Revealing trade o relations among thermal coherences and correlations in Heisenberg XXX model under inhomogeneous magnetic eld

We study the validity of quantum Fisher information (QFI) as a faithful quantum coherence and correlation quanti er by drawing a comparison with subsystem's coherence measure, rst-order coherence (FOC) and the entanglement measure, Negativity to study the behavior of thermal quantum coherence and correlations in two qubit Heisenberg XXX model, placed in independently controllable magnetic eld by systematically varying the coupling parameter, magnetic eld and bath temperature for ferromagnetic and antiferromagnetic case. After carefully observing the pro le of quantum coherence and correlation measures, we propose an inequality relations which shows that there may exist a quantitative relationship between QFI, Negativity and FOC in which, the equality exists at zero temperature. We identify QFI to be a more useful coherence quanti er, as it quanti es coherence of individual subsystems and correlations among the subsystems. On the other hand, FOC identi es coherence present in the individual subsystems only. A reciprocal relationship between Negativity and FOC is also observed in di erent cases. We also observe the existence of entanglement in ferromagnetic case, in contrast to simple Heisenberg XXX model in uniform magnetic eld. We show that in the ferromagnetic case, a very small inhomogeneity in magnetic eld is capable of producing large values of thermal entanglement. This shows that the behavior of entanglement in the ferromagnetic Heisenberg system is highly unstable against inhomogeneity of magnetic elds, which is inevitably present in any solid state realization of qubits. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 November 2021


I. INTRODUCTION
Quantum correlations and quantum coherence [1][2] are two essential aspects of quantum theory, and are extremely important for any quantum information or quantum computation, both in theory and in experiments. While the quantum correlations characterize the quantum features of a system with at least two parties, the quantum coherence is dened for a single as well as multipartite system [1]. Despite this dierence, they are also intimately related to each other [3].
From the practical point of view, quantum correlations and quantum coherence are invaluable physical resources for implementing various quantum information and computation tasks, such as quantum key distribution (QKD) [4], super dense coding (SDC) [5], quantum cryptography (QC) [6], teleportation [7], and exponentially fast computing that requires entanglement [8]. A recently proposed quantum correlation measure called quantum discord (QD) [2] has been studied in several contexts with various operational interpretations and applications, such as quantum state merging (QSM) [9], remote state preparation (RSP) [10], and quantum metrology [11]. The ultimate quest for discovering a physically meaningful, and universal quantier is a long-standing concern of theoretical physics. In search of nding universal resource quantier for quantum correlation and coherence, the theorists have proposed many quantiers, such as quantum Fisher information (QFI) [12], quantum discord (QD) [2], quantum coherence [1], quantum concurrence [8], rst-order coherence (FOC) [13], Shannon-Jenson divergence coherence (SJDC) [14], and many more. The superiority of one quantier over others in capturing quantum correlations and quantum coherence and their applicability, attracts researchers to investigate the dynamics of these quantiers in dierent existing quantum mechanical systems, such as spin chains [15][16][17][18][19][20], optomechanics [21], quantum dots [22][23] and many other quantum states. These investigations are of great importance as they help us retrieve the quantum properties of a system and act as a tool for nding the universal quantum correlation and coherence quantier.
Recently, some theoretical work on the trade o relations between entanglement and quantum coherence have been appeared in the literature as well [24][25][26]. Commonly used coherence measures consider a multipartite quantum system as a whole, omitting its internal mathematical structure and evolution in subsystems. The knowledge of the internal distribution of coherence between subsystems and their correlations becomes necessary for predicting the evolution (migration) of coherence in the studied system. Our focus here is to look at the behavior of FOC, Negativity and QFI in Heisenberg XXX model, as this model is an experimentally more feasible where the trade o relationship between FOC and Negativity can be seen and their connection with QFI is also discussed. In this paper, we collectively study the behavior of thermal FOC, QFI and Negativity in two-qubit Heisenberg's XXX model under inhomogeneous magnetic eld. Two cases, ferromagnetic and antiferromagnetic spin chain are discussed for two-qubit under uniform as well as non-uniform magnetic elds.
The scheme of this paper runs as follows: In section 2 we discuss the two-qubit Heisenberg's XXX model under an inhomogeneous magnetic eld. In section 3, we dene FOC, QFI, Negativity and QM and present the 3D plots of each and quantier in dierent cases.
In section 4, we collectivelty discuss FOC, QFI, QM and Negativity for our model and discuss its dierent cases with the help of obtained 2D plots, and nally, in section 5, we summarized the conclusion with drawn results.

II. THE MODEL
First of all, let us briey review the standard model of Heisenberg spin chain for our study.
Considering two consecutive neighbor interactions, the Hamiltonian for a one-dimensional chain in an n spin 1 2 system can be written as; where σ α k (α = {x, y, z}) is the local spin pauli operators in the Hilbert space of k th qubit, which satisfy σ n+1 = σ 1 . Here the parameters J α are the real coupling constants for the spin interaction. For J x = J y = J z = J(isotropy), J x = J y = J z (partial anisotropy) and J x = J y = J z (anisotropy), the Heisenberg chains are named as XXX,XXZ and XY Z models, respectively. In addition, there exist two special case Heisenberg models, viz. the XY model for which J z = 0 and the Ising model for which J y = J z = 0. From these models, our model of intereset in this article is Heisenberg XXXchain model (In order to handle the number of parameters, we have chose XXX instead of XYZ or XY) in the presence of an inhomogenous magnetic eld b along with the z-direction, the Hamiltonian of which can be written as In which, J < 0 corresponds to the ferromagnetic chain case and J > 0 to the antiferromagnetic one, and b indicates the degree of inhomogeneity in uniform magnetic eld B. The two independently controllable parametrized magnetic elds (B − b) and(B + b) are applied to two qubits respectively. We have also set = k B = 1 and note that we are working in the units in which B, J and b are dimensionless. Furthermore, for sake of convenience we will set J = 1 for antiferromagnetic case whereas for ferromagnetic case we will assume J = −1 The Hamiltonian (2) in the standard basis{|00>, |10>, |01>, |11>} can be mathematically expressed as The spectrum of eigenvectors and corresponding eigenvalues to (3) are as follows where |0 and |1 denotes spin up and down states respectively and a 1 = −b− √ The expression for the thermal equillibrium state of a quantum system is given as In which Z = T r(exp(− H kT )) is the partition function of the system and k is Boltzmann constant. The Partition function and density matrix for the current system can be written The matrix representation of the (13) in standard basis {|00>, |10>, |01>, |11>} can be written as with matrix elements being In the next section, we dene three quantiers such as QFI, FOC and Negativity.

III. QUANTUM CORRELATIONS AND COHERENCE
Here, we briey discuss QFI, FOC and Negativity for our model individually.

A. First order coherence
Another quantier for coherence in optical coherence theory and condensed matter physics is the measure of FOC of bipartite states [13]. It is related to the purity of the individual subsystems of the combined bipartite system. FOC in a multipartite system can be dened where {i = A, B} denotes the i th subsystem. Then, we can obtain the coherence measurement of dierent subsystems when they are considered independently. The amount of coherence present in the system ρ AB can be represented as The state ρ AB can be transformed via a global unitary U to get state ρ AB = U ρ AB U † such that the FOC can vanish, and the two subsystems become strongly correlated. On the other hand, for certain unitary operations U † , the maximum FOC can be obtained, D 2 where, λ i s are the eigenvalues of the state ρ AB in the decreasing order [13]. Furthermore, D 2 Fmax can be called the degree of available coherence since it represents the maximum FOC extracted under a global unitary transform.  Where θ is the parameter to be estimated and operator is G is the generator. Correspondingly the accuracy of estimating θ is limited by Cramer-Rao inequality Where µ is the number of experiments and F (ρ θ ) is QFI. The QFI can be interpreted as the information on parameters encoded in the quantum state. Similarly, for an observer O on a Hilbert space to be estimated, the QFI, i.e., the information involved in ρ for observable O, can be given Here L is the symmetric logarithmic derivative determined by the following equality where [ρ, O] is a commutation relation Ifρ is a pure state, the QFI can be expressed as In the case of mixed states, the calculation of QFI becomes more complicated and will be shown in detail below. Generally, a mixed state ρ can be described as (20) where |m and P m are respectively the eigenvectors and eigenvalues of the m th component of the state ρ. Exploiting (25) and (27) in (24) gives [27] F Here m and n mark respectively the eigen-parameters (|m , |P m and |n and|P n . We take a two-component composite system as an example and discuss the QFI calculation in detail.
Suppose that ρ ij is any bipartite state while A u and B u are respectively arbitrary local orthonormal observable bases for two subsystems in ρ ij . In this case, the QFI encoded in this two-component system with respect to the observables can be evaluated as [27], which is also called the global information of ρ ij . In general, for a two-qubit system, the local ortho-normal observables A u and B u can be taken as: Here, σ i are the Pauli matrices for which i = 1, 2, 3. Consequently, if ρ ij is given, the QFI can be calculated according to Fig. 2 represents the the 3D plots of QFI as function of magnetic eld and temperature in which Fig.2(a) represents uniform antiferromagnetic case, Fig.2 Thus the QM for any arbitrary state ρ is dened as [28]; Where d is the dimension of ρ and 0 ≤ M (ρ) ≤ 1.

D. Negativity
Negativity in quantum information theory is a good measure of quantum entanglement because it is directly derived from PPT criterion for separability and is easy to compute [29].
It is proved to be an entanglement monotone and so is a proper measure of entanglement.
For any density matrixρ, the Negativity of subsystem is dened as Here,ρ T A represents partial transpose of ρ with respect to subsystem A and  While observing from Fig. 5(a) to Fig. 5(d), it is obvious that Negativity has maximum value N (ρ) = 1 at T = 0 and N (ρ) < 1 for T > 0 which is due to the fact that at T = 0, the ground state is in maximally entangled state and with the Increase in T entanglement is lost due to mixing of entangled ground state with its unentangled states.the ground state of system is in entangled state, whereas for |B| > |B c | it is not an entangled state. It is very interesting to note that the QFI behaves in the same as Negativity in the range where |B| < |B c | where FOC is zero whereas In the range |B| > |B c |, QFI shows the same behavior as FOC where Negativity is zero from which we observe that there exists some kind of trade o behavior between entanglement and subsystems coherence which is very prominent for small T values whereas for large T values, this eect cannot be seen because entanglement is not robust and decays with T . So we realize QFI as a faithful quantier which captures both entanglement and hidden coherence of individual spins but it is unable to dierentiate between FOC and Negativity against B. It is also evident from the same Fig. 5 at T = 0 the system is in perfectly pure state that is why QM is zero and with rise in T the peak value of QM increases whereas it can be suppressed by increasing B.
After carefully looking at the plots, we propose an inequality which shows that there may exist a quantitative relationship between QFI, FOC and Negativity as a function of both uniform and non uniform magnetic eld for dierent xed value of T ; Similarly, In these relations equality holds at T = 0.
These set of inequalities works fairly well in anitferromagnetic case when observed against magnetic eld both uniform and non uniform except for ferromagnetic non uniform mangetic eld case. Fig. 6 shows the plot of QFI, Negativity and FOC as a function of T at dierent xed values of B when J = +1. For B = 0 in Fig. 6(a), it is shown that the ground state is maximally entangled and so Negativity N (ρ) = 1. Whereas for dierent xed values of B > 0, peak value of N (ρ) < 1 which shows that the role of B and T are to diminish T are also satised here as well. Again the equality in the proposed inequalities holds at T = 0 (Fig. 7(a)). Furthermore, again here we can see QFI faithfully behaves with both   Fig. 8(a), it is shown that the ground state is maximally entangled and the so Negativity N (ρ) = 1. Whereas for dierent xed values of b > 0 peak value of N (ρ) < 1 which shows that the role of b and T are to diminish N (ρ). It is also abvious that N (ρ) diminishes to zero earlier than QFI and FOC which shows the fragility of entanglement when comes in contact with bath at T > 0. Moreover, It is seen that QM increases with the increase in T which is trivial as we have calculated the QM for thermal state and in contrast to uniform antiferromagnetic case, in this case increase in the value of b generally slows down the the rate of increase in QM with T .
C. Uniform ferromagnetic case   (Fig. 9(a)). Since there is no entanglement in this case therefore the behavior of QFI is exactly similar to FOC at any value of parameters B and T . It is also observed that QM is zero for all the values of B at T = 0. For small T the behavior of QM is localized at small values of B which becomes delocalize for larger values of T .  Fig. 9(a) where b = B = 0). In contrast to antiferromagnetic case, all In contrast to non uniform ferromagnetic case where is no entanglement at all, in this case entanglement exists which is very sensitive to the values of temperature and non uiform magnetic eld b. At T = 0 Negativity is maximum at b = 0 and start decreasing with |b|.
Whereas for T > 0 Negativity is zero at b = 0 and start increasing after certain value of non uniform magnetic eld b means for nite temperature, Negativity start increases and decreases which is unusual. So in this case a very small inhomogeneity is capable of producing large values of thermal entanglement. This shows that the presence of entanglement in the ferromagnetic Heisenberg system is highly unstable against non uniform magnetic elds.
Furthermore, this is only case where our proposed inequalities dont hold.  Fig. 10(a), it is shown that at ground state both FOC and Negativity are zero. Whereas for slight increase in b > 0, FOC and Negativity achieve its peak value (see Fig.12 (b)) which decreases with increase in T . We have studied the individual as well as collective behavior of thermal quantum coherence, and correlations in two-qubit Heisenberg XXX spin model under independently controllable inhomogeneous magnetic eld in terms of FOC, Negativity, and QFI respectively. Generally, it is known that entanglement does not exist in a ferromagnetic spin chain case under a uniform magnetic eld, but in our case, we have shown that introducing a little inhomogeneity in a uniform magnetic eld to spins creates entanglement in this case which is highly unstable. The strength of this behavior is maximum at T = 0, and diminishes with T > 0. We have proposed the inequality relations between QFI and Negativity for our model, that works well for all the situations discussed, except for nonuniform ferromagnetic cases. The reciprocal behavior of FOC and Negativity is also observed in dierent cases, when plotted against a magnetic eld (both uniform and nonuniform). We have realized QFI as a faithful quantier that captures both entanglement and hidden coherence of individual spins, but it is unable to dierentiate among the two when plotted against magnetic eld except for the nonuniform ferromagnetic cases.

AUTHORS CONTRIBUTION STATEMENT
All the authors contributed for the submission of this manuscript in which Asad Ali concieved the idea and perform all the calculation on Mathematica, whereas Mustansar Nadeem and Professor A. H. Toor and Junaid-ul-Haq discussed the results and Shah Ahad helped in writing the manuscript and editing plots.

AVAILABILITY OF DATA AND MATERIAL
The data generated for the paper are available upon personal request to the authors.

CODE AVAILABILITY
The numerical programs are available upon personal request to the authors.