Experimental verification of a quantumness criterion for single systems based on a Bell-like inequality
Jhonny Castrillón, Omar Calderón, David A. Guzmán, Alejandra Valencia, Boris A. Rodrguez
UUdeA-Uniandes–QOptics
Experimental verification of a quantumness criterion for single systems based on aBell-like inequality
J. Castrill´on, ∗ Instituto de F´ısica, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medell´ın, Colombia
O. Calder´on-Losada, D.A. Guzm´an, Alejandra Valencia,
Quantum Optics Laboratory, Universidad de los Andes, A.A. 4976, Bogot´a D.C., Colombia
B. A. Rodr´ıguez
Instituto de F´ısica, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medell´ın, Colombia
In this letter, we propose and experimentally test a quantumness criterion for single systems.The criterion is based on the violation of an already reported classical inequality. This inequal-ity is expressed in terms of joint probabilities that we identify as realizations of preparation-and-measurement schemes. Our analysis points out that superposition is the key notion to grasp thequantum nature of physical systems. We verify our criterion in an all-optical experimental setup withheralded single photons generated via spontaneous parametric down conversion and use exclusivelyprojective preparations and projective measurements.
PACS numbers: 03.65.Ta, 03.65.Ud, 42.50.Dv
Introduction .– Since the early days of quantum me-chanics, it has been clear the necessity of quantum crite-ria as a way to differentiate whether a physical systemcan be considered classical or not [1–3]. This search hasnot been free of controversy [4]. Some of the proposedquantum criteria are based on the violation of inequali-ties as, Bell theorem that applies for spatially separatedentities [5] or Legget-Garg inequality (LGI) that appliesfor successive measurements on single systems [6]. An-other criterion, the Alicki-Van Ryn, is deduced for ex-pectation values of pairs of non-commuting observablesof single systems [7]. In all these criteria, the authorsused assumptions such as realism, locality, non-invasivemeasurability, macrorealism and other ideas motivatedby the intuition taken from the behavior of the macro-scopic world. In the literature, there are several exper-imental realizations with different physical systems thathave demonstrated the validity of each of those criteria[8–10].Any general statement about the classical/quantumfrontier has to deal with the issue of whether a singlesystem reveals quantum features or not. In this paper,we precisely addressed this issue by proposing and testinga quantum criterion for two-level single systems that, dif-ferently from the previously reported criteria, reinforcesthe idea that the key concept when dealing with quan-tumness is superposition. Our criterion is based on aninequality, that we will refer to as CISS, standing forClassical Inequality for Single Systems. This inequalityis expressed in terms of joint probabilities and consti-tutes a simpler and more general result than Bell andLegget-Garg inequalities since it was previously found asan intermediate result by Wigner [11], when deducingBell inequality, and by Lapiedra [12], when deriving avariation of the LGI. In the present work, the joint prob-
Preparation Measurement
Source D D SystemsClassicalQuantum M +1-1+1-1 FIG. 1. Cartoon representing the preparation and measure-ment stages in a P&M scheme for either classical or quantumdomains. Systems are prepared in the +1 value of the prop-erty Π and then measured in the property M . Detectors D and D collect the data for measurement of values +1 and − M . abilities that appear in the CISS will be interpreted interms of the basic operations of preparation and mea-surement . This approach allows to get rid of possibleambiguities since these operations are well defined froma theoretical [13] and an instrumental point of view [14]as can be recognized by the use of a generic “prepare-and-measure” (P&M) scheme at different research agen-das related with quantum mechanics[14–21]. We test ourcriterion in an all-optical experimental setup with her-alded single photons, generated via spontaneous para-metric down-conversion (SPDC) and use exclusively pro-jective preparations and projective measurements. Ad-ditionally, in the supplementary material we present aderivation of the CISS, that instead of using the assump-tions of joint reality and perfect correlations, as Lapiedradoes, considers the existence of a generalized classicalstate , R , that joints all the properties of a system andavoids the use of other ambiguous classical assumptions. CISS and the P&M Scheme. – Let us assume theexistence of an ensemble of individual systems, each a r X i v : . [ qu a n t - ph ] S e p of them characterized by a generalized physical state R = ( a α b β c γ ). This state is a representation of thephysical situation in which three dichotomous proper-ties, a , b and c , have one of its two possible outcomes α = ± , β = ±
1, and γ = ±
1, respectively. The prop-erties of R can be related using a function P ( π i µ j ) thatrepresents the joint probability that a single system hasthe outcome π i for the Π property and the outcome µ j for the M property, regardless the value of the third one.Different joint probabilities can be related to write a clas-sical inequality for single systems. In particular, for thestate R = ( a + b − c − ), it can be demonstrated that whenproperty b is relaxed, P ( a + c − ) must satisfy P ( a + c − ) ≤ P ( a + b − ) + P ( b + c − ) , (1)which is precisely the inequality that we call CISS. Thisequation sets a bound to the distribution of propertiesin an arbitrary ensemble of single systems and we use itto define our quantum criterion. To see this in a clearerway, let us introduce the parameter S defined by S ≡ P ( a + b − ) + P ( b + c − ) − P ( a + c − ) . (2)In the classical scenario in which the CISS is valid, S ≥ S < S depends on considering classicalor quantum systems, we associate S to our quantumnesscriterion.Joint probabilities of the form P ( π i µ j ), that appearon Eq. (2), can be seen from the perspective of a P&Mscheme in which a system is prepared in π i , a possibleoutcome for a property Π, and then measured in µ j , apossible outcome of another property M . Typically, in aP&M scheme, the “preparation” stage is a set of temporalordered interactions aimed to bring the system of interestto a specific state and the “measurement” stage is a setof operations performed on the state in order to deliveran outcome. Fig. 1 represents a situation in which asystem, not yet specified if it is quantum or classical,passes through a P&M scheme in which Π and M aredichotomic properties with possible outcomes +1 and − π + and measured either in µ + by detector D or in µ − by detector D .For the generalized classical state R , P ( π i µ j ) = P ( µ j π i ). This is not surprising given that classically it isirrelevant the order of the preparation and measurementstages of a P&M scheme. This means that preparing π i and measuring µ j ( π i → µ j ) or conversely, prepar-ing µ j and measuring π i ( µ j → π i ) leads to the samevalue of their corresponding joint probabilities. This in-dependence of the order for P&M leads to the conclusionthat the bound given by Eq. (1) has to be a consequenceof choosing a generalized classical state as R that then defines, not just what a single system is, but how thepopulations, or property densities, are distributed in aclassical ensemble. Quantum superposition and behavior of the S param-eter. – In a quantum context, a dichotomic propertyis an observable represented by an operator, ˆ q , witheigenvalues q = ± | q ±(cid:105) .These properties can be incompatible. When this is thecase, the outcomes of the measurement stage dependon the previously prepared state. Therefore, differentlyfrom the classical scenario, in a P&M scheme the or-der of the preparation and measurement operations isrelevant. For instance, in the situation shown in theFig. 1, π + → µ − (cid:60) µ − → π + . This fact implies that P ( π + µ − ) (cid:54) = P ( µ − π + ) evoking a quantum discord, in theterms used by Zurek in the context of mutual informa-tion: “two classically identical expressions (...) generallydiffer when the systems involved are quantum” [22]. Thisfact can be clearly seen considering the case in which thephysical system is an ensemble of single photons, eachof them characterized by the quantum state, | ψ (cid:105) . Thequantum properties, in this scenario, can be differentlinear polarizations. To say that a photon is in state | q + (cid:105) means that it has a polarization oriented at θ q withrespect to the horizontal (w.r.t.h), and to say that a pho-ton is in | q −(cid:105) means that its polarization is oriented at θ ⊥ q = θ q +90 ◦ . In the canonical {| H (cid:105) , | V (cid:105)} basis, in which | H (cid:105) denotes horizontal and | V (cid:105) vertical polarization, | q ±(cid:105) can be written as [23], | q + (cid:105) = cos θ q | H (cid:105) + sin θ q | V (cid:105) (3a) | q −(cid:105) = sin θ q | H (cid:105) − cos θ q | V (cid:105) . (3b)With these definitions, it is possible to calculate con-ditional probabilities as projective operations P ( π i | ψ ) = | (cid:104) π i | ψ (cid:105) | and P ( µ j | π i ) = | (cid:104) µ j | π i (cid:105) | . Considering thesituation of Fig. 1, P ( µ − | π + ) = | (cid:104) µ − | π + (cid:105) | = sin ( θ µ − θ π ) and without loosing generality, one can choose | ψ (cid:105) = | H (cid:105) , in such a way that the marginal probability, P ( π + ),follows P ( π + ) ≡ P ( π + | H ) = | (cid:104) π + | H (cid:105) | = cos θ π .In general joint probabilities satisfy Bayes’ rule, P ( π i µ j ) = P ( µ j | π i ) P ( π i ), in which joint probabilitiesare related with conditional and marginal probabilities.In light of the P&M scheme, Bayes’ rule can be seen asthe product of the probability of preparing a system in π i , P ( π i ), times the probability of measuring the state µ j given that the system was prepared in π i , P ( µ j | π i ).Invoking Bayes’ rule, P ( π + µ − ) = P ( µ − | π + ) P ( π + )= sin ( θ µ − θ π ) cos θ π (4)and, P ( µ − π + ) = P ( π + | µ − ) P ( µ − )= sin ( θ µ − θ π ) sin θ µ (5)since P ( µ − ) ≡ P ( µ − | H ) = | (cid:104) µ − | H (cid:105) | = sin θ µ . Equa-tions (4) and (5) yield P ( π + µ − ) (cid:54) = P ( µ − π + ). Interest-ingly, the origin of this “discord” must be a consequenceof the superposition that exist in the definitions of thequantum states in Eq. (3a) and Eq. (3b), given that in ourdiscussion we are considering single systems and there areno entangled or product states.To see that for a quantum system, S <
0, we chooseas properties three different orientations of polarizationfor the three dichotomic properties, ˆ a , ˆ b and ˆ c . In thissituation, S becomes a function of the orientation anglesof the polarizations, S = S ( θ a , θ b , θ c ). Using Eq. (4), thejoint probabilities in Eq. (2) become P ( a + c − ) = sin ( θ c − θ a ) cos θ a , (6a) P ( a + b − ) = sin ( θ b − θ a ) cos θ a , (6b) P ( b + c − ) = sin ( θ c − θ b ) cos θ b . (6c)By combining these expressions in Eq. (2), it is foundthat S min = − . θ a (cid:39) ◦ , θ b (cid:39) . ◦ and θ c (cid:39) . ◦ . Experimental setup.–
In order to test that the param-eter S can indeed take negative values when consider-ing a quantum scenario, it is necessary to implementthe proper P&M scheme for each joint probability inEq. (2). These three joint probabilities are equivalentin the sense that they consist on the preparation of the FIG. 2. Experimental setup. A diode laser centered at 405nmwas used to pump a Type-II BBO crystal cut to producecollinear down-converted pairs of photons. A high pass fil-ter (HPF) and a dichroic mirror (DM) remove the residualpump beam. The down-converted photons are filtered by an810nm centered, 10nm bandwidth bandpass interference fil-ter (IF). To use the orthogonally-polarized pairs of photonsas a heralded single photon source, photons are coupled into afiber optic polarizing beam splitter (FPBS) with polarization-maintaining fiber pigtails, obtaining one photon per outputfiber. The vertically polarized photon goes directly to a singlephoton detector (D ), while the horizontally polarized pho-ton passes through the preparation and measurement stages,composed by a polarizer (P ), a half waveplate (HWP), a po-larizing beam splitter cube (PBS) and single photon detectors(D , D ). eigenstate with the positive eigenvalue, for the first prop-erty, and the measurement of the state with the negativeeigenvalue for the second one, which is precisely the casedepicted for detector D in Fig. 1. We take advantageof this equivalence and use the experimental setup de-picted in Fig. 2. The whole experiment can be analyzedalong three main stages: (i) source: heralded single pho-tons (HSP) with horizontal polarization | H (cid:105) are gener-ated via type II SPDC [24]. The HSP source is based onthe fact that in the SPDC process two photons, knownas signal and idler, are generated at the same time andtherefore one of the photons can be used to announcethe presence of its twin, that will play the role of thesingle photon during the experiment. For this setup, weused a 0.5 mm BBO crystal in a collinear configuration,pumped by a CW laser at 405 nm. The output from thecrystal was coupled to a fiber polarizing beam splitter(OZ Optics) whose fibers were single mode and polariza-tion maintaining. The vertical polarized photon was usedas trigger when detected by an avalanche photo-diode(SPCM-AQRH-13-FC), labeled as D in Fig. 2. To fur-ther ensure the correct polarization of the HSP, polarizerP was used in front of the fiber output; (ii) prepara-tion: using polarizer P , the different properties ˆ π areprepared in its eigenstate | π + (cid:105) defined by the angle θ π w.r.t.h.; and (iii) measurement: a half-wave plate (HWP)followed by a polarizing beam splitter (PBS) works like alinear polarization analyzer which outcomes define | µ + (cid:105) and | µ −(cid:105) , the eigenstates of the property ˆ µ . DetectorD counts how many of the photons, prepared in thestate | π + (cid:105) , are measured in state | µ −(cid:105) , and detector D counts how many of the photons prepared in state | π + (cid:105) are measured in the state | µ + (cid:105) . Therefore, by countingcoincidences D &D and D &D , one can obtain a mea-surement of P ( π + µ − ) and P ( π + µ + ), respectively. Thephotodetectors outputs are analyzed by means of a FieldProgramable Gate Array (FPGA) set to count singlesand coincidences in a 9 ns window.The angles associated to ˆ π and ˆ µ properties are set byorientation of P ( θ π ) and the HWP ( θ µ ), respectively. Inparticular, θ π and θ µ were varied in the range [0 ◦ , ◦ ]in steps of 6 ◦ . In the actual implementation, the HWPwas scanned between [0 ◦ , ◦ ] in steps of 3 ◦ due to itsworking principle. For all the combinations of these an-gles, the probabilities P ( a + c − ), P ( a + b − ) and P ( b + c − )from equations (6a)–(6c) are obtained in terms of θ a , θ b and θ c , and the parameter S ( θ a , θ b , θ c ) is reconstructed.Figure 3a depicts the theoretical surface, S ( θ a =156 ◦ , θ b , θ c ), and the experimental dots for S when θ exp a =156 ◦ . This value of θ a is chosen because in our experi-ment, it is the closer that we can get to the theoreti-cal prediction θ a (cid:39) ◦ that minimizes S . The plane S = 0 serves to underline the negative non-classical re-gion of the S parameter. In Fig. 3 b, we depict a profileof S , by choosing θ b as the nearest experimental angleto the theoretical predicted value for maximum viola- ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Θ b (cid:31) ° (cid:30) SS Θ c (cid:31) ° (cid:30) a)b) Θ c (cid:31) ° (cid:30) FIG. 3. Theoretical prediction and experimental data for a) S ( θ a = 156 ◦ , θ b , θ c ) and b) S ( θ a = 156 ◦ , θ b = 126 ◦ , θ c ). Thefact that S can reach negative values is clearly observed. tion that can be reach with our experimental apparatus, S ( θ a = 156 ◦ , θ b = 126 ◦ , θ c ). The agreement betweentheory and experiment is evident. The minimum exper-imental value of S is S expmin = − . ± .
03 that violatesCISS by 17 standard deviations.
Discussion and Conclusion.–
We have reported and ex-perimentally verified a quantum criterion for single sys-tems. In this work, we interpreted joint probabilities onthe light of a P&M scheme. The validity of this inter-pretation can be seen by the agreement between our ex-perimental data and the theoretical prediction in Fig. 3.The use of a P&M scheme removes possible controversialissues and allow us to conclude that the key notion whendealing with quantumness is superposition.The criterion we proposed is based on a previously de-rived inequality, that we called CISS, and the definitionof a parameter S . We proved the validity of our criterionby finding S < R used in the derivation ofthe CISS, is no longer useful in the quantum realm. By assuming R for quantum systems it is possible to obtain S <
S < superposition is quantum . Acknowledgments.
The authors acknowledge partial fi-nancial support from Facultad de Ciencias, Universidadde los Andes, Bogot´a, Colombia, from Departamento Ad-ministrativo de Ciencia, Tecnolog´ıa e Innovaci´on (COL-CIENCIAS), and from Universidad de Antioquia, underProjects No. 2014-989 (CODI-UdeA) and Estrategia deSostenibilidad del Grupo de F´ısica At´omica y Molecu-lar. They also thank Juan Pablo Restrepo Cuartas forhelping with the design of the graphs. ∗ [email protected][1] A. Einstein, Yu. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] N. Bohr, Phys. Rev. , 696 (1935).[3] E. Schr¨odinger, Math. Proc. Cambridge , 555 (1935).[4] O. Freire Jr, The quantum dissidents: rebuildingthe foundations of quantum mechanics (1950-1990) (Springer, Berlin, 2015).[5] J. S. Bell, Physics , 195 (1964).[6] A. J. Leggett and A. Garg, Phys. Rev. Lett. , 857(1985).[7] R. Alicki and N. V. Ryn, J. Phys. A - Math. Theo. ,062001 (2008).[8] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. , 1804 (1982).[9] C. Emary, N. Lambert, and F. Nori, Rep. Prog. Phys. , 016001 (2014).[10] G. Brida, I. P. Degiovanni, M. Genovese, F. Piacentini,V. Schettini, N. Gisin, S. V. Polyakov, and A. Migdall,Phys. Rev. A , 044102 (2009).[11] E. P. Wigner, Am. J. of Phys. (1970).[12] R. Lapiedra, Europhys. Lett. , 202 (2006).[13] L. E. Ballentine, Quantum Mechanics: A Modern Devel-opment (World Scientific Publishing Co. Pte. Ltd., Sin-gapore, 1998).[14] A. Peres,
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