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Effectiveness of entangled coherent state in quantum metrology Osamu Hirota and Kentaro Kato ∗ † Quantum ICT Research Institute, Tamagawa University, 6-1-1, Tamagawa Gakuen, Machida 194-8610 Tokyo, JAPAN Dan Murakami ‡ Graduate School of Tamagawa University, 6-1-1, Tamagawa Gakuen, Machida 194-8610 Tokyo, JAPAN (Dated: January 10, 2012) This Letter verifies the potential of several classes of entangled coherent state in well known quantum metrology which includes detection of classical external force, and shows that there is 2 a class of entangled coherent state for the external force detection system without the quantum 1 limit in thedetection of thelight. Inthecase oftheprecision measurement of continuosparameter 0 like phase measurement, the entangled coherent state with perfect entanglement does not provide 2 remarkable benefit, but we provide a concrete example that certain class of entangled coherent n state gives a remarkable sensitive detection scheme in the discrimination of digital signal affected a byexternal force. J 8 PACSnumbers: 42.50.St, 42.50.Dv,03.65.Ta,06.20.Dk ] h I. INTRODUCTION state is p - nt |ΨiJ =hJ(|αJiA|0iB+|0iA|αJiB) (1) a whereh isthenormalizationcoefficient. Thisisaspecial u Agenericinterferometerhasashotnoiselimitedsensi- J q tivitythatscaleswith 1 [1]. HereN istheaveragepho- case of the general entangled coherent state [9]. On the [ √N otherhand,applicationsoftheentangledstatetodiscrete ton number of the light source. This is called standard signalssuchastargetdetectionorquantumreadinghave 4 quantum limit. Quantum communication theory includ- been discussed[10,11]. Thus, it is important that one v ingquantumestimationandquantumdetectionmaypre- examinesapotentialofageneralECSwhichisthirdway 7 dict a possibility to beat such a limit. In fact, the above 1 of entanglement state application. example of the limit corresponds to a subject of a phase 5 ThequasiBellstatebasedonentangledcoherentstates estimationthroughquantummeasurement. Onecandis- 1 have a potential of the perfect entanglement[12], and cusstheprecisionlimitsofquantumphasemeasurements 8. by the Quantum Cramer-Rao inequality[2,3]. Since the applications to quantum teleportation and computation 0 have been discussed[13,14,15,16]. Furthermore the feasi- mathematical treatment of the lower bound in physical 1 bility [17] and experimental demonstration [18] of such problems has been clarified[4], the best resource for the 1 states have been reported. : phase estimation has been discussed[5,6]. Consequently v Thus, we are concerned with the effectiveness of the whenwepreparequantumcorrelationsbetweenthe pho- i entangled coherent state in quantum metrology where X tons such as entangled state of light, the interferometer various new technologies can emerge. In this Letter, we sensitivity can be improved by a factor of √N. That ar is, the sensitivity now scales with 1 by employing the consider the performance of several classes of entangled N coherent state in well known quantum metrology which NOON state[5,6]. This limit is imposed by the Heisen- includesdetectionofclassicalexternalforce,andpropose berg uncertainty principle. The above Heisenberg limit a new method to detect the weak external force. is believed to be the ultimate precision in optical phase estimation. Recently, more attractive feature has been discoveredbyemployingentangledcoherentstateforthe II. THEORETICAL EVALUATION OF phase estimation problem by J.Joo et al [7], following PHYSICAL LIMITATION the pioneeringworkby C.Gerryetal[8]. This workmay openanewway. Infact,theyshowedthataspecifictype We can separate the issue of the theoretical limit on ofentangledcoherentstategivesthe smallestvariancein the ultimate sensitive measurement into signals of con- thephaseparameterincomparisontoNOONstate. The tinuous parameter and discrete parameter. The former is to clarify the precisionof the measurement observable like the phase, and the later is to clarify the minimum errorperformance of the decision for signalsin the radar ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] or weak external force. The quantum Cramer-Rao in- ‡Electronicaddress: [email protected] equality has been formulated[2,3], in which the bound is 2 asymptotically achieved by the maximum likelihood es- SomeofthesequasiBellstatesarenotorthogonaleach timator as well as the classical estimation theory. other. Here, if κ = κ , then the Gram matrix of them ∗ Here let ρ(θ) be the density operator of the system. becomes very simple as follows: The estimation bound is given as follows: First, the following operator equation is defined 1 0 D 0  0 1 0 0 G= (7) ∂ρ(θ) 1 D 0 1 0 = [Aρ(θ)+ρ(θ)A] (2)   ∂θ 2  0 0 0 1 where A is called symmetric logarithmic derivative, where D = 2κ . which is self-adjoint operator. Then the bound is given 1+κ2 The degrees of entanglement for quasi Bell state are by well known as follows: 1 1 (δθ)2 ≥ FQ = Tr(ρ(θ)A2) (3) E(|Ψ1i)=E(|Ψ3i) (8) 1+C13 1+C13 1 C13 1 C13 = log − log − FQ isalsocalledquantumFisherinformation. Theabove − 2 2 − 2 2 inequalitydefinestheprincipallysmallestpossibleuncer- where E() is the entanglement of formation, C = tainty in estimation of the value of phase. ij Ontheotherhand,intheradardetectionorweakforce |hΨi|Ψji|, and E(|Ψ2i) = E(|Ψ4i) = 1. Thus |Ψ2i,|Ψ4i have the perfect entanglement [12]. detection, the problem becomes whether signal exists or not. The formulation of the ultimate detection performance is called quantum detection theory[2,3]. The lim- IV. QUANTITATIVE PROPERTIES OF itation can be evaluated as follows: LIMITATION Pe = ξ0Trρ0Π1+ξ1Trρ1Π0 A. Phase estimation Π0+Π1 =I, Πi 0 (4) ≥ where ξ is a prioriprobability of quantum states ρ Recently, many authors applied the quantum Cramer- i i { } { } of the system, Π is the detection operator, respec- Raobound to clarifythe ultimate estimationofphase or i { } tively. In the case of pure states, the optimum solution phaseshiftoflight. Whenanentangledstateisprepared, is consideringthesituationwithnoloss,aphaseestimation bound is analytically given as follows[6,7]: 1 Pe = 2[1−p1−4ξ0ξ1|<ψ0|ψ1 >|2] (5) FQ =4[TrAρ(aA†aA)2 (TrAρ(aA†aA))2] (9) − In the issue of precision measurement or super sensi- One can see from the above that NOON state gives the tivediscriminationlikedigitalacousticlasermicrophone, Heisenberg limit. When one of four entangled coherent the origins of the signals in our model are described as states is employed as light source, one can directly cal- follows: culatethevarianceandthe quantumFisherinformation. (a) Phase shift:U(θ g)=exp( θa a ) | − A† A As a result, the bound is given by (b) Amplitude shift: D(α g)=exp(α a α a ), s s A† s A | − where g is an external force, a and a are the annihi- A A† lationandcreationoperatorofthe the mode A,whenan 1 δθ = (10) entangled state is applied. 2 [α4K(1 K)+K α2] p| | − | | where III. SHORT SURVEY OF ENTANGLED COHERENT STATE 1 exp( 2α2) K = ± − | | (11) ± 1 exp( 4α2) ± − | | The four entangled state so called quasi Bell state based onentangled coherentstate are defined as follows: where+isfor Ψ1 ,and Ψ3 ,and isfor Ψ2 and Ψ4 . | i | i − | i | i TheseboundsarelargerthanthatofthestateofEq(1), Ψ1 = h1(α A α B + α A α B) regardlessofthe largeentanglement. Infact,the estima-  ||ΨΨ32ii == hh32((||ααiiAA||αiiBα−B||+−−αiiAα||−−A ααiiBB)) (6) tion bound of Eq(1) was given as follows [7]: | i | i |− i |− i | i 1  |Ψ4i = h4(|αiA|−αiB −|−αiA|αiB) δθ = 2α h (α 2+1) h2 α 2 (12) J J J J | | p | | − | | where hi are normalized constant:h1 = h3 = { } 1/ 2(1+κ2), h2 = h4 = 1/ 2(1 κ2), and where wherehJ =1/ 2(1+exp( αJ 2)),whichprovidessim- p p − p −| | α α =κ and αα =κ . ilar performance with NOON state and is superior to it ∗ h |− i h− | i 3 in the case of weak amplitude α under the same total Secondly, we examine the case of the single mode co- J energy constraint. In addition, the energy of the mode herent state. The initial state and affected state are as A of Ψ is less than that of the quasi Bell state. follows: J | i Ontheotherhand,byusingsuperposedcoherentstate, itwasshownthatitallowsdisplacementmeasurementat α(0) = β (16) | i | i the Heisenberg limit[19]. α(1) = β √ǫ (17) | i | − i A reasonof the above benefit may come from the fact that the entangled coherent state of type of Eq(1) is in- When β √ǫ is 0, The inner product of two coherent − terpreted as a superposition of NOON states, while the states is conventional entangled coherent states have more com- β 0 =exp( β 2/2) (18) plicatedstructure. Theprecisephysicalmeaningonsuch h | i −| | a difference will be discussed in the subsequent article. According to Eq(5), we have 1 B. Detection of discrete external force Pe(C) = [1 1 4ξ0ξ1exp( β 2)] 2 −p − −| | P (ECS) = 0 (19) e There is an anther possibility of the precise measurement of the external force by quantum scheme. It is In several applications, we encounter a situation that the discrete external force detection. In this case, we a priori probabilities of the external force as signals are are concerned with whether certain entangled state can unknown. Then the detection scheme becomes quan- overcomethelimitofthecaseofcontinuousprecisemea- tum minimax detection[20]. However, the performance surement or not. The problem goes to the detection of isgivenbyputtingξ0 =ξ1 =1/2intheaboveequations, the event. Here event means the fact in physical phe- because these are the worst a priori probabilities in any nomena like yes or no. There are many such detection case of the binary pure state situation. problems in communication theory, for example, the co- Furthermore,intheexternalforcedetection,weshould herent laser radar or the detection of parameter shift of consider the case of unknown shift by the external force. a light by an external force in interferometer. We suggest the M parallel systems of the above system We here restrict the problem to the binary detection. which are designed by following energy attenuation: Sodetectiontargetsaretwoquantumstates. Wetakean energyshift by externalforce in this Letter. Thatis, the β1 √ǫ1 =0 − problemistodecidetheshiftfromthesteadystatebythe β2 √ǫ2 =0 externalforce. Ourproposaltorealizegoodperformance − . is as follows: .. Let us assume that following entangled coherent state β √ǫ =0 (20) is employed, M − M Ψ(0) =h(0)(α β α 0 ) (13) When one of systems clicks the result of the external A B A B | i | i | i −|− i | i force, one can record the external force detection. We design the system such that the energy of the mode B is reduced when the external force is applied to the system. The energy shift is denoted as ǫ. However, the V. CONCLUSION external force does not affect to the vacuum state. As a result, after the external force is applied to the system, We have discussed some applications of severalclasses the state of the system is as follows: of entangled coherent state to quantum metrology. The Ψ(1) =h(1)(α A β √ǫ B α A 0 B) (14) role of entangled coherent state with the perfect entan- | i | i | − i −|− i | i glement is minor in the precision measurement of the Here we design that β √ǫ=0. So we have the follow- − continuous parameter. However, certain type of entan- ing situation. The inner product between two entangled gledcoherentstateprovidesagreatadvantageinthedis- coherent states becomes crimination of the digital parameteraffected by external Ψ(1)Ψ(0) =h0h(1)( αA β B αA 0B) force. Following the above results, we have given a new h | i h | h | −h− | h | (α 0 α 0 )=0 (15) method to detect the weak external force by using the A B A B × | i | i −|− i | i parallel system of digital detection. Thus, the inner product is independent of the amplitude parameter of the coherent state, and the performance is given by Eq(5) and the above. Thus the detection Acknowledgments performance is independent of the energy of the light source, but ǫ depends on the system design for external We are grateful Bill Munro and T.S.Usuda for helpful force detection. discussions, and acknowledge the president Y.Obara of This type of detection is sometimes called threshold Tamagawa University for special funding. detection problem in the communication theory. 4 [1] C.M.Caves, Phys.Rev., D-23, 1693, (1981). [11] R.Nair, Phys.Rev.A-84, 032312 (2011). [2] C.W.Helstrom, Quantum detection and estimation the- [12] O.HirotaandM.Sasaki, Proceedings ofQCMC-2000, Ed ory, (Academic, New York, 1976). Also, H.P.Yuen and byP.TombesiandO.Hirota,(KluwerAcademic/Plenum, M.Lax, IEEE. Transaction of Information Theory, IT- 2001), arXiv:quant-ph/0101018,(2001). 21, 125, (1973). [13] S.J.van Enk and O.Hirota, Phys. 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