Quantum contextuality in systems of identical bosonic particles is explicitly
exhibited via the maximum violation of a suitable inequality of
Clauser-Horne-Shimony-Holt type. Unlike the approaches considered so far, which
make use of single-particle observables, our analysis involves collective
observables constructed using multi-boson operators. An exemplifying scheme to
test this violation with a quantum optical setup is also discussed.Comment: 4 pages, 1 figure, LaTe

Fabio Benatti

Marco Genovese

Roberto Floreanini

S. Kochen

Stefano Olivares

English

arXiv

BENATTI, FABIO

OLIVARES, STEFANO

Floreanini F.

Genovese M.

Archivio istituzionale della ricerca - Università di Trieste

Quantum contextuality in N-boson systems

Crossref

Physical Review A

Quantum contextuality in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>-boson systems

Quantum contextuality in systems of identical bosonic particles is explicitly exhibited via the maximum violation of a suitable inequality of Clauser-Horne-Shimony-Holt type. Unlike the approaches considered so far, which make use of single-particle observables, our analysis involves collective observables constructed using multiboson operators. An exemplifying scheme to test this violation with a quantum optical setup is also discussed

F. Benatti

R. Floreanini

M. Genovese

S. Olivares

AIR Universita degli studi di Milano

PHYSICAL REVIEW A 84, 034102 (2011)Quantum contextuality in N-boson systemsFabio Benatti,1,2 Roberto Floreanini,2 Marco Genovese,3 and Stefano Olivares11Dipartimento di Fisica, Universita` degli Studi di Trieste, I-34151 Trieste, Italy2Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34014 Trieste, Italy3INRIM, Strada delle Cacce 91, I-10135 Torino, Italy(Received 6 April 2011; published 15 September 2011)Quantum contextuality in systems of identical bosonic particles is explicitly exhibited via the maximumviolation of a suitable inequality of Clauser-Horne-Shimony-Holt type. Unlike the approaches considered sofar, which make use of single-particle observables, our analysis involves collective observables constructedusing multiboson operators. An exemplifying scheme to test this violation with a quantum optical setup is alsodiscussed.DOI: 10.1103/PhysRevA.84.034102 PACS number(s): 03.65.Ud, 03.65.Ta, 05.30.Jp, 42.50.XaA physical theory is called noncontextual if the measure-ment results about a given observable A do not depend onwhether other commuting observables B are simultaneouslymeasured. Quantum mechanics turns out to be a contextualtheory, with Bell’s nonlocality being a particular form ofcontextuality [1–3].Many experimental tests of contextuality have been pro-posed where the commuting observables A and B refer tothe spatial and polarization degrees of freedom of singlespin- 12 particles or photons [4–14]. Most of the tests checkwhether inequalities of Clauser-Horne-Shimony-Holt (CHSH)type [15], applied now to local, single-system settingsE = |〈A(B + B ′)〉 + 〈A′(B − B ′)〉|  2 , (1)are violated or not, with 〈· · · 〉 being the expectation valuesof the involved quantities in a suitable system state. Theseinequalities are a straightforward consequence of assumingthat the values assigned to products AB, AB ′, A′B, and A′B ′ ofobservables A and A′, pertaining to a given degree of freedom,commuting with observables B and B ′ pertaining to a differentdegree of freedom, are products of the values independentlyassigned to the separate observables A, A′, B, and B ′.Note that different approaches to investigate quantumcontextuality, not based on the inequality (1), have also beenconsidered (for instance, see [16–23]). Furthermore, violationof noncontextuality has been exploited as a resource forquantum key distribution and quantum information processingpurposes [24–26]. The contextuality of mesoscopic systemshas been recently addressed in systems consisting of distin-guishable particles [27] and superconductive devices [28], butall involving single-qubit measurements.The question that naturally arises is whether it is possibleto test noncontextuality in systems of N identical particles,such as trapped ultracold bosons, phonons in systems of mi-cromechanical oscillators, and multiphoton states in quantumoptics, i.e., in truly mesoscopic systems. In such cases, the useof collective observables is mandatory; from the theoreticalpoint of view, these observables can be naturally describedin terms of multiboson algebras, which represent a novel toolin this specific context. This analysis of mesoscopic systemsrepresents also an important contribution to the study of thetransition between quantum and classical systems.Motivated by the previous considerations, in this BriefReport, we address quantum contextuality in systems madeof N identical particles (bosons). Our main result is theidentification of a set of collective observables for multi-particle tests of noncontextuality. More specifically, usingmaximally entangled NOON-like states, we first show thatinequalities of CHSH type as in Eq. (1) can not be violatedif constructed with standard single-boson observables; thisis ultimately due to the indistinguishability of the involvedparticles. In order to make quantum contextuality apparent insuch systems, one has to use instead collective observablesbuilt with suitable multiboson operators; indeed, with suchobservables, one can construct CHSH-type inequalities that aremaximally violated by the same NOON-like states. A physicalimplementation based on a quantum optical interferometricscheme able to test these results is also presented anddiscussed.Let us consider a system of N bosons with two degrees offreedom, e.g., a “spatial” and an “internal” one. For the sakeof definiteness, we assume these degrees of freedom to taketwo possible values, labeled by the variables k = 1,2 and α =−,+, respectively. As appropriate for boson systems, we adopta second quantized description and introduce suitable creationa(α)k† (and annihilation a(α)k ) operators for the mode (k,α),satisfying the usual bosonic algebra [a(α)k , a(β)l †] = δk,lδα,β .Out of the vacuum |0〉, they create the (Fock) states of thesystem|n(−)1 ,n(−)2 ,n(+)1 ,n(+)2 〉 =∏k={1,2}∏α={−,+}(a(α)k†)n(α)k√n(α)k !|0〉 , (2)containingn(α)k bosons in the mode (k,α), such that∑k,α n(α)k =N . For the sake of compactness, hereafter, we shall indicateonly the nonvanishing occupation numbers, i.e., those forwhich n(α)k = 0.A basic example of a quantum system exhibiting contex-tuality, that is, violating the expectation that the result of ameasurement be independent of a previous or simultaneousmeasurement of any set of mutually commuting observables,is a single-particle system (N = 1), prepared in the following034102-11050-2947/2011/84(3)/034102(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 034102 (2011)superposition of Fock states:|ψ〉 = 1√2(a(−)1 † + a(+)2 †)|0〉= 1√2(|n(−)1 = 1〉 + |n(+)2 = 1〉) . (3)By taking the variables k and α to label suitable spinlikecomponents, one finds that CHSH-type inequalities as inEq. (1) are indeed maximally violated, as confirmed inexperiments based on single-neutron [6] and single-photoninterferometers [7,8].Similarly, in the case of N bosons, one can consider thefollowing spinlike observables:Jx = 12∑α={−,+}[a(α)1† a(α)2 + a(α)2 † a(α)1], (4a)Jy = i2∑α={−,+}[a(α)2† a(α)1 − a(α)2 † a(α)1], (4b)Jz = 12∑α={−,+}[a(α)1† a(α)1 − a(α)2 † a(α)2], (4c)Sx = 12∑k={1,2}[a(−)k † a(+)k + a(+)k † a(−)k ], (4d)Sy = i2∑k={1,2}[a(+)k † a(−)k − a(−)k † a(+)k ], (4e)Sz = 12∑k={1,2}[a(−)k † a(−)k − a(+)k † a(+)k ]. (4f)The observables Ji and Si both satisfy the SU(2) algebraicrelations ([Ji,Jj ] = iijkJk , and similarly for Si), while com-muting among themselves [Ji,Sj ] = 0. Furthermore, theireigenvalues are the integers between −N/2 and N/2. Then,once rescaled by 2/N , their eigenvalues lie in the interval[−1,1] and the rescaled observables should violate a suitableconstructed inequality of CHSH type of the form (1).In the case of the single-boson system (N = 1), this canbe easily shown by taking the following choices for the fourobservables appearing in the inequality (1):A = 2 Jz, B = 2 Sz(π/4),A′ = 2 Jx, B ′ = 2 Sz(−π/4),where Sz(θ ) = cos θ Sz + sin θ Sx . For the state (3), straight-forward calculations lead to〈Jz ⊗ Sz〉 = 〈Jx ⊗ Sx〉 = 1/4, (5)〈Jz ⊗ Sx〉 = 〈Jx ⊗ Sz〉 = 0, (6)whence E = 2√2 > 2 indeed exhibits the maximum violationof the CHSH inequality allowed by quantum mechanics [29].For systems of bosons with N > 1, the obvious general-ization of these single-particle choices does not lead to anynoncontextuality violation; this is not surprising since, in orderto exhibit quantum contextuality, the CHSH test E > 2 needsto be adapted to the system under study through a carefulchoice of both the state and of the four observables A, A′, B,and B ′. In the case of N bosons, the state (3) can be easilygeneralized:|	〉 = 1√2N ![(a(−)1 †)N + (a(+)2 †)N ]|0〉= 1√2[|n(−)1 = N〉 + |n(+)2 = N〉]; (7)this is a NOON-like state, superposition of two Fock states, thefirst representing N bosons all in the same mode with k = 1,α = − and similarly the other N bosons in the state withk = 2, α = +; it clearly reduces to the state (3) when N = 1.However, the averages of the same observables as before inthis new state |	〉 become〈Jz ⊗ Sz〉 = N2/4 , (8)〈Jx ⊗ Sx〉 = 〈Jz ⊗ Sx〉 = 〈Jx ⊗ Sz〉 = 0; (9)as a consequence, after the necessary rescaling of the spinlikeobservables by 2/N , one finds E = √2 < 2 and therefore noviolation of the noncontextuality test.Consider instead the multiboson operators [30–32]A(α)k = FN(nˆ(α)k) (a(α)k)N, (10)whereFN(nˆ(α)k) = [ nˆ(α)k + NNnˆ(α)k !(nˆ(α)k + N)!]1/2, (11)with nˆ(α)k = a(α)k † a(α)k the number operator relative to the mode(k,α), with the symbol   representing the integer part. TheA(α)k are collective annihilation operators as they diminish byN the number of bosons; indeed, on the orthonormal basis ofFock number states as in Eq. (2), they act as follows:A(α)k∣∣n(α)k 〉 =√n(α)kN ∣∣n(α)k − N 〉 , (12)while the Hermitian conjugate operators A(α)k † act as collectivecreation operatorsA(α)k† ∣∣n(α)k 〉 =√n(α)kN+ 1 ∣∣n(α)k + N 〉 . (13)Then, one easily finds that [A(α)k ,A(β)l †] = δk,l δα,β .In analogy with Eq. (4), with these multiboson annihilationand creation operators, one can now form collective spinlikeoperators relative to the spatial degree of freedom:Jx = 12∑α={−,+}[A(α)1† A(α)2 + A(α)2 † A(α)1], (14a)Jy = i2∑α={−,+}[A(α)2† A(α)1 − A(α)2 † A(α)1], (14b)Jz = 12∑α={−,+}[A(α)1† A(α)1 − A(α)2 † A(α)2]. (14c)034102-2BRIEF REPORTS PHYSICAL REVIEW A 84, 034102 (2011)One checks that they satisfy the SU(2) commutation relations[Ji ,Jj ] = iijk Jk . In the same way, the multiboson, collectivespinlike polarization operatorsSx = 12∑k={1,2}[A(−)k † A(+)k + A(+)k † A(−)k ], (15a)Sy = i2∑k={1,2}[A(+)k † A(−)k − A(−)k † A(+)k ], (15b)Sz = 12∑k={1,2}[A(−)k † A(−)k − A(+)k † A(+)k ] (15c)satisfy [Si ,Sj ] = iijk Sk . Further, as before, the two sets ofoperators mutually commute [Ji ,Sj ] = 0.Further, notice thatA(α)k† A(α)k =nˆ(α)kN, (16)so that, unlike the single-boson spinlike operators in (4), theoperators in Eqs. (14) and (15) have eigenvalues ±1/2. Thus,we can directly consider the inequality (1) relative to theNOON state (7) and to the collective observablesA = 2Jz, B = 2Sz(π/4), (17a)A′ = 2Jx, B ′ = 2Sz(−π/4), (17b)where Sz(θ ) = cos θ Sz + sin θ Sx . One finds〈Jz ⊗ Sz〉 = 〈Jx ⊗ Sx〉 = 1/4, (18)〈Jz ⊗ Sx〉 = 〈Jx ⊗ Sz〉 = 0 , (19)whence E = 2√2 > 2, so that the CHSH inequality (1) resultsmaximally violated and quantum contextuality is manifest.An example of a physical setup able to test, in principle,this violation can be built within quantum optics. In thisframework, the two degrees of freedom labeled by k and αcan be taken to refer to the photon path (momentum) andpolarization: the operator a(α)k † creates from the vacuum asingle photon along the path k with polarization α, while A(α)k †creates N photons in the same state. Similarly, unlike Ji andSi in Eq. (4), the operators Ji in Eq. (14) and Si in Eq. (15)are collective observables, referring to path and polarizationdegrees of freedom, respectively.Notice that the experimental implementation of the test doesnot require the actual realization of the multiboson operatorsA(α)k† and A(α)k , but only of suitable procedures for measuringthe collective observables Ji and Si . In order to achievethis, operations acting on the N -photon states as a whole arenevertheless needed; as a consequence, linear optical passivedevices can not directly be used.The proposed implementation, sketched in Fig. 1, is basedon a Mach-Zehnder-type interferometric scheme; however, theusual beam splitters are replaced with the “quantum beamsplitters” (QBSs) proposed in Ref. [33], where a nonlinearmedium is inserted in one of the two arms. These quantumdevices, realized through the unitary operator UQBS, generateNOON-like states by acting on a Fock state where the N pho-PCHWPPBS QBS|n(−)1 = N|n(−)1 = N M1M2M1M2JzJxSzSxFIG. 1. (Color online) Physical implementation of a quantumoptics setup able to test quantum contextuality with NOON-likestates. Legend of the components: PBS = polarizing beam splitter;QBS = quantum beam splitter; HWP = half-wave plate; PC = photoncounter. See the text for details.tons are all prepared in a given path-polarization configuration(k,α) [33]:UQBS∣∣n(α)k = N 〉 =∣∣n(α)k = N 〉+ (−1)l∣∣n(α)l = N 〉√2, (20)with l = k and k,l = 1,2. Here, k = 1 (k = 2) refers tothe lower (upper) path, while α = −,+ to the two possiblehorizontal and vertical polarization, i.e., H ≡ −, V ≡ +. TheNOON-like state given in Eq. (7) can then be obtained from theappropriate state (20) with k = 1 and α = − by switching thepolarization in the upper (k = 2) path by means of a half-waveplate (HWP). [It is worth noticing that, due to the presenceof the two mirrors which exchange the modes 1 and 2 (seeFig. 1), the lower (upper) path is detected as k = 2 (k = 1); inorder to avoid confusion, in Fig. 1 we call the detectors Mk ,with k referring to the actual followed path.]The operator Jz can then be measured by sending theoutgoing beams directly to the two detector devices Mk(upper-left scheme in Fig. 1); indeed, the detection operationat M1 and M2, for instance, implemented by photon counters,precisely corresponds to the projection onto the eigenstates ofJz. Instead, in the case of the observableJx , one needs to makethe two beams interfere at another QBS (lower-left scheme inFig. 1) before detection at Mk . Using (20), the outgoing stateafter the second QBS reads now as|	out〉 = 1√2(|	(−)2 〉 − |	(+)1 〉), (21)where |	(+)1 〉 = (|n(+)1 = N〉 − |n(+)2 = N〉)/√2 and |	(−)2 〉 =(|n(−)1 = N〉 + |n(−)2 = N〉)/√2 are eigenstates ofJx , the labelk in |	(±)k 〉, k = 1,2, referring now to the upper and lowerexiting paths, beyond the additional QBS. As a result, thedetection operation at M1 and M2 corresponds to projectionsonto these two states.In order to simultaneously determine, together with Jz,i = x,z, also the operators Sz and Sx , as required by theinequality (1), the detection procedures at Mk need to be more034102-3BRIEF REPORTS PHYSICAL REVIEW A 84, 034102 (2011)structured than those provided by simple photon counters.The measurement of Sz can be implemented through apolarizing beam splitter (PBS) and two photon counters(PCs), which allow us to discriminate between |n(−)k = N〉and |n(+)k = N〉 eigenstates of Sz (see the right-upper schemein Fig. 1). Instead, in order to measure Sx , one needs toinsert before the photon counters a further Mach-Zehnderinterferometric circuit, built with additional HWP and QBS(see the right-lower scheme of Fig. 1). From the sameconsiderations described before, in explaining the determi-nation of Jx now applied to the polarization degrees offreedom, one can check that photon detection at the exitarms of this additional interferometer precisely corresponds toprojection onto the eigenstates of Sx , (|n(−)k = N〉 ± |n(+)k =N〉)/√2.Finally, it is worth noticing that the schemes just illustratedfor measuring the relevant Ji and Si observables are such thatall the N photons arrive at one and only one of the photoncounters. Therefore, by suitably composing one of the twodetecting schemes in the right panel of Fig. 1 with one ofthe measurements of the left panel, one can experimentallydetermine all collective observables needed to test the CHSHinequality discussed above.Summarizing, in this Brief Report, we discussed a test ofquantum contextuality in systems composed by N identicalbosonic particles. We showed that, unlike the analysis sofar considered in the literature, based on systems of distin-guishable particles using single-particle observables, in orderto make quantum contextuality apparent in such mesoscopicbosonic systems, measurements of collective observables areneeded. We further remark that the observables we used arebuilt out of collective operators belonging to a multibosonalgebra, a tool that has never been considered in such a context.Incidentally, a state-independent test of noncontextuality withN -boson systems can also be straightforwardly achieved withour collective observables following, e.g., Refs. [11,13].We are confident that these theoretical results will opennew perspectives in the study of quantum contextuality inmesoscopic systems, as trapped ultracold bosons and phononsin systems of micromechanical oscillators, and provide newinsights in the actual realization of further experimental tests,possibly paving the way to new applications in quantumtechnology.[1] S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967).[2] N. D. Mermin, Rev. Mod. Phys. 65, 803 (1993).[3] M. Genovese, Phys. Rep. 413, 319 (2005).[4] S. Basu, S. Bandyopadhyay, G. Kar, and D. Home, Phys. Lett.A 279, 281 (2001).[5] F. Benatti, R. Floreanini, and R. Romano, J. Phys. A: Math. Gen.35, 4955 (2002).[6] Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch,Nature 425, 45 (2005); Y. Hasegawa, R. Loidl, G. Badurek,M. Baron, and H. Rauch, Phys. Rev. Lett. 97, 230401 (2006).[7] M. Michler, H. Weinfurter, and M. ˙Zukowski, Phys. Rev. Lett.84, 5457 (2000).[8] B. R. Gadway, E. J. Galvez, and F. De Zela, J. Phys. B 42,015503 (2009).[9] E. Karimi et al., Phys. Rev. A 82, 022115 (2010).[10] A. K. Pan and D. Home, Phys. Lett. A 373, 3430 (2009); Int. J.Theor. Phys. 49, 1920 (2010).[11] A. Cabello, Phys. Rev. Lett. 101, 210401 (2008).[12] A. Cabello, S. Filipp, H. Rauch, and Y. Hasegawa, Phys. Rev.Lett. 100, 130404 (2008).[13] E. Amselem, M. Radmark, M. Bourennane, and A. Cabello,Phys. Rev. Lett. 103, 160405 (2009).[14] C. Simon, M. ˙Zukowski, H. Weinfurter, and A. Zeilinger, Phys.Rev. Lett. 85, 1783 (2000).[15] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys.Rev. Lett. 23, 880 (1969).[16] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990).[17] G. Kirchmair et al., Nature (London) 460, 494 (2009).[18] H. Bartosik et al., Phys. Rev. Lett. 103, 040403 (2009).[19] B. H. Liu et al., Phys. Rev. A 80, 044101 (2009).[20] Y.-F. Huang, C.-F. Li, Y.-S. Zhang, J.-W. Pan, and G.-C. Guo,Phys. Rev. Lett. 90, 250401 (2003).[21] O. Gu¨hne et al., Phys. Rev. A 81, 022121 (2010).[22] A. R. Plastino and A. Cabello, Phys. Rev. A 82, 022114 (2010).[23] G. McKeown, M. G. A. Paris, and M. Paternostro, Phys. Rev. A83, 062105 (2011).[24] S. Adhikari, D. Home, A. S. Majumdar, and A. K. Pan, e-printarXiv:1101.0660.[25] M. Kleinmann, O. Gu¨hne, J. R. Portillo, J.-A˚. Larsson, andA. Cabello, e-print arXiv:1007.3650.[26] K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki,M. Pawlowski, and M. Bourennane, e-print arXiv:1006.0468.[27] A. Cabello, Phys. Rev. A 82, 032110 (2010).[28] L. F. Wei, K. Maruyama, X.-B. Wang, J. Q. You, and F. Nori,Phys. Rev. B 81, 174513 (2010).[29] B. S. Cirel’son, Lett. Math. Phys. 4, 93 (1980).[30] R. A. Brandt and O. W. Greenberg, J. Math. Phys. 10, 1168(1969).[31] M. Rasetti, Int. J. Theor. Phys. 5, 377 (1972).[32] F. Raffa and M. Rasetti, Int. J. Quantum Inf. 7, 221 (2009); M.Genovese, F. A. Raffa, and M. Rasetti, e-print arXiv:0912.4582.[33] J. Dunningham and T. Kim, J. Mod. Opt. 53, 557 (2006).034102-4

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quantum contextuality in N-boson systems

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