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the integration of quantum mechanics and information theory gave birth to the theory of quantum information .
as another fundamental part of modern physics theories , relativity theory also has significant interrelationship with quantum mechanics and information theory @xcite .
one intriguing example is the relativistic thermodynamics , which was renewed when quantum properties of black holes @xcite were discovered .
the thermodynamics of moving bodies @xcite demonstrate that probability distributions , which is relevant to shannon entropy information , depend on the inertial frame .
most recently , the relationship of relativity theory to quantum information theory has attracted increasing interest . since peres , scudo and terno
find that the spin von neumann entropy is not lorentz invariant @xcite , the effects of lorentz boosts on quantum states and then quantum entanglement @xcite have been widely investigated .
most of these works about relativistic quantum information theory ( rqit ) concentrated on the quantum state itself .
rqit maybe necessary in future practical experiments .
it has been shown that the fidelity of quantum teleportation with a uniformly accelerated partner is reduced due to davies - unruh radiation @xcite .
other possible applications include quantum clock synchronization @xcite , quantum - enhanced communication @xcite and global positioning @xcite .
quantum decoherence is closely related to several fundamental problems in quantum mechanics , e.g. quantum measurement and quantum to classical transition @xcite .
therefore , the investigation of decoherence in combination with special relativity is naturally of interest and importance @xcite , which may help gain new insights into the fundamental issues of modern theoretic physics .
it is known that quantum systems coupled with an external environment will suffer from inevitable decoherence , which is the most severe obstacle to implementing quantum computation .
a famous strategy to fight against decoherence is quantum dynamical decoupling by applying different control operations , including spin echo and bang - bang control @xcite .
given a single spin-@xmath0 dirac electron with the rest mass @xmath1 , we could realize the qubit by the spin up and down along the @xmath2 direction .
if the dirac electron is coupled with an noisy magnetic environment , the coherence of the spin - qubit will be lost . in conventional quantum information theory ( cqit )
, people always assume that the central quantum system is at rest . in this paper
, we present the time evolution of the spin state of a _ moving _ dirac electron .
it can be seen that the decoherence properties is significantly modified by special relativity .
we demonstrate that the dressed environment induced by special relativity will suppress the spin decoherence .
one should note that , the problem we consider here is different from the gravitational decoherence @xcite , which results from quantum metric fluctuations and unruh effect .
the structure of this paper is as follows . in sec .
ii we demonstrate the spin dephasing of a rest electron . in sec .
iii the influences of special relativity on the spin decoherence properties is investigated by presenting its dynamical behavior in the operator - sum representation form . in sec .
iv are conclusions and some discussions .
we start by considering the decoherence process for the spin degree of freedom of an electron with a background magnetic noise . in cqit ,
the electron is always assumed to be at rest .
since the spin magnetic moment of the electron is @xmath3 , where @xmath4 is the magnitude of the electronic charge , its interaction with a magnetic field @xmath5 is described by the following simple hamiltonian @xmath6where @xmath7 are pauli matrices .
if the noisy background magnetic field is @xmath8 in the @xmath2 direction with gaussian probability distribution @xmath9 , and under quasi - static approximation @xcite , we can write the spin state of the electron at time @xmath10 as a completely positive map with an operator - sum representation @xmath11where @xmath12 is the initial spin state , and the parameters @xmath13 @xmath14 , @xmath15 @xmath16 with @xmath17 .
here we have set @xmath18 for simplicity .
this kind of decoherence model , named dephasing , is very important and has been widely investigated in cqit @xcite . in the dephasing process , the diagonal elements of the spin density matrix remain unchanged .
the decoherence is reflected by the off - diagonal elements , which will decay exponentially as @xmath19 until @xmath20 in the long time limit @xmath21 , i.e. the spin state becomes classically mixed .
to deal with the relativistic dirac electron moving at a constant velocity @xmath22 relative to the rest frame @xmath23 , we should adopt the dirac equation for the electron in external homogeneous static fields . after choosing a suitable reference frame @xmath24 ,
the dirac hamiltonian in foldy - wouthuysen representation greiner is @xmath25where @xmath26 , and @xmath27 represents the electric field .
the above hamiltonian is a non - relativistic expansion to order @xmath28 , where @xmath29 is the relative velocity of the electron in the reference frame @xmath24 .
it is convenient for us to investigate the dynamical properties of the dirac electron in the the moving inertial frame @xmath30 with the velocity @xmath31 relative to the rest frame @xmath23 , in which the electron is at rest , i.e. @xmath32 . by virtue of the foldy - wouthuysen representation , we only need to consider the positive energy states . for the spin and momentum eigen state @xmath33 , of which @xmath34 , @xmath35 .
after time @xmath10 , the evolution of the state is @xmath36 .
the momentum phase @xmath37 becomes trivial when tracing out the momentum degree of freedom .    in the rest frame
@xmath23 , the magnetic field is @xmath8 in the @xmath2 direction , thus the fields viewed in the moving frame @xmath30 can be obtained according to the lorentz transformations greiner2 as follows @xmath38where @xmath39 and @xmath40 mean parallel and perpendicular to @xmath31 , the rapidity @xmath41 is defined as @xmath42 . after some straightforward calculations , and note that @xmath43 , we get the effective hamiltonian for the spin degree of freedom of the dirac electron @xmath44 , where @xmath45 , @xmath46 , and @xmath47 .
the above effective interaction hamiltonian means that from the viewpoint of the moving dirac electron , the environment is different compared to the situation when the electron is at rest due to the relativity of the magnetic field . in the following , we will investigate in detail how this kind of effects will modify the dynamical properties of the spin decoherence .
the interaction with the external magnetic field @xmath48 will introduce a rotation on the spin by @xmath49 about the @xmath50 axis as @xmath51where @xmath52 , @xmath53 , with @xmath54 , @xmath55 , and the rotation angle is @xmath56 . in the similar way ,
the noisy background magnetic field @xmath57 is with gaussian probability distribution , and under quasi - static approximation @xcite , we can write the spin state of the moving dirac electron at time @xmath10 as@xmath58where @xmath59 and @xmath12 denotes the initial spin state .
we first examine the diagonal elements by calculating @xmath60 . according to eq.([rt ] ) , it is easy for us to write @xmath61 , where @xmath62 $ ] with two modulation factors defined as @xmath63 and @xmath64 . after integrating over the magnetic field @xmath57 with gaussian probability distribution ,
we obtain @xmath65where @xmath66 .
the first term is just the same as the dephasing process in cqit , i.e. @xmath67 .
however , the second item in eq.([de ] ) indicates that the diagonal elements will change as time , which is a different decoherence source introduced by the effects of special relativity . in the long time
limit @xmath68 , the spin up population will decrease @xmath69 .
now we turn to the off - diagonal elements , in the similar way we can obtain @xmath70\sin \delta $ ] , where @xmath71 $ ] . after integrating over the gaussian magnetic field @xmath57
, we obtain @xmath72the first term of @xmath73 is in a similar form of exponential decay , except that the decay rate changes to @xmath74 .
the effects of special relativity is also reflected by the second term , which implies that when @xmath75 , @xmath76 that is a saturation value other than zero , i.e. the off - diagonals will not vanish . from the evolution of diagonal and off - diagonal elements in eqs.([de],[nde ] ) , we can establish a physical picture of the above analyses by expressing the spin density matrix at time @xmath10 in the operator - sum representation@xmath77 where @xmath78 , @xmath79 and @xmath80 .
the operators @xmath81 are @xmath82^{1/2}(\cos \varphi \sigma _ { x}+\sin \varphi \sigma _ { y})$ ] and @xmath83 .
if the velocity of the electron @xmath67 , the modulation factor @xmath84 , and the above results reduce to the pure dephasing of a rest spin . when we consider a moving electron , the first two terms in eq.([osr ] )
is similar to pure dephasing .
however , the decay rate of the off - diagonal elements is amplified by the factor @xmath85 .
the suppression of dephasing stems from the third term @xmath86 .
the other two operators @xmath87 and @xmath88 in eq.([osr ] ) represent different decoherence mechanisms which makes the spin suffer from other decoherence than pure dephasing .
we could also formulate the evolution of the spin state of the moving electron as in the dressed environment    @xmath89^{\kappa } ( v\rho v^{\dagger } ) [ \exp ( i\frac{\delta _ { 0}}{2}\sigma _ { z})]^{\kappa } \eta ( b)db\right\ } v\ ] ]    where @xmath90 $ ] is the dressing transformation , @xmath91 is the angle between the axes @xmath92 and @xmath2 . from the viewpoint of qubit , after the control operation @xmath93 , the initial state @xmath94 will undergo pure dephasing in the dressed hilbert space , the final reverse operation @xmath95 will recover some coherence information in the original hilbert space .
the modulation factors @xmath96 and @xmath97 are the key ingredients that characterize the influence of special relativity .
fig 2 shows the modulation factor @xmath96 as a function of the rapidity @xmath41 and @xmath98 . for a given rapidity @xmath41 , the modulation factor
@xmath99 $ ] .
we are interested in the maximum value of the modulation factor @xmath96 .
thus we consider the first order partial differential equation @xmath100 , which leads to @xmath101 , and the corresponding maximum value of @xmath96 is@xmath102we plot the maximum value @xmath103 for various rapidity @xmath41 in the following fig 3(a ) .
it can be seen that @xmath103 always increases monotonically as the rapidity @xmath41 grows . in the limit @xmath104 ,
i.e. the velocity is close to the light velocity @xmath105 , the maximum modulation factor @xmath103 reaches the constant value @xmath14 .    for the angle @xmath98 of the velocity @xmath106 that maximize the value of @xmath107 , the corresponding value of the other modulation factor @xmath108 is @xmath109 .
when we consider the relativistic dirac electron , i.e. the velocity @xmath106 is comparable to the light velocity @xmath110 , @xmath111 , which can be neglected in some sense .
_ example _ to explicitly demonstrate the influence of special relativity on the spin decoherence of a moving dirac electron , we consider a spin qubit in a fully coherent initial state @xmath112 . we could have chosen a more general initial spin state with different relative amplitudes .
however , the above state gives rise to all interesting physical features in the situation considered here .
we set the velocity angle @xmath113 to maximize the modulus of the saturation value of off - diagonal elements , thus @xmath114 \\
\rho _ { \uparrow \downarrow } ( t ) & = & \frac{1}{2}[(1-\eta ) e^{-\gamma ^{\prime } t^{2}}+\eta ] \end{aligned}\]]we plot the decay of the off - diagonal elements of the spin density matrix in fig 3(b ) , and compare two cases when the rapidity @xmath115 and @xmath116 .
the velocity angle @xmath98 is set to maximize the modulation factor @xmath96 . in the short time region ,
the off - diagonal elements decay much faster , which is due to the amplification of @xmath117by the factor @xmath118 .
however , as the time is longer , the decay of the off - diagonal elements will be much suppressed . in particular ,
if the time is sufficiently long , for @xmath116 , the spin degree of freedom becomes completely classical ; while for @xmath115 , the off - diagonal elements will reach a nonzero saturation value @xmath119 , and the spin state becomes steady , which suggests that the decoherence process seems halting .
the above discussions are extensible to the situation of several spin-@xmath0 dirac electrons coupled with one common gaussian background magnetic noise , i.e. @xmath120 @xcite .
for example , we consider two electrons in the spin entangled state @xmath121 moving at the same velocity , if the electrons are at rest , the two - qubit entanglement quantified by the concurrence will decay exponentially as @xmath122 @xcite . in order to highlight the effects of special relativity , we assume that @xmath123 , thus the modulation factors are @xmath124 and @xmath125 .
after simple calculations , it is easy to obtain the evolution of the two - qubit entanglement as @xmath126 .
therefore , unlike the situation of a single electron , the effects of special relativity will makes @xmath127 much more quickly .
the existing research in the field of rqit focused on the effects of special relativity on the _ static _ properties of quantum states , e.g. von neumann entropy and quantum entanglement . in this work
, we investigate the _ dynamic _ properties , i.e. decoherence process of a moving dirac electron .
the decoherence mechanisms will be significantly modified due to the _ dressed environment _ , which leads to the intriguing phenomenon that the coherent information will be preserved even as the electron is in the noisy environment for sufficiently long time .
the extension of this work to general decoherence models will enlarge the research scope of relativistic quantum information theory , and may establish fundamental connections between special relativity and quantum mechanics .
we thank prof .
daniel a. lidar and l. lamata for valuable discussions .
this work was funded by national fundamental research program , ncet-04 - 0587 , the innovation funds from chinese academy of sciences , and national natural science foundation of china ( grant no .
60121503 , 10574126 ) .
l. lamata , m.a .
martin - delgado , and e. solano , phys .
97 , 250502 ( 2006 ) ; l. lamata , j. len , and e. solano , phys . rev . a 73 , 012335 ( 2006 ) ; l. lamata , j. len , and d. salgado , phys . rev . a 73 , 052325 ( 2006 ) .
by investigating the evolution of a moving spin-@xmath0 dirac electron coupled with an a background magnetic noise , we demonstrate that the effects of special relativity will significantly modify the decoherence properties of the spin state 
 . the dephasing could be much suppressed , and for sufficiently long time the decoherence even seems to halt . 
 this interesting phenomenon stems from the _ dressed environment _ induced by special relativity .