We show that one single experiment can test simultaneously and independently
both the nonclassicality of states and measurements by the violation or
fulfillment of classical bounds on the statistics. Nonideal measurements
affected by imperfections can be characterized by two bounds depending on
whether we test the ideal measurement or the real one.Comment: 9 pages, 3 figures. Proceedings of 17th CEWQO 201

Alfredo Luis

Amri T

Amri T Laurat J Fabre C

Dodonov V V

Gerry C C

Malbouisson J M C

Mandel L

Scully M O

Sperling J Vogel W

Ángel Rivas

English

arXiv

© 2011 The Royal Swedish Academy of Sciences. AR acknowledges financial support from the EU Integrated Project QESSENCE and the STREP action CORNER. AL acknowledges support from project no. FIS2008-01267 of the Spanish Dirección General de Investigación del Ministerio de Ciencia e Innovación and by project QUITEMAD S2009-ESP-1594 of the Consejería de Educación de la Comunidad de Madrid.We show that a single experiment can test simultaneously and independently both the nonclassicality of states and of measurements by the violation or fulfillment of classical bounds on the statistics. Nonideal measurements affected by imperfections can be characterized by two bounds depending on whether we test the ideal measurement or the real one.Comunidad de MadridUnión Europea (UE)STREP action CORNERMinisterio de Ciencia e Innovación (MICINN), EspañaDepto. de ÓpticaFac. de Ciencias FísicasTRUEpu

Luis Aina, Alfredo

Rivas, Ángel

Docta Complutense

arXiv:1103.5014v1  [quant-ph]  25 Mar 2011Independent nonclassical tests for states andmeasurements in the same experimentAlfredo Luis1 and Ángel Rivas21 Departamento de Óptica, Facultad de Ciencias F́ısicas, Universidad Complutense,28040 Madrid, Spain2 Institut für Theoretische Physik, Universitt Ulm, Ulm D-89069, GermanyE-mail: alluis@fis.ucm.esAbstract. We show that one single experiment can test simultaneously andindependently both the nonclassicality of states and measurements by the violationor fulfillment of classical bounds on the statistics. Nonideal measurements affected byimperfections can be characterized by two bounds depending on whether we test theideal measurement or the real one.Keywords: Nonclassical states, quantum opticsNonclassical tests 21. IntroductionWithin standard quantum theory, quantum states play two dissimilar but complemen-tary roles: (i) they express the state of the system, represented by a density matrix ρ,and (ii) they determine the statistics of measurements, typically by projection of thesystem state on the eigenstates of the measured observable. More precisely, any observ-able event is represented by a nonnegative Hermitian operator ∆ (maybe part of a largerpositive operator-valued measure) that determines the event probability as p = tr(ρ∆),where ρ is the state of the system. In many relevant situations ∆ is proportional to asuitable state, such as photon-number and quadrature measurements in quantum optics.Positive operators playing the role of ∆ can be turned into the role ρ as shown in [1].While referring to nonclassical states is quite common, not so many effort hasbeen devoted to nonclassical measurements [2, 3, 4, 5]. A customary criterion ofnonclassicality for states ρ is the failure of the Glauber-Sudarshan P function to exhibitall the properties of a classical probability density [6]. This occurs when P takes negativevalues, or when it fails to be a proper function becoming more singular than the deltafunction. Accordingly, we may say that the event represented by ∆ in Hilbert spaceis nonclassical when its P phase-space representative takes negative values or is moresingular than the delta function.Although the nonclassicality of states and measurements are different things, bothmay be tested simultaneously within one single experiment in terms of its statistics p.This possibility is addressed in this work by means of a simple example: nonefficientsingle-photon detection in photon-added thermal states [7]. A key point of this exampleis feasibility, since these states have been already generated in experiment [8], and itexplicitly includes typical imperfections such as losses and thermalization.As nonclassicality criteria we will consider the simple and robust practical testsrecently introduced where nonclassicality is revealed by breaking classical bounds onprobabilities satisfied by all classical states and measurements [4, 9] (see [10] for othernonclassicality criteria). The main features of these tests are recalled in section 2.2. Classical bounds on probabilitiesFor definiteness let us focus on a single mode of the electromagnetic field with complex-amplitude operator a. To derive the nonclassical tests we will use the P and Q phase-space representatives associated to any operator AA =∫d2αPA(α)|α〉〈α|, QA(α) =1π〈α|A|α〉, (1)where |α〉 are coherent states, a|α〉 = α|α〉. They are suitably normalized∫d2αPA(α) =∫d2αQA(α) = trA, (2)with d2α = dxdy, where x, y are the real and imaginary parts of α = x+ iy.Nonclassical tests 3Exploiting the ρ ↔ ∆ symmetry, the same probability p = tr(ρ∆) can be expressedby two equivalent formulasp = π∫d2αPρ(α)Q∆(α) = π∫d2αP∆(α)Qρ(α). (3)By using the first equality we are able to derive bounds sensitive to the nonclassicalityof the state ρ, while the second equality leads to bounds sensitive to the nonclassicalityof the measurement ∆. Note that the Q function is always a positive and well behaved.2.1. Nonclassical test for statesFor classical states, i. e., for ordinary nonnegative functions Pρ(α) ≥ 0, we getPρ(α)Q∆(α) ≤ Pρ(α)Q∆,max, (4)where Q∆,max is the maximum of Q∆(α) when α is varied. Applying this to the firstequality in (3), and taking into account (2), we get the following upper bound S for pp ≤ S = πQ∆,max, (5)that holds for every Pρ(α) compatible with classical physics. If this condition is violatedit means that (4) is false and the state is not classical.2.2. Nonclassical test for measurementsFor classical measurements, i. e., for ordinary nonnegative function P∆(α) ≥ 0 it holdsthatP∆(α)Qρ(α) ≤ P∆(α)Qρ,max, (6)where Qρ,max is the maximum of Qρ(α) when α is varied. Applying this to the secondequality in (3) we get the following upper bound M for p, provided that tr∆ is finite,p ≤ M = πQρ,max tr∆. (7)Equation (7) can be violated if P∆(α) fails to be positive or when it becomes a generalizedfunction (this is a nonclassical measurement) since in both cases (6) fails to be true.3. Inefficient photon detection on photon-added thermal statesThe same probability p may serve to test both the nonclassicality of ρ and ∆. Let usdemonstrate this by applying the above formalism to nonefficient single-photon detectionin photon-added thermal states.Nonclassical tests 43.1. Single-photon-added thermal statesThe single-photon-added thermal states read, in the photon-number basis, [7, 8]ρ = (1− ξ)a†ρtca = (1− ξ)2∞∑n=1ξn−1n|n〉〈n|, (8)where ρtc is a thermal chaotic stateρtc = (1− ξ)∞∑n=0ξn|n〉〈n|, (9)with mean number of photonsn̄ =ξ1− ξ . (10)The P representative of ρ is well-behaved but nonpositivePρ(α) =1πn̄3[(n̄ + 1) |α|2 − n̄]exp(−|α|2n̄), (11)while the Q function isQρ(α) =|α|2π(n̄+ 1)2exp(− |α|2n̄ + 1), (12)so that its maximum occurs for |α|2 = n̄ + 1 beingQρ,max =1πe(n̄ + 1). (13)These states present three relevant features for our purposes:(i) Their nonclassical behavior is independent of other typical nonclassical features,since there is no quadrature squeezing, they present super-Poissonian photon-numberstatistics for all n̄ > 1/√2, and have no oscillatory statistics [4].(ii) They can be generated experimentally [8].(iii) Their definition embodies a typical source of practical imperfection such asthermalization.3.2. Ideal single-photon detectionFor ideal single-photon detection we have, in the photon-number basis, ∆ = |1〉〈1| withtr∆ = 1. The P representative is nonclassical being more singular than the deltafunctionP∆(α) =(1 +∂2∂α∂α∗)δ(2)(α). (14)The Q function isQ∆(α) =|α|2πexp(−|α|2), (15)and the maximum occurs at |α| = 1Q∆,max =1eπ. (16)Nonclassical tests 5vacuumFigure 1. Illustration of inefficient single-photon detection.If the measured state is classical, the single-photon probability p is bounded by [4, 9]p ≤ S = 1e. (17)3.3. Inefficient single-photon detectionIn figure 1 we illustrate the case of inefficient single-photon detection. A detector withquantum efficiency η can be modeled by a beam splitter of amplitude-transmissioncoefficient t =√η, mixing the input state ρ with vacuum, placed before an ideal detector∆ with η = 1 [11]. After this model two different routes can be followed:(i) We can test the underlying ideal detection ∆ regarding inefficiency as a handicapof practical origin. This is to say that we have ideal detection on the state ρ̃ afterthe beam splitter that carries the effect of inefficiency, so that the probability isp = tr(ρ̃∆). Since the transformation of coherent states through lossless beam splittersis |α〉 → |√ηα〉 the state ρ̃ isρ̃ =∫d2αPρ(α)|√ηα〉〈√ηα|, (18)where Pρ(α) is in (11). From this expression we get by direct computation theQ functionof ρ̃Qρ̃(α) =1π[(n̄+ 1)η|α|2(ηn̄+ 1)3+1− η(ηn̄+ 1)2]exp(− |α|2ηn̄+ 1). (19)For η 6= 0 its maximum holds for|α|2 = 1 + ηn̄− (1− η)(1 + ηn̄)η(n̄+ 1), (20)leading toQρ̃,max =η(n̄+ 1)π(ηn̄+ 1)2exp[−ηn̄ + 2η − 1η(n̄+ 1)]. (21)(ii) Alternatively, we can test the real measurement embodying the inefficiency aspart of the measuring apparatus. The real measurement is represented by an Hermitiannonnegative operator ∆̃ to be determined such that the probability can be expressed asp = tr(ρ∆̃). From (3), (18) and the equality p = tr(ρ̃∆) = tr(ρ∆̃) we getQ∆̃(α) = Q∆(√ηα) =η|α|2πexp(−η|α|2), (22)Nonclassical tests 6so thatQ∆̃,max = Q∆,max =1πe. (23)From (2), (22) and tr∆ = 1 we readily gettr ∆̃ =1η. (24)Moreover, by expressing the exponential in (22) as exp(−η|α|2) = exp[(1 −η)|α|2] exp(−|α|2) and expanding the first exponential in power series we getQ∆̃(α) =ηπ∞∑n=0(n+ 1)(1− η)n |α|2(n+1)(n+ 1)!exp(−|α|2), (25)that readily provides the expression of ∆̃ in the photon-number basis∆̃ = η∞∑n=0(n + 1)(1− η)n|n+ 1〉〈n+ 1| = η(1− η)a†a−1a†a. (26)The probability p is independent of the interpretations (i) and (ii), being [from (8)and (26) for example]p = tr(ρ̃∆) = tr(ρ∆̃) = η1 + 2n̄− ηn̄(1 + ηn̄)3. (27)The interpretations also do not affect the classical upper bound for states, from (5) and(23),p ≤ S = πQ∆,max =1e. (28)The interpretations (i)/(ii) affect the classical upper bound for measurements. Sincewe are actually considering two different measurements, the ideal ∆ and the real ∆̃, weget from (13), (21), and (24) two different classical bounds M∆ and M∆̃ withp ≤ M∆ = πQρ̃,max tr∆ =η(n̄+ 1)(ηn̄+ 1)2exp[−ηn̄ + 2η − 1η(n̄+ 1)], (29)andp ≤ M∆̃ = πQρ,max tr ∆̃ =1eη(n̄+ 1). (30)In figure 2 we have plotted p, M∆̃, M∆, and S as functions of n̄ for η = 0.4 and0.9, while in figure 3 they are plotted as functions of η for n̄ = 0.2 and 0.7. Severalconclusions can be derived from these plots:(a) M∆̃ is always above M∆, so that it is more difficult to prove the nonclassicalityof the real ∆̃ than of the ideal ∆. This fits with the general idea that inefficienciesdegrade quantum properties.(b) When η increases M∆̃ and M∆ become closer. This may be expected since whenη → 1 we get ∆̃ → ∆.(c) In general p decreases when n̄ increases, enforcing the fulfillment of the classicalbounds. This agrees with common understanding of the effect of thermalization. AnNonclassical tests 7(b)0.5 1 1.5 2 2.5 30.20.40.60.80.90.5 1 1.5 2 2.5 30.20.30.40.50.60.70.80.9(a)0.4Figure 2. Plots of p (solid), M∆̃(dashed), M∆ (dotted), and S (dash-dotted), asfunctions of n̄ for fixed η = 0.4 (a) and η = 0.9 (b).exception occurs when η is rather low, since for small η thermal photons may increasethe probability of photon detection, as illustrated in figure 2(a).(d) For low values of n̄ increasing η favors the violation of the classical bounds byincreasing p and decreasing M∆̃ and M∆, as illustrated in figure 3(a). On the otherhand, for larger n̄ we get that larger η increases the probability of detecting more thana single photon, decreasing p as illustrated in figure 3(b).(e) For large η and small n̄ it is possible to have p > M∆̃,M∆, S simultaneously,so that one and the same measurement can reveal at the same time the nonclassicalcharacter of the ρ, ∆, and ∆̃ as illustrated in figures 2(b) and 3(a).4. ConclusionsWe have shown that the same experiment can test simultaneously and independentlyboth the nonclassicality of states and measurements. This is because the nonclassicalityof states and measurements manifests via the violation of different and independentbounds to the same statistics. We have shown that practical imperfections produce theNonclassical tests 80.3 0.4 0.5 0.6 0.7 0.8 0.9 10.30.40.50.60.70.80.910.3 0.4 0.5 0.6 0.7 0.8 0.9 10.30.40.50.60.7(a)(b)=0.2=0.7Figure 3. Plots of p (solid), M∆̃(dashed), M∆ (dotted), and S (dash-dotted), asfunctions of η for fixed n̄ = 0.2 (a) and n̄ = 0.7 (b).existence of two bounds testing the ideal and real measurements.AcknowledgmentsA. R. acknowledges financial support from the EU Integrated Project QESSENCEand the STREP action CORNER. A. L. acknowledges support from project No.FIS2008-01267 of the Spanish Dirección General de Investigación del Ministerio deCiencia e Innovación, and by project QUITEMAD S2009-ESP-1594 of the Consejeŕıade Educación de la Comunidad de Madrid.References[1] Luis A and Sánchez-Soto L L 1998 Phys. Lett. A 244 211[2] Luis A and Sánchez-Soto L L 1999 Phys. Rev. Lett. 83 3573Nonclassical tests 9Fiurášek J 2001 Phys. Rev. A 64 024102[3] Coldenstrodt-Ronge H B, Lundeen J S, Pregnell K L, Feito A, Smith B J , Mauerer W, SilberhornCh, Eisert J, Plenio M B, and Walmsley I A 2009 J. Mod. Opt. 56 432Lundeen J S, Feito A, Coldenstrodt-Ronge H, Pregnell K L, Silberhorn Ch, Ralph T C, EisertJ, Plenio M B, and Walmsley I A 2009 Nature Physics 5 27[4] Rivas A and Luis A 2009 Phys. Rev. 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Independent nonclassical tests for states and measurements in the same experiment

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Independent nonclassical tests for states and measurements in the same
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Independent nonclassical tests for states and measurements in the same experiment

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