Discrimination of the binary coherent signal Gaussian-operation limit and simple non-Gaussi

a r X i v :0706.1038v 3 [q u a n t -p h ] 14 A u g 2008Discrimination of the binary coherent signal:Gaussian-operation limit and simplenon-Gaussian near-optimal receiversMasahiro Takeoka and Masahide SasakiNational Institute of Information and Communications Technology,4-2-1Nukui-kitamachi,Koganei,Tokyo 184-8795,JapanCREST,Japan Science and Technology Agency,1-9-9Yaesu,Chuoh-ku,Tokyo 103-0028,Japan(Dated:August 14,2008)We address the limit of the Gaussian operations and classical communication in the problem of quantum state discrimination.We show that the optimal Gaussian strategy for the discrimination of the binary phase shift keyed (BPSK)coherent signal is a simple homodyne detection.We also propose practical near-optimal quantum receivers that beat the BPSK homodyne limit in all areas of the signal power.Our scheme is simple and does not require realtime electrical feedback.PACS numbers:03.67.Hk,42.50.DvI.INTRODUCTIONDiscrimination of the binary phase shift keyed (BPSK)coherent states {|α ,|−α }with the minimum error is one of the most fundamental issues in optical commu-nication and quantum signal detection theory.Coherent communication theory has been developed based on semi-classical theory where these signals are detected by ho-modyne measurement.For the signals with equal prior probabilities,the average error probability is given by P err =erfc[1−e −4|α|2)/2whichis called the Helstrom bound [1].Kennedy proposed a simple near-optimal receiver using a coherent local os-cillator (LO)and photon counting [2].Its error rate is only twice larger than the Helstrom bound and is smaller than the homodyne limit when |α|2>0.4.Dolinar then extended this ‘Kennedy receiver’to the optimal one by introducing the adaptive electrical feedback which is enough faster than the optical signal pulse width [3](see also [1,4,5,6,7]).Although the Dolinar’s concept has been demonstrated recently [8],it is still challenging to experimentally beat the homodyne limit with this ap-proach because of its complicated system.From a quantum mechanical point of view,homodyne measurement belongs to the class of Gaussian operations,i.e.described by up to the second order nonlinearity,while photon counting is the non-Gaussian one.The role of Gaussian operations in quantum information protocols [9]or quantum state estimation [10]have been widely in-vestigated.On the other hand,it has also been shown that some of the important protocols cannot be per-formed by only Gaussian operations and classical commu-nication (GOCC)and inevitably requires non-Gaussian operations,e.g.quantum computing [11],entanglementdistillation of Gaussian states [12,13,14],and the opti-mal cloning of coherent states [15].In this paper,we first show that the homodyne mea-surement is the best strategy to discriminate the binary coherent states within GOCC.To our knowledge,this is the first result addressing the Gaussian limit in quantum state discrimination scenario.In the second half of the paper,we propose novel non-Gaussian quantum receivers based on the Kennedy receiver,that beat the Gaussian limit for any |α|2.In particular,we point out that the amount of displacement in the Kennedy receiver is not optimal.Our schemes do not require realtime feedback and are simple and practical to experimentally overcome the homodyne limit with current technology.II.DISCRIMINATION VIA GAUSSIANOPERATIONS AND CONDITIONAL DYNAMICSIn this section,we show that “the minimum error dis-crimination of a set of two coherent states {|α ,|−α }with the prior probabilities {p +,p −}under GOCC is at-tained by the homodyne detection”.For simplicity,αis assumed to be real.Gaussian operation is defined as the operation that maps Gaussian states to Gaussian states.For Gaussian input states,properties of Gaussian opera-tions have been well investigated [13,14].In our problem,however,although each signal state is given by a Gaus-sian state,the signal from the receiver’s viewpoint is an ensemble of these states,ˆρi =p +|α α|+p −|−α −α|,i.e.non-Gaussian.This is because the receiver does not know which state he or she is receiving.We therefore start by revisiting the measurement processes based on GOCC.A.Characterization of the measurements withGOCCIt is known that any Gaussian operation (Gaussian completely positive (CP)map)in optical system can be implemented by adding an ancillary system prepared in2 Gaussian state,applying Gaussian unitary operation onthe whole system(implementable via linear optics andsqueezing),and then discarding and/or performing ho-modyne measurements on a part of the system[14].TheCP map including measurements is not always trace-preserving and the output quantum state might be condi-tioned on the measurement outcome.When Gaussian op-erations are sequentially applied,the measurement out-comes(classical information)are sometimes useful to dy-namically renew each step of quantum operations,whichis called conditional dynamics.In the following,we char-acterize two types of measurements consisting of Gaus-sian operation with and without conditional dynamics.Thefirst one is the measurement with only Gaus-sian quantum operation(without conditional dynam-ics).Here we call it a‘Gaussian measurement’.Ageneric physical model of the Gaussian measurement isdepicted in Fig.1(a),consisting of Gaussian unitary op-eration,Gaussian ancillary states,and homodyne detec-tors.After the Gaussian measurement isfinished,a setof measurement outcome is classically post-processed,inour case,to make the decision which signal is detected.Throughout this section,we consider only‘noise-free’op-erations and measurements,that is we assume that an-cillary states are always pure and the system is never dis-carded.It does not lose generality.Since to discard someof the modes means to lose their information,it is realizedby measuring them via homodyne detectors and ignor-ing the outcomes,where the latter is included in classicalpost-processing.Mixed ancillary states are provided byfirst preparing entangled pure states and then discardinga part of them,therefore,can be prepared by pure statesand post-processing.These imply the generality of thenoise-free model.A Gaussian measurement detecting anL-mode quantum state is mathematically described by apositive operator-valued measure(POVM){ˆΠG(Γ,δ)}δwhereˆΠG(Γ,δ)>0, dδˆΠG(Γ,δ)=ˆI,andˆI is anidentity operator(see Appendix for its derivation fromthe physical model).The operatorˆΠG(Γ,δ)is a Gaus-sian operator,i.e.its characteristic function is describedbyχ(ω)=exp[−13 P±(D M)= ±α|ˆΠ(ΓM,D M)|±α .Applying a conven-tional Bayesian decision strategy as a post-processing,weobtainP(G) e =p+2α+ln(p+/p−)2α+p−2α−ln(p+/p−)2α ,(2)wheree=1+cosh(2r)+sinh(2r)cosϕ2α,0]T and I2L is a2L×2L iden-tity matrix.The conditional operation is described as follows.The initial single-mode state is interacted with M−1Gaussian auxiliary states(M>N)via a Gaussian unitary operation.Without loss of generality,we can set the auxiliary states to be M−1vacua.At the covariance matrix level,Gaussian unitary operation is described by the matrix transformation via a symplectic matrix S and an additional displacement¯d.These transform the coher-ent statesˆρ(I2,±dα)asI2→SI2⊕I2(M−1)S T≡γ,(5)±dα→S ±√B+γMC T,(9)D±=± d A−C1B+γM ¯d B−d M≡±D+¯D M.(10) Note that each ofˆρ(Γout,D±)is a pure state since the operations are noise-free.Let us show thatˆρ(Γout,D±)can be simultaneously transformed to|α′± α′±|⊗ˆρ′aux via Gaussian unitary op-erations.Since each ofˆρ(Γout,D±)is a pure state,there exists a symplectictransformation(i.e.Gaussian unitary operation)S D such that[17]Γout→S DΓout S T D=I2N,(11) where the displacement is also transformed asD±→±S D D+S D¯D M.(12) Note that S D depends only onΓout and thus inde-pendent of d M.Let S D D≡[d1,d2,···,d2N]T and S D¯D M≡[¯d1,¯d2,···,¯d2N]T.We can transform them to[±d′,0,···,0]T and[¯d′1,¯d′2,···,¯d′2N]T,respectively, by some combination of linear optics(beamsplitters and phase shifters)where the covariance matrix I2N is kept to be invariant.Again parameters of the beamsplitters depend only on{d i}i,and independent of{¯d i}i,i.e.free from d M.After these operations,the states are trans-formed to be the desired onesˆρ(Γout,D±)→ˆρ(I2,[±d′+¯d′1,¯d′2]T)⊗ˆρ′aux,(13)4 whereˆρ′aux is a product of N−1coherent states with thedisplacement[¯d′3,¯d′4,···,¯d′2N]T.These are the desiredones.Finally we apply the above scenario onto the initialstate ofˆρi=p+ˆρ(I2,+dα)+p−ˆρ(I2,−dα).Followingthe above procedures,its conditional output after theGaussian operation is given byˆρout=p+P+(d M)ˆρ(Γout,D+)+p−P−(d M)ˆρ(Γout,D−),(14)whereP±(d M)=1det(B+γM)exp −(±d B+¯d B−d M)T ×12(see Eq.(17)),it is alwaysgiven byϕ=0which is irrespective to the values of¯d′1,¯d′2and thus also d M.An optimal strategy for the sec-ond step Gaussian measurement therefore consists of thephotonτ .Here, we extend the Kennedy receiver and consider the setup depicted in Fig.2(a),where the displacementˆD(α)is replaced by a single-mode general Gaussian unitary op-erationˆU G.We will seek its optimal structure.An on/offdetector is described by the POVM {ˆΠoff,ˆΠon}withˆΠoff=e−ν∞m=0(1−η)m|m m|,ˆΠon=ˆI−ˆΠoff,(18) where|m is an m-photon state,ηis the quantum effi-5 ciency,andνis the dark counts.The Gaussian unitaryoperationˆU G consists of phase shift,displacement,andsqueezing,while one can omit the phase shift since theon/offdetection is insensitive to the global phase.Thenthe average error probability is calculated fromP e=12(ζ∗ˆa2−ζˆa†2)]is thesqueezing operator,andζ=re iϕis the complex squeez-ing parameter.After some algebra,one canfind that r,ϕ,andβhave the extreme at the same point,whereϕ=0,βis real, and the optimal displacementβopt and squeezing r opt, are given byβand r satisfying8ηαβ1−e4r−η+(2−η)e−2rη+(2−η)e−2r ,(20)α=βtanh 4ηαβ2−2e−ν(η+(2−η)e2r opt)(η+(2−η)e−2r opt)×exp −2η(α2+β2opt)η+(2−η)e−2r opt .(22) In the following,we call it as the Type-I receiver.It should be noted that if one can use an arbitrarily higher order nonlinear unitary operation instead ofˆU G,it is able to achieve the Helstrom bound rigorously[4].On the other hand,if one is restricted to use only linear unitary operation,that is the displacementˆD(γ),the conditions in Eqs.(20)and(21)are simplified asα=γtanh(2ηαγ),(23) The schematic is shown in Fig.2(b)and we call it the Type-II receiver.Its average error probability is given by P D e=1(b)1−τ via the beamsplitter of the transmittanceτ. 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