Experimental demonstration of coherent state estimation with minimal disturbance

a r X i v :q u a n t -p h /0510195v 2 20 F e

b 2006

Experimental demonstration of coherent state estimation with minimal disturbance

Ulrik L.Andersen,1Metin Sabuncu,1Radim Filip,1,2and Gerd Leuchs 1

1

Institut f¨u r Optik,Information und Photonik,Max-Planck Forschungsgruppe,Universit¨a t Erlangen-N¨u rnberg,G¨u nther-Scharowsky str.1,91058,Erlangen,Germany ?

2

Department of Optics,Research Center for Optics,

Palacky University,17.Listopadu 50,77200Olomouc Czech Republic

(Dated:February 1,2008)We investigate the optimal tradeo?between information gained about an unknown coherent state and the state disturbance caused by the measurement process.We propose several optical schemes that can enable this task,and we implement one of them,a scheme which relies on only linear optics and homodyne detection.Experimentally we reach near optimal performance,limited only by detection ine?ciencies.In addition we show that such a scheme can be used to enhance the transmission ?delity of a class of noisy channels.

PACS numbers:

An observer who receives an unknown quantum state,from a set of nonorthogonal states,cannot perfectly retrodicts which state he got.Furthermore in the act of trying to reveal the identity of the quantum state,the observer will inevitably alter the state [1,2].Obviously there is a tradeo?between maximal extraction of infor-mation and minimal disturbance of the state [3,4,5,6,7].E.g.if the observer applies the best possible measure-ment strategy allowed by quantum mechanics by which he gains maximal information,the state will be maxi-mally disturbed and vice versa.The fundamental and in-triguing features of this tradeo?have fuelled an explosion of research since the early days of quantum mechanics.Recently,however this research has become of practical relevance since it underpins the security of quantum key distribution schemes [8].

The balance between information gain and disturbance has hitherto mainly been studied in ?nite dimensional systems,where inequalities stating the optimal trade-o?have been established for various cases [4,6,10]and demonstrated recently in an experiment [9].In contrast,very little work has been devoted to the study of this tradeo?in in?nitely dimensional systems where quantum information is carried by observables with a continuous spectrum [11],important examples being the canonically conjugate quadrature amplitudes.A particular class of continuous variable states which have played a key role in various experimental realizations of quantum informa-tion protocols is the class of Gaussian states,since they are experimentally easy to produce and manipulate [12].In this Letter we investigate the tradeo?between in-formation gain and state disturbance for completely un-known coherent states.Under the assumption that the Gaussian statistic must be preserved we derive the opti-mal tradeo?,stated in terms of an inequality using ap-propriate measures for the information gain and state disturbance.The optimal tradeo?can be implemented experimentally using a remarkable simple setup requiring only linear optics and homodyne detection.Besides be-

ing fundamentally interesting,this scheme is capable of increasing the transmission ?delity of some noisy chan-nels,an improvement that will be also demonstrated in this Letter.

Completely unknown pure quantum states described in an in?nitely dimensional Hilbert space can be esti-mated only very poorly based on a single measurement.However,in quantum communication systems,a priori knowledge is often given.It is e.g.normally known that the state belongs to a certain set each occurring with a certain a priori probability.In this work we assume that the states are taken from a ?at distribution of coherent states.With this a priori information at hand,it was re-cently proven that the state can be optimally estimated using a setup where conjugate quadratures are measured simultaneously using a symmetric beam splitter and two homodyne detectors [13].However,employing this strat-egy the coherent state is maximally disturbed.On the contrary,if the unknown coherent state is left untouched,our guess will be completely random but the state will be intact.In the following we will investigate the interme-diate cases and hence address the question:For a given information gain what is the minimum disturbance to the coherent state?

Consider a coherent state characterised by the ampli-tude and phase quadrature ?x in and ?p in with [?x in ,?p in ]=2i .The state is injected into a machine with a classical output and a quantum output.In general the function of such a machine,under the assumption that the Gaus-sian statistics is preserved,can be described by a generic linear transformation:

?x out =g (?x in +?n x )?p out =g (?p in +?n p )(1)

?x m =?x in +?m x

?p m =?p in +?m p

where ?x out and ?p out are the amplitude and phase quadra-ture operators of the quantum state after the interaction,

and ?x m and ?p m are the directly measured operators (and thus commuting).g is the gain of the operation,?n x and ?n p are operators associated with the noise added to con-

2 jugate quadratures of the quantum state,and?m x and

?m p are operators responsible for the added noise in the measurement.Now using the commutation relations for the input and output quadratures and assuming that?x in and?p in commutes with the noise operators we easily?nd the following commutation relations:

[?n x,?n p]=2i

1?g2

(?2?m)2?1).

For quanti?cation of the information gain,we use the estimation?delity G,which is the phase space over-lap between the input state and the state that can be prepared based on

the classical information and the a priori information[13,15].The disturbance is quanti-?ed by the transfer?delity F,which quotes the over-lap between the input state and the post measurement state[13,15].With unity gain the?delities are simply given by G=2/(3+?2?m)and F=2/(2+?2?n).In terms of the?delities the tradeo?relation therefore reads[14]

F≤

G

(1?G)(1?2G) (5)

The optimal tradeo?between classical and quantum information can be implemented using various systems.

a)b)

FIG.1:Three methods by which the optimal tradeo?can be es-tablished.a)Teleportation.b)Asymmetric cloning followed by a joint measurement.c)Simple feed forward approach. C(1,2):Clones,AC:Anti-clone,D:Displacer,AM:Ampli-tude modulator,PM:Phase modulator,PBS:Polarizing beam splitter,BS(T):Variable beam splitter with transmission T, AUX(1,2):Auxiliary beams and LO:Local oscillator.

One strategy is to use teleportation[15],where the mea-surement outcomes at the Bell state analyzer yield the classical information whereas the teleported state serves as the quantum state after disturbance(see Fig.1a).The whole range of optimal tradeo?s can then be achieved by tuning the amount of entanglement.Alternatively,a continuous variable asymmetric quantum cloner[16]can be employed as illustrated in Fig.1b:The anti-clone is mixed with one of the clones on a symmetric beam split-ter,and subsequently amplitude and phase quadratures are measured in the two output ports to retrieve some classical information.The other clone is left unchanged and serves as the output quantum state.By tuning the asymmetry of the cloning machine,and the beam split-ting ratio of the measuring beam splitter accordingly,the whole range of maximal tradeo?s can be accessed. Interestingly,a much simpler approach can realise the optimal tradeo?.The scheme,which relies entirely on simple linear optical components and homodyne detec-tors is depicted inside the dashed box of Fig.1c:The quantum state is partially re?ected o?a beam splitter with transmission coe?cient T.The re?ected part of the state is optimally estimated by simultaneously measuring two conjugate quadratures,i.e.?x and?p,and the result-ing classical information is partly used to guess the state and partly used to displace the transmitted part of the quantum state after an appropriate scaling of the classi-cal https://www.docsj.com/doc/4d4438003.html,ing this very simple scheme,the added noises to the quantum state are?2?n=2(1?

√

2?T

F=

1?T

T

(6)

3

which is saturating inequality

(5).

A distinct di?erence between the ?rst two approaches in Fig.1a and 1b (teleportation and cloning)and the simple approach in Fig.1c is that the latter one does not require any nonlinear interaction.Some similari-ties between the teleportation scheme and the feed for-ward scheme were pointed out in ref.[17],and both schemes have been suggested as potential eavesdropping attacks [11,18].However we note that the cloning ap-proach in Fig.1b might be superior for an eavesdropper,since in this protocol the classical information can be ex-tracted at any instance if a quantum memory is available.We now proceed to the experimental demonstration of the optimal tradeo?using the simple approach.The setup (which is schematically shown in Fig.1c)consists basically of three stages:A preparation stage where the coherent state is prepared,a separation stage in which classical information is separated from the quantum in-formation and ?nally a veri?cation stage where the quan-tum state is characterised.To ensure high purity of the input state we de?ne our states to reside at a certain side-band frequency within a certain spectral window.With this de?nition an arbitrary coherent state can be easily generated by modulating the laser beam with an ampli-tude (AM)and a phase modulator (PM).We chose the sideband to have a bandwidth of 100kHz with a center frequency of 14.3MHz,since at this frequency the laser (Nd:YAG oscillating at 1064nm)was quantum noise lim-ited.

After the preparation stage,the coherent state is in-jected into the separation stage.Here the state is divided into two parts by the variable beam splitter (BS(T)),which is composed of a half wave plate and a polariz-ing beam splitter;any transmission is therefore easily accessed by a simple wave plate rotation.The re?ected part is estimated by performing simultaneous measure-ments on the amplitude and phase quadratures as shown in Fig.1c:The signal interferes at a beam splitter with an auxiliary beam (AUX1)with a π/2relative phase shift and balanced intensities,and subsequently the two out-puts are measured and the di?erence and sum currents are generated.These two outputs then provide infor-mation about the phase and amplitude quadratures of the signal,which is then used to displace the remaining quantum state in order to ensure unity gain operation.This is done by modulating an auxiliary beam (AUX2)using an amplitude and a phase modulator,and subse-quently combine this beam at a 99/1beam splitter with the remaining signal [15].Finally,after the information retrieval and displacement,the resulting quantum state is characterised using a standard homodyne detector with a strong local oscillator (LO).The signal and the noise variances of the phase and the amplitude quadratures are measured using a spectrum analyzer with resolution and video bandwidth set to 100kHz and 30Hz,respectively.These variances fully characterize the output state.We

compute the transfer ?delity by comparing the output to the input state,which was measured using the same homodyne detector in order to make a consistent com-parison [15].The estimation ?delity is calculated from the carefully measured re?ectivity of the variable beam splitter.In order to avoid erroneously overestimations of the ?delities the values are corrected to account for detection ine?ciencies.

G

F

FIG.2:Quantum state ?delity as a function of the estimation ?delity.The optimal tradeo?given by Eq.5is represented by the solid curve,whereas the dashed curve is associated with the tradeo?taking into account detection ine?ciencies.The error bars stem from the inaccuracy in determining the detec-tor e?ciencies.

Many di?erent tradeo?s were realised and the results are shown in Fig.2.The solid curve in Fig.2represents the optimal tradeo?given by the saturation of inequal-ity (5).Deviation from optimality is caused by the in-e?ciency of the in-loop detector which partly degrades the classical guess and partly imposes additional non-fundamental noise onto the quantum state after mea-surement.Therefore we paid special attention to the optimization of this measurement:The mode matching e?ciency between the auxiliary beam (AUX1)and the signal state was carefully optimized to yield a visibility of 99%and the quantum e?ciency of the photo diodes were 95%.Despite the high quality of the homodyne setup,the curve for the optimum tradeo?achievable with the experimental setup is slightly shifted and represented by the dashed curve in Fig.2.

In the last part of the paper we discuss how opti-mal partial state estimation can be exploited to enhance some communication tasks.Let us consider the follow-ing protocol.Alice wants to transmit quantum informa-tion which is encoded into a coherent state,and after the transmission Bob receives the quantum state.Such a communication task is always in?icted by loss or noise,hereby corrupting the quantum state and as a result re-ducing the transmission ?delity.Let us ?rst consider a lossy channel characterized by the transmission coe?-cient η.In such a channel the transmission loss must be compensated by an ampli?er in order to ensure maximal transmission ?delity.The highest transmission ?delity is

4

achieved by amplifying the state before it is injected into the lossy channel.

However,by considering the power constraint scenario [19],ampli?cation prior to transmis-sion is not possible and as a result the ampli?er must be placed at Bob’s receiving station.This yields a trans-mission ?delity of F =η.However,by optimally sep-arating the input state into a classical and a quantum channel as demonstrated in this article,the ?delity can be increased:The optimal post measurement quantum state is sent through the lossy channel,whereas the clas-sical information is sent through a classical channel (see Fig.3).At the receiving station Bob displaces the cor-rupted quantum state based on the information he gains from the classical channel.Obviously there is an optimal separation ratio between the classical and the quantum information for a given attenuation in the channel.This ratio is optimized by setting T =ηand we ?nd the opti-mised ?delity to be F =1/(2?η).This ?delity is for all values of ηlarger than the ?delity achievable when only the quantum channel is used.

We demonstrate this idea by inserting an attenuator with η=31%into the channel,which is placed between the variable beam splitter (BS(T))and the displacement operation (D)in Fig.1c (or Fig.3).If an ampli?er is em-ployed after the channel to compensate for these losses the ?delity is F =31%.Now using our strategy of dividing the information into classical and quantum as shown in Fig.3,we measure a quantum state ?delity of F =63±1%,which clearly surpasses the standard ampli-?er approach.In this experimental run we measured the gains to be 1.00±0.01and 1.01±0.01for the amplitude and phase respectively.

We now consider a fully transparent channel which adds noise to the signal.We ?rst assume that the nature of this noise is additive and deterministic.If the added noise of the quantum channel exceeds two vacuum units,pure classical communication maximises the ?delity and is therefore the better alternative.However if the added noise is less than two vacuum units,then pure quantum communication becomes advantageous.Therefore,only the two extreme schemes will be relevant.From here on we assume that the noise in the channel is additive and probabilistic.In this case the intermediate scheme also becomes important [20]:We consider a channel which is perfectly transmitting the signal with probability p and fails to transmit it with probability (1?p ).The aver-age ?delity of such a channel is given by F =p .If we now apply the partial estimation approach in front of the channel as illustrated in Fig.3,then the ?delity is given by F ′=pF +(1?p )G .We ?nd that for 0

the signal and send it through the classical channel,and as p approaches 4/5,it is best to send the entire signal through the quantum channel.For all intermediate cases the optimal partial estimation is advantageous.The op-timal solution,which is found by solving an algebraic equation,is computed numerically,and we ?nd that the maximal improvement appears for p =0.5where the ?-delity improves by approximately 10%if T =0.405.

FIG.3:Schematic illustration of a protocol capable of increas-ing the ?delity of noisy channels.The displacement opera-tion,D,is similar to the one shown in the long shaded box of Fig.1c,and other experimental details about the preparation and veri?cation follow those of Fig.1c.

In this Letter we have extended the discussion on the optimal information-disturbance tradeo?to the contin-uous variable regime and derived a tradeo?relation for coherent states.Furthermore,we have proposed a simple linear optics circuit which saturates this relation,and we have implemented the idea and obtained near optimal performance.Finally,we have demonstrated that our scheme can be used to enhance the transmission ?delity of some noisy channels,rendering our approach as a use-ful tool in future quantum communication networks.We thank Ladislav Mista and Vincent Josse for fruitful discussions.This work has been supported by the EU projects COVAQIAL (project no.FP6-511004).R.F.was supported by the projects:202/03/D239of GACR,MSM6198959213of MSMT CR and by the Alexander von Humboldt foundation.

?

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5

[14]The derivation of inequality(5)is based on the constraint

that the Wigner function of the quantum state remains Gaussian.After the completion of this work,it was re-alised that if one allows for non-Gaussian features of the output state,the?delity can be slightly improved using a clever(but experimentally very challenging)input ancilla state.L.Mista Jr.e-print quant-ph/0510191.

[15]A.Furusawa et al.Science282,706(1998).[16]J.Fiurasek,Phys.Rev.Lett.86,4942(2001).

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