document: quantum entanglement is a feature of quantum mechanics that has captured much recent interest due to its essential role in quantum information processing @xcite . it may be characterized and manipulated independently of its physical realization , and it obeys a set of conservation laws ; 
   as such , it is regarded and treated much like a physical resource . 
   it proves useful in making quantitative predictions to quantify entanglement.when one has complete information about a bipartite system 
   subsystems @xmath0 and @xmath1the state of the system is pure and there exists a well established measure of entanglement  the _ entropy of entanglement _ , evaluated as the von neumann entropy of the reduced density matrix , @xmath2 with @xmath3 . 
   this measure is unity for the bell states and is conserved under local operations and classical communication . 
   unfortunately , however , quantum systems in nature interact with their environment ; states of practical concern are therefore mixed , in which case the quantification of entanglement becomes less clear . 
   given an ensemble of pure states , @xmath4 with probabilities @xmath5 , a natural generalization of @xmath6 is its weighted average @xmath7 . 
   a difficulty arises , though , when one considers that a given density operator may be decomposed in infinitely many ways , leading to infinitely many values for this average entanglement . 
   the density operator for an equal mixture of bell states @xmath8 , for example , is identical to that for a mixture of @xmath9 and @xmath10 , yet by the above measure the two decompositions have entanglement one and zero , respectively . 
   various measures have been proposed to circumvent this problem , most of which evaluate a lower bound . 
   one such measure , the _ entanglement of formation _ , @xmath11 @xcite , is defined as the minimal amount of entanglement required to form the density operator @xmath12 , while the _ entanglement of distillation _ , @xmath13 @xcite , is the guaranteed amount of entanglement that can be extracted from @xmath12 . 
   these measures satisfy the requirements for a physical entanglement measure set out by horodecki _ 
   et al_. @xcite . 
   they give the value zero for @xmath14 , which might be thought somewhat counterintuitive , since this state can be viewed as representing a sequence of random `` choices '' between two bell states , both of which are maximally entangled . 
   this is unavoidable , however , because assigning @xmath15 a non - zero value of entanglement would imply that entanglement can be generated by local operations . 
   the problem is fundamental , steming from the inherent uncertainty surrounding a mixed state : the state provides an incomplete description of the physical system , and in view of the lack of knowledge a definitive measure of entanglement can not be given . 
   an interacting system and environment inevitably become entangled . 
   the problem of bipartite entanglement for an open system is therefore one of tripartite entanglement for the system and environment . 
   complicating the situation , the state of the environment is complex and unknown . 
   conventionally , the partial trace with respect to the environment is taken , yielding a mixed state for the bipartite system . 
   if one wishes for a more complete characterization of the entanglement than provided by the above measures , somehow the inherent uncertainty of the mixed state description must be removed .    to this end , nha and carmichael @xcite recently introduced a measure of entanglement for open systems based upon quantum trajectory unravelings of the open system dynamics @xcite . 
   central to their approach is a consideration of the way in which information about the system is read , by making measurements , from the environment . 
   the evolution of the system conditioned on the measurement record is followed , and the entanglement measure is then contextual  dependent upon the kind of measurements made . 
   suppose , for example , that at some time @xmath16 the system and environment are in the entangled state @xmath17 a partial trace with respect to @xmath18 yields a mixed state for @xmath19 . 
   if , on the other hand , an observer makes a measurement on the environment with respect to the basis @xmath20 , obtaining the `` result '' @xmath21 , the reduced state of the system and environment is    @xmath22    with conditional system state @xmath23    where @xmath24 is the probability of the particular measurement result . 
   thus , the system and environment are disentangled , so the system state is pure and its bipartite entanglement is defined by the von neumann entropy , eq . 
   ( [ eq : von - neumann ] ) . 
   nha and carmichael @xcite apply this idea to the continuous measurement limit , where @xmath25 executes a conditional evolution over time . 
   in this paper we follow the lead of nha and carmichael , also carvalho _ et al . _ 
   @xcite , not to compute their entanglement measure _ per se _ , but to examine the entanglement _ dynamics _ of a cascaded qubit system coupled through the oneway exchange of photons . 
   the system considered has been shown to produce unconditional entangled states  generally a superposition of bell states  as the steady - state solution to a master equation @xcite . 
   for a special choice of parameters ( resonance ) , a maximally entangled bell state is achieved @xmath26 except that the approach to the steady state takes place over an infinite amount of time .    here 
   we analyze the conditional evolution of the qubit system to illuminate the dynamical creation of entanglement in the general case , and to explain , in particular , the infinitely slow approach to steady - state in the special case . 
   we demonstrate that in the special case the conditional dynamics exhibit a distinct bimodality , where the approach to the bell state is only one of two possibilities for the asymptotic evolution : the second we call an _ entangled - state cycle _ , where the qubits execute a sustained stochastic switching between two bell states . though involving just two qubits and elementary quantum transitions , the situation is similar to that of a bimodal system in classical statistical physics in the limit of a vanishing transition rate between attractors .    the physical model of the cascaded qubit system 
   is presented in sec . 
   [ subsec2_1 ] and the quantum trajectory unraveling of its conditional dynamics in sec . 
   [ subsec3_1 ] . in sec . 
   [ subsec4_1 ] we analyze the quantum trajectory equations to demonstrate bimodality and the existence of entangled - state cycles . 
   finally , a discussion and conclusions are presented in sec . 
   [ subsec5_1 ] . 
   in this section we briefly outline the physical model for the cascaded qubit system to be analyzed . a more detailed description , together with the techniques and assumptions used to derive the model master equation presented here , is available in @xcite . 
   the system considered consists of two high - finesse optical cavities , each containing a single tightly - confined atom , the cavities arranged in a cascaded configuration with unidirectional coupling from cavity 1 to cavity 2 ( fig . 
   [ fig : fig1 ] ) . for simplicity 
   , we consider the cavity modes to be identical , with resonance frequency @xmath27 and field decay rate @xmath28 . 
   inefficiencies and losses in the coupling between the cavities are modeled by a real parameter @xmath29 , @xmath30 , with perfect coupling corresponding to @xmath31 . 
   the atoms are assumed to have five relevant electronic levels , of which two ground states , @xmath32 and @xmath33 , represent an effective two - state system , or qubit .     and 
   @xmath33 , to three excited states , @xmath34 , @xmath35 , and @xmath36.,scaledwidth=40.0% ]    for each atom , the cavity field in combination with auxiliary laser fields ( incident from the side of the cavity ) drives two separate resonant raman transitions between states @xmath32 and @xmath33 . 
   an additional laser field coupled to the @xmath37 transition provides a tunable light shift of the energy of state @xmath32 . 
   all fields are assumed far detuned from the atomic excited states , so these states may be adiabatically eliminated and atomic spontaneous emission ignored . under the further assumption that the cavity field decay rate is much larger than the transition rates between @xmath32 and @xmath33 , the cavity fields may also be adiabatically eliminated to yield a master equation for the reduced two - atom density matrix  @xmath12 , @xmath38}+{\left[\hat r_2,\rho\hat r_1^\dag\right]}\right)},\end{aligned}\ ] ] with @xmath39 where @xmath40 , and @xmath41 and @xmath42 are the rates of @xmath43 and @xmath44 transitions , respectively .    by virtue of the cavity output , 
   the system is an open system and solutions to master equation ( [ eq : me ] ) generally describe mixed states . under appropriate conditions 
   , however , the system evolves to a pure and entangled steady state . 
   if the coupling between cavities is perfect ( @xmath31 ) and the parameters of the subsystems are the same ( @xmath45 , @xmath46 ) then the steady state is the pure state @xmath47 where we use the abbreviated notation @xmath48 and @xmath49 . 
   then when @xmath50 , which we shall refer to as the _ resonance _ condition , the steady state is a maximally - entangled bell state . 
   this may seem to be ideal , but a problem arises when we consider the eigenvalues of the operator @xmath51 . specifically , the characteristic time for the system to reach steady state , @xmath52 , where @xmath53 denotes the eigenvalue of @xmath51 with smallest ( in magnitude ) non - zero real part , approaches infinity as the resonance condition is approached . 
   this is shown by the plot in fig . 
   [ fig : fig2 ] . 
   thus the master equation itself , in particular its steady state , offers limited insight into the behavior of the system at resonance . 
   we wish to learn more about this special case ; in particular , how does the entanglement develop dynamically . 
   also , if additional information is factored into the description , by making measurements on the environment , can we better characterize the long term behavior , or possibly find perfect entanglement after a finite time ? 
   we demonstrate that quantum trajectory theory can provide answers to these questions .     the relaxation time @xmath54 plotted as a function of @xmath55 . 
   note the singularity at resonance , @xmath56.,scaledwidth=40.0% ] 
   as with any open system , the first step in unraveling the master equation is to identify the points of coupling to the environment . 
   the first is obvious  the output from cavity 2 . to measure this output , 
   let us assume the existence of an ideal photon detector in the path of the output from cavity 2 ; we call it _ detector 1_.    the second point of coupling to the environment is more subtle 
   . our model does not assume the inter - cavity coupling to be perfect ; only a fraction @xmath29 of the output photon flux from cavity 1 makes it into cavity 2 . 
   physically , this loss may be caused , for example , by non - ideal transmissivity of the faraday isolators or by absorption in the cavity mirrors . 
   these imperfections cause photons to be scattered into the environment in some uncontrollable fashion . 
   formally , though , this is equivalent to assuming that the apparatus is ideal , except that there exists a beamsplitter between the cavities , as drawn schematically in fig . 
   [ fig : fig3 ] . 
   we therefore further assume the existence of a second photon detector to collect photons reflected by this beamsplitter ; we call it 
   _ detector 2_.        we now proceed to develop the quantum trajectory formalism for the cascaded qubit system . in this approach the system is described by a pure state which is dependent on ( conditioned on ) the counting histories , or records , of detectors 1 and 2 . 
   firstly , we rewrite the master equation in a form suitable for translation into the quantum trajectory language . 
   we reexpress eq . 
   ( [ eq : me ] ) in the form @xmath57 with    @xmath58-\frac{1}{2}\sum_{i=1,2 } 
   \left(\hat c_i^{\dag}\hat c_i \rho+\rho\hat c_i^{\dag}\hat c_i\right),\\ \mathcal{s}\rho&=\sum_{i=1,2}\hat c_i \rho\hat c_i^{\dag},\end{aligned}\ ] ]    where    @xmath59    then , within quantum trajectory theory , the evolution of the system is described by a pure state @xmath60 which evolves under the non - hermitian effective hamiltonian @xmath61 the continuous evolution interrupted at random times by quantum jumps , @xmath62 , where the jumps occur with probability @xmath63 in time interval @xmath64 . 
   physically , the jump operators @xmath65 and @xmath66 account for the reduction of the state of the system , given a photon count is recorded by detector  1 or detector  2 , respectively . 
   thus , within the quantum trajectory description of the coupled cavity system , we consider an experiment in which ideal detectors are employed , such that every scattered photon is detected and recorded . given the history of detector ` clicks ' , one has complete information about the system state , in the sense that that state is always pure ; hence , although the solution to the master equation is generally mixed , one is able to characterize the entanglement in an unambiguous ( conditional ) fashion @xcite . 
   consider the special case where the coupling between the cavities is optimal ( @xmath31 ) . in this case 
   there is only one output from the system , that from cavity 2 , recorded by detector 1 . 
   standard numerical algorithms @xcite have been used to simulate typical quantum trajectories for various values of @xmath55 . 
   specifically , we consider the evolution of the conditional expectation of the operator product @xmath67 , where @xmath68 is the pauli operator diagonal in the @xmath69representation , @xmath70 this expectation has a number of convenient properties ; for example , the steady - state value @xmath71 regardless of the value of @xmath55 , which makes it easy to compare rates of convergence to the steady state for different system parameters .    -0.5 
   cm    figure  [ fig : fig4 ] contrasts the solution to the master equation and a single quantum trajectory . 
   the solution to the master equation exhibits a completely smooth evolution that tends asymptotically towards the steady state . 
   the quantum trajectory , on the other hand , undergoes a sequence of switches between two extreme values of @xmath72 , which occur at each photon detection . provided the parameters 
   are chosen away from resonance , the photon detections eventually stop and the trajectory settles into the steady state ( [ eq : ss ] ) , with @xmath73 ; the steady state is clearly a dark state . at resonance , however , the photon detections may continue indefinitely . 
   physically , this seems plausible , since it simply implies that the atoms continue to switch between states @xmath32 and @xmath33 , scattering one photon with each transition . at resonance , apparently 
   , a unique equilibrium dark state can not be established . 
   the cyclic behavior that replaces it is completely invisible if we consider only the ensemble average  a vivid demonstration of how single quantum trajectories can provide additional insight into the evolution of an open quantum system . 
   the oscillatory behavior featured in fig . 
   [ fig : fig4 ] hints at a simple cyclic process . 
   in fact , it is simple enough that we can understand why it occurs without resorting to numerics . in this section 
   we formulate a graphical description of individual trajectories . 
   figure  [ fig : fig4 ] demonstrates that the conditional expectation @xmath72 is conserved during the periods of evolution between quantum jumps . 
   the positively and negatively correlated subspaces @xmath74 are coupled only through quantum jumps . 
   noting that @xmath75 are each @xmath76-dimensional ( assuming real amplitudes without loss of generality ) , we manage to break up a @xmath77-dimensional space into two @xmath76-dimensional planes , linked to one another by the quantum jumps . 
   we refer to this representation as the _ cascaded system phase space_. trajectories within it can be viewed as lines moving continuosly within either plane and jumping discontinuously between the planes . 
   we use phase space portraits within @xmath78 and @xmath79 to characterize the behavior of the system , where for the sake of simplicity , and without loss of generality , we are assuming @xmath80 and @xmath81 to be real . 
   we define @xmath82 and scale time by setting @xmath83 . 
   the master equation then takes the form ( @xmath31 ) @xmath84 where @xmath85 the resonance condition is now @xmath86 . 
   it is useful to convert to a matrix notation , such that a pure state @xmath60 of the system is represented by a @xmath77-vector , @xmath87 and system operators are written as @xmath88 matrices , e.g. , @xmath89 and @xmath90 the evolution of @xmath60under @xmath91 is written as a linear differential equation in four variables , @xmath92\ !   { |\phi\rangle}\nonumber\\ & = \left(\begin{array}{cccc } -2&0&0&2r\\ 0&-(1+r^2)&2r^2&0\\ 0&2&-(1+r^2)&0\\ 2r&0&0&-2r^2\\\end{array}\right ) \label{eqn : basicde}\!{|\phi\rangle}.\end{aligned}\ ] ] as noted above , this evolution is constrained within either @xmath78 or @xmath79 . 
   thus we can write @xmath60 as a vector sum of two orthogonal components @xmath93 and @xmath94 , @xmath95 , to obtain the decoupled dynamics    @xmath96    eigenvectors of the two dynamical matrices correspond to states of the system that are preserved under the evolution between quantum jumps . 
   note , however , that it does not necessarily follow that such a state is a steady state of the quantum trajectory evolution as a whole ; it must eventually experience a quantum jump if its norm decays  i.e . 
   , the corresponding eigenvalue is not zero . 
   recall from quantum trajectory theory that the probability for a state not to jump prior to time @xmath16 is given by its norm @xcite . 
   for the systems of equations given above we find the following ( unnormalised ) eigenstates and eigenvalues :    1 . 
   @xmath97 ,  @xmath98 ; this is the steady state of the system for @xmath99 . 
   2 .   @xmath100 ,  @xmath101 ; this state in @xmath78 is orthogonal to @xmath102 and must eventually jump to a state in @xmath79 . 
   3 .   @xmath103 ,  @xmath104 ; this state in @xmath79 must eventually jump to a state in @xmath78 unless @xmath86 ; in the latter case it plays no role once an entangled - state cycle is initiated ( see below ) . 
   4 .   @xmath105 ,  @xmath106 ; this state in @xmath79 must eventually jump to a state in @xmath78 .    in the special case of resonance , @xmath86 , 
   there are two independent steady states , @xmath102 and @xmath107 , which helps to explain the failure of the master equation evolution to approach a unique steady state . 
   it also suggests a fundamental feature of the indefinite switching , the cyclic behavior , revealed by individual quantum trajectories : during such an _ entangled - state cycle _ , the system state must remain orthogonal to @xmath102 and @xmath107 . 
   we verify this shortly , after examining the trajectory evolution away from resonance , where the steady state @xmath102 is always reached for perfect inter - cavity coupling . 
   typical quantum trajectories for @xmath108 are shown in figs . 
   [ fig : fig5 ] and [ fig : fig6 ] , where the @xmath78 and @xmath79 subspaces are drawn as circular planes . 
   normalized states are located on the circumferences of the circles . 
   the bell states    @xmath109    lie at intersections of the circumference with the dotted lines as shown .    between quantum jumps , under the influence of the non - hermitian hamiltonian @xmath91 
   , the norm of the state decays and the point representing it within the phase space moves to the interior of one of the circles . 
   quantum jumps cause a switch from @xmath78 to @xmath79 or vice - versa . 
   they are represented by the lines connecting the two planes , where for illustrative purposes , the system state is renormalized after each quantum jump ; thus jumps terminate at points on the circumference of the circles . 
   we restrict ourselves to separable initial states located in one or other of the two subspaces ; for example , the states @xmath110 and @xmath111 , respectively , are considered in figs . 
   [ fig : fig5 ] and [ fig : fig6 ] . 
   the action of the jump operator @xmath65 on states located in @xmath78 ( with renormalization ) is    @xmath112    while the action of @xmath65 on states in @xmath79 is @xmath113 thus , when a quantum jump occurs , any state within @xmath78 collapses onto the bell state @xmath114 in @xmath79 , while any state within @xmath79 collapses onto the state @xmath115 in @xmath78 . 
   consider an initial normalized state in @xmath78 , @xmath116 , for some ( real ) @xmath117 . 
   given that @xmath102 is a steady state of the evolution between quantum jumps , the probability of an eventual quantum jump to @xmath79 is @xmath118 while with probability @xmath119 the system evolves to the steady state @xmath102 without any photon emissions . 
   if a jump from @xmath79 to @xmath120 has just occurred , then by the same argument one shows that the probability of a future quantum jump to @xmath79 is @xmath121 , or , alternatively , the probability of reaching the steady state after such a jump is @xmath122 ^ 2 $ ] . 
   consider now an initial state in @xmath79 , @xmath123 , for some ( real ) @xmath124 . 
   owing to the instability of both @xmath107 and @xmath125 for @xmath99 , an eventual quantum jump is guaranteed ; thus , @xmath126 armed with this information , we move to an explanation of the quantum trajectories displayed in figs . 
   [ fig : fig5 ] and [ fig : fig6 ] .      in fig . 
   [ fig : fig5 ] we plot three typical phase - space trajectories for @xmath108 and @xmath127 . 
   [ fig : fig5](a ) illustrates the case where the system evolves directly to the steady state @xmath102 . 
   the probability of this event is @xmath128 , so it is the most likely occurrence for the chosen parameters . 
   if a first quantum jump does occur , then typical trajectories are shown in figs . 
   [ fig : fig5](b ) and ( c ) . following the jump to @xmath129 in @xmath79 , a second jump returning the state to @xmath78 is guaranteed . for @xmath108 , 
   this leaves the system in the state @xmath130 , from which the probability of a further cycle of jumps is @xmath131 . 
   thus , after a first quantum jump cycle , it is most likely that further cycles will follow , as seen in figs . 
   [ fig : fig5](b ) and ( c ) , where in both cases a total of five cycles ( ten photon detections ) occur before the system finally reaches the steady state .    in fig . 
   [ fig : fig6 ] we plot three typical phase - space trajectories for @xmath108 and @xmath132 . in this case , at least one quantum jump is certain to occur , following which the probability of further jumps is @xmath131 , as above . 
   so for this initial condition , the most likely outcome is a sequence of quantum jump cycles following a first guaranteed photon detection . in fig . 
   [ fig : fig6 ] ( a ) only the first detection occurs , while in figs . 
   [ fig : fig6](b ) and ( c ) this detection is followed by a sequence of cycles before the steady state is eventually achieved . 
   the case @xmath86 is of particular interest . 
   the normalized eigenstates of the evolution between quantum jumps are the bell states @xmath133 , @xmath134 , @xmath135 , and @xmath136 . 
   the eigenvalues are @xmath137 and @xmath138 . 
   the action of the jump operator @xmath65 on states within @xmath78 simplifies to    @xmath139    and its action on states within @xmath79 to @xmath140 for @xmath86 , photon detections , if they occur , are associated with collapses onto one of two maximally - entangled bell states .    for initial states @xmath141 and @xmath142 in @xmath78 and @xmath79 , respectively , 
   the system evolves continuously , without the emission of any photons , to @xmath133 and @xmath143 , with probabilities @xmath144 and @xmath145 in @xmath79 or @xmath146 in @xmath78 . in this case , as both terminal states are unstable under the between - jump evolution , a second detection and quantum jump must follow . according to eqs . 
   ( [ eq : plusjump ] ) and ( [ eq : minusjump ] ) this simply exchanges @xmath147 for @xmath146 and vice - versa . 
   hence , a perpetual switching between bell states @xmath147 and @xmath146 occurs . 
   we designate this behavior an _ entangled - state cycle_.    thus , at resonance we find a distinctly bimodal behavior . 
   the system either evolves into a maximally - entangled bell state without emitting photons , or an entangled - state cycle is initiated under which the system switches indefinitely between orthogonal bell states while emitting a continual stream of photons . as an aside , such behavior can be regarded as a quantum measurement that distinguishes the bell states @xmath148 from @xmath149 . 
   the two alternative outcomes of the quantum trajectory evolution are illustrated in figs . 
   [ fig : fig7 ] and [ fig : fig8 ] for the initial states @xmath150 in @xmath78 and @xmath132 in @xmath79 , respectively . with this choice of initial states 
   there are equal probabilities for reaching the steady states , @xmath151 [ fig . 
   [ fig : fig7](a ) ] and @xmath152 [ fig .  [ fig : fig8](a ) ] , and for commencing an entangled - state cycle [ figs . 
   [ fig : fig7](b ) and [ fig : fig8](b ) ] . note that once an entangled - state cycle is initiated , the trajectory remains in a plane orthogonal to the lines defining @xmath151 and @xmath152 ; the cycle continues indefinitely      our original model allowed for the possibility of imperfect intercavity coupling , through the parameter @xmath29 and the jump operator @xmath66 which describe the effects of photon loss in propagation between the two cavities . 
   focusing on the resonant case ( @xmath86 ) , we now consider the situation in which @xmath153 . 
   typical trajectories for @xmath154 are shown in figs . 
   [ fig : fig9](a ) and [ fig : fig10](a ) , with the two photon count records shown in frames ( b ) and ( c ) of the figures . 
   remarkably , entangled - state cycles persist , but now the system settles into one or other of two distinct cycles , involving either the symmetric or antisymmetric bell states .    to understand the behavior , consider the forms of the operators involved ; in particular , for @xmath86 , we have effective hamiltonian @xmath155 and jump operators    @xmath156    and @xmath157    significantly , these operators commute with one another , @xmath158=[\hat 
   c_1,\hat h_{\rm eff}]=[\hat c_2,\hat h_{\rm eff}]=0.\end{aligned}\ ] ] their operation upon the bell states is given by    @xmath159    and    @xmath160    thus , the bell states are eigenstates of @xmath91 , and the jump operators interchange bell states in @xmath78 and @xmath79 : each jump operator converts the symmetric ( antisymmetric ) bell state in @xmath78 to the symmetric ( antisymmetric ) bell state in @xmath79 and vice - versa . 
   now , let us consider a particular quantum trajectory for which a total of @xmath161 jumps occur , separated by the time intervals @xmath162 . 
   for an initial state @xmath163 , the ( unnormalized ) state at the conclusion of the @xmath161 jumps is written as @xmath164 where each @xmath165 is either @xmath65 or @xmath66 . 
   since all operators in the string acting on @xmath166 commute , this expression can be rewritten in a variety of forms , two of which prove to be especially useful in explaining the distinct behaviors illustrated by figs . 
   [ fig : fig9 ] and [ fig : fig10 ] . 
   in the first case , we may write    @xmath167    passing all @xmath168 ocurrences of @xmath65 to the right and all @xmath169 occurrences of @xmath66 to the left ( @xmath170 ) ; in the second we write @xmath171 where all jump operators are passed to the left .    the arbitrary ( pure ) initial state can be expressed as a superposition of bell states , @xmath172 where @xmath173 , @xmath174 , @xmath175 , and @xmath176 are expansion coefficients , generally complex . substituting this expansion into eqs . 
   ( [ phit1 ] ) and ( [ phit2 ] ) , and using eqs . 
   ( [ heff1])([c2psi])assuming for simplicity that @xmath168 and @xmath169 are even  the two forms for the state @xmath177 are    @xmath178    where    @xmath179    observe now that the ratio of the eigenvalues satisfies @xmath180 it follows that @xmath181 and @xmath182 allow us to predict quite distinct asymptotic behaviors for the system state . for sufficiently large @xmath168 , the contribution to @xmath181 from the symmetric bell states is negligible compared with the contribution from the antisymmetric bell states , in which case , using eqs . 
   ( [ eq : cycle1 ] ) and ( [ eq : cycle1prime ] ) , @xmath183 the system is locked into a cycle between the two antisymmetric bell states , the situation illustrated in fig .  [ 
   fig : fig9 ] ( for @xmath184 , @xmath185 ) . 
   in contrast , for sufficiently large @xmath16 , the contribution to @xmath182 from the antisymmetric bell states is negligible compared with that from the symmetric bell states , and using eqs . 
   ( [ eq : cycle2 ] ) and ( [ eq : cycle2prime ] ) , @xmath186 the system is locked into a cycle between the two symmetric bell states , as shown in fig . 
   [ fig : fig10 ] .    which of the two cycles is chosen in a particular realization of the photon counting record is random , as is the time taken to settle into the cycle . 
   effectively , the decision is the outcome of a competition between the periods of evolution between quantum jumps and the jumps themselves  specifically , those associated with photon counts at detector 1 . considering eqs . 
   ( [ eq : cycle1prime ] ) and ( [ eq : ratio ] ) , we see that every count at detector  1 results in an increased probability to find the system in one of the antisymmetric bell states . on the other hand , from eqs . 
   ( [ eq : cycle2prime ] ) and ( [ eq : ratio ] ) , the periods of evolution between counts have the reverse effect 
   they increase the probability for the system to be found in a symmetric bell state . 
   the critical factor that decides which tendency wins is the number of photon counts occuring at detector  1 over a given ( substantial ) interval of time . 
   if there are many , as in fig . 
   [ fig : fig9](b ) , the entangled - state cycle between antisymmetric bell states wins out ; if there are few , fig . 
   [ fig : fig10](b ) , the cycle between symmetric bell states occurs . 
   the same decision mechanism is observed in other examples @xcite . 
   note that counts at detector 2 are not involved  not directly at least . 
   they do figure indirectly as a mechanism reducing the average number of counts at detector 1 ; indeed , they are the ultimate source of the asymmetry reflected in the ratio @xmath187 .    as the system approaches a particular cycle the quantum trajectory evolution tends to reinforce the establishment of the cycle . 
   close to the antisymmetric cycle , the evolution between jumps is dominantly governed by @xmath188 and is therefore relatively fast . 
   this leads to frequent photon counts at detector 1 [ fig . 
   [ fig : fig9](b ) ] . 
   close to the symmetric cycle , the between - jump evolution is dominantly governed by @xmath189 , hence is relatively slow . 
   photon counts at detector 1 become much less frequent [ fig . 
   [ fig : fig10](b ) ] .    from the dramatic difference in count rates at detector  1 for the two cycles , 
   it is clear that one can determine which entanglement cycle the system evolves to for a particular realization . 
   however , without knowledge of the record of photon counts at detector 2 , which by definition we do not have , one can not know where on the cycle the system is , i.e. , whether the state is in @xmath78 or @xmath79 . 
   thus , the ensemble average state of the system is mixed , described by one of the density operators @xmath190 
   consider a thought experiment where the cascaded qubit system , set to resonance , evolves freely and its entire output is collected and stored inside a black box . at some time the lasers driving the raman transitions are turned off , so the evolution ceases . 
   the box and qubits are separated and moved to causally disconnected regions of space time . 
   let alice and bob be standard observers of the qubits , and give eve jurisdiction over the box . 
   we can now ask , how much entanglement exists between the qubits of alice and bob ? while this is simply a roundabout way of asking how entanglement evolves , it helps elucidate some of the key concepts behind the quantum trajectory measure of entanglement . 
   conventional entanglement measures are based upon an analysis of the density matrix at this time . 
   they throw away the box and look at the system of qubits alone  they disregard eve and view the system from the perspective of alice and bob .    yet in general 
   every interaction between two objects entangles them , and as the qubit system and box interacted in the past , their states are intertwined . neither 
   possess an independent reality , and neither , considered alone , can be completely described . 
   eve s box contains information , which , if discarded , adds entropy to the qubit system of alice and bob . 
   this entropy is the source of ambiguity in the quantification of entanglement . from this point of view , as noted in the introduction , the problem of bi - partite entanglement in an open system relates to that of tri - partite entanglement in a closed one . to completely characterize the entanglement of the present example , in addition to the entanglement between alice and bob 
   , we must consider their entanglement with eve . 
   a quantum description of the box is impractical , but it is feasible to extract classical information about what it contains , through measurement . 
   quantum trajectories facilitate this , and allow us not to discard the box completely . in turn 
   , the system state retains its purity , conditional on the classical information extracted from the box . with this extra information 
   , we can extract more entanglement from the cascaded qubit system . 
   working from the master equation for the cascaded system @xcite , previously it was assumed that the system evolved gradually into a pure state , whereby entanglement was generated . 
   the behaviour at resonance , however , was unclear , since there the master equation had two zero eigenvalues and no well - defined steady state . by considering the conditional evolution we have shown that , at resonance , asymptotically the system is either in the bell state @xmath151 or oscillating ( stochastically switching ) between two bell states , @xmath146 and @xmath147 . 
   from the density matrix point of view , the latter is an equal mixture of bell states and would yield no entanglement under any mixed state measure ; physically , alice and bob , without collaboration from eve , can not extract any entanglement from their qubits . 
   suppose , however , that eve opens her box to count the number of photons inside . 
   seeing whether the count is even or odd , she is able to deduce exactly which bell state alice and bob s system is in . 
   thus , her measurement unravels the density operator , creating entanglement , despite the fact that the measurement is not causally connected to alice and bob s qubits . 
   it is tempting to say that the entanglement was always there , as a matter of fact , until one realizes that there are many other ways in which eve could choose to measure her state , each producing a different unravelling of the qubit system and yielding a different value of entanglement . 
   the entanglement facilitated by eve s measurements is _ contextual _ in this sense . 
   this thought experiment demonstrates why any attempt to quantify the entanglement of an open system from the density operator alone can not be considered complete . 
   the density operator should not be treated as a fundamental object , as it does not provide a complete description of the physical state . 
   we have presented a simple example where oscillations between maximally entangled states are hidden within a separable density operator . the fact that the density operator contains entropy , implies that information about its entanglement with an external system was discarded at some time . in studying such a mixed state , there is benefit from considering , not only the mixed state itself , but the process through which it was generated , and the access this potentially gives to a conditional dynamics . 
   the results of this paper could be extended by employing quantum trajectories in a broader sense . in cases where the results of environmental interactions can not be measured , such as coupling loss , wiseman and vaccaro @xcite 
   have shown that only certain unravelings can be physically realized . a conceivable measure of entanglement would take the minimum of all physically realizable unravelings . 
   alternatively , one might take the maximum of all physically realizable unravellings , which would measure the maximum distillable entanglement when local measurements on the environment are taken into account . 
   h.  j. carmichael , p. kochan , and l. tian , `` coherent states and open quantum systems : a comment on the stern - gerlach experiment and schrdinger cats , '' in _ coherent states : past , present , and future _ , eds . 
   d.  h. feng , j.  r. klauder , and m.  r. strayer ( world scientific , singapore , 1994 ) , pp .  75 - 91 .
summary: a system of cascaded qubits interacting via the oneway exchange of photons is studied . while for general operating conditions the system evolves to a superposition of bell states ( a dark state ) in the long - time limit , under a particular _ resonance _ condition no steady state is reached within a finite time . 
   we analyze the conditional quantum evolution ( quantum trajectories ) to characterize the asymptotic behavior under this resonance condition . 
   a distinct bimodality is observed : for perfect qubit coupling , the system either evolves to a maximally entangled bell state without emitting photons ( the dark state ) , or executes a sustained entangled - state cycle  random switching between a pair of bell states while emitting a continuous photon stream ; for imperfect coupling , two entangled - state cycles coexist , between which a random selection is made from one quantum trajectory to another .