Gaussian states under coarse-grained continuous variable measurements
aa r X i v : . [ qu a n t - ph ] J un Gaussian states under coarse-grained continuous variable measurements
Jiyong Park, Se-Wan Ji, Jaehak Lee, and Hyunchul Nha
1, 2 Department of Physics, Texas A&M University at Qatar, Education City, P.O.Box 23874, Doha, Qatar School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea
The quantum-to-classical transition of a quantum state is a topic of great interest in fundamentaland practical aspects. A coarse-graining in quantum measurement has recently been suggested as itspossible account in addition to the usual decoherence model. We here investigate the reconstructionof a Gaussian state (single mode and two modes) by coarse-grained homodyne measurements. Tothis aim, we employ two methods, the direct reconstruction of the covariance matrix and the max-imum likelihood estimation (MLE), respectively, and examine the reconstructed state under eachscheme compared to the state interacting with a Gaussian (squeezed thermal) reservoir. We clearlydemonstrate that the coarse-graining model, though applied equally to all quadrature amplitudes, isnot compatible with the decoherence model by a thermal (phase-insensitive) reservoir. Furthermore,we compare the performance of the direct reconstruction and the MLE methods by investigatingthe fidelity and the nonclassicality of the reconstructed states and show that the MLE method cangenerally yield a more reliable reconstruction, particularly without information on a reference frame(phase of input state).
PACS numbers: 03.65.Ta, 42.50.Dv, 03.65.Ud
I. INTRODUCTION
The discrepancy between quantum and classical me-chanics over the description of physical phenomena haslong been an object of interest and controversy. Althoughquantum mechanics has been successful in describing andmanipulating a microscopic world, a macroscopic worldcan interestingly be explained by classical mechanics thathas different premises and framework from quantum me-chanics. There has thus been much interest in accountingfor the quantum-to-classical transition, and in particular,the decoherence by environmental interactions is nowa-days perceived as one of the most promising models inthis respect [1, 2].Recently, there have also been some different attemptsto explain the quantum-to-classical transition [3–6]. Incontrast to the decoherence program, these focus on theinefficiency of quantum measurement, namely, coarse-grained outcomes by imperfect detectors [3, 4] or impre-cise control of target operations [5, 6]. Comparing theseapproaches to the usual decoherence model is thus im-portant to extending our understanding of the quantum-to-classical transition.In this paper, we investigate single-mode and two-mode Gaussian states under the coarse-graining in thehomodyne measurement. Gaussian states and operationsprovide crucial elements of quantum information process-ing for continuous variables and have been extensivelystudied both theoretically and experimentally [7]. Ourcoarse-graining model is similar to the Ehrenfest’s ideaof coarse graining [8, 9], and recently the same modelhas been considered in the context of the uncertainty re-lation [10, 11] and the entanglement detection [12]. Un-like the last of these [10–12], where the obtained data donot fully characterize the state under investigation, weare interested in quantum state tomography: the processof inferring the prepared quantum state from the mea- sured data [13]. Reconstructing the density matrix or thephase-space distribution of a quantum state, the processendeavors to provide the maximal information about thegiven state, which can also be used to verify nonclassi-cal features, e.g. negativity in phase-space and entan-glement. Using the coarse-grained data from homodynedetection, we may reconstruct a Gaussian state and com-pare it to the same state under a Gaussian noisy channel(squeezed thermal environment), thereby comparing thecoarse-graining model and the decoherence model in viewof the quantum-to-classical transition.In quantum optics, the inverse Radon transformationof the marginal distribution acquired from homodyne de-tection was theoretically proposed [14] and experimen-tally implemented [15, 16] to reconstruct the Wigner dis-tribution of a given state. However, the direct applicationof the inverse Radon transformation yields an unphysi-cal state due to the unavoidable process of data binning[17]. To assure the legitimacy of the reconstructed state,quantum state estimation, which is to determine the mostprobable physical state from the measured data, was pro-posed [18] and has been employed in experiments [19–22].We here employ two methods for state reconstructionunder coarse-graining, namely, a direct reconstruction ofthe covariance matrix and a maximum likelihood estima-tion (MLE) [18]. The coarse graining is equally appliedto the homodyne measurement of each quadrature am-plitude, and is therefore isotropic in phase space. Onemight then expect that there can exist an equivalent de-coherence model by a thermal reservoir, more precisely, aphase-insensitive Gaussian reservoir. We, however, showthat it is not the case.Furthermore, we investigate the performance of tworeconstruction methods by examining the fidelity be-tween an input state and the reconstructed state and thenonclassicality (squeezing or entanglement) of the recon-structed state. In a realistic situation, sharing the ref-erence frames between the preparer and the verifier canbe a critical issue. We thus study how this issue canparticularly affect the performance of the direct recon-struction method by considering cases with and withoutinformation on the phase of the input state.
II. PRELIMINARIES
To begin with, we first introduce our coarse-grainingmodel with homodyne mesurements and the decoherencemodel with an environmental interaction, respectively.
A. Homodyne measurement under coarse-graining
A homodyne detector measures the quadrature ampli-tude ˆ X ϕ = (ˆ a † e iϕ + ˆ ae − iϕ ) / a (ˆ a † ) is the annihilation (creation) operator and ϕ is thephase determined by a local oscillator. The probabilitydistribution P ( x ϕ ) of the amplitude x ϕ is given by [14] P ( x ϕ ) = 1 π Z ∞−∞ dkC ( λ = ike iϕ ) e − ikx , (1)where C ( λ ) is the characteristic function of the state ρ , C ( λ ) = tr[ ρ ˆ D ( λ )] , (2)with the displacement operator ˆ D ( λ ) = exp( λ ˆ a † − λ ∗ ˆ a ).The characteristic function C ( λ ) contains the full infor-mation on the state ρ . In turn, a complete set of homo-dyne measurements over all phase angles ϕ ∈ [0 , π ] canbe used to construct the density matrix ρ or equivalentlyits phase-space distributions.Suppose now that the homodyne measurement doesnot yield a smooth continuous distribution due to the in-efficiency of photodetectors. More precisely, if the mea-surement cannot distinguish the values of x ϕ within aninterval of size σ , similar to the Ehrenfest’s idea of coarse-graining [8, 9], we obtain a coarse-grained probabilitydistribution as P D ( x ϕ ) = ∞ X m = −∞ P σ [ m, ϕ ]rect (cid:18) xσ − m (cid:19) . (3)Here, rect( x ) is a step functionrect( x ) = (cid:26) | x | > / , | x | ≤ / , (4)and P σ [ m, ϕ ] represents the coarse-grained (averaged)probability in the region of x ∈ [( m − ) σ, ( m + ) σ ]as P σ [ m, ϕ ] ≡ σ Z ( m + ) σ ( m − ) σ dxP ( x ϕ ) , (5)using P ( x ϕ ) from Eq. (1). As an example, Fig. 1 il-lustrates how the coarse-graining process transforms an - - p H x L FIG. 1: (Color online) Illustration of the coarse-graining pro-cess in Eq. (3). A Gaussian probability distribution (red solidline) is transformed to a piecewisely flat distribution (blackdashed line) under the coarse-graining of size σ = 0 . original distribution P ( x ϕ ) to a piecewise flat distribu-tion P D ( x ϕ ).In general, it is known that this coarse-grainedmarginal distribution cannot be directly used to recon-struct a density matrix or its phase-space distributionsbecause the output does not correspond to a physi-cal state [17]. To reconstruct a legitimate quantumstate from the coarse-grained homodyne measurement,we thereby employ an MLE method that is designed tofind the most probable physical state by maximizing thelog-likelihood estimator L = Z dµP D ln P E , (6)where µ is a probability measure, P D the probabilitydistribution obtained from measurement, and P E is theprobability distribution from an estimated state. Fromthe perspective of information theory, the method can beseen as the minimization of the relative entropy of twodistributions D ( P D || P E ) ≡ Z dµP D ln P D P E , (7)that is, we optimize P E for a given P D to obtain a min-imal value of D ( P D || P E ). The relative entropy becomeszero if and only if P D = P E , that is, only when the ob-tained data can correspond to a certain physical state. B. n -mode Gaussian states under Gaussianreservoirs An n -mode Gaussian state is fully identified by its firstand second moments (for a review, see Ref. [7]). It hasa Gaussian characteristic function in the form C ( λ ) ≡ tr[ ρ n Y i =1 ˆ D i ( λ i )]= exp( − λ Γ λ T + i √ h ˆ R i λ T ) , (8)where ˆ R ≡ (ˆ q , ˆ p , ..., ˆ q n , ˆ p n ) is related to the quadra-ture amplitudes ˆ X i, = ˆ q i / √ X i,π/ = ˆ p i / √ i ∈ { , ..., n } , λ = ( ℑ [ λ ] , −ℜ [ λ ] , ..., ℑ [ λ n ] , −ℜ [ λ n ]),with Re[ λ i ] and Im[ λ i ] the real part and the imaginarypart of λ i , respectively. Γ is the covariance matrix whoseelements areΓ ij = 12 h ˆ R i ˆ R j + ˆ R j ˆ R i i − h ˆ R i ih ˆ R j i , (9)where h ˆ o i ≡ tr( ρ ˆ o ) is the expectation value of the opera-tor ˆ o .A Gaussian process transforms a Gaussian state intoanother Gaussian state, and a typical Gaussian processis the environmental interaction with Gaussian (thermalsqueezed) reservoirs, which usually leads to decoherence.This decoherence process can be described by a masterequation˙ ρ ( t ) = n X i =1 γ i { ( N i + 1) L [ˆ a ] + N i L [ˆ a † ] − M ∗ i D [ˆ a ] − M i D [ˆ a † ] } ρ ( t ) , (10)where γ i is the interaction strength for the i -th mode, and L [ˆ o ] ρ = 2ˆ oρ ˆ o † − ˆ o † ˆ oρ − ρ ˆ o † ˆ o and D [ˆ o ] ρ = 2ˆ oρ ˆ o − ˆ o ˆ oρ − ρ ˆ o ˆ o are Lindblad superoperators. The covariance matrix ofthe reservoir interacting with the i -th mode is given by Γ i,r = (cid:18) + N i + ℜ [ M i ] ℑ [ M i ] ℑ [ M i ] + N i − ℜ [ M i ] (cid:19) , (11)with N i and M i representing the mean thermal photonnumber and the squeezing parameter of the reservoir, re-spectively. The master equation in Eq. (10) can be con-verted into a differential equation for the characteristicfunction ∂∂t C ( λ , t ) = − n X i =1 γ i A i + B i ) C ( λ , t ) . (12)where A i = (1 + 2 N i ) | λ i | − M i ( λ ∗ i ) − M ∗ i λ i ,B i = λ ∗ i ∂∂λ ∗ i + λ i ∂∂λ i . (13)The solution to Eq. (12) can be represented in terms ofthe covariance matrix Γ ρ ( t ) of the state at time t Γ ρ ( t ) = √ G [ Γ ρ (0) − Γ r ] √ G + Γ r , (14)where G = L ni =1 exp( − γ i t ) and Γ r = L ni =1 Γ i,r . III. SINGLE-MODE GAUSSIAN STATEESTIMATION
In this section, we investigate single-mode Gaussianstates reconstructed from the coarse-grained homodynedata. In general, as mentioned before, one can recon-struct a given state by measuring the probability dis-tributions of quadrature amplitudes for all (practicallyspeaking, many) phase angles and then relying on theRadon transformation [14]. We adopt this approach un-der the coarse grained measurement together with theMLE method.
A. Direct reconstruction of covariance matrix
On the other hand, since a Gaussian state is completelyidentified by its first and second moments, one can alsoreconstruct the given state by determining only those mo-ments, which will be another approach, namely, a directreconstruction of the covariance matrix. For the caseof ideal homodyne detection, the moments can be de-termined by measuring only three different quadraturesˆ X ϕ =0 , ˆ X ϕ = π/ , and ˆ X ϕ = π/ asΓ = h ˆ q i − h ˆ q i = 2 h ˆ X ϕ =0 i − h ˆ X ϕ =0 i , Γ = h ˆ p i − h ˆ p i = 2 h ˆ X ϕ = π/ i − h ˆ X ϕ = π/ i , Γ = 12 h ˆ q ˆ p + ˆ p ˆ q i − h ˆ q ih ˆ p i =2 h ˆ X ϕ = π/ i − h ˆ X ϕ =0 i − h ˆ X ϕ = π/ i− h ˆ X ϕ =0 ih ˆ X ϕ = π/ i , (15)where the n -th moment of the quadrature ˆ X ϕ is givenby h ˆ X nϕ i = R dx ϕ x nϕ P ( x ϕ ) with a relevant probabilitydistribution P ( x ϕ ).An arbitrary single-mode Gaussian state can be ex-pressed as a displaced squeezed thermal state in the form ρ = ˆ D ( α ) ˆ S ( r, φ i ) ρ th (¯ n ) ˆ S † ( r, φ i ) ˆ D † ( α ) . (16)Here ˆ S ( r, φ i ) = exp[ − r { exp(2 iφ i )(ˆ a † ) − exp( − iφ i )ˆ a } ]is the squeezing operator with the squeezing strength r ,the angle φ i of the squeezing axis, and ρ th (¯ n ) is the ther-mal state with the mean photon number ¯ n : ρ th (¯ n ) = ∞ X n =0 ¯ n n (¯ n + 1) n +1 | n ih n | . (17)For a squeezed thermal state, the covariance matrix isgiven byΓ = (¯ n + 12 )[cosh(2 r ) − sinh(2 r ) cos 2 φ i ] , Γ = (¯ n + 12 )[cosh(2 r ) + sinh(2 r ) cos 2 φ i ] , Γ = − (¯ n + 12 ) sinh(2 r ) sin 2 φ i , (18)and its characteristic function can be expressed as C ( λ ) = exp( − Γ λ r − Γ λ i + 2Γ λ r λ i ) , (19)with λ r = ℜ [ λ ] and λ i = ℑ [ λ ]. The corresponding homo-dyne distribution is then given by P ( x ϕ ) = r π ∆ exp (cid:18) − x ϕ (cid:19) , = (¯ n + 12 )[cosh(2 r ) − sinh(2 r ) cos(2 ϕ − φ i )] . (20)Inverting relations in Eq. (18), we obtain¯ n = √ det Γ − ,r = 12 arcsinh r γ det Γ ! , φ i = − arcsin √ γ ! for Γ ≤ Γ ,π + arcsin √ γ ! for Γ > Γ , (21)with det Γ = Γ Γ − Γ and γ = (Γ − Γ ) + 4Γ .Using Eq. (21), we can determine the parameters(¯ n, r, φ i ) characterizing a single-mode Gaussian state [Eq.(16)] by simply measuring three quadratures in Eq. (15),which will be used under coarse-grained measurements.From now on, we assume that our measurement settingsare fixed to measure three quadratures corresponding tothe angles ϕ = 0 , π/ , π/ φ i in Eq. (16) is unknown to anexperimenter.Note that the finite precision σ of homodyne measure-ment under coarse-graining induces noise to the variousmoments in Eq. (15), thereby degrading the informationon the elements Γ ij of the covariance matrix. Specifi-cally, using Eqs. (3) to (5), the variance of the measuredquadrature can be decomposed into∆ X σ = σ
12 + ∆ X m (22)where the first term σ represents the variance of a flatdistribution of size σ and the second term the variance ofthe discretized probability distribution centered at eachvalue x m ≡ mσ [10, 11]. That is,∆ X m ≡ ∞ X m = −∞ x m P ( x m ) − ∞ X m = −∞ x m P ( x m ) ! , (23)where the discrete distribution is given by P ( x m ) = σP σ [ m ] using Eq. (5). For the case of the initial Gaussiandistribution with variance V (0), we have P σ [ m ]= 12 σ n erf h(cid:16) m + 12 (cid:17) σ N √ i − erf h(cid:16) m − (cid:17) σ N √ io , (24) - - - - q p H a L - - - - q p H b L - - - - - - - - - - - - Π Π Π Π Π Π Π Π Σ Φ i H c L FIG. 2: (Color online) (a) Original squeezed state (b) re-constructed state with coarse-graining σ = 0 . σ and the in-put squeezing angle φ i . In all plots, the input state has theparameters (¯ n, r ) = (0 , with the normalized coarse-graining size σ N ≡ σ/ p V (0).Under the coarse-grained homodyne detection, therefore,the characterization of the output state using the directreconstruction method is affected by way of Eq. (22) inconjunction with Eqs. (15) and (21). From Eq. (20),the variance of coherent states is given by ∆ = 1 / σ in our consideration is such that σ = 1 takes the homodyne data within the range 2∆ ofcoherent states into a single bin.First, as an illustration, we plot an original Gaus-sian (squeezed) state [Fig. 2(a)] and the reconstructedstate under coarse-graining [Fig. 2 (b)]. We can clearlysee that the degree of squeezing is degraded due tothe inevitable noise introduced by the coarse-graineddata. Furthermore, we also see that the squeezing axisis slightly rotated as a result of the coarse-grained con-struction through Eq. (15).In Fig. 2 (c), we plot the difference in the squeezing an-gle between an input state and its estimated state undercoarse-graining as a function of the coarse-graining size σ and the input squeezing angle φ i . We have used inputsqueezed thermal states with (¯ n, r ) = (0 , n, r ). This impliesthat the information on the reference frame (squeezingdirection) of an input state is important in estimatingthe given state. (See also the plots in Figs. 4 and 5.)Note that the rotation of the squeezing axis does notoccur for the input squeezing angles φ i = kπ ( k =1 , , , , ,
7) regardless of σ [Fig. 2 (c)]. This can beexplained by looking into Eq. (21), where the angle φ i isdetermined by the ratio | (Γ − Γ ) / | .(i) For φ i = π or π , i.e., the input squeezing is along thedirection half way between the q and p axes, we obviouslyobtain ∆ ˆ X ϕ =0 = ∆ ˆ X ϕ = π/ from homodyne measure-ments. This leads to Γ − Γ = 0 regardless of σ .(ii) For φ i = π or π , we obtain ∆ ˆ X ϕ =0 = ∆ ˆ X ϕ = π/ using Eq. (20), which must be true even with the coarse-graining of the homodyne data. Then, from Eq. (15), wehave 2Γ = 2∆ ˆ X ϕ =0 − ˆ X ϕ = π/ = Γ − Γ .(iii) For φ i = π or π , we obtain ∆ ˆ X ϕ = π/ =∆ ˆ X ϕ = π/ using Eq. (20). Then, from Eq. (15), wehave 2Γ = − ˆ X ϕ =0 + 2∆ ˆ X ϕ = π/ = − Γ + Γ .The above relations do not change even with added noisesdue to coarse-graining, and the ratio | (Γ − Γ ) / | is unchanged. B. Maximum-likelihood-estimation Method
Next we compare the reconstructed coarse-grainedGaussian states with the same input states under a Gaus-sian reservoir to see if there can be correspondence be-tween the two models. The decoherence by a thermalreservoir adds noise isotropically to all quadratures inphase-space, so it does not change the squeezing direc-tion of the input state. We thus immediately see thatour coarse-graining model based on direct reconstructionis not compatible with the decoherence model by a ther-mal reservoir, and to find out an equivalence, we haveto look into the case of a phase-sensitive reservoir, i.e.,a squeezed thermal reservoir. Mathematically, note thatEq. (14) can be simplified to a convex sum of two covari-ance matrices, Γ ( t ) = y Γ (0) + (1 − y ) Γ r , (25)where y = exp( − γt ) ( ∈ [0 , Σ y FIG. 3: (Color online) The fraction y in Eq. (28) to makeequal the state estimation process and the decoherence pro-gram with an isotropic (thermal) reservoir as a function ofcoarse-graining size σ . The input squeezed thermal states arecharacterized by (¯ n, r ) = (0 ,
1) (red solid line), (¯ n, r ) = (1 , n, r ) = (0 ,
2) (brown dot-dashedline). rect reconstruction based on only three quadrature distri-butions in Eq. (15), we may avoid some negative featureslike the state rotation in phase space if we obtain a fullset of homodyne data and employ the MLE method. Inthis case, the MLE works for the optimization of L = Z π dϕ Z ∞−∞ dxP D ( x ϕ ) ln P E ( x ϕ ) , (26)where P D ( x ϕ ) and P E ( x ϕ ) are the coarse-grained homo-dyne distribution and an estimated one, respectively. Ifthe estimation process and the decoherence model are tobe equivalent, there must exist a solution y ∈ [0 ,
1] ofEq. (25) for each estimated state as y = (2¯ n e + 1) sinh(2 r e ) − (2¯ n r + 1) sinh(2 r r )(2¯ n i + 1) sinh(2 r i ) − (2¯ n r + 1) sinh(2 r r ) , (27)where the subscripts e and i represent the estimatedstate and the input state, respectively. If the reservoiris isotropic, that is, a thermal reservoir with no squeez-ing r r = 0, Eq. (27) can be simplified to y = (2¯ n e + 1) sinh(2 r e )(2¯ n i + 1) sinh(2 r i ) . (28)In Fig. 3, we plot the value of y in Eq. (28) as a functionof coarse-graining size σ for the input squeezed thermalstates with the parameters (¯ n, r ) = (0 , n, r ) = (1 , n, r ) = (0 , y >
1, there-fore, the coarse-graining model cannot be made equiva-lent to the decoherence model with an isotropic (phase-insensitive) thermal reservoir.On the other hand, if we consider a squeezed ther-mal reservoir with r r >
0, we can find a solution y inthe range of y ∈ [0 , n e + 1) sinh(2 r e ) > (2¯ n i + 1) sinh(2 r i ) as clearly seenfrom Fig. 3 and Eq. (27), we readily derive the squeezingcondition r r to have a legitimate solution to Eq. (27) assinh(2 r r ) > n e + 12¯ n r + 1 sinh(2 r e ) . (29)The value of r r can be made arbitrarily small by increas-ing n r , but cannot be zero. This points out that al-though our coarse-graining model is isotropic in the sensethat the coarse-graining applies equally to each quadra-ture in phase-space, it is equivalent only to a phase-sensitive (squeezed) reservoir. Moreover, the effect ofcoarse-graining is state dependent whereas a Gaussianreservoir affects input states all equally. We may thussay that the quantum-to-classical transitions due to de-coherence program and the coarse-grained measurement,respectively, entail unequal features in general. C. Fidelity and nonclassicality
From now on, we compare the performance of two esti-mation methods, the direct reconstruction of the covari-ance matrix and the MLE, by investigating the fidelitybetween an input state and its estimated state and thenonclassicality of the estimated state. The fidelity be-tween two single-mode Gaussian states with the samemeans is given by [23] F = 1 √ ∆ + Λ − √ Λ , (30)where ∆ = det( Γ + Γ ) and Λ = 4 det( Γ + i J ) det( Γ + i J ) with the symplectic matrix J = (cid:18) − (cid:19) . (31)A single-mode Gaussian state is nonclassical, i.e.squeezed, when r > r c ≡ ln(2¯ n + 1) [24]. This con-dition can also be related to the entanglement potential( P ent ) of a single-mode Gaussian state, P ent = max[0 , r − r c ln 2 ] , (32)the amount of two-mode entanglement that can be pro-duced by injecting the given state into one mode of 50:50beam-splitter [25].In Fig. 4, we plot the fidelity F and the nonclassi-cal squeezing r nc ≡ P ent ln 2 of the estimated squeezedthermal state as a function of the coarse-graining size σ ,for the input squeezed thermal states with the param-eters (¯ n, r ) = (0 , n, r ) = (1 , n, r ) = (0 , σ becomes larger with increasing input squeezing. : Σ F H a L > Σ r nc H b L FIG. 4: (Color online) (a) Fidelity F between an input stateand its reconstructed state and (b) nonclassical squeezing r nc of the reconstructed state as functions of the coarse-graining size σ , for the input squeezed thermal states with(¯ n, r ) = (0 ,
1) [green dot-dashed line, red solid and bluedashed lines, the second curves from the top for (a) and (b)],(¯ n, r ) = (1 ,
1) [pink dot-dashed line, orange solid and purpledashed lines, the first curves from the top for (a) and thethird curves from the top for (b)], and (¯ n, r ) = (0 ,
2) [graydot-dashed line, brown solid and black dashed lines, the thirdcurves from the top for (a) and the first curves from the bot-tom for (b)]. Solid curves represent the case of the MLEmethod, dot-dashed (dashed) curves the direct reconstruc-tion method with (without) information on the input phase,respectively. For the plots of dashed curves, each point rep-resents an averaged value over the whole range of the inputsqueezing angles. See main text.
For the case of direct reconstruction, we also see thatthe information on the reference frame (phase φ i of theinput state) plays a crucial role in characterizing thegiven state. If we have access to the phase information,we may adjust our measurement settings in the directreconstruction, in which only three measurement anglesare chosen in an interval of π/ φ i , the combination oftwo variances in Eq. (22), which involves σ in a nontriv-ial way, e.g., error function, leads to non-homogeneousbehavior in the variance of the reconstructed state, andsubsequently the fidelity (dot-dashed curves), as a func-tion of σ . On the other hand, if we have no access tosuch phase information, the state characterization gen-erally becomes worse. The dashed curves in Fig. 4show the results averaged over the squeezing angle φ i of the input state, which show worse results than thedot-dashed curves. In those cases, we see that therealso exists a rather counter-intuitive regime that a lessnonclassical state is more robust to the increment of thecoarse-graining, i.e., the crossover of two dashed curvesaround σ = 1 for the nonclassicality in Fig. 4 (b).The input squeezed thermal state with the parameters(¯ n, r ) = (0 ,
1) retains nonclassical squeezing even whenthe input state with a larger squeezing, i.e., (¯ n, r ) = (0 , σ becomes rather large, the performance of the MLE is sig-nificantly better than that of the direct reconstructionwithout information on the input phase. Thus, to haveaccess to a reference frame is an important issue in prac-tical situations. We note, however, that in an experimen-tally achievable regime with current technology ( σ = 0 . < r nc < . IV. TWO-MODE GAUSSIAN STATEESTIMATION
In this section, we extend our study to two-mode Gaus-sian states reconstructed from the coarse-grained homo-dyne data. We again investigate the fidelity and the non-classicality, now entanglement, of the output two-modestate by two coarse-grained methods, the direct recon-struction of the covariance matrix and the MLE. Weconsider as our input state a two-mode squeezed ther-mal state (TMST) in the form ρ = ˆ S ( r, φ )[ ρ th (¯ n ) ⊗ ρ th (¯ n )] ˆ S † ( r, φ ) , (33) where S ( r, φ ) = exp[ − r { exp( iφ )ˆ a † ˆ a † − exp( − iφ )ˆ a ˆ a } ]is the two-mode squeezing operator with the squeezingstrength r and the squeezing angle φ . Its covariance ma-trix is given by Γ = a ℜ [ c ] ℑ [ c ]0 a ℑ [ c ] −ℜ [ c ] ℜ [ c ] ℑ [ c ] b ℑ [ c ] −ℜ [ c ] 0 b , (34)where a = ¯ n cosh r + ¯ n sinh r + 12 cosh 2 r,b = ¯ n sinh r + ¯ n cosh r + 12 cosh 2 r,c = −
12 (¯ n + ¯ n + 1) exp( iφ ) sinh 2 r. (35)The characteristic function of TMST is given by C ( λ , λ ) = exp( − a | λ | − b | λ | − c ′ | λ || λ | ) , (36)where c ′ = | c | cos( ϕ + ϕ − φ ) and ϕ i = − i ln( λ i / | λ i | ) ( i ∈{ , } ). From this, we obtain the homodyne distributionas P ( x ,ϕ , x ,ϕ )= 1 π p ab − | c | exp (cid:20) − bx + ax + 2 c ′ x x ab − | c | (cid:21) . (37)Similar to the single-mode case, the covariance matrixof a two-mode state can also be constructed by measur-ing three different quadratures for each mode. The localmatrix elements are just the same as in Eq. (15), and thecorrelation elements are given byΓ = Γ = 2 h ˆ X , ˆ X , i − h ˆ X , ih ˆ X , i , Γ = Γ = 2 h ˆ X , ˆ X ,π/ i − h ˆ X , ih ˆ X ,π/ i , Γ = Γ = 2 h ˆ X ,π/ ˆ X , i − h ˆ X ,π/ ih ˆ X , i , Γ = Γ = 2 h ˆ X ,π/ ˆ X ,π/ i − h ˆ X ,π/ ih ˆ X ,π/ i . (38)Using these matrix elements and the relations in Eq. (35),we can determine the output two-mode state with theparameters¯ n i = ( − i +1 ( a − b ) − √ γ ′ ,r = 12 arcsinh | c |√ γ ′ ! ,φ = arctan ℑ [ c ] ℜ [ c ] ! for ℜ [ c ] ≥ ,π − arctan ℑ [ c ] ℜ [ c ] ! for ℜ [ c ] < , (39)with i ∈ { , } and γ ′ = ( a + b ) − | c | .The fidelity between two two-mode Gaussian stateswith same means is given by [23] F = 1 √ Σ + √ Λ − q ( √ Σ + √ Λ) − ∆ , (40)where ∆ = det( Γ + Γ ), Λ = 16 det( Γ + i J ) det( Γ + i J ) and Σ = 16 det[( JΓ )( JΓ ) − ] with J = M i =1 J i , J i = (cid:18) − (cid:19) ( i = 1 , . (41)On the other hand, the entanglement of a two-modeGaussian state can be measured by the logarithmic neg-ativity [29] as E N = max[0 , − log (2˜ ν − )] , (42)where the smaller symplectic eigenvalue ˜ ν − of the par-tially transposed state is given by2˜ ν − = f − p f − g , (43)with f = a + b + 2 | c | and g = ab − | c | for the statesin our consideration.In Fig. 5, we plot the fidelity F and the logarith-mic negativity E N of the estimated two-mode squeezedthermal state as a function of the coarse-graining size σ , for the input two-mode squeezed thermal states with(¯ n, r ) = (0 , n, r ) = (1 , n, r ) = (0 , n = ¯ n = ¯ n . Theseplots show a tendency similar to the plots for the single-mode case in Fig. 4. The fidelity and the logarithmicnegativity decrease with the coarse-graining size σ , andthe degrading rate is larger for a more nonclassical (en-tangled) initial state. However, each scheme shows adifferent performance for the characterization of outputstates.The direct reconstruction method without access tothe information on the input phase of two-mode squeez-ing (dashed curves) can generally yield a worse outputthan that with the information (dot-dashed-curves). Forthe two-mode squeezed thermal states, the local homo-dyne distribution for each mode is isotropic as it has nobearing on the phase of two-mode squeezing, so that theestimated mean photon numbers are invariant even whenthe phase information is not available. Only correlationparts vary under the rotation of the reference frame. Ascan be seen from Fig. 5, the difference in the performancebetween the two methods is thereby relatively less thanthat in Fig. 4. In addition, the MLE method employ-ing a full set of homodyne measurements again showsperformance at the intermediate level. However, the dis-tinctions are not very prominent, and in particular, thosethree methods yield almost the same results with the cur-rently accessible coarse-graining ( σ = 0 .
1) [22, 26–28].On the other hand, one can readily see that if an asym-metric input state with local squeezings is considered,the availability of the phase information can affect theresults more significantly than here. : Σ F H a L > Σ E N H b L FIG. 5: (Color online) (a) Fidelity F between an input stateand its reconstructed state and (b) logarithmic negativity E N of the reconstructed state as functions of the coarse-grainingsize σ , for the input squeezed thermal states with (¯ n, r ) =(0 ,
1) [green dot-dashed line, red solid and blue dashed lines,the second curves from the top for (a) and (b)], (¯ n, r ) = (1 , n, r ) = (0 ,
2) [gray dot-dashed line,brown solid and black dashed lines, the third curves from thetop for (a) and the first curves from the bottom for (b)]. Forsimplicity, we assume that the thermal photon number of twomodes are the same, ¯ n = ¯ n = ¯ n . Solid curves representthe case of the MLE method, dot-dashed (dashed) curves thedirect reconstruction method with (without) information onthe input phase, respectively. For the plots of dashed curves,each point represents an averaged value over the whole rangeof the input squeezing angles. V. SUMMARY AND DISCUSSION
In this paper, we investigated the reconstruction of aquantum state by a coarse-grained homodyne measure-ment. Employing both the direct reconstruction methodof the covariance matrix and the MLE method, we ex-amined single-mode and two-mode Gaussian states tosee how those states undergo quantum-to-classical tran-sition. The reconstruction method has been comparedto the decoherence model typically employed to accountfor the quantum-to-classical transition. In particular, asour coarse-graining models produce a Gaussian outputstate from a Gaussian input state, those models havebeen compared to the decoherence by a Gaussian reser-voir, i.e., thermal squeezed reservoir. We have clearlyshown that the coarse-graining model is not compati-ble with the decoherence model in addressing the stateevolution and that the effects (added noise) of coarse-grained reconstruction are particularly state dependentin contrast to the decoherence program. Even thoughthe coarse-graining applies equally to all quadrature am-plitudes, i.e., isotropic in phase-space, it turns out thatits effect on the state can be made equivalent only by aphase-sensitive reservoir with nonzero squeezing.Furthermore, we also compared the performance be-tween the direct reconstruction and the MLE in terms ofthe fidelity and the nonclassicality of the output states.In general, the direct reconstruction method employinghomodyne measurement of only three quadratures, there-fore practically less demanding, can yield a better outputthan the MLE method employing a full set of homodynemeasurements. However, this is possible only when one has access to the information on the phase of the inputstate. If the phase information is not available, the MLEmethod yields better results than the direct reconstruc-tion. In a practical regime of, e.g., σ = 0 .
1, all thosemethods yield almost identical results.As a concluding remark, the reconstruction under acoarse-grained homodyne measurement generally yieldsa non-Gaussian distribution, i.e., the piecewise flat dis-tribution in Fig. 1. Therefore, even though we know thatthe input state is a Gaussian state, it will be interesting tostudy how the characteristics of the reconstructed statecan be modified if the MLE method is applied with ref-erence to a set of non-Gaussian states. That is, we takethe estimated states to be non-Gaussian and investigatethe fidelity and the nonclassicality of the output states,which will be left for future study together with the caseof non-Gaussian input states.
Acknowledgments
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