You are an expert at summarizing long articles. Proceed to summarize the following text: 
the degree of understanding of quantum correlations ( qc ) underlies our current picture of nature @xcite . it has been recently found that there exist important manifestations of the quantumness of correlations in composite systems that are different from those of entanglement - origin ( eo ) and that may be relevant in quantum information technologies @xcite . the quantifier of these non - eo correlations is called the quantum discord ( qd ) @xmath1 and is based , for a bipartite system , on shannon s mutual information . we are thus speaking of an information - theoretic ( it ) tool . for pure states qd does not add any qcs , but that is not the case for mixed states . the @xmath6concept , advanced in the pioneer paper by ollivier and zurek @xcite , quantifies : i ) the minimum change in the state of the system and ii ) the information on one of its parts induced by a measurement of the other one . if the state has only classical correlations , @xmath1 vanishes , which implies that the quantum discord concept somehow quantifies the `` correlational - quantumness '' . it has been evaluated for several families of states both in its original form and in variously altered versions and generalizations . a particularly compelling instance expresses the qd - notion in terms of conditional density operators @xcite . interesting operational qd - interpretations have also been advanced @xcite . evaluating qd requires a rather involved optimization procedure , analytical expressions being known in just a few instances @xcite . @xmath1 is built up as an entropic difference , the difference between a quantum entropic measure and its classical counterpart , which is derived from local measurements on one or both of the participant subsystems . its amount is a new feature @xmath7 of the quantum state @xmath8 , which in turn induces a @xmath6amount `` ordering '' for states of the form @xmath9 , for instance . now , it - tools come in many distinct varieties . given the immense body of literature that has been generated in the past two decades concerning physically motivated statistical formalisms based on generalizations of shannon s information measure ( see @xcite and references therein ) , it seems both natural and necessary to tackle the qd issues from this generalized angle , in the hope of gaining interesting insights , and , in particular , so as to establish the invariance or not of the discord - induced order under a change of the prevailing statistics , from shannon s to its many rivals .    to show that this is indeed a fruitful endeavor is the aim of the present paper , in which a generalization of the quantum discord _ concept _ , in the context of generalized statistics , will be advanced and the `` ordering - question '' answered . in section 2 we introduce our conceptual qd - extension , discussing its main properties in sect . next we present some results for general bipartite states , focusing attention on analytical results . we also perform numerical simulations for random bipartite states ( sect . finally , some conclusions are drawn in section 5 . tsallis power - law @xmath11-entropy was introduced in @xcite as an extension of the shannon entropy as follows @xcite    @xmath12    where the @xmath11-logarithm is defined by @xmath13 , @xmath14 is the probability distribution of the pertinent random variable @xmath15 , and @xmath11 is any nonnegative real number . tsallis s entropy converges to shannon s in the limit @xmath16 . @xmath17 plays a fundamental role in recent developments of statistical mechanics @xcite . the generalization has indeed received a lot of attention in the last years , with about 2000 papers containing interesting results and useful applications , many of them in the complex systems area @xcite but also in connection with a variegated family of quantum mechanical settings ( see , for example @xcite ) . in what follows we retrace the developments of ref . @xcite in a tsallis , @xmath18context . thus , just by setting @xmath19 we recover the ollivier - zurek quantities . we begin then with the mutual information , defined as    @xmath20    and the following chain rule holds @xcite :    @xmath21    where the conditional entropy reads @xmath22 the chain rule gives the relation between a conditional entropy and a joint entropy . using this relation we can define another , classically equivalent , expression for the mutual information    @xmath23    the @xmath24 difference is of the essence for ollivier - zureck goals , after expressing the two quantifiers in quantal fashion . let us then do the same with @xmath25 and @xmath26 . @xmath25 can be easily generalized defining appropriate density matrices for the quantum systems , @xmath27 , @xmath28 , and @xmath29 , and applying then the @xmath11-generalization of von neumann s entropy @xmath30 . one has    @xmath31    to generalize the @xmath32expression , following @xcite , we focus on a perfect measurements of @xmath28 defined by a set of projectors @xmath33 such that @xmath34 . accordingly ,    @xmath35    where    @xmath36    with the state of @xmath8 given , once the measurement is performed , by    @xmath37    and @xmath38 .    the two classical expressions for the standard mutual information we have presented above are identical , but this is not necessarily so in the quantum case actually , the quantum discord is defined as the _ minimum _ possible difference between the two possibilities , given by an _ optimum _ set of @xmath33 @xcite . thus , we are to be concerned here with what happens to the expressions @xmath25 and @xmath26 . introduce the quantity    @xmath39    we define now our quantum q - discord as the difference    @xmath40    this quantum q - discord is the minimum of the difference between eq . ( [ imutua ] ) and eq . ( [ jmutua ] ) . we normalize this measure via a trivial re - scaling in order to compare , in an adequate way , different quantities :    @xmath41    for similar reasons , @xmath42 denotes the logarithm of base 2 . in what follows @xmath43 . note that @xmath44 . we see that @xmath45 for @xmath46 , and this might be related to the concavity of the @xmath11-conditional entropy @xmath47 with respect to @xmath29 . indeed , @xmath48 becomes negative if @xmath11 grows from @xmath49 , negativity increasing with @xmath11 . quite convenient is the particular case @xmath50 , since it requires only to compute the traces of @xmath51 , and matrix - diagonalization is avoided , making computations more efficient . taking the limit @xmath52 of the normalized measure we obtain    @xmath53    to study the positivity of the q - discord we consider separately two cases : pure states and mixed states .      for pure states , that is @xmath54 , the q - discord takes the form @xmath55 and the quantum q - discord coincides with the reduced ( quantum ) tsallis entropy . we can easily verify this fact by casting @xmath56 in its schmidt decomposition form @xmath57 . thus , since the q - entropy is positive for all @xmath11 ,    @xmath58      in the case of mixed states our q - discord is positive only for values of @xmath11 in @xmath59 . in order to demonstrate the positivity of the q - discord for mixed arbitrary states we follow @xcite and consider the proposition : @xmath60 with @xmath61 . now , @xmath62 is block diagonal and as in the @xmath19 case , doing a block by block analysis the proposition can be proved . now we need to verify the inequality :    @xmath63    by recourse of the previous preposition we can establish the following relation for any measurement @xmath33    @xmath64    and by the ( conjectured ) concavity ( see below ) of the conditional entropy ( @xmath65 ) with respect to @xmath66 for @xmath46 we are led to    @xmath67      the set of states in an @xmath69-dimensional hilbert space can be regarded as a product - space of the form @xcite ,    @xmath70 where @xmath71 stands for the family of all complete sets of ortho - normal projectors @xmath72 ( @xmath73 the identity matrix ) , and @xmath74 is the convex set of all real @xmath68tuples of the form @xmath75 any state in @xmath76 takes the form @xmath77 in order to explore @xmath78 we introduce an appropriate measure @xmath79 on this space . such a measure is required to compute volumes within @xmath78 , as well as to determine what is to be understood by a uniform distribution of states on @xmath80 . an arbitrary state @xmath66 of our @xmath69-dimensional hilbert space can always be expressed as a product of the form    @xmath81 u^{\dagger}.\ ] ]    here @xmath82 is an @xmath83 unitary matrix and @xmath84 $ ] is an @xmath83 diagonal matrix whose diagonal elements are , precisely , our above defined @xmath85 . the group of unitary matrices @xmath86 is endowed with a unique , uniform measure , known as the haar s measure , @xmath87 @xcite . on the other hand , the @xmath69-simplex @xmath74 , consisting of all the real @xmath69-uples @xmath88 appearing in ( [ udot ] ) , is a subset of a @xmath89-dimensional hyperplane of @xmath90 . consequently , the standard normalized lebesgue measure @xmath91 on @xmath92 provides a measure for @xmath74 . the aforementioned measures on @xmath86 and @xmath74 lead to a measure @xmath79 on the set @xmath93 of all the states of our quantum system @xcite ,    @xmath94    if one needs to randomly generate mixed states , this is to be done according to the measure ( [ memu ] ) .      here we attempt a numerical verification of the concavity of the conditional entropy for @xmath46 , that is ,    @xmath96+(1-t)[s_q(\xi)-s_q(\xi_a)]\}\geq   0,\ ] ]    where @xmath97 , @xmath27 is the reduced density matrix corresponding to @xmath66 , @xmath98 ( @xmath99 ) the reduced density matrices of @xmath100 ( @xmath101 ) and , finally , @xmath102 .     for different values of @xmath103 for which the concavity is verified . inset : @xmath104 for different @xmath105 . the curves are left- shifted . all curves were constructed using of the order of @xmath106 ( numerically generated ) states . all depicted quantities are dimensionless.[pdfs ] ]    the concavity of the standard conditional entropy ( @xmath19 ) was proved in @xcite by assuming the validity of a lemma by lieb @xcite . the proof is rather difficult even in this case @xcite . as a first step we evaluate numerically the inequality ( [ deltaq ] ) by generating random states in an @xmath69-dimensional hilbert space . in order to assess , for these randomly generated states , how the concavity - requirement is satisfied , we evaluate ( [ deltaq ] ) for a large enough number of simulated states ( @xmath100 and @xmath101 ) . we set @xmath107 for the dimension of the state - space in all simulations and we randomly generate @xmath108 $ ] .    we investigate the positivity of @xmath109 , upon which the concavity of the conditional q - entropy is based , by constructing the probability distributions for the values of @xmath110 . the corresponding distributions , for different values of @xmath11 , are depicted in fig . [ pdfs ] . in the inset we plot the probability distribution of @xmath109 for @xmath50 and @xmath111 . the curves are constructed using of the order of @xmath106 states . these simulations provide us with strong evidence about the validity of the conjecture advanced above on the concavity of the conditional q - entropy and , consequently , on the positivity of the quantum q - discord for @xmath112 . let us now investigate the relation between the q - discord and its original counterpart for different sorts of states . we begin with bell diagonal states . these are two - qubit states with maximally - mixed reduced - density matrices and have the form    @xmath113    where @xmath114 are real constants constrained by certain conditions ( in order to have a well defined density operator @xmath66 ) and @xmath115 s are the pauli operators . let @xmath116 $ ] , @xmath117 be the eigenvalues of @xmath66    @xmath118    the marginal states of @xmath66 are @xmath119 and @xmath120 . thus , the quantum q - mutual information of @xmath66 is    @xmath121    and    @xmath122,\ ] ]    where @xmath123 . we find , for a general ( bell - diagonal ) two - qubit state ,    @xmath124+\sum_i\lambda_i^q\ln_q\lambda_i.\ ] ]    let us specialize ( [ ana1 ] ) to the particular instance @xmath125 , i.e. , the celebrated werner states ,    @xmath126,\ ] ]    with @xmath127 . by following @xcite one easily obtains    @xmath128\nonumber\\&+&3\left(\frac{1-c}{4}\right)^q\ln_q\frac{1-c}{4}+\left(\frac{1 + 3c}{4}\right)^q\ln_q\frac{1 + 3c}{4},\end{aligned}\ ] ]    and , as seen in fig . [ wernerstates ] , positivity prevails for the prototype - mixed state . in fig . [ wernerstates ] we plot the ( normalized ) q - discord as a function of the state - parameter @xmath129 for different values of @xmath11 and also as a function of the mixedness - degree as given by the linear entropy @xmath130,\ ] ] trivially related to the purity @xmath131 of a state via @xmath132 as expected , an inverse relationship between mixedness - degree and quantum correlations is displayed . we remark on the single - valuedness of the werner - relation between q - discord and mixedness , even for @xmath19     for werner state , as a function of @xmath129 , for different values of @xmath11 . ( b ) @xmath2 for werner states , as a function of the degree of mixedness measured by the linear entropy , for different values of @xmath11 . all depicted quantities are dimensionless.[wernerstates],title="fig : " ]   for werner state , as a function of @xmath129 , for different values of @xmath11 . ( b ) @xmath2 for werner states , as a function of the degree of mixedness measured by the linear entropy , for different values of @xmath11 . all depicted quantities are dimensionless.[wernerstates],title="fig : " ]      we also will study the quantum q - discord for the following one - parameter states     for @xmath133 states , as a function of @xmath133 , for different values of @xmath11 . ( b ) @xmath2 for @xmath133 states , as a function of the degree of mixedness , for different values of @xmath11 . all depicted quantities are dimensionless.[alphastates],title="fig : " ]   for @xmath133 states , as a function of @xmath133 , for different values of @xmath11 . ( b ) @xmath2 for @xmath133 states , as a function of the degree of mixedness , for different values of @xmath11 . all depicted quantities are dimensionless.[alphastates],title="fig : " ]    @xmath134    where @xmath135 . let @xmath136 . the q - discord becomes    @xmath137 +   2\left(\frac{1-\alpha}{2}\right)^q\ln_q\frac{1-\alpha}{2}+\alpha^q\ln_q\alpha.\ ] ]    in fig . [ alphastates ] we depict the quantum q - discord as a function of the state s parameters for different values of @xmath11 and also plot it as a function of the linear entropy . positivity again prevails . the single - valuedness between discord and mixedness is lost for these states .       of two @xmath133 states as a function of the parameter @xmath11 . a non trivial ordering relation is found . all depicted quantities are dimensionless.[alphasigno ] ]    in fig . [ alphasigno ] we display the difference between the q - discord of two @xmath133-states ( corresponding to @xmath139 and @xmath140 , respectively ) , as a function of @xmath11 . this difference takes negative or positive values depending on the range of @xmath11 . this is indeed a novel feature . a relation of order for quantum states based on the discord - concept can not univocally be established , because it depends on the entropic quantifier one chooses to employ . in other words , the quantal correlations that the discord quantifies are seen in different manners by distinct entropic quantifiers . _ this lack of uniqueness is the leit - motif of the present considerations_.      as a last particular kind of special state to be analyzed , consider the two - parameters state    @xmath142    where @xmath135 and @xmath143 . we display the @xmath144discord versus the @xmath145discord for this state in fig . [ abstate ] . a strong correlation is exhibited between the two @xmath18measures . this could be taken as evidence that changing @xmath11 from its original @xmath145value does not per se modify the overall manner in which q - discord quantifies quantum correlations . ( q=2 ) vs @xmath1 for @xmath146-like states . all depicted quantities are dimensionless.[abstate ] ]      here , we focus our discussion on general ( pure or mixed ) states of two qubits . for such system we can parametrize the basis of the measurement by @xmath147 and @xmath148 ,    latexmath:[\[\begin{aligned }   |\psi\rangle&=&\cos(\theta)|0\rangle+e^{i\phi}\sin(\theta)|1\rangle\\\nonumber      as a function of @xmath1 for randomly generated two - qubit states ( a ) @xmath150 , ( b ) @xmath50 . all depicted quantities are dimensionless.[qdranstates],title="fig : " ]   as a function of @xmath1 for randomly generated two - qubit states ( a ) @xmath150 , ( b ) @xmath50 . all depicted quantities are dimensionless.[qdranstates],title="fig : " ]    we numerically search the @xmath147 - @xmath148 space for the set of values that maximize eq . ( [ optimization ] ) . the resultant density operator @xmath151 , when such measurements are performed on subsystem @xmath152 , is    @xmath153    where each complete set , composed of two elements , of possible measurements is defined as follows ,    @xmath154    . the curves were constructed using @xmath155 generated random states . all depicted quantities are dimensionless . [ qdpdf ] ]    we randomly generate states uniformly distributed according to the measure @xmath156 and by recourse of the previously described optimization procedure . of course , we numerically search for @xmath147 and @xmath148 and compute the q - discord for these states . in fig . [ qdranstates ] we display the correlation between the discord and the q - discord for different values of @xmath11 . negative values of the q - discord are depicted in the plot for the case @xmath105 ( @xmath50 ) . figure [ qdpdf ] depicts the probability distribution of finding a given value of q - discord in the whole space of two - qubit states for @xmath157 . we also compute the difference between the q - discord and the discord between pairs of randomly generated states @xmath66 and @xmath100 . in fig . [ qddif ] we plot the resultant differences of the q - discord versus similar differences ( for the same pair of states ) corresponding to 1-discord . this plot depicts the pertinent results . overall , our numerical simulations confirm the conclusions reached by the analysis of special kinds of states .     as a function of @xmath158 ( a ) @xmath150 , ( b ) @xmath50 . all depicted quantities are dimensionless . [ qddif],title="fig : " ]   as a function of @xmath158 ( a ) @xmath150 , ( b ) @xmath50 . all depicted quantities are dimensionless . [ qddif],title="fig : " ] we have introduced a new family of quantum discord measures that quantifies quantum correlations based in the chain rule relating the i ) conditional- and ii ) joint - tsallis entropies . via two types of study        1 . there is a strong correlation between the `` new '' q - discord and the original one of ollivier and zureck . 2 .   however , an order - relation for quantum states based on discord lacks unicity because it definitely depends on the quantifier one chooses to employ . this means that @xmath11-discord functionals corresponding to different values of @xmath11 measure different aspects of the non - classicality ( quantumness ) of correlations . this work was partially supported by the projects fqm-2445 and fqm-4643 of the junta de andalucia and the grant fis2011 - 24540 of the ministerio de innovacin y ciencia ( spain ) . a.p.m . acknowledges support by genil through ytr - genil program . acknowledges support from the senior grant cei bio - tic genil - spr . l. amico , l. fazio , a. osterloh , v. vedral , rev . mod . phys . * 80 * ( 2008 ) 517 . i. bengtsson , k. zyczkowski , geometry of quantum states : an introduction to quantum entanglement , cambridge univ . press , cambridge , 2006 . v. vedral , phys . lett * 90 * ( 2003 ) 050401 . j. maziero , l.c . celri , r.m . serra , v. vedral , phys . a * 80 * ( 2009 ) 044102 . j. oppenheim , m. horodecki , p. horodecki , r. horodecki , phys . lett * 89 * ( 2002 ) 180402 . m. ali , a.r.p . rau , g. alber , phys . a * 81 * ( 2010 ) 042105 . s. furuichi , j. math . phys . * 47 * ( 2006 ) 023302 . a. wehrl , rev . * 50 * ( 1978 ) 221 . e. lieb , adv . math . * 11 * ( 1973 ) 267 . k. zyczkowski , p. horodecki , a. sanpera and m. lewenstein , phys . a * 58 * ( 1998 ) 883 .
the notion of quantum discord introduced by ollivier and zurek [ phys . 
 rev . 
 lett * 88 * , 017901 ( 2001 ) ] ( see also henderson and vedral [ j. phys . a * 34 * , 6899 ( 2001 ) ] ) has attracted increasing attention , in recent years , as an entropic quantifier of non - classical features pertaining to the correlations exhibited by bipartite quantum systems . here 
 we generalize the notion so as to encompass power - law q - entropies ( that reduce to the standard shannon entropy in the limit @xmath0 ) and study the concomitant consequences . 
 the ensuing , new discord - like measures we advance describe aspects of non - classicality that are different from those associated with the standard quantum discord . 
 a particular manifestation of this difference concerns a feature related to order . 
 let @xmath1 stand for the standard , shannon - based discord measure and @xmath2 for the @xmath3 one . 
 if two quantum states @xmath4 are such that @xmath5 , this order - relation does not remain invariant under a change from @xmath1 to @xmath2 .