Dynamical entanglement versus symmetry and dynamics of classical approximations

PHYSICAL REVIEW A 73, 052111 �2006�

Dynamical entanglement versus symmetry and dynamics of classical approximations

Nikola Burić*Department of Physics and Mathematics, Faculty of Pharmacy, University of Beograd, Vojvode Stepe 450, 11000 Beograd, Yugoslavia

�Received 1 November 2005; published 24 May 2006�

It is shown that dynamical entanglement between two qubits depends on the symmetry of the quantummodel. On the other hand, the latter is reflected in the qualitative properties of the dynamics of a classicalapproximation of the quantum system. For generic separable pure initial states, the dynamical entanglement islarger if the system is less symmetric and its classical approximation is chaotic. The influence of different typesof Markov environments on the established relation between the dynamical entanglement, symmetry and theclassical dynamics is also studied.

DOI: 10.1103/PhysRevA.73.052111 PACS number�s�: 03.65.Ud

I. INTRODUCTION

A compound quantum system could be brought into anentangled state by a measurementlike process or by unitaryevolution owing to some interaction between the subsystems.In the latter case, it is sad that the entanglement is dynami-cally generated. It is conceivable that the important proper-ties of the dynamical entanglement, like its rate, should de-pend on the interaction between the subsystems as well as onthe internal dynamics of each of the subsystems. In particularit is plausible that the symmetries of the system and theexistence of the conserved quantities should make the dy-namics of the entanglement very dependent on the initialstate.

The dynamical entanglement is a phenomenon whichcould occur in course of the evolution of a quantum systemwhich is obtained by a quantization of a classical systemwith more than one degree of freedom. Dynamics of suchclassical systems could be qualitatively very different. If thequantum system is obtained by quantization of a classicalmechanical model, or even if there is only some formal clas-sical approximation, like in the case of the spin, the questionof the relation between the qualitative properties of the clas-sical dynamics and the properties of the dynamical entangle-ment is an important aspect of the correspondence principle.

It is our goal to study the dependence of the dynamicalentanglement on the geometrical symmetries of the systemand the existence of the global constants of motion for animportant quantum system in the deeply quantum regime.

Various lines of research, during the last decade, contrib-uted to a better understanding of the relation between thedynamical entanglement and qualitative properties of the dy-namics. Strong impetus to the study of all aspects of quan-tum entanglement came from the theory of quantum compu-tation �1�. In particular, various quantities which measure adegree of entanglement in a state of a quantum system havebeen designed. Quantization of classical nonintegrable sys-tems, and various characteristic properties of resulting quan-tum systems, have been studied for a long time. As a resultof these studies, there is a number of examples of the quan-tum systems, with nonintegrable classical limit, with well

*
Electronic address: [email protected]
1050-2947/2006/73�5�/052111�10� 052111

understood dynamics and means to study it �2�. The depen-dence of the dynamical entanglement, between a quantumsystem and its environment, on the qualitative properties ofthe dynamics of the system or the environment was studiedindirectly, within the theory of environmental decoherence�3�. Direct analysis of the relation of the rates of dynamicalentanglement and the qualitative properties of the dynamicswas, to our knowledge, initiated in Ref. �4�. Various aspectsof this relation have been studied �5–14�. However, it is fairto say that the relation is still far from being fully and con-sistently described. In particular, in all the systems studied inRefs. �4–14� at least one of the entangled subsystems is inthe semiclassical regime, so that the properties of classicaldynamics �of this subsystem� are clearly seen in the exactquantum dynamics, for example, in its Husimi representa-tion. None of these works study the relation between thequalitative properties of the dynamics and the dynamical en-tanglement in the deeply quantum domain, which is our goaland is appropriate for the system of qubits.

The plan of the paper is as follows. In Sec. II we formu-late the class of quantum systems that will be analyzed, andrecapitulate the main properties of the classical approxima-tions, as provided by the corresponding coherent states.Commonly used measures of entanglement for pairs of qu-bits in pure or mixed states, that shall be used also in thispaper, are briefly recapitulated in Sec. III. Results of ouranalysis for isolated systems are reported in Sec. IV. Thegeneral conclusion is that for generic separable pure initialstates, the dynamical entanglement is larger if the system isless symmetric and its classical approximation is chaotic,rather than integrable. The influence of different types ofMarkov environments on the established relation betweenthe dynamical entanglement, symmetry and the classical dy-namics are reported in Sec. V, summary and discussion aregiven in Sec. VI.

II. THE SYSTEMS OF COUPLED QUBITS AND THEDYNAMICS OF ITS CLASSICAL APPROXIMATION

Entanglement is of crucial importance for quantum com-putation, and a system of two interacting qubits in an exter-nal field is sufficient to have a realization of a universalquantum processor �15,1�. Obviously, the results of the qua-
siclassical approximation are not applicable for the system of
©2006 The American Physical Society-1

http://dx.doi.org/10.1103/PhysRevA.73.052111

NIKOLA BURIĆ PHYSICAL REVIEW A 73, 052111 �2006�

qubits. However, such a system could display qualitativelydifferent dynamics, depending on the symmetry of the rela-tion between the external field and the interaction betweenthe qubits. These were the main reasons for us to study con-nection between the dynamical entanglement and the prop-erties of the dynamics using the system described by thefollowing Hamiltonian:

H = b1�1 + b2�2 + ��1�2, �1�

where ��i�i��x�x1�x

2+�y�y1�y

2+�z�z1�z

2 and �x,y,zi are the

three Pauli matrices of the ith qubit, and satisfy the usualSU�2� commutation relations. The parameter bi could beconsidered as a local magnetic field, and � represents theinteraction between the first and the second qubit. These pa-rameters are usually functions of time, bringing an explicit

time dependence in H, but we shall consider the model withtime-independent values of the parameters. The Hamiltonianof the universal quantum processor, realized, for example, byan array of quantum dots, is of the form �1� �16�. Of course,the Hamiltonians of the form �1� have been used to modelvarious systems in the solid state physics for a long time. Inparticular, and in relation to potential applications in quan-tum information processing �QIP�, a variety of interactingcharge qubits based on Josephson junctions are described bythe Hamiltonian of the form �1�, with the corresponding in-terpretation of the parameters bj and �. Various theoreticaland experimental results have been reviewed recently in Ref.�17�.

In order to study the relation between the qualitative prop-erties of a typical Hamiltonian dynamical system and thedynamical entanglement in the related quantum system, adescription of the quantum kinematics and dynamics interms of an appropriate Hamiltonian dynamical system on aphase space is needed. Furthermore, the Hamiltonian systemshould be nonintegrable. However, the dynamical part of thephase space description of an interacting multiparticle quan-tum system is a rather difficult problem. On the one hand,full details of the irregular dynamics on the phase space areseen only in terms of the typical Hamiltonian flow of pointsin the phase space, and on the other hand, such a descriptionof the quantum dynamics is necessarily approximate. Thequalitative properties of the dynamics of different approxi-mations could be different, which complicates the study ofthe relation to the dynamical entanglement in the quantumsystem. We shall now describe a particular type of the phasespace approximation of the system of interacting qubits, thatis used in this paper.

The construction of the phase space of a qubit is based onthe theory of the continuous group representations and thegeneralized coherent states. In the case of the qubit the rel-evant group is SU�2� group represented by the two-dimensional representation, and the resulting phase space is atwo-dimensional �2D� sphere S2 with the corresponding sym-plectic structure. We shall not repeat here the well-knownconstruction �19,18� but the most relevant formulas have tobe recapitulated.

A special class of functions on S2 represents observablesof the qubit. For our purpose the Q representation of the

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quantum operators is the relevant one. Let B be an arbitrarylinear operator acting on the representation space V2. The Q

symbol of the operator B is a function on S2 defined by

BQ��*,�� = ��B�� B�Q, �2�

where �� is an SU�2� coherent state, which can be param-etrized by the most convenient coordinates. For example,there are local canonical coordinates, i.e., such that the Pois-son bracket between the two functions on S2 has the canoni-cal form

f ,g =�f

�q

�g

�p−

�f

�p

�g

�q. �3�

We shall need the Q symbols of the operators �±,z for thesingle qubit system,

��±�Q = �q � ip��2 − q2 − p2,

��z�Q = �q2 + p2 − 1� . �4�

The linear combinations of operators �±,z are expressed asthe same linear combinations of the corresponding Q sym-bols, however a product of two operators is represented bythe convolution of the corresponding symbols. For example,

��z2�Q = ��z�Q��z�Q + 1

2 �p2 + q2��2 − p2 + q2� . �5�

Let us now define the classical Hamiltonian dynamicalsystem corresponding to the system of coupled qubits withthe Hamiltonian of the form �1�, that will be used in thispaper. As the phase space for the two qubits we take thedirect product of the spheres S2, with the geometric structureof the product: M=S2�S2. The points of the phase spaceM, are the coherent states of two qubits at some fixed timet0, ��1� � ��2�. They are disentangled �separable� statesof the system. Entangled �nonseparable� pure states are de-scribed by delocalized distributions on the product of 2spheres. This completes the kinematical part of the classicalmodel.

The Hamiltonian dynamical equations of the classical ap-proximation are given, in terms of the four canonical coor-dinates, as follows:

dqj =�

�pjHQ�q,p� ,

dpj = −�

�qjHQ�q,p�, j = 1,2, �6�

where

�q,p�H�q,p� � HQ�q,p�

is the Q symbol of the Hamiltonian of the form �1�. Theequations �6� come from the zero stationary exponent ap-proximation of the path integral form of the coherent staterepresentation of the propagator, i.e., G��1 , t ;�0 , t0��1�G�t , t0��0� where ��1,2� are arbitrary two-qubit coher-

ent states, and G�t , t0� is the propagator for the Schrödinger
equation with the Hamiltonian �1� �19�.
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DYNAMICAL ENTANGLEMENT VERSUS SYMMETRY¼ PHYSICAL REVIEW A 73, 052111 �2006�

The equations �6� complete the definition of the classicalapproximate model of the two-qubit quantum system �1�.Notice that the Hamilton’s function �H�Q is the exact coher-ent state representations of the quantum Hamiltonian. ThisHamiltonian dynamical system is different from the classicalapproximation which is obtained in the limit when the di-mension of the representation space goes to infinity. Obvi-ously, owing to the quantum correlations, the evolution ac-cording to the Hamiltonian system �6� of Q symbols of theoperators that are nonlinear expressions of any of the gen-erators �±,z

1,2, or that involve products of these operators forboth qubits, is different from the exact quantum evolution, sothat the Hamiltonian system �6� is only an approximatemodel of the quantum system. Nevertheless, the importantpoint for our analyzes will be that the geometrical symme-tries of the quantum systems �1� are preserved in the classicalapproximation.

For example, the quantum systems with the Hamiltonians

H�/� = �z1 + �z

2 + ��x�x1�x

2 + �y�y1�y

2 + �z�z1�z

2� �7�

with

�: �z � 0, �x = �y = 0

and

�: �x = �y = �z � � � 0

are symmetric with respect to SO�2� rotations around the zaxis, and the corresponding classical models

H�/� = �p12 + p2

2 − 1� + �p12 + p2

2 − 1� + ��yp1p2 + �xq1q2�

��2 − p12 − q1

2��2 − p22 − q2

2�

+ �z�p12 + q1

2 − 1��p22 + q2

2 − 1� �8�

with the corresponding values of �x, �y, and �z are inte-

grable. Indeed, the commutator between H of the form �7�and �z= �z

1� 12+11 � �z

2 is given by

�H,�z� = 2i��x − �y��x1

� �y2 + �y

1� ax

2� �9�

and the Poisson bracket between �H�Q and ��z�Q are given by

�H�Q,��z�Q = 2��x − �y��x1�Q��y

2�Q + ��y1�Q��x

2�Q� .

�10�

Thus the quantum and the classical systems simultaneouslyhave �z and ��z�Q, respectively, as conserved quantities when

�x=�y =� �or zero�, which together with �H�Q , ��x1�Q��x

2�Q

+ ��y1�Q��y

2�Q+ ��z1�Q��z

2�Q and ��z1�z

1�Q gives enough inte-grals of motion in involution for integrability of the classicalsystem. Notice that the last function ��z

1�z1�Q is not equal to

��z1�Q��z

1�Q which is not an integral.On the other hand, the Hamiltonian �7� with

�: �x � 0, �z = �y = 0

does not have any such symmetry and the correspondingclassical model �8� is nonintegrable. In this case, for a suffi-ciently large coupling �x the dynamics appears quite chaotic.
For intermediate values of �x the classical model displays
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typical properties of the mixed phase space. Thus, the quan-tum systems with the same values of the coupling betweenthe qubits, but with different symmetry are modelled by clas-sical systems with qualitatively different dynamics. We shallsee that this difference is clearly demonstrated in the proper-ties of the dynamical entanglement.

III. MEASURES OF ENTANGLEMENT

Before we proceed with the analysis of the model �1� wewould like to recapitulate the most common measures of thedegree of entanglement, that are applicable for pairs of qu-bits, and will be used in this paper. The von Neuman entropyof the reduced density matrix is a valid measure of entangle-ment if the total system is in a pure state and is considered asconsisting of two subsystems �1�. It is based on the probabil-ity that the total system undergoes from any separable initialstate 0 into an entangled state . The corresponding relativeentropy between the states 0 and , sometimes calledKullback-Leibler distance �20,21�, is given by

S��0� = Tr��ln − ln 0�� . �11�

Minimum of the relative entropy over all separable states 0gives the relative entropy of entanglement. If the compoundsystem is in a pure state, i.e., 2= �and 0

2=0�, the aboverelative entropy of entanglement is equal to the von Neu-mann entropy of the reduced density matrix pertaining toeither of the subsystems, given by

E�� − Tr�1 ln 1� = − Tr�2 ln 2� = min0�S��0�� ,

�12�2 = , 0

2 = 0,

where 1=Tr2�� and 2=Tr1��The measure of the degree of entanglement for a pair of

qubits which is not in a pure state, called the entanglement offormation �22,23�, is defined by minimization of the expres-sion kpkE��k�� over all possible convex expansions of thestate = kpk�k� where vectors �k� form a basis of purestates of the compound system. The definition is meaningfulgeneralization of the von Neumann entropy for arbitrary sys-tems, but only in the case of a pair of qubits the entangle-ment of formation can be calculated by simple algebraic pro-cedure. First, a quantity called concurrence is calculated bythe following formula:

C�� = max0,��1 − ��2 − ��3 − ��4 , �13�

where �1� ¯�4 are the eigenvalues of the matrix ��1,y

� �2,y��1,y � �2,y� where is the complex conjugate of calculated in the standard bases. The entanglement of forma-tion is then given via the function

h�x� = − x ln x − �1 − x�ln�1 − x�

by the following formula:

E�� = h�1 + �1 − C��2

� . �14�

In our analyses of the dynamics of entanglement, we shall
use the von Neumann entropy for isolated pairs of qubits and
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the entanglement of formation for the pair of qubits interact-ing with the environment.

IV. ENTANGLEMENT VS SYMMETRY

We now turn to the comparison of the dynamical en-tanglement in the systems H�,�,�. We start with an initialseparable pure state of the two qubits, and in order to mea-sure the evolution of the entanglement between the qubits,we calculate the evolution of the von Neumann entropy

E�t� � E„1�t�…

= − Tr1�1�t�ln 1�t�� = − r11�t�ln�r1

1�t�� − r21�t�ln�r2

1�t��15�

of the reduced state 1�t�=Tr2��t�� or analogously for2�t��, where r1,2

1 �t� are the eigenvalues of 1�t�.In all our calculation we used the initial states of the fol-

lowing form:

= �1� � �2� , �16�

where

�1,2� = exp�i�1,2/2�sin� 1,2/2��↓�

+ exp�− i�1,2/2�cos� 1,2/2��↑� , �17�

so that the initial states are parametrized by two pairs ofspherical angles � 1 ,�1� and � 2 ,�2�.

We shall first describe a choice of relevant results of nu-merical calculations, and then try to give a plausible inter-pretation. Typical behavior of the von Neumann entropy forthe first qubit for the three models is illustrated in Fig. 1 fortwo values of the parameter �. All time series are expressedin terms of the dimensionless t /T0, where T0 is the period ofan isolated single qubit. We also plot the orbit, in Fig. 2, of

ˆ 1 ˆ 1 ˆ 1 ˆ 1
the Bloch vector �� Tr�1�t��x� ,Tr�1�t��y� ,Tr�1�t��z�,
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of the average values of the first qubit. The entanglement inthe symmetric cases is characterized by simple periodic os-cillations with relatively small amplitude. The motion of theBloch vector is periodic and possible values of ��1� arebounded to a relatively small domain of the ball B1 repre-senting the state space of the first qubit. On the other hand, inthe nonsymmetric case the Bloch vector fills a 2D surfacethat is extended over much larger domain of B1. The timeseries given by the evolution of the von Neumann entropyappears much more complicated, has much larger amplitude,and the frequency of appearance of values close to zero ismuch smaller than in the symmetric cases. Furthermore,these observations are equally true for values of the couplingparameter such that the nonintegrable system is strongly cha-otic �say �=1.8� or with very small domains of the chaoticdynamics ��=0.9�. In fact, the amplitude of E�t� for thesymmetric systems does not depend on the value of �, andfor the nonsymmetric system the amplitude of E�t� increaseswith �x, at least for a generic initial state. The initial stateswith special properties will be discussed later.

The dependance of the dynamical entanglement on theinitial state, the symmetry of the system and the couplingparameter can be quantified by the time average of the vonNeumann entropy S(1�t�) along the orbits 1�t�,

E„1�t0�… = limT→�

1

T�

0

T

E„1�t�…dt . �18�

In order to systematically explore the dependence on the ini-tial states we fix the second qubit in some state and rotate thestate of the first qubit through the different sequences ofstates of the form �17� with 1

i = �i /21�2�, i=1,3 , . . ., 19, 20and �1 fixed. Thus, the states in one sequence are character-ized by fixed 2, �2, and �1. The series �i.e., the fixed angles 2, �2, and �1� used in Fig. 3 are chosen such as to represent

FIG. 1. Dynamical entangle-ment E�t� �15� for two values of�=0.9 �a,c� and �=1.8 �b,d�, andfor the integrable H� �a,b� and thenonintegrable H� �c,d�. The initialstate is the same typical separablevector.

typical results. We can make several observations. First, the

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averaged dynamical entanglement �ADE� �18� is lager for thenonsymmetric case then in the symmetric cases, with thesame values of the parameters, for a majority of the initialstates. ADE is much less dependent on the initial state for thenonsymmetric system then for the symmetric ones. ADE forthe symmetric cases has a large and sharp peak around somespecial initial states, when it has almost the same value as inthe nonsymmetric case, and is otherwise much smaller. Ataround the special states, the ADE for a symmetric systemcan even be larger than for the nonsymmetric one. Thedependence of ADE in the symmetric and the nonsymmetriccases on the value of the coupling parameter is also verydifferent. The ADE for the symmetric cases is numericallyindependent on the values of �. The results for �=0.9��z=0.9� and �=1.9 ��z=1.9� cannot be distinguished inFig. 3. This is clear, since in the symmetric systems theamplitude of E�t� does not depend on �, and only the fre-quencies of oscillations are increased with �, which is aver-aged out in ADE. On the other hand, ADE for the nonsym-metric case increases with �x for all initial states, except forthe special states, when the dependence of ADE on �x isnegligible.

The previous observations lead to the following conclu-sions. The dynamical entanglement mostly depends on theextension of the orbit ��1� in the sense that the dynamical

1

FIG. 2. The dynamics of Tr1�1�1� with the sa

entanglement is smaller if all states on the orbit �� are

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close, and larger if the vector ��1� wonders over a largerdomain. The symmetry of the system, and the �non�existenceof the constants of motion, is very important for this aspectof the dynamical entanglement. In the nonsymmetric case theorbit �� covers a larger domain and the ADE is larger if thecoupling �x is larger. However, there are some initial statessuch that the dynamical entanglement mostly depends on theinitial state, and is almost independent of the symmetry anddynamics or the values of the coupling parameters. Theseseparable states lead to strongly entangled states, and sincethe dynamics is periodic in the symmetric case this can in-duce larger ADE than in the nonsymmetric case when thedynamics is quasiperiodic. On the other hand, the quantita-tive properties of the dynamics of the classical model in thenonsymmetric case �“degree of chaoticity”�, induced by dif-ferent values of �x, are paralleled only by the quantitativeproperties of the dynamical entanglement, and no qualitativedifferences happen.

V. DYNAMICS OF ENTANGLEMENTFOR THE OPEN SYSTEM

The coupling to the environment in general results in de-coherence and this affects the dynamics of entanglement be-tween the subsystems of the open system.

itial state, and for the same cases as in Fig. 1.
me in
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There have been many theoretical and experimental stud-ies of the influence of the environment on the system ofqubits, especially of the single qubit �24�, and many nonin-teracting qubits �17�, in interaction with a random environ-ment. However, interacting qubits in a random environmentare comparatively much less studied. Examples of such stud-ies are Refs. �25–32�. In general, the states and the dynamicsof an open quantum system are most commonly described byconsidering the reduced density matrix and the correspond-ing master equation �33,34�. If the dynamics of the environ-ment can be considered as satisfying the Markov assumptionthen the evolution of �t� is described by a semigroup ofcompletely positive maps. Furthermore we shall assume thateach of the two qubits interacts with its local environment.The master equation satisfied by the density matrix of thewhole two-qubit system then has the Linblad form �35,34�

d�t�dt

= − i�H,� −1

2 i

�Li,Li†� + �Li,Li

†� , �19�

where the Linblad operators Li describe the influence of theenvironment on each of the qubits. The equation �19� is thegeneral form of equations satisfied by generators of semi-groups of completely positive maps.

We have studied the dynamics of entanglement betweeninteracting qubits under the influence of two common typesof environments. These are the purely dephasing and thethermal environments. Purely dephasing environment, whichmodifies only the off-diagonal elements of , is described byLinblad operators

Li = �d�+i �−

i , i = 1,2, �20�

where �d is a parameter. The thermal environment is de-scribed by

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L1 =�T�n + 1�

2�−

1, L2 =�T�n + 1�

2�−

2 ,

�21�

L3 =�Tn

2�+

1, L4 =�Tn

2�+

2

and induces dissipation �L1 , L2� and excitation �L3 , L4� pro-cesses between the two levels of each of the qubits.

Our major goal was to study the influence of the twoenvironments on the relation between the dynamical en-tanglement on the one side and the symmetry of the isolatedsystem and the dynamics of its Hamiltonian model on theother side. To this end we have calculated the time depen-dence of the entanglement of formation E�t� �we use thesame symbol E to denote �12� and �14��, starting from sometypical separable pure initial states, discussed in the preced-ing section. Results of the calculations are illustrated in Figs.4 and 5. Irrespective of the dynamics and the type of theenvironment, the entanglement of formation exhibits dumpedoscillations and becomes equal to zero for all times aftersome �. The dumping rate is much faster for the case of thethermal environment than for the dephasing, with � of thesame order and n=1. If the initial separable state is such thatthe averaged entanglement of the isolated systems is clearlylarger in the symmetric system �Figs. 4�a�, 4�d�, 5�a�, and5�d�� then this relation is preserved during the disentangle-ment, in the sense that E�t� for the nonsymmetric case re-mains most of the time larger than for the symmetric sys-tems. However if the difference between the averagedentanglement for isolated symmetric and nonsymmetric sys-tems is small �Figs. 4�b�, 4�c�, 4�e�, 4�f�, 5�b�, 5�c�, 5�e�, and5�f�� then after some time �about one-half of �� the clearrelation between the �small values of the� entanglement offormation for the two types of systems is lost.

FIG. 3. Some typical values ofthe averaged dynamical entangle-ment for the integrable � �boxes�,� �diamonds� and the noninte-grable � �circles� systems. In theintegrable cases the two values ofthe parameter �=0.9,1.8 are in-distinguishable, and in the nonin-tegrable case �=0.9 is repre-sented by filled circles and �=1.8 by open circles. The initialstates are parametrized by fixed1=2=0, 2= �20/21�2� �a�, 1

=0, 2=� /2, 2= �20/21�2� �b�,1=� /2, 2=0, 2= �5/21�2��c�, and 1=2=� /2, 2

= �5/21�2� �d�.

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as in


Thus, the state disentangles into the separable statethat can be represented as a classical statistical mixture:�t�= kpk1

k�t� � 2k�t�, where each of the qubits need not be

in a pure state. The state �t� could display different types ofcorrelations between the qubits even when �t� has achievedzero entanglement of formation due to decoherence. In Fig. 6we show typical time series M(R1�t�)�Tr��x

1�+Tr��y1�

+Tr��z1� for the symmetric H� �a,c� and nonsymmetric

H� �a,c� and for the thermal �a,b� and the dephasing �c,d�environments. In the case of the thermal environment and thesymmetric dynamics �Fig. 6�a�� M(R1�t�) converges to 1 in-dicating that the disentangled state is close to a product ofpure states. This does not happen with the dephasing envi-ronment irrespective of the dynamics, and with the thermalenvironment and the nonsymmetric dynamics. Small, sto-chastic variations for larger t in the curves in Fig. 6 are dueto the method of calculations, as is explained in the nextparagraph.

In our computation of �t� we did not use the masterequation �19� but its stochastic equivalent, as provided in themethod of quantum state diffusion �QSD� �36–38�. The den-sity matrix can be written, in different ways, as a convexcombination of pure states of the pair of qubits. Each ofthese results in a stochastic differential equation for a purestate ��t�� in the Hilbert space of the system. Such stochas-tic Schrödinger equations �SSE� are called stochastic unrav-eling �38,39,34� of the Lindblad master equation for the re-duced density matrix �t�. They are all consistent with therequirement that the solutions of �19� and of the SSE satisfy

�t� = M��t��t�� , �22�

where M��t��t�� is the mean value over the realizations

FIG. 4. Dynamics of the entanglement of formation with the the� �d,e,f� and the nonintegrable � �a,b,c�. Initial states are the same

of the stochastic process ��t��.

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The QSD equation is given by the following formula �38�:

�d� = − iH��dt + l

�2�Ll†�Ll − Ll

†Ll��t��dt

− l

�Ll†��Ll��t��dt +

l

�Ll − �Ll��t��dW ,

�23�

where � � denotes the quantum expectation in the state ��t��and dWl are independent increments �indexed by l� of com-plex Wiener c-number processes Wl�t�. They satisfy

M�dWl� = M�dWldWl�� = 0,

M�dWldWl�� = �l,l�dt , �24�

Wl is the complex conjugate of Wl. The equation �38� repre-sents a symbolic form of an Ito stochastic differential equa-tion on the Hilbert space of the open system.

There are several advantages of the description by theSSE �23� compared to the description by the density matrixand Eq. �19�. The numerical calculations of the evolution ofthe mean values like Tr��t��x,y,z

1,2 � using the SSE to calculate��t��x,y,z

1,2 ��t�� and then the mean value over different real-izations of the stochastic process are more efficient than di-rect calculation of �t� using the Linblad equation �19�. Allour calculations of �t� have been done using the QSD for-malism, and the QSD package �40�, with only 100 realiza-tions. Visualization of the dynamics of ��t��x,y,z

1,2 ��t�� onthe Bloch spheres S1 or S2 of the first and the second qubit isinformative �41�. This provides an illustration of the effects

environment for �=1.8, and �T=0.1, n=1, for the integrable caseFig. 5�a� and i=3 in 1

i � �a ,d� and i=5 in �b,e� and i=9 in �c,f�.
rmal
of the coupling between the qubits and the environment.

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in F


For example, in parallel with the results presented in Fig. 6,the pure states ��t�� of the compound system, satisfying�23�, display different types of correlations between the qu-bits for different environments and symmetry. In Fig. 7 weshow the time series R1�t�= ��t��x

1��t��2+ ��t��y1��t��2

+ ��t��z1��t��2, where ��t�� is a two-qubit state of a typical

realization. Again, in correspondence with the results forM(R1�t�) �Fig. 6� the qualitative behavior of R1�t� is quitedifferent in the case of the thermal environment and the sym-

FIG. 5. Dynamics of the entanglement of formation with the de�d,e,f� and the nonintegrable � �a,b,c�. Initial states are the same as

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metric dynamics �Fig. 7�a�� as compared to all other cases. Inthe first case, the disentanglement is clearly reflected on R1�t��or R2�t�� and, as soon as E�t� becomes zero the states ofeach of the qubits are such that R1�t��1. On the other hand,with the dephasing environment irrespective of the dynam-ics, and with the thermal environment and the nonsymmetricdynamics, the time series R1�t� oscillates most of the timethrough all values in the interval �0,1�, and converges only inthe mean over many different realizations.

ing environment for �=1.8, and �d=0.1, for the integrable case �ig. 5�a� and i=3 in 1

i � �a ,d� and i=5 in �b,e� and i=9 in �c,f�.

FIG. 6. M(R1�t�) �see the maintext� for the thermal environmentand the integrable � case �a�, andthe nonintegrable � case �b�.M(R1�t�) for the dephasing envi-ronment and the integrable caseand nonintegrable cases are shownin �c� and �d�, respectively.

phas

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ases


VI. SUMMARY AND DISCUSSION

We have studied the time dependence of entanglement, inisolated or open systems, of two interacting qubits, which areinitially in a pure separable state. Our primary goal was toinvestigate relations between the dynamical entanglement onthe one side and the symmetry and the qualitative propertiesof the dynamics on the other side. SU�2� symmetry of thequantum system is preserved by the classical approximationand is clearly manifested in the qualitative properties of theclassical dynamics. Existence or lack of the conserved quan-tity, corresponding to the SU�2� symmetry, makes the classi-cal system completely integrable or nonintegrable.

Our main conclusion is that there is a clear relation be-tween the symmetry and the capability of the dynamics togenerate entangled states. The following picture emergesfrom our analyses: Starting from an initial pure separablestate the system oscillates through states with varying degreeof entanglement, in such a way that, in general, more en-tangled states appear more often if the system is nonsymmet-ric. The oscillations of the entanglement are dependent onthe initial state and on the domain of the state space that isvisited by the state during the evolution. The latter cruciallydepends on the symmetry, i.e., on the existence of the corre-sponding conserved quantity. If there is no such integral thequantum dynamics is quasiperiodic and the orbit can go

FIG. 7. R1�t� �see the main text� for the thermal environment anddephasing environment and the integrable case and nonintegrable c

through a large part of the state space. The dynamics of the

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classical approximation is then nonintegrable. In this case,larger value of the coupling increases the averaged entangle-ment. For the symmetric system the dynamics of the en-tanglement mostly depends on the initial state and the depen-dence on the value of the coupling is negligible. The finelocal structure of the mixed phase space dynamics of theclassical approximation for medium values of the coupling isnot reflected in the dynamical entanglement of the quantumsystem �in the deeply quantum regime�.

The influence of the environment is in this paper studiedonly for the quantum system. The entangling due to the in-teraction between qubits and the disentangling due to theinteraction with the environment compete for some time, andfinally the two qubits evolve only through separable states.The relation between the symmetry and the entangling due toqubits interaction is, in general, preserved under the influ-ence of the environment.

It would be interesting to compare the dynamics of theopen quantum system with the dynamics on the phase spaceof the classical approximation of the open system �42�.

ACKNOWLEDGMENTS

This work is partly supported by the Serbian Ministry ofScience Contract No. 141003. The author acknowledges the

integrable � case �a�, and the nonintegrable � case �b�. R1�t� for theare shown in �c� and �d�, respectively.

the

support and hospitality of the Abdus Salam ICTP.

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Dynamical entanglement versus symmetry and dynamics of classical approximations