseudoadditivity of the tsallis entropy indexed by q the concept of conditional entropy is generalized to the case of nonadditive nonextensive composite systems the proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context indicating its utility for characterizing quantum entanglement a criterion deduced from it for separability of density matrices for validity of local realism is examined in detail by employing a bipartite spin 1 2 system it is found that the strongest criterion is obtained in the limit q

enumeration of algebras close to absolutely free algebras and binary trees abstract we study three classes of algebras absolutely free algebras free commutative non associative and free anti commutative non associative algebras we study asymptotics of the growth for free algebras of these classes and for their subvarieties as well mainly we study finitely generated algebras also the codimension growth for varieties in theses classes is studied for these purposes we use ordinary generating functions as well as exponential generating functions the following subvarieties are studied in these classes solvable completely solvable right nilpotent and completely right nilpotent subvarieties the obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees we enumerate these trees using generating functions for solvable and right nilpotent algebras the generating functions are algebraic for completely solvable and completely right nilpotent algebras the respective functions are rational it is known that these three varieties of algebras satisfy schreier s property i e subalgebras of free algebras are free for free groups there is schreier s formula for the rank of a subgroup of a free group we find analogues of this formula for these varieties they are written in terms of series as an application we study invariants of finite groups acting on absolutely free algebras

nonadditive entropies and quantum entanglement abstract generalized quantum information theory is developed based on the nonadditive tsallis entropy indexed by q to see how this formalism is superior to the ordinary additive theory with the von neumann entropy realized in the limit q 1 the problem of quantum entanglement of mixed states is discussed it is shown that for the parameterized werner popescu type state of a general nn system i e an n partite n level system the present approach leads to the limitation on validity of local realism which is stronger than that derived from the additive theory

quantum work fluctuations versus macrorealism in terms of non extensive entropies abstract 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

generalized disorder measures and the detection of quantum entanglement abstract we discuss the main properties of general non additive entropic forms based on arbitrary concave functions and their relation with the theory of majorization as application we examine the generalized entropic criterion for separability of mixed states of composite quantum systems where non additive disorder measures play a fundamental role these features are then analyzed in two qubit and two qudit systems where it is seen that the von neumann based criterion is especially weak for detecting entanglement in contrast with those based on the tsallis and other non additive disorder measures

a nonextensive statistical approach to the kinetics of phase transformation abstract some limitations of johnson mehl avrami kolmogorov jmak equation which is used widely for describing kinetics of phase transformation were demonstrated using probabilistic analysis and monte carlo simulations acta mater 48 2000 4217 as well known jmak equation predicts correctly the real transformed fraction only if the number of the growing nuclei in the controlled volume is large however jmak equation seems to fail if the number of growing nuclei is small no matter how large the volume of the controlled volume is in this study we propose a different equation for describing kinetics of phase transformation using a nonextensive formalism namely tsallis thermostatistics which has been commonly employed to study the various physical systems for a decade

generalized conditional entropy in bipartite quantum systems we analyze for a general concave entropic form the associated conditional entropy of a quantum system a b obtained as a result of a local measurement on one of the systems b this quantity is a measure of the average mixedness of a after such measurement and its minimum over all local measurements is shown to be the associated entanglement of formation between a and a purifying third system c in the case of the von neumann entropy this minimum determines also the quantum discord for classically correlated states and mixtures of a pure state with the maximally mixed state we show that the minimizing measurement can be determined analytically and is universal i e the same for all concave forms while these properties no longer hold for general states we also show that in the special case of the linear entropy an explicit expression for the associated conditional entropy can be obtained whose minimum among projective measurements in a general qudit qubit state can be determined analytically in terms of the largest eigenvalue of a simple 3 3 correlation matrix such minimum determines the maximum conditional purity of a and the associated minimizing measurement is shown to be also universal in the vicinity of maximal mixedness results for x states including typical reduced states of spin pairs in xy chains at weak and strong transverse fields are also provided and indicate that the measurements minimizing the von neumann and linear conditional entropies are typically coincident in these states being determined essentially by the main correlation they can differ however substantially from that minimizing the geometric discord

optimal control of the convective velocity coefficient in a parabolic problem abstract we consider optimal control of the coefficient of the first spatial derivative in a parabolic equation for our convective velocity coefficient h x t we treat the case of one spatial dimension and the control depends on space and time control of such a coefficient requires more regularity on the control set than controlling the coefficient of a zeroth order term the existence of an optimal control is proved and the optimality system is derived we obtain the characterization of an optimal control through an elliptic partial differential equation thus the optimality system will consist of two parabolic equations coupled with an elliptic variational inequality

quasicanonical gibbs distribution and tsallis non extensive statistics abstract we derive and study quasicanonical gibbs distribution function which is characterized by the thermostat with finite number of particles quasithermostat we show that this naturally leads to tsallis non extensive statistics and thermodynamics with tsallis parameter q is found to be related to the number of particles in the quasithermostat we show that the chi square distribution of fluctuating temperature used recently by beck can be partially understood in terms of normal random momenta of particles in the quasithermostat also we discuss on the importance of the time scale hierarchy and fluctuating probability distribution functions in understanding of tsallis distribution within the framework of kinetics of dilute gas and weakly inhomogeneous systems

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

bipartite separability of symmetric n qubit noisy states using conditional quantum relative tsallis entropy abstract in any bipartition of a quantum state it is proved that the negative values of the conditional version of sandwiched tsallis relative entropy necessarily imply quantum entanglement for any n the separability ranges in the 1 n 1 partition of symmetric one parameter families of noisy n qubit w ghz w w states are determined using the conditional quantum relative tsallis entropy approach the 1 n 1 separability range matches exactly with the range obtained through positive partial transpose criterion for all n the advantages of using non commuting version of q conditional relative tsallis entropy are brought out through this and other one parameter families of states

when non extensive entropy becomes extensive abstract tsallis non extensive entropy s q enables us to treat both power and exponential evolutions of underlying microscopic dynamics on an equal footing by adjusting the variable entropic index q to proper one q we propose an alternative constraint of obtaining the proper entropic index q that the non additive conditional entropy becomes additive if and only if q q in spite of the fact that the associated system cannot be decomposed into statistically independent subsystems long range time correlation expressed by the q exponential function is discussed based on the nature that the q exponential function cannot be factorized into independent factors when q 1

conditional entropies and their relation to entanglement criteria we discuss conditional renyi and tsallis entropies for bipartite quantum systems of finite dimension we investigate the relation between the positivity of conditional entropies and entanglement properties it is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion moreover we show that the entanglement of werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

quantum q divergence the quantum q divergence is the nonadditive relative entropy associated with the tsallis entropy indexed by q here its basic properties including the q deformation and quantum group structure convexity and monotonicity are summarized then it is applied to measuring the degree of state purification it is shown that the quantum q divergence is always well defined if the entropic index q is in the range 0 1 whereas the ordinary quantum relative entropy is singular when the reference state is a pure state thus the additive limit q 1 0 does not exist in general

generalization of shannon s theorem for tsallis entropy by using the assumptions that the entropy must i be a continuous function of the probabilities pi pi 0 1 i only ii be a monotonic increasing function of the number of states w in the case of equiprobability iii satisfy the pseudoadditivity relation sq a b k sq a k sq b k 1 q sq a sq b k2 a and b being two independent systems q r and k a positive constant and iv satisfy the relation sq pi sq pl pm pl qsq pi pl pm qsq pi pm where pl pm 1 pl i 1wlpi and pm i wl 1wpi we prove along shannon s lines that the unique function that satisfies all these properties is the generalized tsallis entropy sq k 1 i 1wpiq q 1

tsallis entropy for assessing quantum correlation with bell type inequalities in epr experiment abstract a new bell type inequality is derived through the use of the tsallis entropy to quantify the dependence between the classical outcomes of measurements performed on a bipartite quantum system as typical of an epr experiment this new inequality is confronted with standard correlation based bell inequalities and with other known bell type inequalities based on the shannon entropy for which it constitutes a generalization for an optimal range of the tsallis order the new inequality is able to detect nonlocal quantum correlation with measurements from a larger set of quantum observables in this respect it is more powerful and also complementary compared to the previously known bell type inequalities

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators

a new nonlinear quantizer for image processing within nonextensive statistics abstract in this study we introduce a new nonlinear quantizer for image processing by using tsallis entropy lloyd max quantizer is commonly used in minimizing the quantization errors we report that the new introduced technique works better than lloyd max one for selected standard images and could be an alternative way to minimize the quantization errors for image processing we therefore hopefully expect that the new quantizer could be a useful tool for all the remaining process after image quantization such as coding lossy and lossless compression

information theoretical properties of tsallis entropies a chain rule and a subadditivity for the entropy of type which is one of the nonadditive entropies were derived by daroczy in this paper we study the further relations among tsallis type entropies which are typical nonadditive entropies the chain rule is generalized by showing it for tsallis relative entropy and the nonadditive entropy we show some inequalities related to tsallis entropies especially the strong subadditivity for tsallis type entropies and the subadditivity for the nonadditive entropies the subadditivity and the strong subadditivity naturally lead to define tsallis mutual entropy and tsallis conditional mutual entropy respectively and then we show again chain rules for tsallis mutual entropies we give properties of entropic distances in terms of tsallis entropies finally we show parametrically extended results based on information theory