Quantum contextuality for a three-level system sans realist model
aa r X i v : . [ qu a n t - ph ] J un Quantum contextuality for a three-level system sans realist model
A. K. Pan ∗ and K. Mandal † National Institute of Technology Patna, Ashok Rajhpath, Bihar 800005, India
Recently, an interesting form of non-classical effect which can be considered as a form of contextu-ality within quantum mechanics, has been demonstrated for a four-level system by discriminating thedifferent routes that are taken for measuring a single observable. In this paper, we provide a simplerversion of that proof for a single qutrit, which is also within the formalism of quantum mechanicsand without recourse to any realist hidden variable model. The degeneracy of the eigenvalues andthe L ¨ u der projection rule play important role in our proof. I. INTRODUCTION
Quantum mechanics(QM) is a statistical theory and the occurrence of a definite outcome in an individual measure-ment of a dynamical variable cannot be ensured within the formalism. The realist hidden variable models of QM aimto provide the complete state of a quantum system by including hidden variables along with the quantum wave func-tion. In such hypothetical models, the individual outcomes of a dynamical variable are desired to be predeterminedby the suitable values of the hidden variables(say, λ s). In his path-breaking work, Bell [1] first showed that in orderto reproduce all QM statistics by a realist model, a constraint needs to be imposed at the level of individual measuredvalue - a concept, widely known as nonlocality. Another similar constraint on the hidden variable model known ascontextuality, was first put forward independently by Bell [2] and by Kochen and Specker [3] who demonstrated aninconsistency between the predictions of the QM and the relevant noncontextual hidden variable model.Lat us first encapsulate the essence of noncontextuality assumption in QM and in a realist hidden variable model.Let an observable b A be commuting with b B and b C , while b B and b C being non-commuting. Given a wave function,QM ensures that the measured statistics of b A is independent of, whether the measurement is performed togetherwith b B or b C . This feature is considered as noncontextuality at the level of QM statistical results. The assumption of‘noncontextuality’ in any realist model assumes that the predetermined individual outcome of any relevant observable,for a given λ , is the same , whatever be the compatible way that observable is measured. In a non-contextual model,let v ( A ) be the individual measured values of b A , as specified by a λ , and let v ( B ) and v ( C ) be the individual measuredvalues of the observables b B and b C respectively, that are also predetermined by the same λ . Now, if the individualmeasured value of the observable is assumed to follow the same context-independence as QM, the crucial question inthis regard is, as to what extent the assumption of noncontextuality is compatible with the statistical prediction ofQM. It is shown [2, 3] that, if the dimension of the Hilbert space is greater than two, the assignment of values bynoncontextual realist models is inconsistent with the QM for all possible set of experiments, thereby requiring thehidden variable models to be contextual to reproduce all the predictions of QM.Various forms of demonstrations of contextuality have been given by providing a variety of elegant proofs, (see,for example, [4–12]) and a flurry of experiments have been reported [13–18]. The original proof by Kochen andSpecker was demonstrated by using 117 different rays for three-level system, and later simpler versions have beenput forward [4, 5, 12] by reducing the number of rays. By using general observables, Peres [6] demonstrated aninconsistency between QM and the non-contextual realist models for an entangled state for a pair of two qubits whichwas later extended to the state-dependent one by Mermin [7]. Using a three-level system, an elegant proof is providedby Klyachko et al. [11] using only five observables, which has experimentally been tested [18]. A curious form ofcontextuality within the formalism of QM has also been demonstrated [9] without using any realist model.Note that, the usual proofs of contextuality are demonstrated by first assuming the non-contextual value assign-ments by a realist model for a set of compatible observables that are being co-performed and then by showing thecontradiction of such value assignments with QM. In other words, such proofs can be regarded as the violation ofnoncontextual realist model by QM. Recently, for a two-qubit system an interesting non-classical character of QM hasbeen demonstrated by Sala Mayato and Muga [19]. Specifically, the authors have shown that within the formalism ofQM, it is possible to discriminate the different routes that are adopted for measuring a given observable by analysingthe reduced density matrices after the measurements.In this paper, we provide a simpler proof of such effect than that is given in Ref.[19]. Our argument is based on ∗ [email protected] † [email protected] using a single observable in a qutrit system instead of tensor product observable in four-level system used in Ref.[19].One may argue that, since different experimental contexts are employed for measuring a given observable that producetwo distinct reduced density matrices, such a non-classical effect may be considered as a kind of contextual characterinherent in the formalism of QM . Note that, such a form of contextuality is without recourse to any realist hiddenvariable model. In this demonstration, the degeneracy of the eigenvalues plays an important role and for this, vonNeuman projection rule needs to be replaced by L ¨ u der projection rule.The paper is organized as follows. In Section II, we provide a simpler proof of contextuality by discriminatingthe routes of measuring a given observable for a three-level system. In Section III, we provide a brief sketch aboutthe possible experimental realization of our scheme by explicitly using the probe state. We conclude and discuss ourresults in Section IV. II. ’QUANTUM CONTEXTUALITY’ FOR A SINGLE QUTRIT
Before presenting our simpler scheme, let us recapitulate the essence of the argument given in Ref.[19]. The use ofL ¨ u der projection rule instead of von Neumann rule is crucial because the eigenvalues of the different observables aredegenerate.Let an observable ˆ M has discrete eigenvalues m , m , m , ... having degeneracies g , g , g , .. respectively, consider P in is the projection operator associated with each eigenvector and ρ is the initial density matrix of the system. Insuch a scenario, the von Neumann projection rule prescribes the reduced density matrix to be ρ ′ = X n,i P in ρP in (1)where P in = | χ in ih χ in | . (2)However, using the L ¨ u ders projection rule the reduced density matrix can be obtained as ρ ′ = X n P n ρP n (3)where P n = g n X i =1 | χ in ih χ in | (4)Thus, for an observable with degenerate eigenvalues, the L ¨ u ders rule provides the reduced density matrix to bepartially mixed but the von Neumann rule produces a maximally mixed state. For example, for the measurement of ˆ I d where degeneracy is d , the von Neumann rule provides the reduced density matrix I d /d but according to L ¨ u der rulethe state will remain unchanged, as expected. In contrast to the von Neumann projection rule, the L ¨ u ders rule doesnot reduce the state to an eigenstate, it can then be considered a kind of incomplete measurement. The conceptualdifference between L ¨ u der and von Neumann projection rule is discussed in Ref.[20].In Ref.[19], the authors have considered three observables that were used in Peres-Mermin proof [7] of contextuality.Those observables are ˆ M = ˆ σ x ⊗ I , ˆ M = I ⊗ ˆ σ y and ˆ M = ˆ σ x ⊗ ˆ σ y where ˆ M = ˆ M ˆ M . Since ˆ M , ˆ M and ˆ M aremutually commuting, they have common eigenstates. Now, one can perform the measurement of the observable ˆ M by using two different routes; first by directly measuring ˆ M and second, by measuring ˆ M and then by successivelymeasuring ˆ M . It has been shown in Ref.[19] that the reduced density matrix obtained for these two different routesof measurements are not the same and thus constitutes a proof of contextuality.In this paper, we provide a simpler proof that also uses three commuting observables but in a three-level systeminstead of four-level system used in Ref.[19]. In order to demonstrate it, we choose two observables ˆ A and ˆ B andanother observable ˆ C , so that, ˆ C = ˆ A ˆ B is satisfied. The spectral decompositions of the observables ˆ A , ˆ B and ˆ C areas follows; A = | φ ih φ | + | φ ih φ | (5)whose eigenvalues are , and with the corresponding eigenvectors | φ i = (1 , , T , | φ i = (0 , , T , | φ i = (0 , , T respectively. B = | φ ih φ | + | φ ih φ | (6)whose eigenvalues are , and and corresponding eigenvectors are | φ i , | φ i and | φ i respectively. C = | φ ih φ | (7)whose eigenvalues are , and and corresponding eigenvectors are | φ i , | φ i and | φ i respectively.Let the initial state of the three-level system is considered to be | η i = α | φ i + β | φ i + γ | φ i (8)where α, β, γ in general complex satisfying | α | + | β | + | γ | = 1 .If the observable ˆ C is directly measured, the reduce density matrix ρ C can be obtained by using L ¨ u der rule givenby Eq.(3), is given by ρ C = ( | φ ih φ | + | φ ih φ | ) ρ ( | φ ih φ | + | φ ih φ | ) + ( | φ ih φ | ) ρ | ( φ ih φ | )= | α | | φ ih φ | + | β | | φ ih φ | + | γ | | φ ih φ | + α ∗ γ | φ ih φ | + γ ∗ α | φ ih φ | (9)Now, if the observable ˆ A is measured first and one employs the L ¨ u der projection rule before measuring the observable ˆ B , the reduce density matrix can be obtained as ρ A = ( | φ ih φ | + | φ ih φ | ) ρ ( | φ ih φ | + | φ ih φ | ) + ( | φ ih φ | ) ρ | ( φ ih φ | )= | α | | φ ih φ | + | β | | φ ih φ | + | γ | | φ ih φ | + αβ ∗ | φ ih φ | + βα ∗ | φ ih φ | (10)Using ρ A as initial state, if the observable ˆ B is measured the reduce state ρ AB can be obtained by again using L ¨ u derprojection rule is given by ρ AB = ( | φ ih φ | + | φ ih φ | ) ρ A ( | φ ih φ | + | φ ih φ | ) + ( | φ ih φ | ) ρ A | ( φ ih φ | )= | α | | φ ih φ | + | β | | φ ih φ | + | γ | | φ ih φ | (11)Similarly, if another route is taken by first measuring the observable ˆ B followed by ˆ A , the reduce density matrix ρ BA is given by ρ BA = | α | | φ ih φ | + | β | | φ ih φ | + | γ | | φ ih φ | (12)Since ρ BA = ρ C and ρ AB = ρ C , it is in principle possible to distinguish the routes of the measurements of theobservable b C .We thus provided a proof of quantum contextuality for three-level system in terms of discriminating the routes ofmeasuring a given observable. Note that, this demonstration provides a true form of quantum contextuality withoutreference to any realist hidden variable model. The proof presented here is much simpler than the earlier one [19]which uses entangled state and tensor product observables in a two-qubit system.Note here that, by using von Neumann projection rule given by Eq. (1), one obtains, ρ C = ρ AB = ρ BA = | α | | φ ih φ | + | β | | φ ih φ | + | γ | | φ ih φ | and dose not constitute any proof of contextuality.We would also like to point out that the degeneracy of the eigenvalues plays a crucial role in our proof. If theobservables ˆ C , ˆ A and ˆ B all are non-degenerate no such proof of contextuality can be shown. To give an example, weconsider three mutually commuting observable b D , b D ,and b D , such that, all have non-degenerate eigenvalues and b D = b D . b D are as follows; b D = (1 + √ | d ih d | + (1 − √ | d ih d | (13)whose eigenvalues are (1+ √ , (1 −√ and and corresponding unnormalized eigenvectors are | d i = (1 , − √ , T , | d i = (1 , − − √ , T and | d i = ( − , , T . b D = √ | d ih d | − √ | d ih d | + | d ih d | (14)whose eigenvalues are −√ , √ and and the corresponding unnormalised eigenvectors are | d i , | d i and | d i . Theeigenvalues of b D are √ , − √ , and corresponding eigenvectors are | d i , | d i and | d i .If one takes an initial state of the form given by | γ ′ i = a | d i + a | d i + a | d i (15)where a , a and a are in general complex. Now following the prescription used before, we can calculate the reduceddensity matrices ρ D D , ρ D D and ρ D and show that ρ D D = ρ D D = ρ D = | a | | d ih d | + | a | | d ih d | + | a | | d ih d | (16)which implies that the degeneracy plays the crucial role for showing the contextuality in the earlier case.Note that, the above calculation is done by considering the system state only. In the next section, by introducingthe relevant probe states we provide a brief sketch to show that how the non-classical effect we have demonstratedcan experimentally realized. III. POSSIBLE EXPERIMENTAL REALIZATION OF THE NON-CLASSICAL EFFECT
In any measurement scenario the probe state plays the important role, because by looking at the probe pointerposition the value of the system is inferred. We provide a measurement model by explicitly using the probe state.In an ideal quantum measurement scenario, after the measurement interaction, the system and probe states becomeentangled, so that, a perfect one-to-one correspondence between system and probe states is established. But, themeasurement may not be completed without mentioning how the observable probabilities can be obtained from suchan entangled state, thereby requiring the final step (non-unitary) of quantum measurement - a notion, widely knownas collapse of the wave function. For a measured system observable having degenerate eigenvalues the final step isto use L ¨ u der projection rule, otherwise the von Neumann rule is sufficient. Note that inclusion of probe state shouldretain the same non-classical effect demonstrated above.Let us again consider the initial system state is given by Eq.(8) and assume the initial probe state is | ψ i , so that, thetotal state is | Ψ i = | ψ i| η i . The system-probe entangled state after a suitable measurement interaction for measuring ˆ A is given by | Ψ A i = α | ψ i| φ i + β | ψ i| φ i + γ | ψ i| φ i (17)where | ψ i and | ψ i are the post-interaction probe states corresponding to the eigenvalues and respectively. Forideal measurement situation h ψ | ψ i = 0 needs to be satisfied. The entangled state given by Eq.(17) is then subject tomeasurement of the observable ˆ B . After this measurement interaction, a different system-apparatus entangled stateis obtained is given by | Ψ AB i = α | ψ i| φ i + β | ψ i| φ i + γ | ψ i| φ i (18)where | ψ i is the post-interaction probe state corresponding to the eigenvalue of the observable ˆ B . Again, for anideal measurement, h ψ | ψ i = 0 and h ψ | ψ i = 0 .Next, the order of measurement is swapped by first considering the interaction for measuring ˆ B followed by ˆ A . Thesystem-probe entangled state after the measurement of ˆ B can be written as | Ψ B i = α | ψ i| φ i + β | ψ i| φ i + γ | ψ i| φ i (19)Now measuring ˆ A using | Ψ B i as initial state, one obtains | Ψ BA i = α | ψ i| φ i + β | ψ i| φ i + γ | ψ i| φ i (20)If the observable ˆ C is directly measured, the system-apparatus entangled state is given by | Ψ C i = α | ψ i| φ i + β | ψ i| φ i + γ | ψ i| φ i (21)Now, the observable probe signals can be obtained by suitably using the L ¨ u ders projection rule in the Eqs.(18),(20) and (21) corresponding to the three routes of the measurements.One may also check that inclusion of probe state does not effect the contextuality argument presented in the abovesection. For this, by taking the partial trace over the probe state, the reduced density matrix ρ ′ AB = T r p [ | Ψ AB ih Ψ AB | ] of the system for the first route is equal to ρ AB . Similarly, it can be shown that ρ ′ C = T r p [ | Ψ C ih Ψ C | ] = ρ C . IV. SUMMARY AND CONCLUSIONS
John Bell [2] had remarked that “The result of an observation may reasonably depend not only on the state ofthe system .... but also on the complete disposition of the apparatus". Along the same vain, we provided a curiousform of nonclassical effect by showing that the results of the measurement of a particular observable may reasonablydepend not only on the state but also on the contexts in which the observable is being measured. Specifically, wehave shown that the reduced density matrices that are produced by employing the different routes for measuring agiven observable can be empirically distinguished. Such a nonclassical effect can then be considered as a contextualitywithin the formalism of QM. This form of contextuality was first demonstrated in Ref.[19] for a four-level system. Inthis paper, we demonstrated a simpler proof for a qutrit system which is also within the formalism of QM withoutrecourse to any realist hidden variable model. Note that, any proof of quantum violation of noncontextual realistmodel requires at least four observables. Here, we use a single observable but employed two different routes tomeasure it. As regards the relation to our proof of contextuality with the existing usual proof in terms of the violationof non-contextual realist model, it is trivial that any realist hidden variable model which reproduces the effect ofnon-classicality demonstrated here should be contextual.
Acknowledgments
AKP thanks Prof. R. Sala Mayato for helpful discussions. AKP acknowledges the support from RamanujanFellowship research grant. KM gratefully acknowledges the summer student fellowship from TEQIP grant and thelocal hospitality from NIT Patna. [1] J. S. Bell, Physics, 1, 195 (1964).[2] J. S. Bell, Rev. Mod. Phys., 38, 447 (1966).[3] S. Kochen, and E. P. Specker, J. Math. Mech. 17, 59 (1967).[4] M. Kernaghan, J. Phys. A 27, L829 (1994); M. Kernaghan, and A. Peres, Phys. Lett. A, 198, 1 (1995).[5] A. Cabello, J. M. Estebaranz, and G. Garcia-Alcaine, Phys. Lett. A, 212, 183 (1996).[6] A. Peres, Phys. Lett. A, 151, 107 (1990); J. Phys. A, 24, L175 (1991);
Quantum Theory: Concepts and Methods (Kluwer,Dordrecht, 1993), pp. 196-201.[7] N. D. Mermin, Phys. Rev. Lett., 65, 3373 (1990); Rev. Mod. Phys., 65, 803 (1993).[8] A. Cabello, Phys. Rev. Lett., 101, 210401 (2008).[9] A. K. Pan and D. Home, Phys. Lett., A, 373, 3430(2009); A.K. Pan and D. Home, Int. J. Theor. Phys., 49, 1920(2010).[10] A. K. Pan, EPL, 90, 40002 (2010); A. K. Pan and D. Home, Eur. Phys. Jour. D, 66, 62 (2012).[11] A. A. Klyachko, M. A. Can, S. Binicioglu and A. S. Shumovsky, Phys. Rev. Lett., 101, 020403 (2008).[12] S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012).[13] Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, Nature 425, 45 (2003).[14] H. Bartosik et al.
Phys. Rev. Lett., 103, 040403 (2009).[15] G. Kirchmair et. al , Nature, 460, 494 (2009).[16] B. H. Liu et al. , Phys. Rev. A, 80, 044101 (2009).[17] E. Amselem, M. Radmark, M. Bourennane and A. Cabello, Phys. Rev. Lett., 103, 160405 (2009).[18] R. Lapkiewicz, et al.,et al.,