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We investigate the entanglement dynamics of an anisotropic two-qubit Heisenberg XYZ system with Dzyaloshinskii-Moriya (DM) interaction in the presence of both inhomogeneity of the external magnetic field b and intrinsic decoherence which has been studied. The behavior of quantum correlation and the degree of entanglement between the two subsystems is quantified by using measurement-induced disturbance (MID), negativity (N) and Quantum Discord (QD), respectively. It is shown that in the presence of an inhomogeneity external magnetic field occur the phenomena of long-lived entanglement. It is found that the initial state is the essential role in the time evolution of the entanglement.

Nowadays, correlated systems represent one of the most important partners in the context of quantum communication [

The intrinsic decoherence noise is one of the most important types of noise [

In this paper, we will investigate the quantum correlations base on MID in our model. As we probably are aware, the quantum entanglement of dense issue frameworks is a vital developing field as previously. Individuals have made a few examinations of quantum entanglement of thermal equilibrium states of spin chains subject to an external magnetic field at finite temperature [

The rest of this paper is organized as follows. In Section 2, we introduce the Hamiltonian of the Heisenberg model with different DM interaction and present the exact solution of the model. In Section 3, is devoted to investigating the dynamics of entanglement and quantum correlation by means of negativity, measurement-induced disturbance [

The Hamiltonian H for a two-qubit anisotropic Heisenberg model with z-component interaction parameter D z is

H = J ( 1 + γ ) σ 1 x σ 2 x + J ( 1 − γ ) σ 1 y σ 2 y + J z σ 1 z σ 2 z + D z ( σ 1 x σ 2 y − σ 1 y σ 2 x ) + ( B + b ) σ 1 z + ( B − b ) σ 2 z (1)

where J and J z are the real coupling coefficients, γ is the anisotropic parameter. D z is the z-component DM interaction parameter, and σ i ( i = x , y , z ) are Pauli matrices. B is the homogeneous part of the magnetic field and b describes the inhomogenity. The external magnetic fields and Dzyaloshinskii Moriya interaction are assumed to be along the z-direction. All the parameters are dimensionless. We get on the eigenvalues of the Hamiltonian H are given by

E 1 = J z + 2 B 2 + J 2 γ 2 E 2 = J z − 2 B 2 + J 2 γ 2 E 3 = − J z + 2 b 2 + D z 2 + J 2 E 4 = − J z − 2 b 2 + D z 2 + J 2 (2)

We get on the eigenvectors of the Hamiltonian H are given by

| ψ 1 〉 = 1 1 + μ 2 ( μ | 11 〉 + | 00 〉 ) (3)

| ψ 2 〉 = 1 1 + ν 2 ( ν | 11 〉 + | 00 〉 ) (4)

| ψ 3 〉 = 1 1 + X ( N | 10 〉 + | 01 〉 ) (5)

| ψ 4 〉 = 1 1 + Y ( M | 10 〉 + | 01 〉 ) (6)

where,

μ = B + B 2 + J 2 γ 2 J γ , θ = b + b 2 + D z 2 + J 2

ν = B + B 2 + J 2 γ 2 J γ , ϕ = − b + b 2 + D z 2 + J 2

X = θ 2 D z 2 + J 2 , Y = ϕ 2 D z 2 + J 2

N = θ ( − I D z + J ) D z 2 + J 2 , M = ϕ ( I D z − J ) D z 2 + J 2 (7)

The master equation describing the intrinsic decoherence under the Markovian approximations is given by

d ρ ( t ) d t = − i [ H , ρ ( t ) ] − Γ 2 [ H , [ H , ρ ( t ) ] ] (8)

where Γ is the intrinsic decoherence rate. The formal solution of the above master equation can be expressed as

ρ ( t ) = ∑ k = 0 ∞ ( Γ t ) k k ! M k ρ ( 0 ) M + k (9)

where ρ ( 0 ) is the density operator of the initial system and M k is defined by

M k = H k e − i H t e − Γ t 2 H 2 (10)

According to Equation (9) it is easy to show that, under intrinsic decoherence, the dynamics of the density operator ρ ( t ) for the above-mentioned system which is initially in the state ρ ( 0 ) is given by

ρ ( t ) = ∑ m n exp [ − Γ t 2 ( E m − E n ) 2 − i ( E m − E n ) t ] × 〈 ψ m | ρ ( 0 ) | ψ n 〉 | ψ m 〉 〈 ψ n | (11)

where E m , E n , | ψ m 〉 , | ψ n 〉 are the eigenvalues and the corresponding eigenvectors of H. In the standard basis { | 00 〉 , | 01 〉 , | 10 〉 , | 11 〉 } the time evolution of the density operator of the system will be obtained for two different initial states as:

The two qubits are initially in an entangled state ρ ( 0 ) = | ϕ 〉 〈 ϕ | , | ϕ 〉 = cos ( α ) | 01 〉 + sin ( α ) | 10 〉 , we get

ρ ( t ) = ( 0 0 0 0 0 ρ 22 ρ 23 0 0 ρ 32 ρ 33 0 0 0 0 0 ) (12)

ρ 22 = 1 4 y 2 [ ( ( b + y ) ( y − b cos ( 2 α ) + J sin ( 2 α ) ) ) − ( e − 4 i y t − 8 y 2 t Γ ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J + i D z y ) sin ( 2 α ) ) ) − ( e 4 i y t − 8 y 2 t Γ ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J − i D z y ) sin ( 2 α ) ) ) + ( ( − b + y ) ( y + b cos ( 2 α ) − J sin ( 2 α ) ) ) ]

ρ 23 = 1 4 y 2 [ ( ( − i D z + J ) ( y − b cos ( 2 α ) + J sin ( 2 α ) ) ) − ( ( − i D z + J ) ( y + b cos ( 2 α ) − J sin ( 2 α ) ) ) ] + 1 4 ( i D z + J ) ( y 2 ) × ( ( e − 4 i y t − 8 y 2 t Γ ( b + y ) ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J + i D z y ) sin ( 2 α ) ) ) + ( e 4 i y t − 8 y 2 t Γ ( b − y ) ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J − i D z y ) sin ( 2 α ) ) ) )

ρ 32 = 1 4 y 2 [ ( ( i D z + J ) ( b 2 + D z 2 + J 2 − b cos ( 2 α ) + J sin ( 2 α ) ) ) − ( ( i D z + J ) ( y + b cos ( 2 α ) − J sin ( 2 α ) ) ) ] + 1 4 ( i D z + J ) ( y 2 ) × [ ( e − 4 i y t − 8 y 2 t Γ ( b − y ) ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J + i D z y ) sin ( 2 α ) ) ) + ( e 4 i y t − 8 y 2 t Γ ( b + y ) ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J − i D z y ) sin ( 2 α ) ) ) ]

ρ 33 = 1 4 y 2 ( b + y ) [ ( D z 2 + J 2 ) ( y − b cos ( 2 α ) + J sin ( 2 α ) ) ] + 1 4 y 2 ( − b + y ) [ ( D z 2 + J 2 ) ( y + b cos ( 2 α ) − J sin ( 2 α ) ) ] + 1 4 y 2 [ ( e − 4 i y t − 8 y 2 t Γ ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J + i D z y ) sin ( 2 α ) ) ) + ( e 4 i y t − 8 y 2 t Γ ( ( D z 2 + J 2 ) cos ( 2 α ) + ( b J − i D z y ) sin ( 2 α ) ) ) ]

There is some systems could be entangled but have a zero negativity and non-zero values of quantum correlations. This means that there are some quantum correlation cannot be predicted by using the negativity as a measure of entanglement. However, by using the measurement-induced disturbance as a measure of quantum correlation, one can quantify the unpredicted quantum correlation [

By using Π k = Π i a ⊗ Π j b and Π i a , Π j b are complete projective measurements consisting of one-dimensional orthogonal projections for parties a and b, we can apply local measurement { Π k } ( Π k Π k ′ = δ k k ′ Π k ) and ∑ k Π k = 1 , to any bipartite state ρ (of course, including thermal state). After the measurement, we get the state Π ( ρ ) = ∑ i j ( Π i a ⊗ Π j b ) ρ ( Π i a ⊗ Π j b ) which is a classical state. If the measurement Π is induced by the spectral resolutions of the reduced states ρ a = ∑ i p i a Π i a and ρ b = ∑ i p i b Π i b , the measurement leaves the marginal information invariant and is in a certain sense the least disturbing. In fact, Π ( ρ ) is a classical state that is closest to the original state ρ since this kind of measurement can leave the reduced states invariant. One can use any reasonable distance between ρ and Π ( ρ ) to measure the quantum correlation in ρ . In this article, we will use Luos method i.e., quantum mutual information difference between ρ and Π ( ρ ) , to measure quantum correlation in ρ . The total correlation in a bipartite state ρ can be well quantified by the quantum mutual information I ( ρ ) = S ( ρ a ) + S ( ρ b ) − S ( ρ ) , and I ( Π ( ρ ) ) , quantifies the classical correlations in ρ since Π ( ρ ) is a classical state. Here S ( ρ ) = − ∑ i λ i log 2 λ i denotes the von Neumann entropy. So the quantum correlation can be quantified by the measurement induced disturbance Q ( ρ ) = I ( ρ ) − I ( Π ( ρ ) ) . where

Π ( ρ ) = ( ρ 11 0 0 0 0 ρ 22 0 0 0 0 ρ 33 0 0 0 0 ρ 44 ) (13)

Quantum discord is based on the difference between the quantum mutual information and the classical correlation. For a two-qubit quantum system, the total correlation is measured by their quantum mutual information L ( ρ a b ) = S ( ρ a ) + S ( ρ b ) − S ( ρ a b ) , where ρ a ( b ) and ρ a b denote the reduced density matrix of a ( b ) and the density of the bipartite system respectively, and S ( ρ ) = − t r ( ρ log 2 ρ ) is the von Neumann entropy. Quantum discord, which quantifies the quantumness of correlation between A and B, is then defined as the difference between the total correlation and classical correlation. For the X state described by the density matrix quantum discord (QD) is given as

Q D = min [ D 1 , D 2 ] (14)

with

Q D j = Γ ( ρ 11 + ρ 33 ) + ∑ i = 1 4 λ i log 2 λ i + R j

and

R 1 = − Γ ( ρ 11 + ρ 33 ) − ∑ i = 1 4 ρ i i log 2 ρ i i ,

R 2 = Γ ( p ) (15)

p = 1 + [ 1 − 2 ( ρ 33 + ρ 44 ) ] 2 + 4 ( | ρ 14 | + | ρ 23 | ) 2 2

where λ i being the four eigenvalues of the density matrix ρ and Γ ( x ) = − x log 2 x − ( 1 − x ) log 2 ( 1 − x ) . As the density matrix of our system in Equation (9) is X states, quantum discord can be evaluated by substituting from Equation (9) into Equation (11) after straightforward calculation, quantum discord reads

Q D = min [ D 1 , D 2 ] (16)

with

D 1 = ( 1 / log 2 [ 4 ] ) ( − ( ρ 22 + ρ 33 ) log 2 [ 4 ] − 2 ρ 22 log 2 [ ρ 22 ] − 2 ρ 33 log 2 [ ρ 33 ] + ( ρ 22 − 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 )

× log 2 [ ρ 22 − 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ] + ( ρ 22 + 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ) × log 2 [ ρ 22 + 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ] )

D 2 = 1 log 2 [ 4 ] ( 4 p tanh − 1 [ 1 − 2 p ] + 4 ρ 33 tanh − 1 [ 1 − 2 ρ 33 ] − ( ρ 22 + ρ 33 ) log 2 [ 4 ] − 2 log 2 [ 1 − p ] − 2 log 2 [ 1 − ρ 33 ] + ( ρ 22 − 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 )

× log 2 [ ρ 22 − 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ] + ( ρ 22 + 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ) × log [ ρ 22 + 4 ρ 23 ρ 32 + ( ρ 22 − ρ 33 ) 2 + ρ 33 ] )

In

and decay faster before arrive at a steady-going non-zero value for the long-time case for different values of the DM interaction in

In the presence of both external magnetic field and intrinsic decoherence, we have treated the entanglement dynamics of an anisotropic two-qubit Heisenberg XYZ system with Dzyaloshinskii-Moriya interaction which has been studied. We found that the initial state of the system plays an important role in the time evolution of the entanglement. The magnetic field has an effective role in maintaining the intertwining for a long time and non-analysis. The negativity, MID, and QD for different DM interaction will arrive at a steady-going non-zero value for the long-time case, which means that the external magnetic field b is a positive component to the entanglement when the partial anisotropic parameter of the system is at a fixed non-zero value.

Mohammed, A.R. and El-Shahat, T.M. (2017) Study the Entanglement Dynamics of an Anisotropic Two-Qubit Heisenberg XYZ System in a Magnetic Field. Journal of Quantum Information Science, 7, 160-171. https://doi.org/10.4236/jqis.2017.74013