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Quantum Correlation with Sandwiched Relative Entropies: Advantageous as Order Parameter in Quantum Phase Transitions

Quantum discord is a measure of quantum correlations beyond the entanglement-separability paradigm. It is conceptualized by using the von Neumann entropy as a measure of disorder. We introduce a class of quantum correlation measures as differences between total and classical correlations, in a shared quantum state, in terms of the sandwiched relative R\'{e}nyi and Tsallis entropies. We compare our results with those obtained by using the traditional relative entropies. We find that the measures satisfy all the plausible axioms for quantum correlations. We evaluate the measures for shared pure as well as paradigmatic classes of mixed states. We show that the measures can faithfully detect the quantum critical point in the transverse quantum Ising model and find that they can be used to remove an unquieting feature of nearest-neighbor quantum discord in this respect. Furthermore, the measures provide better finite-size scaling exponents of the quantum critical point than the ones for other known order parameters, including entanglement and information-theoretic measures of quantum correlations.


INTRODUCTION
Characterization and quantification of quantum correlation [1,2] play a central role in quantum information. Entanglement, in particular, has been successfully identified as a useful resource for different quantum communication protocols [3] and computational tasks [4]. Moreover, it has also been employed to study cooperative quantum phenomena like quantum phase transitions in many-body systems [5,6]. However, in the recent past, several quantum phenomena of shared systems have been discovered in which entanglement is either absent or does not play any significant role. Locally indistinguishable orthogonal product states [7] (c.f. [8]), and the model of deterministic quantum computation with one quantum bit [9][10][11] are prominent examples where entanglement does not play an important role. Such phenomena motivated the search for concepts and measures of quantum correlation independent of the entanglement-separability paradigm. Introduction of quantum discord [12,13] is one of the most important advancements in this direction and has inspired a lot of research activity [2]. It has thereby emerged that quantum correlations, independent of entanglement, can also be a useful ingredient in several quantum information processing tasks [2]. Other measures in the same direction include quantum work deficit [14], measurement-induced nonlocality [15], and quantum deficit [16]. These measures can be generally considered to be quantum correlation measures within an "information-theoretic paradigm".
In classical as well as quantum information theory, one of the most important pillars is the framework of entropy [17], which quantifies the ignorance or lack of information in the relevant physical system. Moreover, it helps to understand information theory from a thermodynamic perspective. Almost all the quantum correlation measures incorporate entropic functions in various forms. And, most of the quantum correlation measures are defined by using the von Neumann entropy [18]. The operational significance of von Neumann entropy has been widely recognized in numerous scenarios in quantum in-formation theory. Nonetheless, there are classes of generalized entropies like the Rényi [19] and Tsallis [20] entropies, which are also operationally significant in important physical scenarios. Both the Rényi and Tsallis entropies reduce to the von Neumann entropy when the entropic parameter α → 1. For α ∈ (0, 1), the relative Rényi entropy appears in the quantum Chernoff bound which determines the minimal probability of error in discriminating two different quantum states in the setting of asymptotically many copies [21]. In Ref. [22], it was shown that the relative Rényi entropy is relevant in binary quantum state discrimination, for the same range of α. The concept of Rényi entropy has also been found to be useful in the context of holographic theory [23]. It has also been found useful in dealing with several condensed matter systems [24]. The significance of the Tsallis entropy in quantum information theory has been established in the context of quantifying entanglement [25], local realism [26], and entropic uncertainty relations [27] (see also [28]). Both the Rényi and Tsallis entropies have important applications in classical as well as quantum statistical mechanics and thermodynamics [29].
While there are important interpretational and operational breakthroughs that have been obtained by using the concept of quantum discord, there are also several intriguing unanswered questions and thriving controversies [2,30]. It is therefore interesting and important to look back upon the conceptual foundations of quantum discord and inquire whether certain changes, subtle or substantial, in those concepts lead us to a better understanding of quantum correlations. Towards this aim, we introduce measures of the total, classical, and quantum correlations of a bipartite quantum state in terms of the entire class of relative Rényi and Tsallis entropy distances. We show that the measures satisfy all the required properties of bipartite correlations. We then evaluate the quantum correlation measure for several paradigmatic classes of states. As an application, we find that the quantum correlation measures, via relative Rényi and Tsallis entropies, can indicate quantum phase transitions and give better finite-size scaling exponents than the other known order parameters.

arXiv:1406.5065v1 [quant-ph] 19 Jun 2014
There are at least two distinct ways in which the relative Rényi and Tsallis entropies are defined, and are usually referred to as the "traditional" [31] and "sandwiched" [32,33] varieties. The sandwiched varieties incorporate the noncommutative nature of density matrices in an elegant way, and it is therefore natural to expect that it will play an important role in fundamentals and applications. Indeed, the sandwiched relative Rényi entropy has been used to show that the strong converse theorem for the classical capacity of a quantum channel holds for some specific channels [32]. Moreover, an operational interpretation of the sandwiched relative Rényi entropy in the strong converse problem of quantum hypothesis testing is noted for α > 1 [34]. On the other hand, the sandwiched relative Tsallis entropy has recently been shown to be a better witness of entanglement [35] than the traditional one [25]. The relative minand max-entropies [36][37][38], which can be obtained from the sandwiched relative Rényi entropy for specific choices of α, play significant roles in providing bounds on errors of one-shot entanglement cost [39], on the one-shot classical capacity of certain quantum channels [40], and in several scenarios in non-asymptotic quantum information theory [41].
The paper is organized as follows. In Sec. II, we discuss the relative Rényi and Tsallis entropies. In Sec. III, we talk about the usual quantum discord. The Rényi and Tsallis quantum correlations are defined in Sec. IV, where we also derive their properties and evaluate them for paradigmatic classes of bipartite quantum states. Some special cases like the "linear", "min-", and "max-" quantum discord are also formulated and discussed there. The quantum correlation measure is then applied for detecting quantum phase transition in a quantum many-body system in Sec. V. We present a conclusion in Sec. VI.

II. RELATIVE RÉNYI AND TSALLIS ENTROPIES
The Rényi [19,42] and Tsallis [20,43] entropies of a density operator ρ are given respectively by Here, the parameter α ∈ (0, 1)∪(1, ∞), unless mentioned otherwise. All logarithms in this paper are with base 2. Both the entropies reduce to the von Neumann entropy [18], S(ρ) = −tr(ρ log ρ), when α → 1. The Tsallis entropy for α = 2 is called the linear entropy, S L (ρ), given by The traditional quantum relative Rényi entropy between two density operators ρ and σ is defined as Note that all the quantum relative entropies, traditional or sandwiched, discussed in this paper, are defined to be +∞ if the kernel of σ has non-trivial intersection with the support of ρ, and is finite otherwise. S R α (ρ||σ) reduces to the usual quantum relative entropy [44], S(ρ||σ), when α → 1, where Recently, a generalized version of the quantum relative Rényi entropy (called "sandwiched" relative Rényi entropy) has been introduced, by considering the non commutative nature of density operators [32,33]. It is defined asS Note thatS R α (ρ||σ) also reduces to S(ρ||σ) when α → 1. In Ref. [32,33,[45][46][47] several interesting properties of the sandwiched Rényi entropy have been established. Here, we mention some of them (for two density operators ρ and σ) which we will use later in this paper. 4.S R α (ρ||σ) is invariant under all unitaries U , i.e., S R α (U ρU † ||U σU † ) =S R α (ρ||σ) . The traditional quantum relative Tsallis entropy between two density operators ρ and σ is defined as The sandwiched relative Tsallis entropy between two density operators ρ and σ is given by [35] S T α (ρ||σ) = tr σ BothS T α (ρ||σ) andS T α (ρ||σ) also reduce to S(ρ||σ) when α → 1. It can be easily verified that the properties (1-4), satisfied byS R α (ρ||σ) are also satisfied byS T α (ρ||σ). In this paper, we will predominantly use the sandwiched version of both the relative entropies. Hereafter, by relative entropy, we will mean the sandwiched form of the relative entropies, unless mentioned otherwise. Some of the important special cases of the Rényi and Tsallis relative entropies are given below. a. Relative Linear Entropy: At α = 2,S T α (ρ||σ) gives the relative linear entropy, The relative linear entropy has also been defined in the literature by using the traditional version of the relative entropy at α = 2. However, in this paper, we will use the relative linear entropy defined only through the sandwiched relative entropy (at α = 2). b. Relative Collision Entropy: At α = 2,S R α (ρ||σ) has been called the relative collision entropy [36], c. Relative Min-and Max-Entropies: In [48], it is pointed out that at α = 1 2 ,S R α (ρ||σ) gives relative minentropy [38], Note that where σ| is the fidelity between the states ρ and σ. It is shown in [33], that the relative max-entropy [37] is nothing but relative Rényi entropy, when α → ∞ i.e. where III. QUANTUM DISCORD Quantum discord is a measure of quantum correlations of bipartite quantum states that is independent of the entanglement-separability paradigm [12,13]. It can be conceptualized from several perspectives. An approach that is intuitively satisfying, is to define it as the difference between the total correlation and the classical correlation for a bipartite quantum state ρ AB . The total correlation is defined as the quantum mutual information of ρ AB , which is given by where ρ A and ρ B are the local density matrices of ρ AB . The mutual information I(ρ AB ) can also be expressed in terms of the usual quantum relative entropy as This follows from the fact that and the non-negativity of relative von Neumann entropy between two density matrices. Therefore, the quantum mutual information is the minimum usual relative entropy distance of the state ρ AB from the set of all completely uncorrelated states, σ A ⊗ σ B , whence we obtain a ground for interpreting the quantum mutual information as the total correlation in the state. Further evidence in this direction is provided in [49][50][51]. The classical correlation is given in terms of the measured conditional entropy, and is defined as [12,13] where is the conditional entropy of ρ AB , conditioned on measurements at B with rank-one projection-valued mea- where I A is the identity operator on the Hilbert space of A. J (ρ AB ) can also be defined in terms of the mutual information as where The classical correlation can therefore be seen as the minimum relative entropy distance of the state ρ AB from all uncorrelated states, maximized over all rank-one projective measurements on B, and is given by The maximization in Eq. (21) or in Eq. (17) ensure that J (ρ AB ) quantifies the maximal content of classical correlation present in the bipartite state ρ AB . Hence, if we subtract J (ρ AB ) from the total correlation, the remaining correlation is "purely" quantum, and is defined as [12,13]

IV. TOTAL, CLASSICAL, AND QUANTUM CORRELATIONS AS RELATIVE ENTROPIES
In this section, we define the total, classical, and quantum correlation in terms of the sandwiched relative Rényi and Tsallis entropies. We discuss the properties of these measures and evaluate them for several important families of bipartite quantum states. In the final subsection, we also compare the results with those obtained with traditional relative entropies.

A. Generalized Mutual Information as Total Correlation
We define the generalized mutual information of ρ AB as Here, the minimum is taken over all density matrices, σ A and σ B . The relative entropy, although not a metric on the operator space, is a measure of the distance between two quantum states.S Γ α (ρ AB ||σ A ⊗ σ B ) is a distance between the quantum state ρ AB and a completely uncorrelated state σ A ⊗ σ B . Here, and hereafter, the superscript Γ is either R or T , depending on whether it is the Rényi or Tsallis variety that is considered. The corresponding minimum distance can be interpreted as the total correlation present in the system. The generalized mutual information I Γ α (ρ AB ) becomes equal to the usual quantum mutual information I(ρ AB ) when α → 1:

B. Classical and Quantum Correlation
The Rényi or Tsallis version of the classical correlation, denoted by J Γ α (ρ AB ), is defined as where ρ AB is obtained by performing rank-1 projective measurements as in the definition of original classical correlation (in Eq. (20)). Therefore, quantum correlation using generalized entropies is defined as with α ∈ [ 1 2 , 1) ∪ (1, ∞). By using the data processing inequality, which holds in this range of α, one can prove the non-negativity of the quantum correlation [45]. We now look into the properties of D Γ α (ρ AB ), which provide independent support for identifying the quantities as correlation measures.
The proof for the vanishing of total correlations follows by noting that the product state in the argument itself is the state which gives the optimal relative entropy distance. A similar argument, but for the measured state, holds for the classical correlation. Property 3. I Γ α , J Γ α remain invariant under local unitaries, which follow from the fact thatS R α (ρ||σ) is invariant under all unitaries U . Hence, D Γ α is also invariant under local unitaries. Property 4. I Γ α , J Γ α are non increasing under local operations, which follow from the data processing inequality,S R α (ρ||σ) ≥S R α (E(ρ)||E(σ)), for any CPTPM E. Property 5. D Γ α is non-negative, as J Γ α is upper bounded by I Γ α . The latter statement is due to the fact that J Γ α is obtained by performing a local measurement on ρ AB , and we know from the data processing inequality thatS Γ α is monotone under CPTPM. The classical correlation measure that we have defined here, satisfies all the plausible properties for classical correlation proposed in Ref. [12], except the one which states that for pure states, the classical correlation reduces to the von Neumann entropy of the subsystems. We wish to mention that this property is natural for the measure which involves the von Neumann entropy, and is not expected to be followed by the measures with generalized entropies. This is because the definition of classical correlation in terms of the relative entropy reduces naturally to the one in terms of the conditional entropy in the case of the von Neumann entropy.
We use the convention that each of the definitions of I Γ α , J Γ α and D Γ α also incorporates a division by log 2 bits, whence all the definitions can be considered to be dimensionless.
We note here that there has been previous attempts to define quantum discord by using Tsallis entropies [52][53][54]. These definitions however do not always guarantee positivity of the quantum discord, so defined. Also, the corresponding total and classical correlations are not necessarily monotonic under local operations. Further works in Refs. [55] and [56] define quantum correlations by using Tsallis and Rényi entropies respectively, whose approaches however are different from ours.

Linear Quantum Discord
The relative linear entropy can be used to define the "linear quantum discord", given by where I T 2 (ρ AB ) and J T 2 (ρ AB ) are defined by using the relative linear entropy, given in Eq. (9).

Min-and Max-Quantum Discords
We also define the "min-and max-quantum discords" by considering relative min-and max-entropies as and

D. Pure States
Any bipartite pure state of two qubits can be written, using Schmidt decomposition, as where λ i are non-negative real numbers satisfying i λ i = 1. Since a bipartite pure state is symmetric, it is plausible that the state σ A ⊗ σ B , which minimizes the relative entropy of |ψ AB with uncorrelated states, is also symmetric. Numerical studies support this view. Moreover, numerical results indicate that for arbitrary |ψ AB , the state σ A ⊗ σ B which gives the minimum, is diagonal in the Schmidt basis of |ψ AB . Therefore, the minimum σ A or σ B is given by where a i are non-negative real numbers satisfying i a i = 1. With these assumptions, the total correlation of |ψ AB is given by where a 0 = a, a 1 = 1 − a, λ 0 = λ, λ 1 = 1 − λ. The value of a is obtained from the condition for α ∈ (2/3, 1) ∪ (1, ∞). For 1 2 ≤ α ≤ 2 3 , the minimization in Eq. (32) yields For pure states, numerical searches indicate that the classical correlation is independent of the measurement basis. We consider measurement performed in the Schmidt basis for calculating the classical correlation of the original state. Just like the total correlation in the original state, the σ A ⊗ σ B , which minimizes the relative entropy of the post-measurement state with uncorrelated states, is symmetric, since we perform the projective measurement in the Schmidt basis. Moreover, from numerical results, we find that σ A ⊗ σ B is again diagonal in the Schmidt basis of |ψ AB . The Rényi classical correlation of |ψ AB is therefore given by The value of a is obtained from the condition The linear quantum discord for |ψ AB is given by The min-quantum discord is vanishing for every twoqubit pure state. The max-quantum discord for |ψ AB is given by In Fig. 1, we plot the Rényi quantum correlation of |ψ AB for various values of α.
where |ψ − = 1 √ 2 (|01 − |10 ), I denotes the identity operator on the two-qubit Hilbert space, and 0 ≤ p ≤ 1. Suppose the σ min A and σ min B are the optimal σ A and σ B which minimizes the relative Rényi entropy of ρ W with uncorrelated states. Using the fact that the Werner state is symmetric and local unitarily invariant, we can take where a i are non-negative real numbers satisfying i a i = 1. It is now possible to perform the minimization for α ∈ [ 2 3 , 1) ∪ (1, ∞). In this range, the relative Rényi entropy distance corresponding to the total correlations is minimum for a 0 = a 1 = 1 2 . Therefore, the Rényi total correlation of the Werner state for α ≥ 2 3 (α = 1) is given by Just like for the case of pure bipartite states, the Rényi classical correlation is again independent of measurement basis, as is expected from the property of rotational invariance of the Werner state. Numerical observations also suggest that for α ≥ 1 2 (α = 1) and for any p, the relative Rényi entropy is minimum at σ A ⊗σ B = I 4 for the post-measurement state corresponding to the Werner state. So the Rényi classical correlation, in this range of α, is given by Hence, the Rényi quantum correlation of the Werner state for α ≥ 2 3 (α = 1) is given by For 1 2 ≤ α < 2 3 , we find the Rényi quantum correlation for the Werner states by numerical evaluation. In Fig. 2, we exhibit the Rényi quantum correlation for the Werner states for different values of α.
The Rényi quantum correlation is maximum for the Werner state at p = 1 for α ≥ 2 3 . The singlet state, and states that are local unitarily connected with it, is therefore maximally Rényi quantum correlated in that range of α, among the Werner states. However, for 1 2 ≤ α < 2 3 , the Bell states are not the maximally Rényi quantum correlated states. In this range of α, we get maximal quantum correlation among the Werner states, for a value of p that is different from unity. For example, for α = 0.6, we find that the state, ρ W , with mixing parameter p ≈ 0.96 has the maximal quantum correlation among all Werner states. For α = 1/2, the same is at p ≈ 0.88. For α = 1 2 , i.e., for min-entropy, the singlet has zero quantum correlation. Indeed, all pure states have vanishing min-quantum discord. We will visit this issue again in Sec. IV F.
The linear quantum discord for the Werner state is The max-quantum discord can also be calculated similarly for the Werner state and is given by We have numerically evaluated the min-quantum discord for the Werner state (see Fig. 2).
(ii) Bell Mixture: We consider a mixture of two Bell states, given by for α ≥ 2 3 (α = 1). Hence, in this range of α, We have found numerically that if one performs measurement in the {|0 , |1 } basis, the relative entropy of the post-measurement state with I 4 gives the Rényi classical correlation for the entire range of α, i.e., for α ∈ ( 1 2 , 1) ∪ (1, ∞), and it is equal to unity for any p and α. Hence for α ≥ 2 3 (α = 1), The linear quantum discord for this state is given by Similarly, In Fig. 3, the Rényi quantum correlations for ρ BM is depicted for different values of α. (iii) Mixture of Bell state and a Product State: Consider the state given by The Rényi quantum correlation is calculated numerically, and in Fig. 4, we plot it for ρ BN , for different values of α.

F. Sandwiched vs Traditional Relative Entropies
Until now, in this section, we have used the sandwiched relative entropy distances to define the Rényi and Tsallis quantum correlations. We now briefly consider the traditional variety for defining quantum correlation, and discuss some of its implications. In the preceding subsections, we have observed anomalous behavior of the Rényi quantum correlation in the range 1 2 ≤ α < 2 3 for pure states, as well as in certain families of mixed states in the neighborhood of pure states. In these cases, we have, e.g., seen that the Bell states are not the maximally Rényi quantum correlated state for α < 2 3 and at α = 1 2 , i.e, for the min-entropy, all pure states have vanishing quantum correlations.
We can also define quantum correlations with the traditional relative Rényi and Tsallis entropies. The properties (1-4) discussed in Sec. II, are also followed by both the traditional relative entropies [57], but the data processing inequality holds for α ∈ [0, 1) ∪ (1, 2] [58]. We can therefore define quantum correlation with traditional relative entropy distances for this range of α. If we consider the traditional relative entropies, then we do not see any anomalous behavior of the Rényi quantum correlation. But from the traditional version of the relative Rényi entropy, we do not get the min-entropy. Moreover, in [34], the authors have argued that the sandwiched relative Rényi entropy is operationally relevant in the strong converse problem of quantum hypothesis testing for α > 1, but for α < 1, the traditional version is more relevant from an operational point of view. The anomalous behavior of the quantum correlation with the sandwiched relative entropy distances seems to indicate that to define quantum correlation for α < 1, the more appropriate candidates are the traditional relative entropies. Here we discuss about the traditional Rényi quantum correlation for two-qubit pure states and the Werner state. (i) Pure States: Numerical observations similar to the case with the sandwiched variety, give us that the total correlation of a two-qubit pure state, |ψ AB = 1 i=0 λ i |i A i B , for traditional relative Rényi entropy, with α ∈ ( 1 2 , 1), is given by where 0 ≤ a ≤ 1, and the value of a is obtained from the condition For the same range of α, the classical correlation for the traditional one is same as the sandwiched relative Rényi entropy. For α ∈ [0, 1 2 ], the traditional Rényi quantum correlation is independent of α and is given by In Fig. 5, we have plotted the D T R α (|ψ AB ), for different values of α. No anomalous behavior can be seen, and the maximally entangled states have maximal quantum correlations.
(ii) Werner States: Like in the sandwiched version, exploiting the rotational invariance and symmetry of the Werner state, it can be shown analytically that the total correlation of the Werner state for the traditional relative Rényi entropy, for α ∈ [ 1 2 , 1), is given by The classical correlation of the Werner state is also measurement basis independent for the traditional version, like the sandwiched one. We get that the classical correlation, in this range, is given by The forms of the total and classical correlations, in this case, are equivalent to those in the sandwiched version. But here, the range of α is different. Hence, for α ∈ [ 1 2 , 1), the traditional Rényi quantum correlation for the Werner state is given by For α < 1 2 , we have numerically evaluated the traditional Rényi quantum correlation for the Werner state and no anomalous behavior is noticed for α ∈ (0, 1). In Fig. 6, we have plotted the D T R α (ρ W ), for different values of α.

V. APPLICATION: DETECTING CRITICALITY IN QUANTUM ISING MODEL
In this section, we show that the Rényi and Tsallis quantum correlations can be applied to detect cooperative phenomena in quantum many-body systems. Let us consider a system of N quantum spin-1/2 particles, described by the one-dimensional quantum Ising model [59]. Such models can be simulated by using ultracold gases in a controlled way in the laboratories [5,60], and is also known to describe Hamiltonians of materials [61]. The Hamiltonian for this system is given by where J is the coupling constant for the nearest neighbor interaction, σ's are the Pauli spin matrices, and h represents the external transverse magnetic field applied across the system. Periodic boundary condition is assumed. The Hamiltonian can be diagonalized by applying Jordan-Wigner, Fourier, and Bogoliubov transformations [59]. At zero temperature, it undergoes a quantum phase transition (QPT) driven by the transverse magnetic field at λ ≡ h J = λ c ≡ 1 [59]. Such a transition has been detected by using different order parameters [59,62], including quantum correlation measures like concurrence [63], geometric measures [64][65][66], and quantum discord [67].
We now investigate the behavior of the Rényi and Tsallis quantum correlations of the nearest neighbor density matrix at zero temperature, near the quantum critical point. Note that we have reverted back to the sandwiched version of the relative entropies in this section. The nearest neighbor bipartite density matrix, ρ AB , of the ground state of the Hamiltonian given by Eq. (55), represented by ρ AB , can be written [59] in terms of the diagonal two-site correlators and the average magnetization in z-direction. The density matrix, ρ AB , in the thermodynamic limit of N → ∞, is given by T ij = tr(σ i ⊗ σ j ρ AB ) and M z = tr(I A ⊗ σ z ρ AB ). The correlations and transverse magnetization, for the zero-temperature state, are given by [59] T xx (λ) = G(−1, λ), T yy (λ) = G(1, λ), and Here and Note that λ is a dimensionless variable. The Rényi and Tsallis quantum correlations are calculated for the state, ρ AB , for different values of α. In Fig. 7, we plot the Rényi quantum correlation as a function of λ for different values of α. The QPT corresponds to a point of inflexion in the D Γ α versus λ curve. Finite-size scaling : The Rényi and Tsallis quantum correlations are shown in Fig. 7 to detect phase transitions in infinite systems. Ultracold gas realization of such phenomena, however, can simulate the corresponding Hamiltonian for a finite number of spins [68]. The quantum Ising model, which has been briefly described earlier in this section, can also be solved for finite-size systems [59]. We calculate the quantum correlations of nearest neighbor spins for finite spin chains using both the Tsallis and Rényi entropies. We find that the quantum correlations detect the transition in finite-size systems too. We perform finite-size scaling analyses for the Rényi and Tsallis quantum correlations near the QPT point, for several different values of α, and obtain the corresponding scaling exponents. The exponent is a measure of the rapidity with which the QPT point, λ N c , in a   finite size system of size N , approaches the QPT point, λ c , of the infinite system, as a function of N . Table I exhibits the scaling exponents for both D R α and D T α for some values of α. It is found that for α = 2, the scaling exponents are much higher for both D R α and D T α than any other known measures. In particular, the scaling exponents for transverse magnetization, fidelity, concurrence, quantum discord, and shared purity are respectively -1.69, -0.99, -1.87, -1.28, and -1.65 [63,[69][70][71][72].

VI. CONCLUSIONS
Quantum discord is a quantum correlation measure, belonging to the information-theoretic paradigm, and it has the potential to explain several quantum phenomena that cannot be explained by invoking the concept of quantum entanglement. In this paper, we have defined quantum correlations with generalized classes of entropies, viz. the Rényi and the Tsallis ones. The usual quantum discord incorporates the von Neumann entropy in its definition. We first defined the generalized mutual information in terms of sandwiched relative entropy distances. Using this definition of generalized mutual information, we introduced the generalized quantum correlations, and have shown that they fulfill the intuitively satisfactory properties of quantum correlation measures. We have evaluated the generalized quantum correlations for pure states and some paradigmatic classes of mixed states.
As an application, we find that the generalized quantum correlations can detect quantum phase transitions in the transverse quantum Ising model. Interestingly, a finite-size scaling analysis reveals that the scaling exponents obtained for the generalized quantum correlations can be significantly higher than the usual quantum discord as well as other order parameters, like transverse magnetization and concurrence, at the same critical point. This aspect can lead to the usefulness of these measures in quantum simulators in ultracold gas experiments, potentially realizing finite versions of quantum spin models.