M.Jafarpour,F.Kazemi Hasanvand,and D.Afshar

Physics Department,Shahid Chamran University of Ahvaz,Ahvaz,Iran

1 Introduction

Quantum entanglement,[1]the most studied quantum correlation,has several applications in quantum information processing,including teleportation,[2?3]quantum cryptography[4]and quantum computation.[5]However,it has been revealed that there are other quantum correlations like measurement-induced disturbance[6]and discord,[7?8]which are useful in this regard as well.Moreover,correlations deteriorate under decoherence processes due to the interaction with the environment;therefore,it is vital to study the behavior of such correlations under decoherence.[9?10]There are abundant works on the subject of the decoherence of the qubit-qubit[11?22]and qutrit-qutrit[21?24]systems.However,several chain compounds similar to ACu(PbaoH)(H2o)3 nH2o,where A=Co,Ni,Zn,Fe,with two different local spins(1/2,S),have been already synthesized and their magnetic properties studied.[25?28]Therefore,this has also motivated some researchers to study the qubit-qutrit system decoherence due to different environments,including dephasing,[29?36]bit-and qutrit- fl ip,[31?36]depolarizing[31?32,34?37]and amplitude damping;[38]however,investigations regarding the decoherence due to spin chains,are very rare.[39]Moreover,Ref.[39],the only one we have found,presents the decoherence properties of a hybrid qubit-qutrit system,due to a spin chain with short range interactions,in the presence of Dzyaloshinsky Moriya interaction.Here,we also study the qubit-qutrit system decoherence due to a spin chain environment embedded in a transverse magnetic field;however,some new features have been introduced into the problem.We consider an Ising chain with long range interactions instead,which presents a better simulation of the real physical systems in some cases.[40?41]Moreover,this choice also renders the application of the mean field method advantageous.[41?44]Our goal is to study and compare the dynamics of negativity[45?47]and the measurementinduced disturbance[6,9]for this hybrid problem.

The organization of the rest of this paper is as follows.In Sec.2 we introduce the model Hamiltonian.Measures of correlations,negativity and measurement-induced disturbance,are explained in Sec.3.In Secs.4 and 5 we introduce our initial x-state[29]and p-state[30]respectively,obtain their corresponding time dependent density matrices,and calculate the measures negativity and measurementinduced disturbance,introduced in Sec.3.Finally,Sec.6 is devoted to conclusions and discussion.

2 Hamiltonian and Time Dependent Density Matrix

We consider a qubit-qutrit spin(1/2,1)system which its components do not interact with each other,but are coupled to an environment composed of an Ising chain,embedded in a transverse magnetic field.Following Refs.[41–44]with some modi fications,the total Hamiltonian of the system may be expressed as follows

where,HEand HSEdenote the Hamiltonian of the Ising chain and the interaction between the system and the en-vironment,respectively.andare the system qubit and qutrit operators along the Z direction,respectively;andandare the environmental qubit operators along the Z and x direction,in that order.J and J0are the exchange coupling constants,λ is the strength of the transverse magnetic field,Nis the total number of qubits in the environmental chain and f is the qubit and qutrit interaction discrepancy factor.We note that the environment represents a long range Ising interaction whose coupling has been scaled toN;this will guarantee the extensivity of the energy of the system.The environmental thermal density matrix is given by

where,T is the temperature and Z is the partition function given by

The total density matrix is expressed by

and the state of total system at time t is given by

where,U=e?iHtis the evolution operator.The system time-dependent density matrix may be found by tracing the degrees of freedom of the environment out.We have

To calculateps(t)it will be convenient to get rid of the nonlinear term in Eq.(3);therefore,we assume a large numberNof the qubit environment and apply the mean field method.That is,we replace HEwith its mean field expression given by[41?44]

where,the absolute value of m ranges from 0 to 1/2 and may be obtained from the equality

with

Using Eqs.(4),(5),(7)and(8)we obtain the density matrix of the system as follows

where

Finally,we find

where

Here,uμ,uνare given by

and

3 Measures of Correlations in Qubit-Qutrit System

We use measured-induced disturbance(M)and negativity(N)to quantify the quantum correlation and entanglement,respectively.M is given by[6]

where,I is the mutual quantum information given by

and

andare sets of orthogonal one-dimensional eigenprojection operators for systems A and B,respectively andis a complete orthogonal one for the bipartite system.

We also use negativity as a measure of entanglement of the system;it is de fi ned by[45?47]

where,pTA(B)is the partial transpose of the density matrixpwith respect to system A or B,and ∥∥ denotes the trace norm.

4 p-State as an Initial State

We consider the following mixed qubit-qutrit initial pstate[30]

where,p is a parameter which is restricted to the range 0 ≤ p ≤ 1/2 to guarantee the positivity condition ofp(0).It is straightforward to check that the initial statep(0)is entangled in the mentioned range,except at p=1/3.The correlation dynamics of p-state has been studied in a dephasing environment previously.[30]Using Eq.(22)in Eq.(12),the time dependent state is expressed by

where,the decoherence factors are given by

We also may verify easily that the eigenprojections for the reduced density operatorpAp(t)are given by

and for the reduced density matrixpBp(t)are given by

For largeN,the mean field method is a good approximation and Eq.(24)reduces to the following result

We need the following decoherence factors in our subsequent calculations.

Now using Eqs.(21),(23),(28),and(29)we obtain the negativity for the p-state as follows

Also using Eqs.(18)–(20),(23),and(28)–(29),we derive the following expression for the measured-induced disturbance of the p-state

In Figs.1 and 2 we have presentedNpand Mpversus scaled time J0t for different values of the temperature T.It is noted that both measures vanish for long enough time;however,the higher the temperature the faster these measures die down.

Fig.1 Npversus J0t.T=0.25(solid line),T=0.35(dotted line),T=0.5(dashed line),T=0.75(dashdotted line);λ=0.1,f=1,p=0.2,J=2.

Fig.2 Mpversus J0t.T=0.25(solid line),T=0.35(dotted line),T=0.5(dashed line),T=0.75(dashdotted line);λ=0.1,f=1,p=0.2,J=2.

In Figs.3 and 4 we have presented the three-dimensional plot ofNpand Mpversus the scaed time J0t and f.We observe that in both cases the measures attain the maximum value for f=1/2 at any time,but fade out to zero as the value of f deviates from 1/2 in any direction.

Fig.3 Npversus J0t and f.λ=0.1,p=0.2,T=0.35,J=2.

Fig.4 Mpversus J0t and f.λ=0.1,p=0.2,T=0.35,J=2.

Fig.5N pversus J0t. λ1=0.1(solid line),λ2=1(dotted line),λ3=1.5(dashed line);f=1,T=0.35,p=0.2,and J=2.

In Figs.5 and 6 we have presentedNpan Mpversus scaled time J0t for different values of the field strength λ.It is observed that both measures vanish for long enough time;however,an interesting and valuable phenomenon emerges;the higher the field strength,the slower these measures die down.That is,the decoherence may be controlled and slowed down by the transverse magnetic field.

Fig.6 Mpversus J0t. λ1=0.1(solid line),λ2=1(dotted line),λ3=1.5(dashed line);f=1,T=0.35,p=0.2,and J=2.

5x-State as an Initial State

Now,we consider one more state as the initial one,which we call x-state[29]and it is also a mixed one given by

where,the positivity of the density matrix requires that 0≤x≤1/4.One may check easily that the initial xstate is entangled for 1/8≤x≤1/4;however,Mxis an increasing function of x and non-vanishing for all values of x.The entanglement properties,including the entanglement sudden death of the x-state,have also been studied in a dephasing environment.[29]The time dependent density matrixpABx(t)is expressed by

where,the decoherence factor F16is given by Eq.(28).We also note that the eigenprojectors forpAx(t)andpBx(t)are again given by Eqs.(25)and(26)respectively.Now following the same procedure as the previous section,we obtainNxand Mxfor the x-state as follows

The measuresNxand Mxare depicted versus the scaled time J0t,in Figs.7 and 8,for several values of the temperature,respectively.It is observed that both measures approach to zero after a fi nite time;however,the higher the temperature,the faster this approach occurs.

Fig.7 Nxversus J0t.T=0.25(solid line),T=0.35(dotted line),T=0.5(dashed line),T=0.75(dashdotted line);λ=0.1,f=1,x=0.2,J=2.

Fig.8 Mxversus J0t.T=0.25(solid line),T=0.35(dotted line),T=0.5(dashed line),T=0.75(dotdashed line);λ=0.1,f=1,x=0.2,J=2.

Figures 9 and 10 displayNxand Mxversus scaled time J0t,for several values of the parameter f,respectively.Both measures approach to zero;however,die out faster for larger f values.

Fig.9 Nxversus J0t.f1=0.2(solid line),f2=0.5(dotted line),f3=1(dashed line);λ=0.1,T=0.35,x=0.2,and J=2.

Figures 11 and 12 displayNxand Mxversus scaled time J0t,for several values of the parameter λ,respectively.The same phenomenon as in the case of the p-state is observed here too;the transverse magnetic field may be used to control and slow down the decoherence process.

Fig.10 Mxversus J0t.f1=0.2(solid line),f2=0.5(dotted line),f3=1(dashed line);λ=0.1,T=0.35,x=0.2,and J=2.

Fig.11 Nxversus J0t. λ1=0.1(solid line),λ2=1(dotted line),λ3=1.5(dashed line);f=1,T=0.35,x=0.2,and J=2.

Fig.12 Mxversus J0t. λ1=0.1(solid line),λ2=1(dotted line);λ3=1.5(dashed line);f=1,T=0.35,x=0.2,and J=2.

6 Conclusions and Discussions

Considering two instances of the initial states and using the mean field method,we have studied entanglement and measured-induced disturbance of a qubit-qutrit system under decoherence due to a qubit Ising chain with long range interactions,embedded in a magnetic field.We have observed that both quantities die down eventually and the fading time is a decreasing function of temperature.However,an interesting phenomenon emerges;the external magnetic field delays the decoherence process and the fading time is an increasing function of it.That is,the transverse field may be used to control and slow down the decoherence process.We also have observed that contingent on the initial state,the size of discrepancy in the interaction parameters of qubit and qutrit with the environmental qubits plays a substantial role in the speed of the coherence fade out.

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