We show that there are extremely simple signal detection schemes where the
finiteness of energy resources places no limit to the resolution. On the
contrary, larger resolution can be obtained with lower energy. To this end the
generator of the signal-dependent transformation encoding the signal
information on the probe state must be different from the energy. We show that
the larger the deviation of the probe state from being minimum-uncertainty
state, the better the resolution.Comment: 5 pages, 3 figure

Alfredo Luis

Braginsky

Braunstein

Giovannetti

Hall

Luis

Lynch

Maldonado-Mundo

Napolitano

Rivas

Tsang

Zozor

English

arXiv

© 2012 Elsevier Inc. A.L. acknowledges Drs. A. Rivas and M.J.W. Hall for enlightening comments. This work was supported by Project No. FIS2008-01267 of the Spanish Direccion General de Investigacion del Ministerio de Ciencia e Innovacion, and from Project QUITEMAD S2009-ESP-1594 of the Consejeria de Educacion de la Comunidad de Madrid.We show that there are extremely simple signal detection schemes where the finiteness of energy resources places no limit on the resolution. On the contrary, larger resolution can be obtained with lower energy. To this end the generator of the signal-dependent transformation encoding the signal information on the probe state must be different from the energy. We show that the larger the deviation of the probe state from being the minimum-uncertainty state, the better the resolution.Comunidad de MadridMinisterio de Ciencia e Innovación (MCINN), EspañaMinisterio de Economía y Competitividad (MINECO), EspañaDepto. de ÓpticaFac. de Ciencias FísicasTRUEpu

Luis Aina, Alfredo

Docta Complutense

arXiv:1203.4325v2  [quant-ph]  16 Jan 2013Signal detection without finite-energy limits to quantum resolutionAlfredo Luis∗Departamento de Óptica, Facultad de Ciencias F́ısicas, Universidad Complutense, 28040 Madrid, Spain(Dated: January 17, 2013)We show that there are extremely simple signal detection schemes where the finiteness of energyresources places no limit to the resolution. On the contrary, larger resolution can be obtained withlower energy. To this end the generator of the signal-dependent transformation encoding the signalinformation on the probe state must be different from the energy. We show that the larger thedeviation of the probe state from being minimum-uncertainty state, the better the resolution.PACS numbers: 03.65.Ca, 03.65.Ta, 42.50.St, 42.50.DvINTRODUCTIONPrecise detection provides a readily practical applica-tion of quite fundamental quantum ideas, such as quan-tum statistics, nonclassical states, and uncertainty rela-tions. Quantum metrology is framed by two widespreadbeliefs: (i) for fixed mean energy, quantum uncertaintylimits the resolution in the detection of signals with un-avoidable bounds valid for all system states, and (ii) thesebounds depend on the energy resources employed so thatbetter resolution requires larger amounts of energy. Thisis the case of the well-known standard quantum limit andHeisenberg limit [1–11].Here we show that there are practical schemes whereenergy finiteness places no limit to the resolution. On thecontrary, lower energy favors larger resolution. Moreover,we also show that the larger the deviation of the probestate from being minimum-uncertainty state, the betterthe resolution, contrary to the more intuitive idea thatimproved resolution would require probes with minimumuncertainty. These results may have deep implications inthe development of new detection technologies free fromquantum limits. This may also provide new insights inthe still open vivid debate on quantum metrology limits[2–9].The structure of any signal-detection scheme is quiteuniversal. The information about some signal χ is en-coded in a system state by a signal-dependent transfor-mation Uχ = exp(iχG) acting on a probe previously pre-pared in a known state |ψ〉, where G is the generatorof the transformation. The transformed state Uχ|ψ〉 ismonitored by a measurement M whose outputs providean estimator χ̃ of the signal χ with some uncertainty∆χ̃ depending in general on χ, G, M , |ψ〉, and the dataanalysis followed.In practical terms, G is essentially an interactionHamiltonian HI coupling the system probe to the exter-nal variables to be monitored. In an impulsive regime,the interaction Hamiltonian HI is so strong, and the in-teraction time τ so short, that the dynamics is controlledentirely by HI . Thus the signal χ depends on the cou-pling between the system and the external variables andthe interaction time.Most approaches assume that the generator of thetransformation coincides with the energy H of the probesystem in absence of signal, G ∝ HI ∝ H . This is typi-cally the case of phase shifts in harmonic oscillators gen-erated by the number operator G ∝ a†a, where a is thecomplex amplitude operator. The importance in physicsof harmonic oscillators may suggest that this is an univer-sal link due to very fundamental quantum features suchas uncertainty relations. The examples developed in thiswork shows that this is not so, and when G 6= H thefiniteness of energy resources may place no limit on theresolution.The condition G 6= H leads naturally to the so calledquantum nondemolition strategies [12, 13], in which theobservable M , in absence of signal, is a constant of themotion [H,M ] = 0. Since observable signal-inducedtransformations require [G,M ] 6= 0, quantum nondemo-lition [H,M ] = 0 implies that G 6= H , which is exactlywhat we are looking for. There are two typical quantumnondemolition measurements: (i) linear momentum fora free particle M = p, H ∝ p2, and (ii) energy itselfM = H , for example for an harmonic oscillator. Nextwe examine both possibilities from the perspective of po-tential ultimate quantum limits caused by finiteness ofresources.The performance improvement available when G 6= His at the heart of previous proposals of beating quan-tum limits by nonlinear detection schemes where, roughlyspeaking, G ∝ Hk [10]. This has been recently confirmedexperimentally [11].FREE PARTICLEProbe stateLet us consider that the probe system in absence ofsignal is the one-dimensional dynamics of a free particle.The signal is a shift of linear momentum to be inferredby a measurement of the momentum. This is to say H =p2, G = x, and M = p, where x and p are suitablyscaled dimensionless position and momentum operatorswith [x, p] = i. Following some previous works [14–16]2P(p)pFIG. 1: Plot of the momentum statistics P (p) of the initialstate of the probe as a function of p for α = 2 (solid line),α = 10 (dashed line), and α = 20 (dotted line), all with γchosen so that 〈H〉 = 1/3.we consider the probe to be prepared in the pure statein the momentum representation|ψ〉 =∫∞−∞dpψ(p)|p〉,ψ(p) = 〈p|ψ〉 =√α21/α2γΓ(1/α) exp (− |p/γ|α) , (1)where |p〉 are the momentum eigenstates, Γ is the gammafunction, and α, γ are real nonnegative parameters. Themomentum statistics in absence of signal isP (p) = |ψ(p)|2=α21/α2γΓ(1/α)exp (−2 |p/γ|α) . (2)We will consider α to be even integers in order to avoidcontinuity problems of derivatives at p = 0. For α = 2these are Gaussians, while for α → ∞ they are superGaussians that tend to be a square distribution (see Fig.1). These probes may be produced by letting free par-ticles pass consecutively through rotating slits with suit-able width and separation.Signal-induced transformation in Heisenberg andSchrödinger picturesThe momentum shift may be created by a large forceacting an extremely short time, i. e., an impulse. Thisforce results from an interaction Hamiltonian of the formHI = −λx, where λ is a suitable constant. In the Heisen-berg picture, the equations of motion areṗ = −i[p,H+HI ] = λ, ẋ = −i[x,H+HI ] = 2p, (3)with H +HI = p2 − λx. After an interaction time τ thetransformed position x̃ and momentum p̃ in terms of theoriginal ones x, p, arep̃ = λτ + p, x̃ = λτ2 + 2τp+ x, (4)where, roughly speaking, τ is given by the time spent bythe particle within the interaction region. In an impul-sive regime the interaction HI is strong compared to freeenergy H , and is acting during a time interval τ shortcompared with typical evolution times of the free sys-tem. Roughly speaking, λ → ∞ and τ → 0, with finiteχ = λτ . In such a regime the effect of the free term His negligible and the transformation after the interactiontime τ reads approximatelyp̃ = χ+ p = U †pU, x̃ ≃ x = U †xU, (5)whereχ = λτ, U = exp(−iHIτ) = exp(iχx), (6)and the signal-dependent transformation is U =exp(iχG) with G = x.On the other hand, in the Schrödinger picture thetransformed probe state is|ψ̃〉 = U |ψ〉 =∫ ∞−∞dpψ(p)U |p〉 =∫ ∞−∞dpψ(p)|p+ χ〉=∫ ∞−∞dpψ(p− χ)|p〉, (7)so that the momentum statistics P̃ (p) of the transformedprobe state isP̃ (p) = P (p− χ) =α21/α2γΓ(1/α)exp(−2∣∣∣∣p− χγ∣∣∣∣α). (8)Examples of interaction HamiltonianAs a first suitable practical implementation of the de-sired interaction Hamiltonian HI = −λx we may con-sider the interaction between a charged particle and aclassical electric field E in the dipole approximation withjust a nonvanishing component along the x axisHI = −qxE, χ = qEτ, (9)where q is the electric charge. In this case the signal χmay represent either an electric field, the charge, or theinteraction time.As a second example we may consider an Stern-Gerlachdevice with the interaction between a magnetic momentµ and an inhomogeneous magnetic field with just a non-vanishing component pointing in the z axis Bz ≃ B0x sothatHI = −µzB0x, χ = µzB0τ. (10)In this case the signal χ may represent either the gra-dient of a magnetic field, a component of the magneticmoment, or the interaction time.Within a quantum field framework, sudden momentumkicks arise from momentum conservation in the absorp-tion/emission of photons during the interaction betweenthe charge and the field. Thus χ represents the strengthof the scattering of photons by the charge.3Signal uncertainty and main resultsSignal resolution is estimated in the standard way viathe Cramér-Rao lower bound providing a minimum forthe estimator uncertainty ∆χ̃ [17, 18](∆χ̃)2≥1NF≥1NFQ, (11)whereN is the number of repetitions of the measurement,F is the Fisher informationF =∫ ∞−∞dp1P (p|χ)(dP (p|χ)dχ)2, (12)and P (p|χ) is the momentum statistics conditioned tothe signal value χ. FQ is the quantum Fisher informa-tion, that for pure states reads FQ = 4(∆G)2 = 4(∆x)2.Note that free evolution does not affect the statistics sincemomentum is preserved. This is a key feature of this ap-proach as a quantum nondemolition scheme.Since the signal-induced transformation is a momen-tum shift we have P (p|χ) = P (p − χ) and the Fisherinformation does not depend on χ. We readily obtainF = FQ =α222/αΓ(2− 1/α)γ2Γ(1/α), (13)so that the scheme is efficient in the sense of reachingthe maximum resolution allowed by the quantum Fisherinformation. The amount of energy involved in the mea-surement can be conveniently expressed by the meanvalue〈H〉 = (∆p)2 =γ2Γ(3/α)22/αΓ(1/α), (14)where we have taken into account that 〈p〉 = 0.In order to analyze the role played by finiteness ofenergy let us fix the value of α, expressing γ in termsof 〈H〉 to get a one-parameter family of probe statesparametrized by 〈H〉. This allows us to get the followinglower bound to the signal uncertainty(∆χ̃)2≥〈H〉Γ2(1/α)Nα2Γ(2− 1/α)Γ(3/α). (15)In Fig. 2 we have represented N(∆χ)2/〈H〉 as a func-tion of α, showing that the best scenario holds whenα ≫ 1, so that the momentum statistics tends to besquare. In such a case the following approximation holds(∆χ̃)2≥3〈H〉Nα. (16)It can be appreciated in Fig. 2 that the approximationworks quite well even for small α.This provides us with the main results of this work:N( ) /<H>∆χ 2α2 6 100.20.40.60.811.21.414 18FIG. 2: Plot of N(∆χ̃)2/〈H〉 as a function of α (solid line)along with its approximation in Eq. (16) (dashed line), re-stricted to even values of α.i) The lower bound on the signal uncertainty (∆χ̃)2is proportional to 〈H〉. Thus, the lower the energy thebetter the resolution. This behavior can be clearly andsimply explained. Roughly speaking, the inferred signaluncertainty is determined by the momentum uncertaintyof the probe. For a free particle with 〈p〉 = 0 momen-tum uncertainty equals mean energy, so that lower 〈H〉improves resolution.ii) For fixed mean energy 〈H〉 the uncertainty (15) canbe arbitrarily reduced by a proper choice of the probestate, this is simply by increasing α approaching a rect-angular momentum statistics. This lack of limit may beascribed to the increasingly sharp edges of P (p) as α in-creases.These results are in sharp contrast with the typical sit-uation arising in harmonic-oscillator detection schemes,where there is an unavoidable lower bound to uncertaintyinversely proportional to mean energy that applies to ev-ery probe state (see Section 3.1 below).Finally note that for α 6= 2 the lower bound in Eq.(15) deviates from the uncertainty δχ̃ that may be de-rived from a much more simple analysis in terms of errorpropagation(δχ̃)2 =(∆M)2N∣∣∣∂〈M〉∂χ∣∣∣2 =〈H〉N. (17)This is because for α 6= 2 the statistics is no longer Gaus-sian and the simple mean χ̃ =∑j pj/N need not be anefficient estimator.Numerical simulation and uncertainty relationsThe practical approaching of the lower bound dependson the number of repetitions N of the measurement. Anestimation of the number of repetitions NB required is4P( | )χ χχFIG. 3: Plot of a simulation of the posterior distributionP (χ̃|χ) for the estimator χ̃ for χ = 0, 〈H〉 = 1/3, N = 50and α = 20.given by [17]NB ≃2F 2∫ ∞−∞dp[1P(d2Pdp2)2−13P 3(dPdp)4]− 2,(18)leading in our case toNB ≃2Γ(2− 3/α)Γ(1/α)Γ2(1− 1/α)− 2 ≃ 2α (19)where the last approximation holds for α ≫ 1. We havethat the value of NB increases as resolution increases,but we get always accessible NB even for α values corre-sponding to almost perfect square momentum statistics.A simple numerical simulation confirm the result (15).In Fig. 3 we have represented a simulation of the poste-rior distribution P (χ̃|χ) for the signal estimator χ̃P (χ̃|χ) ∝ ΠNj=1P (pj |χ̃) ∝ exp−2N∑j=1∣∣∣∣pj − χ̃γ∣∣∣∣α ,(20)for χ = 0, 〈H〉 = 1/3, and α = 20, conditioned to N = 50random outcomes pj of the measurement that have beensimulated from an uniform distribution between −1 and1, which is very close to the actual distribution P (p|χ =0) for α = 20 shown in Fig. 1. It can be appreciated thatP (χ̃|χ) closely resembles a Gaussian distribution. In thisexample the variance of χ̃ in the distribution P (χ̃|χ) is(∆χ̃)2 = 10−3, which coincides with the lower boundin Eq. (15) for these parameters. This coincidence isnatural since after Eq. (19) the lower bound should beapproached for NB ≃ 40 repetitions.Finally we examine the role played by the position-momentum uncertainty relation. Taking into accountthat, for 〈p〉 = 0, we have 〈H〉 = (∆p)2 and FQ =4(∆x)2, we can express the signal uncertainty in Eq. (11)as(∆χ̃)2 ≥〈H〉N14 (∆x)2 (∆p)2, (21)where 4(∆x)2(∆p)2 ≥ 1 by uncertainty relations. Notethat, contrary to what might be assumed at first sight,the minimum signal uncertainty ∆χ̃ is obtained forprobe estates departing from being minimum uncertaintystates. This rather paradoxical behavior was already no-ticed in a different context in Ref. [19].HARMONIC OSCILLATORFinite-energy bound for G = x, M = pFor harmonic oscillators (in suitable units) H = p2 +ω2q2, so that neither p nor q are constants of the motion.Therefore, for the same conditions as above, i. e., G = x,M = p, we get very different conclusions. The meanenergy, taking into account ∆x∆p ≥ 1/2, is〈H〉 = (∆p)2 + 〈p〉2 + ω2(∆x)2 + ω2〈x〉2≥ ω2(∆x)2 +14(∆x)2≃ ω2(∆x)2, (22)where, since FQ = 4(∆x)2 we consider ∆x≫ 1, so that(∆χ̃)2≥ω24N〈H〉. (23)The dependence of ∆χ̃ on the mean energy 〈H〉 is now inthe denominator, leading to an energy-depending boundthat decreases for increasing energies. Moreover, thesame bound holds unavoidably for every probe state withthe same 〈H〉. The situation can be reverted in the free-particle limit ω → 0, so that the approximation in Eq.(22) no longer holds and must be instead replaced by〈H〉 ≥ 1/[4(∆x)2].Mean number shiftsNevertheless, within the same harmonic-oscillator con-text we can provide a more sophisticated example byconsidering a different choice for G and M . Let us con-sider that the signal is encoded via number shifts. Thisshould correspond to a transformation generated by thephase G = φ and monitored by a number measurementM = n. However, the proper quantum translation ofthe harmonic-oscillator phase φ finds many difficulties[20], so there is no simple transformation of the formUχ = exp(iχφ). To avoid this difficulty we can focus onthe following transformation Uχ for small enough signals,Uχ|n〉 ≃ |n〉+√χn+ 1|n+ 1〉, (24)5where |n〉 are the eigenstates of the number operator a†a,a†a|n〉 = n|n〉. This transformation produces a shift ofthe mean number of photons〈n〉 → 〈n〉+ χ. (25)If the input probe state is |n〉 and the measured observ-able is the number operator a†a we get that the Fisherinformation and the signal uncertainty are:F ≃1χ(n+ 1), (∆χ̃)2≥χ (〈n〉+ 1)N. (26)We can appreciate that also within an harmonic-oscillator system we can get (∆χ̃)2 ∝ 〈H〉. The prizeto be paid is that the example is more abstract and lesspractical than the free-particle case discussed in Section2. This has been introduced here merely to illustratethat results valid for phase shifts of harmonic oscillatorsdo not exhausts all possibilities in quantum metrology.CONCLUSIONSWe have shown that there are simple detection schemeswhere resolution is not limited by finiteness of energy re-sources, contrary to the standard examples in harmonic-oscillator systems. Moreover, we have shown that probestates deviating from minimum uncertainty provide bet-ter resolution. In practical terms this means that usingschemes other than harmonic oscillators may be the keyto improve resolution beyond currently established lim-its.AcknowledgmentsA. L. acknowledges Drs. A. Rivas and M. J. W. Hall forenlightening comments. This work has been supportedby Project No. FIS2008-01267 of the Spanish DirecciónGeneral de Investigación del Ministerio de Ciencia e In-novación, and from Project QUITEMAD S2009-ESP-1594 of the Consejeŕıa de Educación de la Comunidadde Madrid.∗ Electronic address: alluis@fis.ucm.es[1] V. Giovannetti, S. Lloyd, L. Maccone, Science 306 (2004)1330-1336.[2] A. Rivas, A. Luis, New J. Phys. 14 (2012) 093052.[3] V. Giovannetti, S. Lloyd, L. Maccone, Phys. Rev. Lett.108 (2012) 260405.[4] M. J. W. Hall, D. W. Berry, M. Zwierz, H. M. Wiseman,Phys. Rev. A 85 (2012) 041802(R).[5] M. Tsang, Phys. Rev. Lett. 108 (2012) 230401.[6] V. Giovannetti, L. Maccone, Phys. Rev. Lett. 108 (2012)210404.[7] M. J. W. Hall, H. M. Wiseman, New J. Phys. 14 (2012)033040.[8] Y. Gao, H. Lee, J. Phys. A 45 (2012) 415306.[9] R. Nair, arXiv:1204.3761 [quant-ph].[10] A. Luis, SPIE Reviews 1 (2010) 018006.[11] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood,R. J. Sewell, M. W. Mitchell, Nature 471 (2011) 486-489.[12] V. B. Braginsky, F.Ya. Khalili, Phys. Lett. A 147 (1990)251-256.[13] V. B. Braginsky, F. Ya. Khalili, Rev. Mod. Phys. 68(1996) 1-11.[14] A. Luis, Opt. Lett. 31 (2006) 3644-3646.[15] A. Luis, Phys. Rev. A 75 (2007) 052115.[16] S. Zozor, M. Portesi, C. Vignat, Physica A 387 (2008)4800-4808.[17] S. L. Braunstein, Phys. Rev. Lett. 69 (1992) 3598-3601.[18] S. L. Braunstein, C. M. Caves, Phys. Rev. Lett. 72 (1994)3439-3443.[19] D. Maldonado-Mundo, A. Luis, Phys. Rev. A 80 (2009)063811.[20] R. Lynch, Phys. Rep. 256 (1995) 367-436.

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Signal detection without finite-energy limits to quantum resolution

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Signal detection without finite-energy limits to quantum resolution

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