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in optical communication systems , information is often carried by states with gaussian statistics ; the coherent state is a simple and very important example . the processing of these states in the quantum domain is now a major concern toward attaining ultimate capacity of bosonic channels @xcite , and also the construction of quantum networks and computers @xcite . toward these applications , it is important to generate and manipulate superpositions of pure gaussian states . in fact , certain types of such superposition states can be used as ancillary states to build up a universal set of operations to construct an arbitrary processor  @xcite . ( see also ref  @xcite for its experimental perspectives . ) these states are essentially non - gaussian states , and are easily destroyed even by a small amount of losses . in practice , it is not easy to prepare in labs the state that preserves enough purity and useful non - gaussian properties . therefore , one needs to develop the methods how to stabilize and/or purify such states . a simple and more experimentally feasible example is a superposition of two coherent states with small amplitudes . such a state can be conditionally generated from a squeezed vacuum ( gaussian state ) by using the photon number measurement  @xcite . recently , a proof - of - principle experiment of this idea was successfully presented in ref  @xcite , where an on / off - type detector ( which clicks upon the arrival of a photon ) was used rather than a photon number resolving detector . there is , however , still a distance between the observed state and the ideal one due to various losses .    in this paper , we develop a feasible scheme to purify such coherent superposition states ( csss ) that has undergone linear dissipation . we consider a specific scenario of the purification , but it meets the current experimental situations and is based only on the use of a single beam splitter and a single homodyne detector , which probabilistically selects out favourable events . the scenario we treat is the following ; consider a css of an equal superposition of two distinct coherent states , @xmath0 , where the amplitude @xmath1 and the phase @xmath2 are a priori known and @xmath1 is relatively small . this state decoheres into a mixed state by the linear loss and represented by a classical mixture of a css and two coherent states . our task is then to amplify the probability of the css as efficiently as possible . the css itself has also been discussed as a resource of various optical quantum information processing protocols @xcite . as a related topic to our study , glancy _ et al . _ have proposed a method to restore a css , whose weights and phase information of the superposition are unknown , by applying a three - bit quantum error correcting code @xcite . however , this proposal requires almost perfect photon number resolving detectors and pure csss as ancillary states . in our study we make no use of non - gaussian ancillary states in the purifaction method . the schematic of our proposal is shown in fig . [ fig : scheme ] . a fraction of the input state is tapped off by a beam splitter and subsequently measured with a homodyne detector . from the measurement outcomes we try to distinguish the css from the mixture of coherent states through the differences of their wave functions . based on the success of this estimation , the coherence of the css is retrieved through selection of favourable events . we show that homodyne measurement is one of the optimal measurements for the most efficient purification given such a simple setup .    using the purification protocol in fig .  [ fig : scheme ] , the amplitude of the coherent state components in the ccs is inevitably degraded due to the conditional selection of events . so we then address the possibility to recover the amplitude by using the css amplification scheme proposed in refs @xcite , where it has been shown that a css with small amplitudes of its constituents is equivalently generated by squeezing a single photon state and the amplitudes can be amplified using linear optics and on / off detectors . this paper is organized as follows . in section [ sec:2 ] , the influence of linear dissipation to the css is briefly reviewed . in section [ sec:3 ] , we discuss the purification scheme in general terms without specifying the exact measurement strategy applied to the tapped off state . in section [ sec:4 ] , we consider the example where a homodyne detector is used and show that it is one of the optimal measurement strategies . in section [ sec:5 ] , we discuss the amplification of css using two csss that has suffered from linear loss , and finally in section [ sec:6 ] we conclude .    . ] in this section we briefly summarize the evolution of css s under linear dissipation . let us first define a superposition of two coherent states ( css ) with the amplitudes @xmath3 and the relative phase @xmath2 as @xmath4 where @xmath5 we assume @xmath6 to be real without loss of generality .    in practice , the most dominant loss in optical channels is due to a linear interaction between the signal mode and the vacuum environment . this is modeled by coupling the signal mode to the environment , letting them evolve for some time and finally trace out the states of the environment . a completely positive ( cp ) map describing such a linear interaction between the signal and the environment is given by @xcite @xmath7 , \end{aligned}\ ] ] where @xmath8 and @xmath9 are the superoperators defined as @xmath10 for an arbitrary operator @xmath11 . the transmission of the environment is denoted by @xmath12 and thus the linear loss is given by @xmath13 . using the transformation @xmath14 \nonumber\\ & &     \times| \alpha_1 \sqrt{\eta } \rangle \langle \alpha_2 \sqrt{\eta } | .\end{aligned}\ ] ] we find that the css , @xmath15 where @xmath16 is a mixed state of two coherent states @xmath17 . here the fraction of the original state in this mixed state is given by @xmath18 as expected , the linear lossy interaction with the environment transforms the original pure css state into a classical mixture . the aim of this paper is to develop a method by which the decohered state can be probabilistically purified to produce a smaller number of states with higher purity , or in other words ; the aim is to increase the value of @xmath19 . finally , we note that the purity defined by @xmath20 $ ] is not a suitable figure of merit for our purpose . one can easily show that , for the mixed css in eq . ( [ eq : decohered_cat ] ) , a smaller amplitude gives a higher purity of the state even for small @xmath19 . this is because such a state is close to a vacuum state which is completely pure . however , it is obvious that it does not mean the state is close to a pure css . our proposed scheme for probabilistic purification of csss is illustrated in fig . [ fig : scheme ] . the state under interrogation is mixed with the vacuum at a beam splitter with the transmission coefficient @xmath21 . the reflected beam is subsequently measured using a positive operator - valued measure ( povm ) generally described by a set of positive operators @xmath22 with @xmath23 . depending on the outcome of this measurement , the transmitted state is probabilistically purified : if the outcomes fall within a certain detection range , the transmitted state is purified , otherwise the state turns out to be worse and is discarded . let us first describe the function of the purification scheme when an already pure css @xmath24 is injected into it . in this case , the output of the beam splitter ( described by the unitary operator @xmath25 ) is given by the entangled state @xmath26 where @xmath27 is the reflectivity of the beam splitter . if we now consider only one output and trace out the other , the pure css is transformed into a mixed state as a result of the coupling to the vacuum environment . the purification protocol now works by making a measurement on the `` environmental '' mode ( mode b ) and selecting the outcomes that map @xmath28 back onto a pure css at the output . it implies that the measurement outcomes for successful events have to satisfy @xmath29 where the superscript @xmath30 denotes the successful events . the unnormalized conditional output for such outcomes is given by @xmath31 ) , it is possible to recover the purity of the input css . let us now consider the purification of a decohered css . the input to the purifier is now the mixture @xmath32 the reflected part of the state is measured using the above mentioned povm and events corresponding to condition ( [ eq : condition ] ) are selected . the resulting density operator of the output state conditioned on this measurement is given by @xmath33 where @xmath34   \nonumber\\   & = & \left|\langle\pi_j^s|\sqrt{r}\alpha\rangle\right|^2   \frac{1+\cos(\varphi+\theta_j ) \ , e^{-2t\alpha^2 } } { 1+\cos\varphi \ , e^{-2\alpha^2 } } \end{aligned}\ ] ] and @xmath35   \nonumber\\ & = & \left|\langle\pi_j^s|\sqrt{r}\alpha\rangle\right|^2    \label{eq : det_probability_0}\end{aligned}\ ] ] correspond to the probability distributions for detecting the outcomes @xmath36 in the states @xmath37 and @xmath38 , respectively . as a consequence , one finds the fraction of the css after the purification as @xmath39 the condition for successful purification @xmath40 is then given by @xmath41 the maximum purification efficiency is achieved when @xmath42 is minimized . there are two free parameters controlling the magnitude of this fraction , namely @xmath43 and @xmath21 . first we note that the phase @xmath43 is optimized , that is @xmath42 is minimized when @xmath44 . second , it is clear that the fraction ( [ eq : purif_codition ] ) decreases with decreasing transmissions @xmath21 . this , on the other hand , means that the amplitude of the css is decreasing . therefore , there is a trade - off between the efficiency of the purification protocol and the amplitude of the purified css . since large amplitude csss are desired for many applications , we consider the possibility of amplifying the degraded css in sec .  v. the optimal povm for the purification protocol is therefore the one that obeys eq . ( [ eq : condition ] ) and simultaneously minimizes the fraction ( [ eq : purif_codition ] ) by satisfying the constraint @xmath45 . these conditions provide a nice flexibility to chose among different measurement strategies . two well known measurement strategies that satisfy the requirements are photon number resolving detectors and homodyne detectors . if however the photon number resolving detector is used to execute the purification protocol , successful demonstration will only be obtained if the phase between the coherent state components in the css , is @xmath46 or @xmath47 . furthermore , these detectors are experimentally very challenging to construct and thus , to date , they are not an integral part of the standard lab .    fortunately , homodyne detectors ( which have matured much further than photon number resolving detectors ) also meet the requirements for successful purification . in fact by using homodyne detectors there are no restrictions on the phase @xmath2 . in the following section we investigate in greater details the purification of css using homodyne detection . a homodyne detector measures a quadrature of the electromagnetic field , with the specific quadratures being determined by the phase , @xmath48 , of the intrinsic local oscillator . it is mathematically described by an operator @xmath49 which projects the state onto a continuous set of quadrature eigenstates @xmath50 with @xmath48 denoting the quadrature phase . specific and important examples are the position and momentum eigenstates corresponding to @xmath51 and @xmath52 , respectively . the probability amplitude associated with the detection of a quadrature eigenvalue , @xmath53 , from the coherent state @xmath54 by homodyne detection is given by @xmath55.\end{aligned}\ ] ] the necessary condition for our purification scheme to work as quoted in eq . ( [ eq : condition ] ) is fulfilled when @xmath52 , that is when the momentum eigenstate is measured . in that case we obtain @xmath56 from which we see that the phase shift introduced by the measurement depends on @xmath57 as @xmath58 note that the homodyne measurement always allows us to choose the appropriate @xmath57 to optimize @xmath59 for any given @xmath2 .    in the following we set @xmath60 . the unnormalized conditional output corresponding to eq . ( [ eq : cond_output ] ) is then given by @xmath61 and one obtains the probability distributions for detecting the @xmath62 and @xmath63 as    @xmath64    and @xmath65 respectively .     with @xmath66 ( solid line ) and a mixture of coherent states @xmath67 ( dotted line ) . @xmath68 and @xmath69 . the purification protocol is successful when the measurement outcome falls within the interval indicated by the dashed vertical lines . ]     with @xmath70(solid line ) and a mixture of coherent states @xmath67 ( dotted line ) . @xmath68 and @xmath69 . the purification protocol is successful when the measurement outcome falls outside the interval indicated by the dashed vertical lines . ] [ fig : cond_k_p ] and [ fig : cond_k_m ] show these probability distributions for @xmath46 and @xmath47 , respectively . the probability distributions @xmath67 , which is represented by the dashed curves , are gaussian , while the probability distributions for the css , @xmath71 , ( represented by solid lines ) are non - gaussian due to the quantum coherence between the two coherent state components . as discussed above , these distinct differences between the probability distributions enable the purification protocol : it succeeds when the absolute quadrature value falls within a certain threshold for @xmath66 and outside a certain threshold for @xmath72 , since in these regions the probability for a css to occur is larger than the probability for @xmath63 to occur . the efficiencies of the purification protocol for @xmath46 and @xmath73 are illustrated in figs . [ gainp ] and [ gainm ] , respectively . in case of @xmath46 , the purification condition of eq . ( [ eq : condition ] ) is satisfied in the range given by @xmath74 and the efficiency of the purification protocol is optimised for @xmath75 . the protocol works more efficiently when the initial css has the phase @xmath73 . in that case , purification is achieved in the range given by @xmath76 and is maximized at @xmath77 where the initial input phase @xmath73 is transformed into @xmath46 after purification . the maximum increase of @xmath78 as a function of @xmath79 for @xmath80 and @xmath81 is depicted in fig . [ gain - p ] , and we see that the improvement for these parameters is highest around @xmath82 .    in figs . [ gain - alpha ] and [ gain - t ] , we show how the purification efficiency depends on the amplitude @xmath1 and the transmittance of the beam splitter @xmath21 , respectively , for an input state with @xmath83 . from fig . [ gain - alpha ] , we see that the purification of states with @xmath66 ( represented by the solid curve ) is most effective around @xmath84 whereas for states with @xmath70 the efficiency increases with smaller amplitudes . in fig . [ gain - t ] , where the purification efficiency is plotted versus the transmission coefficient , it is evident that small transmission coefficients are advantageous . however , as already pointed out above , small transmission coefficients are on the other hand not desirable since it degrades the amplitude of the state . note that , obviously the purification does not work at @xmath85 . although , in fig .  [ gain - t ] , @xmath86 is still larger than 1 for @xmath73 at @xmath85 , the probability amplitude @xmath87 goes to zero and thus this event never happens . these results , therefore , show that our scheme can be used to purify small amplitude csss , which are exactly the ones that recently has been produced @xcite . however , the reduction of amplitude of the css is unavoidable in our scheme and it is therefore desirable to amplify the amplitude of the csss after purification . this is the topic of the next section .    before ending this section , we address the effect of practical losses in homodyne detection . in practice , the efficiency of homodyne detection is less than unity due to the imperfect quantum efficiency of photodiodes , misalignment , mode mismatch , losses of optics etc . such non - ideal homodyne detection can be modeled by a lossy quantum channel defined in eq . ( [ eq1.1 ] ) plus a perfect homodyne detector @xcite . the effect of this imperfection in our scheme is estimated by replacing @xmath12 with @xmath88 in eq . ( [ eq : p ] ) where @xmath89 is the effective efficiency of the homodyne detection . however , it should be noted that the recent quantum network experiments in cw domain at the wavelength of si - photodiode can achieve sufficiently high homodyne effciency , e.g. @xmath90 in @xcite . for example , with @xmath73 , @xmath68 , @xmath69 , and @xmath83 , the purified outputs will be @xmath91 and @xmath92 for @xmath93 and @xmath94 , respectively . ( although the experiment in @xcite was performed in pulsed regime , the feasibility of non - gaussian experiments with cw sources has also been discussed in @xcite for example . )     conditioned on the homodyne detection outcomes . the phase of the local oscillator is chosen to measure the quadrature @xmath60 . @xmath68 , @xmath69 , and @xmath83 . ]     conditioned on the homodyne detection outcomes . the phase of the local oscillator is chosen to measure the quadrature @xmath60 . @xmath68 , @xmath69 , and @xmath83 . ]     on @xmath79 for the css with @xmath46 ( solid line ) and @xmath73 ( dashed line ) where the outputs are conditioned on the measurement outcomes @xmath95 and @xmath96 , respectively . @xmath68 and @xmath69 . the dotted line denotes @xmath97 as a reference . ] ( solid line ) and @xmath73 ( dashed line ) . @xmath68 and @xmath83 . the measurement outcome is selected to satisfy the optimal condition @xmath98=0 . ]     ( left ) and the probability density @xmath99 ( right ) on the transmittance of the tapping beam splitter @xmath21 for @xmath46 ( solid line ) and @xmath73 ( dashed line ) . @xmath83 and @xmath81 . the measurement outcome @xmath100 is selected to satisfy the optimal condition @xmath101 . ] lund _ et al . _ have proposed a scheme to conditionally produce a large amplitude css from two small amplitude csss by using two 50/50 beam splitters , an ancillary coherent state and two on / off photon detectors @xcite , and later on the scalability of this scheme was studied in details  @xcite . they pointed out that when one uses a weakly squeezed single photon state corresponding to a small css ( which in practice inevitably consists of a mixture between the single photon state and a vacuum state ) it is possible to reduce the squeezed vacuum component during the amplification process . in this section , we discuss the application of such a scheme to obtain a larger amplitude css from two decohered csss . we show that , the decoherence is , unfortunately , in most cases increased as a result of the amplification process . the scheme proposed in ref . @xcite is illustrated in fig . [ fig : amplification ] . two copies of the csss with small amplitude are combined on a 50/50 beam splitter . then one of the outputs is mixed with a coherent state with amplitude @xmath102 on a 50/50 beam splitter . subsequently the two outputs are measured using two on / off photon detectors . if both detectors click , the remaining quantum state is projected onto a larger css . in addition to the amplification of the amplitude , the projection process also filter out the squeezed vacuum component when the inputs are two mixed states of weakly squeezed single photon and vacuum .    now , suppose that the input is two copies of the decohered css @xmath103 defined in eq . ( [ eq : initial_state ] ) . then , the input state is described by the tensor product @xmath104 and in the conditioned amplified output state , only the first term is mapped onto a large css whereas all other terms are transformed into a mixture of coherent states . as a result , the fraction of a css in the conditioned output states are approximately degraded as @xmath105 . more precisely , the output probabilities @xmath106 and @xmath107 with inputs @xmath108 and @xmath109 , respectively , are given by @xmath110 note that the conditioned output css with the probability @xmath111 is always @xmath112 @xcite . the parameter region where the inputs are purified during the conditional process can be evaluated from the condition @xmath113 . first , we find that if the phase @xmath2 is @xmath114 ( that is @xmath108 is amplified ) , the inequality can not be satisfied . therefore , in this case , the output probability @xmath106 is always smaller than the input probability , or in other words , the decoherence is always amplified . however , if the amplitude of decohered css with @xmath70 ( that is @xmath109 ) are amplified , then the inequality @xmath115 can be satisfied if @xmath116 this implies that the average photon number of the input css must be @xmath117 for successful amplification of the amplitude without amplifying the decoherence . finally , we investigate the possibility to repeat the purification process by concatenating this amplification scheme . the situation considered here is illustrated in fig . [ fig : concatenation ] . two initial states @xmath118 are purified in parallel by the purification processes with @xmath80 . once the purified outputs @xmath119 from both processes are simultaneously obtained , the outputs are injected into the amplification process to obtain the final output which has the original amplitude @xmath1 . let us now determine whether the output probability @xmath78 is larger than the input probability @xmath79 for such a machine : since the phase of the css obtained from the amplification process is always @xmath120 , we assume the initial states to have @xmath46 . we then look for parameters where @xmath40 by replacing @xmath79 in eq . ( [ eq : amp_out ] ) by @xmath121 ( which is the output probability from the purification step ) . however , unfortunately , a parameter interval fulfilling such criterion does not exist , and we conclude that for all possible amplitudes @xmath1 , the decoherence is amplified along with the amplitudes . it suggests that when the goal is to generate large amplitude csss from smaller ones by using the conditional processes with linear optics proposed in ref . @xcite , we have to be careful about the effects of linear loss even if the initial states contain only a very small fractions of mixtures of coherent states . as we have shown , the decoherence is in most cases amplified during the conditional generation of large csss . in this paper , we have investigated a simple scheme to recover the quantum coherence of decohered coherent state superpositions . the scheme we have examined is based on partial estimation of the state by tapping off a part of it using a beam splitter , measuring the reflected part by homodyne detection and finally select out favourable events that maps the decohered state onto a purer state . we derived the condition for the measurement to maximize the efficiency of the purification and showed that homodyne measurement is one of the optimal strategies . since homodyne measurement is a well developed technique , our purification scheme is experimentally easier to implement than previously proposed purification protocols . we therefore believe that our scheme can be directly implemented in current on - going experiments on the generation of css to compensate for losses and increase the nonclassicality of the states . in addition , we showed that this simple measurement - induced scheme inevitably causes further amplitude degradation and is not suitable for the purification of large amplitude css with @xmath122 . moreover , we have discussed the possibility to restore the degraded amplitude by the linear optical scheme proposed in @xcite . if the initial csss are decohered due to a linear loss , the scheme is increasing not only the amplitude but also the degree of the mixture . we showed that , even combining this scheme with the purification process , it is still hardly possible to recover the coherence of css . note that , also in the context of entanglement purification , it has been suggested in @xcite that the purification of entangled coherent states decohered by linear loss also seems to be difficult by using only linear optics and photon counting without ancillary csss . these results emphasize the importance of the further investigations of the purification scheme .
we present a simple protocol to purify a coherent state superposition that has undergone a linear lossy channel . 
 the scheme constitutes only a single beam splitter and a homodyne detector , and thus experimentally feasible . in practice 
 , a superposition of coherent states is transformed into a classical mixture of coherent states by linear loss , which is usually the dominant decoherence mechanism in optical systems . 
 we also address the possibility of producing a larger amplitude superposition state from decohered states , and show that in most cases the decoherence of the states are amplified along with the amplitude .