Meng Li(李萌), Yun-Feng Huang(黄运锋),and Guang-Can Guo(郭光灿)

Key Laboratory of Quantum Information,University of Science and Technology of China,CAS,Hefei 230026,China

Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of

China,Hefei 230026,China

1 Introduction

Quantum correlations are part of the key points in quantum mechanics.They not only are so different from classical correlations that make us take an interested in them,but also offer some advantages in quantum information processing and so on.Usually,there are several kinds of quantum correlations such as quantum entanglement,[1]quantum discord,[2]quantum deficit,[3−5]quantumness of correlations,[6]and quantum dissonance,[7]and so on.Among them,quantum entanglement and quantum discord are two main kinds of quantum correlations.Quantum entanglement[1]constitutes one of the most important phenomena in quantum world.[8]For a long time,people have been focused on several fundamental problems of quantum mechanics using entanglement,such as quantum nonseparability,nonlocality and the violation of Bell’s inequalities,[9]Kochen–Specker theorem,[10]and so on.However,in recent years,it began to be viewed as a kind of resource which can be explored and used.The applications of it in quantum computation,[11]quantum key distribution[12−13]and quantum teleportation,[14]etc,have been demonstrated.However,quantum entanglement is so fragile that may easily be destroyed by the interaction with the environment,such as collision for atoms and birefringence for photons.On the other hand,entanglement is not the only type of quantum correlations which can be used in quantum technology.In 2001,Olliver and Zurek introduced another quantum correlation which is called quantum discord.[2]Quantum discord is fundamentally distinct from entanglement.It comes from classical mutual information.In a bipartite system,it means the difference of the quantum mutual information and classical correlations.Recently,it was found that without entanglement,the quantum system may also reveal quantum nonlocality.[15−17]Thus when a quantum system is working during the quantum information processing,the evolution of quantum discord may differ from that of quantum entanglement.So it is essential that we investigate the evolution of the quantum entanglement and quantum discord during the quantum information processing and communication.

There are many interesting phenomena in quantum entanglement evolution such as entanglement sudden death,entanglement revival,[18]etc.Among those phenomena,there is one which people did not do much researches on,that is the asymmetry of the entanglement evolution.The asymmetry of entanglement evolution in the quantum channel has been studied in Ref.[18]and the time evolution equation is given in Ref.[19].In that work they found that when a two qubits quantum system is in a special state,the entanglement evolution of qubitAdecaying in the quantum channel while qubitBis not,is different from qubitBdecaying in the quantum channel while qubitAis not.Further,they find that the subsystemwhich is called “quantum”is more robust than the subsystem which is called “classical” when interacting with the environment.So what will take place in other kinds of quantum channels is an interesting question.However,the general work of the asymmetry of quantum entanglement evolution in various quantum channels is blank.To quantum discord,Mahdian,Yousefjani,and Salimi have shown the time evolution of three qubits in Greenberger–Horne–Zeilinger(GHZ)state andWstate in several kinds of dissipation quantum channels.[20]Quantum channels they investigated including Pauli channels and depolarization channels.However,when the channel becomes more complicated,what will happen to quantum discord is still awaiting us to study.

In this paper,we investigate the asymmetry of entanglement and quantum discord evolution in diverse kinds of quantum channels.We consider a simple bipartite system consisting of two qubits,such as two photonic polarization qubits.One qubit transmits through the quantum channel,the second one transmits through the space freely.Another case to consider is that we send the second qubit into the quantum channel and let the first qubit fly.The evolution of quantum entanglement and quantum discord in the two cases will be different in some special situations.That is the asymmetry of the quantum correlations evolution.We also investigate different kinds of asymmetry in diverse kinds of quantum channels and find some interesting results.The paper is structured as follows.In Sec.2 we introduce the model of time evolution of the quantum state in the quantum channels.In Sec.3 we show that the von Neumann entropy of one qubit of the qubit pairs will violate the inequalities of von Neumann entropy of a quantum state with classical correlations.In Sec.4 we show different kinds of asymmetry of quantum entanglement evolution and quantum discord evolution in different quantum channels.The conclusion of the paper is given in Sec.5.

2 Models of Time Evolution

In this paper,we consider the evolution of two qubits quantum system in the environment.Letρbe the density operator of a quantum system of which the Hilbert space is finite-dimensional.The general form of the transformation of the quantum channel can be written as the evolution of superoperator:

whereρiniis the initial density operator,ρfinis the final density operator,Mµis the superoperator.A simple but important quantum channel is random external fields.[21]They can be written as

whereAi,i=1,2,...,kare unitary operators and→p=(p1···pk)is the normalized vector of probabilities,i.e.,

Here the unitary operators of the quantum channel are defined as

whereσi,i=1,2,3 are Pauli operators.In general,we consider seven types of random external fields channels.The difference between them is that they have different vectors of probabilities:

Of course,there are many kinds of quantum channels that are different from random external fields.We shall consider two types of decaying channels.One is the amplitude damping channel:

where

The other is the phase-damping channel:

where

Now we consider two qubits subjected to the interaction with the environment:only one of them decays in the quantum channel while the other is not.In the first case,qubitAis sent into the quantum channel while qubitBis not.In the second case qubitBis sent into the quantum channel whileAis not.We investigate the asymmetry of the entanglement evolution and the quantum discord evolution in the two situations.

3 The Initial State

In this paper,the initial state of the bipartite quantum system is designed as[22]

where

The matrix form of the initial state is

From the initial state given above,we can get the reduced density matrices of the two subsystems

For a quantum state with classical correlations,the information that we get from measuring any of the subsystems is not greater than that from measuring the entire system.Both of them satisfy the two inequalities of the von Neumann entropy[23]

The definition of the von Neumann entropy is

However,in Ref.[18],they mention that whena2＞q＞1/2 the first inequality is violated,while the second one is not.So there must be some non-classical correlations in the state of this condition.That makes us to investigate the classical correlations and quantum correlations,to quantify them and show the evolution of them for this state in different kinds of quantum channels.

4 Asymmetry of the Entanglement Evolution

From the discussion before,we discover that the entropy of the state we investigated has some special properties and such results make us to investigate the classical correlations and quantum correlations of the state.Here we will show that the asymmetry of the entanglement evolution and quantum discord evolution will happen when different subsystem is sent into the quantum channel.In this paper,we investigate nine types of quantum channels.Seven of them are random external fields channels.The other two are the amplitude damping channel and phasedamping channel respectively.The metrology of the entanglement is concurrence,[24−25]while the metrology of quantum discord follows Mazhar Ali,Rau,and Alber.[26]What we will study is how the entanglement and quantum discord evolute when we change the intensity of the quantum channel,and the comparison of the difference between the two situations of sending different subsystems into the quantum channel.The parameters of the initial state areq=3/5,a2=3/4.

First let us study the asymmetry of the evolution of the entanglement inp3channel.

Following the evolutionary model in Ref.[18],we write a program withMathematicaand calculate the change of the concurrence when the intensity of the channel is increased.

The corresponding dynamical process is denoted in Fig.1.

Fig.1 (Color online)The dynamics of quantum correlations in p3channel.The correlations of the state changed when we change the intensity of the quantum channel in two cases.The purple dotted line and green dotted line are for the case of sending the subsystem A into the channel while subsystem B not.Purple line shows the evolution of the quantum entanglement and the green line shows the evolution of the quantum discord.The blue dotted line and the khaki dotted line are for the case of sending the subsystem B into the channel while subsystem A not.Blue dotted line shows the evolution of the quantum entanglement and khaki dotted line shows the evolution of quantum discord.

From Fig.1 we can see that the asymmetry of the entanglement evolution happens inp3channel.When subsystemAis sent into the channel,the concurrence decreases with the increasing of the intensity of the channel.The concurrence also reduces when subsystemBis sent into the channel,but it decreases faster than the case of sending subsystemAinto the channel.When the concurrence becomes zero,it will not change when we increase the intensity of the channel in two cases.

Like quantum entanglement,the asymmetry of quantum discord also happens inp3channel.But an important difference between quantum entanglement and quantum discord is that quantum entanglement goes down monotonously and never goes up.However,quantum discord not only goes down much slower than quantum entanglement,but also can revive after it becomes zero.That is an important feature.It tells us quantum discord can be nonzero when entanglement is zero.So when quantum entanglement is zero,the quantum system may also reveal some quantum non-localily because the quantum discord is nonzero.Moreover,the fact that the evolution of quantum discord is slower than that of quantum entanglement shows that quantum discord is more robust than quantum entanglement.As quantum entanglement is fragile,whether quantum discord can be another kind of more robust quantum resource is still an open question which is worthy of investigating.

Fig.2 (Color online)The quantum correlations evolution in p1a,p1b,p1cchannel.In p1achannel,the evolutions are symmetric.The asymmetries only happen in the p1band p1cchannels.

Figure 2 shows the entanglement evolution and the quantum discord evolution inp1a,p1b,p1cchannels.For entanglement,there are two significant differences when we send either of the two subsystems into thep1a,p1b,p1cchannels.First,entanglement evolution of the case that we send subsystemAinto thep1achannel is the same as the case of sending subsystemBinto thep1achannel,so it means that the entanglement evolution in thep1achannel is symmetric.The asymmetry only happens in thep1b,p1cchannels.Second,when we increase the intensity of the channel,the concurrence first goes down,then becomes zero,and finally it revivals in all the three channels.At the third stage,the concurrence increases when we increase the intensity of the channel.

The evolutions of quantum discord inp1a,p1b,p1cchannels have many distinct features than before.Inp1achannel,the evolution of quantum discord is similar to that of quantum entanglement as they are symmetric.However,the difference is also obvious because that the evolution is unsmooth at two points.That makes us a bit surprise because the intensity of the quantum channel increases smoothly and there is no unsmooth point in the evolution of quantum entanglement.It shows that at such points the properties of the quantum state change suddenly,and this change cannot be discovered from entanglement.This phenomenon tells us that quantum discord shows the correlations of quantum system from another aspect which is different from quantum entanglement.Inp1bchannel,the evolution of quantum discord is asymmetric.Moreover,when subsystemAis sent into the quantum channel,the quantum discord changes very little,though it descends at first,and then rises up.However,when subsystemBis sent into the channel,with the increase of the channel intesity,the quantum discord descends obviously at first,and then rises clearly.Inp1cchannel,the tendency of the quantum discord evolution is similar to that inp1achannel,but the evolution is asymmetric,and they also have unsmooth point.We can see that the change of sending subsystemBinto the channel is faster than that of sending subsystemAinto the channel.

From Fig.3,we can see that the entanglement evolution in thep2a,p2b,p2care similar.Asymmetries of the entanglement evolutions happen in all the three channels and there is no revival.

The quantum discord evolutions inp2a,p2b,p2cchannels are all asymmetric.Inp2achannel,when subsystem A is sent into the channel,quantum discord descends at first,then rises up.They also have unsmooth points during evolution.However,when subsystemBis sent into the channel,the main tendency of the quantum discord evolution is decreasing,although it rises a little bit in the middle.Inp2bchannel,for both cases,quantum discord mainly descends,the difference is that no matter which subsystem is sent into the quantum channel,the quantum discord will not go up,it decreases monotonously,and they also have unsmooth points.The evolutions inp2cchannel are similar to those inp2bchannel.

Fig.3 (Color online)The quantum correlations evolution in the p2a,p2b,p2cchannels.All of them are asymmetric.

From Fig.4,we can see that no matter which subsystem is sent into the channel,the concurrence decreases much slower in amplitude damping channel than in random external fields channel.Moreover,an important difference from the channels before is that the decrease of the concurrence when we send the subsystemBinto the channel is slower than that when we send subsystemAinto the channel.This is an important feature of this quantum channel,it shows that we cannot only change the absolute speed of the entanglement evolution,but also the relative speed of the entanglement evolution when we change the quantum channel.

The evolution of quantum discord is also asymmetric in amplitude damping channel.With the increasing of the intensity of the channel,quantum discord goes down monotonously.Similar to the evolution of quantum entanglement in amplitude damping channel,the evolution of quantum discord is slower when subsystemBis sent into the channel.That is very different from all the situations before.

Fig.4 (Color online)The quantum correlations evolution in the amplitude damping channel.

Figure 5 shows that the entanglement evolutions in the phase-damping channel are symmetric and there is no revival.The concurrences of the two situations decrease to zero almost linerly and never go up.

Fig.5 (Color online)The quantum correlations evolution in the phase-damping channel.

The evolutions of quantum discord are also symmetric in phase-damping channel.They descend monotonously when we increase the intensity of the quantum channel.There is likewise unsmooth point during evolution.

5 Conclusion

We investigate the quantum correlations evolution,including quantum entanglement evolution and quantum discord evolution,with a simple model in several kinds of quantum channels,including random external fields,amplitude damping channel and phase-damping channel.We numerically calculate the asymmetry of the entanglement evolution and quantum discord evolution in different kinds of quantum channels and show the change tendency.We find that the asymmetry only happens in some special channels,and in some particular channels the entanglement or quantum discord will go down at first and then revive.

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