Quantum mechanical effect of path-polarization contextuality for a single photon
aa r X i v : . [ qu a n t - ph ] M a y Quantum mechanical effect of path-polarization contextuality for a single photon
Alok Kumar Pan ∗ and Dipankar Home † CAPSS, Department of Physics, Bose Institute, Sector-V, Salt Lake, Calcutta 700091, India
Using measurements pertaining to a suitable Mach-Zehnder(MZ) type setup, a curious quantummechanical effect of contextuality between the path and the polarization degrees of freedom of apolarized photon is demonstrated, without using any notion of realism or hidden variables - aneffect that holds good for the product as well as the entangled states. This form of experimentalcontext-dependence is manifested in a way such that at either of the two exit channels of the MZsetup used, the empirically verifiable subensemble statistical properties obtained by an arbitrarypolarization measurement depend upon the choice of a commuting(comeasurable) path observable,while this effect disappears for the whole ensemble of photons emerging from the two exit channelsof the MZ setup.
PACS numbers: 03.65.Ta
I. INTRODUCTION
Investigations of the implications of a possible ‘incom-pleteness’ of quantum mechanics(QM) have resulted instriking discoveries of fundamental constraints which anyrealist model has to satisfy in order to be compatiblewith the empirically verifiable predictions of QM. Oneof such constraints is, of course, the comprehensivelystudied incompatibility between QM and the local real-ist models of quantum phenomena discovered using Bell’stheorem[1, 2] and its variants[3] - for a comparatively re-cent review of investigations in this area, see, for example,[4]. The other constraint that, of late, has also been at-tracting an increasing attention is the one concerning theinconsistency between QM and the noncontextual real-ist(NCR) models(the Bell-Kochen-Specker theorem[5, 6]and its variants[8-26]). It is this latter strand of studywhich leads to the present paper. For this, we proceed byfirst recapitulating the essence of what is usually meantby the notion of ‘noncontextuality’.Given any realist hidden variable model of quantumphenomena, the individual measured values of any dy-namical variable are predetermined by the appropriatevalues of hidden variables( λ ’s) which are used in a re-alist model for a ‘complete specification’ of the state ofan individual quantum system. Now, the condition of‘noncontextuality’, in its most general form underlyingits usual use, stipulates that the predetermined individ-ual measured value of any dynamical variable, for a given λ , is the same whatever be the way the relevant dynami-cal variable is measured. The question as to what extentthis putative condition is compatible with the formalismof QM has been subjected to two different lines of studyby exploring the implications of two separate facets ofthis condition.One of these is contingent upon assuming that the pre-determined individual measured value of a given dynam- ∗ [email protected] † [email protected] ical variable is independent of whatever be the choice ofthe other commuting(comeasurable) observable that ismeasured along with it. This condition has led to theformulation of a testable Bell-type inequality[13] that isderived as a consequence of the NCR models, but is vio-lated by QM for the entangled states by a finite amount,thereby enabling an empirical discrimination betweenQM and the NCR models[12, 14, 15]. Subsequently, theexperimental investigation along this line has been en-riched by further studies[17].The other line of study on the issue of contextuality vis-a-vis QM is based upon a feature characterising the NCRmodels that can be expressed as follows: For an individ-ual measurement, the definite outcome obtained for anobservable(say, A ), as specified by a given hidden variable λ , be denoted by v ( A ) . Now, let B be any other commut-ing(comeasurable)observable whose measured value in anindividual measurement, as fixed by the same given λ , bedenoted by v ( B ) . Then, if one denotes an individual out-come of a holistic measurement of the product observable AB by v ( AB ) which is determined by the same valueof the hidden variable λ , the notion of noncontextualityis taken to imply the following condition(known as the‘product rule’) v ( AB ) = v ( A ) v ( B ) (1)which is assumed to hold good independent of the exper-imental procedure(context) used for measuring the jointobservable AB in a holistic way, and is also independentof the way the individual separate measurements of A and B are performed separately. This feature of noncon-textuality was elegantly used by Mermin[7] for formu-lating a proof of quantum incompatibility with the NCRmodels for two spin-1/2 particles that holds for any state.Later, Cabello[22] cast Mermin’s proof[7] in the form of atestable inequality involving the statistically measurablequantities - this inequality being violated by QM by a fi-nite measurable amount for an arbitrary two-qubit state.Subsequently, the state-independent quantum violationof this inequality has been experimentally corroboratedusing the polarization and the linear momentum degreesof freedom of a single photon[23].In contrast to the above two directions of study, inthis paper we explore a third line of probing, initiatedin a recent paper[26] that used a suitable path-spin en-tangled state of a spin-1/2 particle, in which the issueof contextuality is probed within the framework of QM,devoid of any reference to the NCR models. With aview of extending the ambit of this new line of study,the present paper reveals that that it is indeed possibleto show a form of state-independent contextuality within QM for any state that is an entangled or a product statein the four dimensional space. By using photons andan appropriate setup we show that a statistically dis-cernible effect of the path-polarization interdependence ismanifested in terms of the operationally suitably defined subensemble statistical properties of an arbitrary polar-ization measurement that depend upon the choice of acomeasurable(commuting)path observable. Interestingly,this context-dependence gets obliterated for the statisti-cal results pertaining to the whole ensemble of photonsemerging from the setup used for the polarization mea-surement, whatever be the choice of the comeasurablepath observable. Let us now proceed by first explainingthe specifics of the setup(Figure 1) that is required forour demonstration.
II. THE SETUP FOR PREPARING THEREQUIRED PRODUCT OR AN ENTANGLEDSTATE
In order to formulate our argument, the required path-polarization product state can be prepared using a 50:50beam splitter(BS1), while an entangled path-polarizationstate can be prepared by using a 50:50 BS1 in conjunctionwith a half wave plate(HWP) that is placed in one of theoutput ports of the BS1(Fig.1). The relevant path andthe polarization measurements will be considered sepa-rately for these prepared states.Let us consider that an ensemble of photons havinghorizontal polarized state | H i be incident on a 50:50beam-splitter(BS1). Any given incident photon can thenemerge along either the reflected or the transmitted chan-nel corresponding to the state designated by | r i or | t i respectively. Here note that for any given lossless beam-splitter, arguments using the unitarity condition showthat for the photons incident on BS1, the phase shift be-tween the transmitted and the reflected states of photonsis essentially π/ [27]. The prepared path-polarizationproduct state after emerging from BS1 can then be writ-ten as | Ψ i pr = 1 √ | t i + i | r i ) | H i (2)On the other hand, for preparing an entangled path-polarization state, photons in the channel correspondingto | t i are passed through a HWP that flips the polariza-tion | H i to | V i . The state emerging from HWP can then be written as | Ψ i en = 1 √ | t i | V i + i | r i | H i ) (3)which is an entangled state between the path and thepolarization degrees of freedom.In writing both Eqs.(2) and (3) we have taken into ac-count a relative phase shift of π/ between the states | r i and | t i that arises because of the reflection from BS1.Eqs.(2) and (3) represent the prepared states on whichwe will separately consider the relevant path and the po-larization measurements for the purpose of showing thepath-polarization interdependence for the product as wellas for the entangled state. For this, we proceed as follows. M1HWP |t / > Polarizer
BS1 BS2 |H > |r > PS |r > |r / > Polarizer M2 Figure 1:
Horizontally polarized (denoted by | H i ) photons en-ter the indicated Mach-Zehnder type setup through a 50:50 beam-splitter BS1, and pass through the channels corresponding to | t i and | r i thereby generating the path-polarization product state.For generating an entangled path-polarization state, a half waveplate (HWP) that flips the horizontally polarized state | H i intothe vertically polarized state | V i is placed along one of the chan-nels | t i . Subsequently, for the required measurements pertaining tothe path observable, a phase-shifter(PS) is placed along the chan-nel | r i that creates a relative phase shift between the channels | t i and | r i . The two channels are then recombined at a 50:50 beam-splitter BS2. The PS in conjunction with BS2 serves the purpose ofpath measurement(see the text). Finally, the measurement in anarbitrary polarization basis is made by using two polarizers thatare placed along the two output channels | t ′ i and | r ′ i of the BS2.The path-polarization interdependence can then be demonstratedby considering the subensemble mean values associated with eachof the two output channels | t ′ i and | r ′ i . III. THE QM DEMONSTRATION OFPATH-POLARIZATION INTERDEPENDENCE
After passing through the mirrors M1 and M2, photonsare subjected to a phase shifter(PS) along the channel | r i which introduces a relative phase shift of φ between thepath channels | r i and | t i . The two path states are thenrecombined at BS2, and the output path states | t ′ i and | r ′ i are respectively given by | t ′ i = 1 √ (cid:0) e iφ | r i − i | t i (cid:1) (4a) | r ′ i = 1 √ (cid:0) − ie iφ | r i + | t i (cid:1) (4b)Eqs.(4a,4b) show that, for a given linear combinationof | t i and | r i , using the different values of φ , one canunitarily generate at the output of BS2 various linearcombinations of | t i and | r i that correspond to differentprobability amplitudes of finding photons in the chan-nels corresponding to | t ′ i and | r ′ i . This, in turn, impliesthat PS in conjunction with BS2 can be regarded as cor-responding to different choices of the path observables b β φ = | t ′ i h t ′ | − | r ′ i h r ′ | with eigenvalues ± . Such ob-servables, in terms of actual measurements, correspondto different relative counts registered by the detectorsplaced along the channels represented by | t ′ i and | r ′ i .Using Eqs.(4a,4b) the path observable is of the form β φ = (cid:18) ie iφ − ie − iφ (cid:19) (5)that can be written as the following linear combinationof the Pauli matrices β φ = − sinφ b σ x + cosφ b σ y = ~σ.~n φ (6)where ~n φ = − sinφ b i + cosφ b j .It is, therefore, evident from Eq.(5) that the path ob-servable b β φ which is represented by the vector componentgiven by Eq.(6) varies according to the magnitude of thephase shift φ , i.e., different choices of φ provide differentcontexts pertaining to the polarization measurements.Next, we consider the measurement of an arbitrarilychosen polarization variable, say, b δ which is given by b δ = | H ′ i h H ′ | − | V ′ i h V ′ | (7)whose eigenstates are | H ′ i = cosα | H i + sinα | V i and | V ′ i = sinα | H i − cosα | V i , where α denotes the orien-tation of the two polarizers that are placed separatelyalong the channels | t ′ i and | r ′ i .Now, we will consider the expectation value of the po-larization observable b δ that involves contributions fromboth the output subensembles corresponding to polar-izers separately placed along the channels | t ′ i and | r ′ i .The subensemble mean values of b δ measured in each ofthe two output channels, calculated using either the pre-pared product or an entangled path-polarization stategiven by Eqs.(2) and (3) respectively, are denoted by (¯ δ ) t ′ and (¯ δ ) r ′ , whence Db δ E Ψ pr/en = (¯ δ ) t ′ + (¯ δ ) r ′ (8)where the subscript Ψ pr/en represents the prepared en-tangled or product states given by Eq.(2) or (3) respec-tively. Note that all the three quantities occurring in theequality given by Eq.(8) have the same operational sta-tus as far as their statistical reproducibility is concerned.But there is a crucial distinction between the left andthe right hand sides of Eq.(8) with respect to the issue ofpath-polarization interdependence. The quantity on theleft hand side of Eq.(8), the expectation value of h b δ i Ψ pr/en pertaining to the whole ensemble, is independent of whichpath observable is measured along with it. On the otherhand, each of the quantities on the right hand side ofEq.(8), the subensemble mean values denoted by (¯ δ ) t ′ and (¯ δ ) r ′ are contingent upon the choice of the comea-surable path observable. This can be seen by consideringthe polarization measurement outcomes relevant to anyone of the two output subensembles.In order to display the manifestation of this form ofcontext-dependence within QM, we consider two differ-ent experiments involving measurements of the path ob-servable b β φ and the polarization variable b δ , first for theprepared product state given by Eq.(2), and then for theprepared entangled state given by Eq.(3). A. Path-polarization context-dependence for aproduct state
We first consider the path-polarization product stategiven by Eq.(2) as the input state for which the statethat emerges from BS2 can be written as | Φ i pr = 12 (cid:2) i | t ′ i (cid:0) e iφ (cid:1) + | r ′ i (cid:0) − e iφ (cid:1)(cid:3) | H i (9)Then it follows that the expectation value of the po-larization observable b δ pertaining to the whole ensemble of photons emerging from BS2 is of the form Db δ E Ψ pr = cos α (10)which comprises the respective subensemble polariza-tion mean values calculated from Eq.(9) given by (¯ δ ) t ′ = (1 + cosφ ) cos α δ ) r ′ = (1 − cosφ ) cos α (11)Next, we come to the crux of our argument indicatedas follows that hinges on two different choices of φ , andwhere the superscript β ( β π/ ) is used to denote thechoice of the path observable specifying the given con-text: ( a ) Taking φ = 0 , this implies the choice of a particularpath observable b β = ~σ.~n where ~n = b j . In this case,using Eq.(11), we obtain (¯ δ ) ( β ) t ′ = cos α ; (¯ δ ) ( β ) r ′ = 0 (12)while, using Eq.(2), the polarization expectation valuefor the whole ensemble, h b δ i Ψ pr = cos α . ( b ) Taking φ = π/ , this implies the choice of a dif-ferent path observable b β π = ~σ.~n π/ where ~n π/ = − b i .Consequently, using Eq.(11), we obtain (¯ δ ) ( β π/ ) t ′ = cos α δ ) ( β π/ ) r ′ = cos α (13)while, using Eq.(2), the polarization expectation value forthe whole ensemble remains the same , h b δ i Ψ pr = cos α .It is then evident from Eqs.(10-13) that, while thequantum expectation value h b δ i Ψ pr of the observable b δ pertaining to the whole ensemble remains the same forboth the choices of b β and b β π/ , the path-polarizationcontext-dependence gets manifested in terms of the subensemble polarization mean values given by thetestable quantities (¯ δ ) ( β ,β π/ ) t ′ and (¯ δ ) ( β ,β π/ ) r ′ . To put itprecisely, in our example, the interdependence betweenthe path and the polarization degrees of freedom has thefollowing operational meaning (¯ δ ) ( β ) t ′ = (¯ δ ) ( β π/ ) t ′ ; (¯ δ ) ( β ) r ′ = (¯ δ ) ( β π/ ) r ′ (14)i.e., the subensemble mean value of the polarizationvariable b δ depends upon which of the path observables b β or b β π/ is comeasured, where both b β and b β π/ commutewith b δ . B. Path-polarization context-dependence for anentangled state
Now, we consider the path-polarization entangled stategiven by Eq.(3) as the input state for which the state thatemerges from BS2 can be written as | Φ i en = 12 (cid:2) i | t ′ i (cid:0) | V i + e iφ | H i (cid:1) + | r ′ i (cid:0) | V i − e iφ | H i (cid:1)(cid:3) (15)It follows from Eq.(15) that corresponding to the pre-pared path-polarization entangled state | Ψ i en given byEq.(3), the expectation value of the polarization variable b δ for the whole ensemble of photons emerging from thebeam-splitter BS2 is given by h b δ i Ψ en = 0 (16)which is made up of the respective subensemble polariza-tion mean values calculated from Eq.(15), which are ofthe form (¯ δ ) t ′ = sin α cosφ δ ) r ′ = − sin α cosφ (17)Then, similar to the argument given above for the in-put product state, in this case too Eq.(14) holds good, i.e., the path-polarization context-dependence gets man-ifested in terms of the subensemble polarization meanvalues given by the testable quantities (¯ δ ) ( β ,β π/ ) t ′ and (¯ δ ) ( β ,β π/ ) t ′ , while the quantum expectation value h b δ i Ψ en pertaining to the whole ensemble remains the same forboth the choices of b β and b β π/ . IV. THE SIGNIFICANCE AND OUTLOOK
The essence of what is demonstrated in this paper isas follows. A statistically discernible signature of inter-dependence between the path and the polarization de-grees of freedom of polarized photons is revealed - aneffect which is quantum mechanically calculable in termsof the measured values of a polarization variable pertain-ing to the operationally well-defined subensembles thatcomprise the final output ensemble at the two exit chan-nels of our setup. The subensemble polarization meanvalue registered at either of the two exit channels varies according to the choice of the comeasurable(commuting)path observable(whose choice is fixed by the magnitudeof the phase shift that is introduced by PS in the chan-nel | r i ) But, importantly, such a variation takes place by preserving the context-independence of the polarizationexpectation value that is defined for the whole outputensemble.In other words, we show that for an arbitrarily pre-pared state that can be either a path-polarization entan-gled or a product state, a form of ‘parameter dependence’is displayed in this example in a way that is restricted tothe subensemble statistics . This is quite distinct fromthe issue of ‘parameter independence’[33] for the EPR-Bohm type states involving the polarization variables oftwo spatially separated photons where the ‘parameterindependence’ is in the sense that no statistically dis-cernible effect in any form can be detected in any one ofthe two wings of the EPR-Bohm pair that depends uponthe measurement setting in the other wing - a possibilitywhich is ruled out by the much-discussed ‘no-signaling’condition[34]. But, in contrast, in our example, for anarbitrary input state considered in our example for apolarized photon, there is no such constraint which for-bids the statistical manifestation of an intraparticle path-polarization context-dependence. While such an effectdoes occur in our example, curiously, it is confined tothe subensemble statistics in such a way that the ‘path-polarization interdependence’ disappears for the statis-tics of the whole ensemble.In terms of the two distinct aspects of the NCR mod-els as discussed earlier, we may recall that the statisti-cally verifiable inequalities for the entangled as well asthe product states have been analyzed, both theoreti-cally and experimentally, thereby highlighting an incom-patibility between QM and the NCR models. However,it needs to be stressed that these earlier demonstrationsnecessarily involve the assumption of the notion of real-ism that is used in tandem with the feature of noncontex-tuality at the level of individual measurement outcomes.In contrast, the statistically verifiable manifestation ofthe quantum mechanical ‘path-polarization interdepen-dence’ shown in this paper is entirely independent of anynotion of ‘realism’ or ‘hidden-variables’, because this ef-fect is demonstrated essentially within the ambit of QMand, crucially, is independent of the nature of the inputstate which can be either a path-polarization product oran entangled state. A significant point to stress here isthat a similar effect of interdependence showing a nonlo-cal connection between the entangled variables of the spa-tially separated photons cannot be demonstrated withinQM unless one takes recourse to the notion of ‘realism’ or‘hidden-variables’. This fundamental distinction betweenquantum nonlocality and contextuality brought out bythe example analyzed in this paper as well as in an ear- lier work[26] call for careful probing. Such investigationsmay provide useful insights into the type of constraintsthat would restrict the nonlocal realist models in the lightof the recent studies[35–37] based on Leggett’s work[38]. Acknowledgements
Authors thanks H. Rauch and Y. Hasegawa for theuseful interactions that served as a prelude to this work.AKP recalls the discussions, in particular, with A. Ca-bello during his visit to Benasque Center for Science,Spain. AKP acknowledges the Research Associateshipof Bose Institute, Kolkata. DH acknowledges the projectfunding from DST, Govt. of India and support from Cen-ter for Science and Consciousness, Kolkata. [1] Bell J S 1964
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