Bell measurements as a witness of a dualism in entanglement
E. Moreva, G. Brida, M. Gramegna, S. Bose, D. Home, M. Genovese
aa r X i v : . [ qu a n t - ph ] M a r Bell measurements as a witness of a dualism in entanglement
E. Moreva , , G. Brida , M. Gramegna , S. Bose , D. Home , M. Genovese ∗ INRIM, strada delle Cacce 91, 10135 Torino,Italy International Laser Center of M.V.Lomonosov Moscow State University, 119991, Moscow, Russia Department of Physics and Astronomy, Univestity College London,Gower Street, London WCIE 6BT, United Kingdom CAPSS, Physics Department, Bose Institute, Salt Lake, Sector V, Kolkata 700097, India ∗ corresponding author: [email protected] We show how a property of dualism, which can exist in the entanglement of identical particles,can be tested in the usual photonic Bell measurement apparatus with minor modifications. Twodifferent sets of coincidence measurements on the same experimental setup consisting of a Hong-Ou-Mandel interferometer demonstrate how the same two-photon state can emerge entanglementin the polarization or the momentum degree of freedom depending on the dynamical variables usedfor labeling the particles. Our experiment demonstrates how the same source can be used as botha polarization entangled state, as well as a dichotomic momentum entangled state shared betweendistant users Alice and Bob in accordance to which sets of detectors they access. When the particlesbecome distinguishable by letting the information about one of the variables to be imprinted in yetanother (possibly inaccessible) system or degree of freedom, the feature of dualism is expected tovanish. We verify this feature by polarization decoherence (polarization information in environment)or arrival time difference, which both respectively destroy one of the dual forms of entanglement.
I. INTRODUCTION
Quantum sources of entangled particles represent afundamental component for different applications [4] suchas quantum computing [1–3], quantum communicationbased protocols such as dense coding [5], teleportation[6], quantum cryptography [7] and quantum metrology[8–11]. Likewise, the quantum indistinguishability ofidentical particles has varied applications in informationprocessing [12–15] and is currently quite a topical issue[16]. It is quite natural to expect that the entangledstates of identical particles will have curious features byvirtue of their quantum indistinguishability. One suchproperty, dubbed “entanglement dualism” [17] was in-troduced recently for indentical quantum particles. Itcharacterizes the entanglement of two identical particlesby different interchangeable variables. Recently, one ob-servation of this property [19] has been made and a novelapplication called “entanglement sorting” has been for-mulated [20] .A pure bipartite state described by the wave function | Ψ( x , x ) i with variables x , x of particles 1 , | Ψ( x , x ) i 6 = | φ ( x ) i| ϕ ( x ) i . (1)Variables x , x can be discrete, like polarization of theparticle or spin projection, or continuous, like direction ofmomentum. Thus, the maximally entangled polarizationstate of two identical photons, occupying only one of twospatial modes each k or − k can be written in the form | Ψ i = √ ( | H i k | V i − k − | V i k | H i − k ) (2)The propagation directions (and photon’s momentum)are commeasurable with the photon’s polarization, there- fore the state | Ψ i can be equally rewritten into a form | Ψ i = √ ( | − k i H | k i V − | k i H | − k i V ) (3)where roles of polarization and momentum coordinatevariables were rearranged. Equations (2) and (3) cap-ture the property of dualism: we can use the momenta k and − k as the “which particle” label and then the polar-ization of the two photons will be found to be entangled(2). Alternatively, one can use the polarization of thetwo photons as the “which particle” label, in which case,their momenta will be entangled (3) [17]. In contrastto “hyper-entanglement” [18], in which into a quantumstate more than one variable is simultaneously entangled,dualism emerges only from the interchangeability of dif-ferent dynamical variables used for labeling the particlesand has a property of complementarity in sense that onecannot observe entanglement on both variables at thesame time. Following the general schematic of the ap-paratus of Ref.[17] an experimental observation of theduality has been made in a photonic experiment [19].However, it has not yet been noticed that a different,and potentially much simpler, experimental setup canalso test the same duality.In this paper, we show how the entanglement dualitycan be tested by minimally modifying the usual Bell statemeasurement apparatus and demonstrate this scheme ex-perimentally. For this purpose in our set-up we evalu-ate the entanglement using witness operator for the po-larization and momentum degrees of freedom and showthat only for undistinguishable particles the entangle-ment manifest its feature irrespective of variables usedfor labeling the particles. Moreover, an interesting be-haviour of the entanglement of the states displaying thedualism has not yet been fully tested. This is the factthat even if some form of distinguishability or decoher-ence reduces the entanglement in one of the two degreesof freedom involved in the duality, the entanglement inthe other degree of freedom may still be fully retained.This can be regarded as a type of “robustness” of theentanglement in the states displaying the duality. It mayhave the practical use that if there is a decoherence (oreavesdropping) in one degree of freedom, the state maystill retain some usefullness – communicating parties Al-ice and Bob may then resort to using the undecohereddegree of freedom for quantum communications. Theprevious experiment on dualism [19] has tested only forthe disappearence of the momentum entanglement due toa distinguishability of the photons, while the polarizationentanglement was retained. Here we add to this by alsoshowing that even when the polarization entanglementis destroyed due to decoherence, the momentum entan-glement remains a useful resource. For the detection ofentanglement we use a set of local measurements and cal-culate an entanglement-witness operator. Different fromquantum tomography, this method does not provide afull reconstruction of the quantum state but allows one,with a minimal number of local measurements, to checkif the entanglement is really present. The entanglement-witness operators that we will use are: for the polariza-tion degree of freedom W p = | + 45 i| + 45 ih +45 |h +45 | + | − i| − ih− |h− |− ( | R i| L ih R |h L | + | L i| R ih L |h R | ) , (4)where 1,2 are numbers of the photons and eigenstates | x ± , i , | y ± , i are defined as | ± , i = √ ( | H , i ± | V , i ) , | R , i = √ ( | H , i + i | V , i ) , | L , i = √ ( | H , i − i | V , i ) , (5)and for the momentum degree of freedom W m = | K +1 i| K +2 ih K +1 |h K +2 | + | K − i| K − ih K − |h K − |−− ( | K +1 i| K − ih K +1 |h K − | + | K − i| K +2 ih K − |h K +2 | ) , (6)where | K ± , i = √ ( | k , i ± i | − k , i ) . (7)To see the above witness operators in a slightly differ-ent notation, one can define the Pauli operator σ x = | H ih V | + | V ih H | , σ y = i ( | V ih H | − | H ih V | ), σ z = | H ih H | − | V ih V | for the polarization degree of freedom.Then the witness operator W p = 0 . σ x σ x + σ y σ y ),where 1 , σ x = | K ih K | + | K ih K | , σ y = i ( | K ih K | − | K ih K | ) for momentum degrees of free-dom, we have W m = 0 . σ x σ x + σ y σ y ). Defining thevalue of entanglement/momentum witness like a mean ofthe modulus of expectation value of the witness operatorit is easy to verify that | < W p,m > | > . | < W p,m > | ≤ . II. THE SETUP
Let us consider a pure entangled state of two photons,created via spontaneous parametric down-conversion innonlinear crystal with phase matching conditions of typeII. If we consider the entangled state with polarizationas the entangled variable, and momentum as the “which-particle” label, then such state can be written as (2),while a state with momentum as the entangled variableand polarization as the indexing has a form (3). Thedualism in entanglement can be expressed as: √ ( | H i k | V i − k −| V i k | H i − k ) = √ ( |− k i H | k i V −| k i H |− k i V )(8)To confirm the entanglement duality we use experi-mental setup depicted in Fig. 1. The setup consistsof a Mach-Zehnder interferometer with non-polarizingbeamsplitter (BS), number of half-wave and quarter-waveplates (HWP, QWP) and two polarizing beamsplitters(PBS) at the outputs of the interferometer, which sharephotons between distant users Alice and Bob. Such con-figuration is commonly used for polarization Bell-statesmeasurements, implemented due to the presence of thesymmetry of the momenta states. We, in turn, alsodemonstrate their entanglement, and show how the wit-ness of a dualism can be probed.A cw argon-ion laser is used to pump 2 mm β -barium-borate (BBO) crystal. The crystal is cut for type-II phasematching and produces pairs of polarization-entangledphotons with central wavelength 702 . nm propagatedin two spatial modes k , − k (selected by 3 mm iris di-aphragms, 1m away from the crystal). The resultingwalk-off effect arising due to crystal’s birefringence iscompensated by a combination of a half-wave plate at45 ◦ and a 1 mm BBO crystal. The initial setting ofthe setup is fixed on generating the output state | Ψ i = √ ( | H i k | V i − k − | V i k | H i − k ). Polarization analysis pro-vides a visibility of interference curves ≈
97% in the
H/V basis and ≈
96% for the +45 / −
45 basis. This shows thatwe have produced a high-quality polarization-entangledphotons source.The photon pair is then sent to a Mach-Zehnder in-terferometer, formed by BBO crystal and BS, via thedifferent input ports − k and k . The relative path align-ment ∆ x to within the coherence length ( l ≈ µm ) iscontrolled by the optical trombone P1 with high pre-cision. After the BS photons are separated accordingto their polarization on PBSs, directed to Alice or Bobparts and detected on single photon avalanche photodi-odes (SPADs) D1,D2,D3,D4 equipped with 5nm FWHMbandwidth interference filter and iris diaphragms.For initial path length adjustment, the two-photon in-terference at BS is registered by monitoring the coinci-dence counts between detectors D , D ◦ in one arm of the interferometer andvarying ∆ x by means of piezo drive. FIG. 1: Experimental setup for the measurement of the en-tanglement dualism. Blocks a and b serve for measuringthe polarization witness and momentum witness correspond-ingly. Experimental scheme allows one to switch betweentwo types of measurements. In scheme - Ar laser: argoncw laser with wavelength 351 nm, M: mirror, V: vertical ori-ented Glan-Thompson prism, BBO: nonlinear barium boratecrystals, IF: interfilter, P , optical trombones λ/ λ/
4: half-wave, quarter-wave plates, QP: thick quarts plates, BS: beam-splitter, PBS: polarization beamsplitters, D i : single photonavalanche photodetectors, CC: coincidence circuit. Fig.2 shows a standard Hong-Ou-Mandel (HOM) an-ticoincidence dip for the overlap of two photons at a BS.
III. EXPERIMENTAL RESULTS
For demonstration of the duality in entanglement wedo two types of measurements with the same experimen-tal setup (blocks a and b in Fig. 1). We start at first fromthe polarization correlation measurements on quantumstate of two photons and verify their entanglement. Forthis we investigate two-photon interference fringes by us-ing HWPs at the front of the single photon detectors inFig. 1 (block a). In the measurements of the two-photoninterference fringes, we fix the HWP in one arm at 0 or22 . ◦ and measure the coincidence count rates while ro-tating the orientation angle of HWP in the second arm.Clear two-photon interference fringes with high visibilitywere measured.The next step is measuring the polarizationentanglement-witness operator W p . As we have men-tioned before, the value of polarization entanglement-witness operator W p takes value 1 for the maximumentangled state and falls to 0 for mixed states. Forpure non entangled states value W p does not exceed0.5. W p can be locally measured by choosing cor-responding correlated measurements. For example, | + 45 i| + 45 ih +45 |h +45 | is measured via correlations FIG. 2: Coincidence count rate as a function of the pathlength difference ∆ x . For perfect overlap destructive interfer-ence takes place, the observed visibility is (90 . ± . between detectors D , D | − i| − ih− |h− | between the pair D , D
3, at HWPs at 22 . ◦ in both arms.Operators | R i| L ih R |h L | and | L i| R ih L |h R | aremeasured at QWPs at 45 ◦ in both arms by coincidencebetween detectors D , D D , D W m givenby Eq.(7). Consider two photons born in the SPDC pro-cess and propagating in two directions − k and k . Fouroutput ports from BS+PBSs are labelled by k , k , k , k and registered at the detectors D , D , D , D | k ih k | = ( | k H i + i |− k H i ) √ h k H | + i h− k H | ) √ | k ih k | = ( | k V i + i |− k V i ) √ h k V | + i h− k V | ) √ | k ih k | = ( | k V i− i |− k V i ) √ h k V |− i h− k V | ) √ | k ih k | = ( | k H i− i |− k H i ) √ h k H |− i h− k H | ) √ (9)Then momentum entanglement-witness operator W m is defined through the correlated measurements betweenfour pairs of detectors: D , D D , D D , D D , D W m = | k i| k ih k |h k | + | k i| k ih k |h k |− ( | k i| k ih k |h k | + | k i| k ih k |h k | ) (10)The value W m > . ∆ x , µm W m W p . ± .
03 0 . ± . . ± .
03 0 . ± . . ± .
03 0 . ± . p W p W m . ± .
02 0 . ± . . . ± .
02 0 . ± .
031 0 . ± .
02 0 . ± . x after readjusting the arm lengthsof the interferometer.FIG. 4: Experimental results display the decrease of momen-tum entanglement witness W m as the path delay increases,while the inserted graphs show that polarization interferencefringes remain unaffected for different values of W m . (3). The data from the momentum entanglement mea-surements are shown in Fig. (3)By moving path length different ∆ x in the one arm ofthe interferometer we measure coincidences between fourpairs of detectors: D , D D , D D , D D , D
4, andthe total number of events for different arriving time ofphotons on the BS. From these data we can reconstructthe momentum entanglement witness. For ∆ x = 0 ittakes maximum value 0 . ± .
03, while for a nonzerodelay the quantum state (3) becomes mixed by momen-tum degree of freedom due to different arrival time onbeamsplitter and the value of the momentum witness W m falls below 0.5 (Fig.4). However, when we consider thepolarization degree of freedom for the same state we ob-serve a similarly large amount of the polarization witness W p . Two inserted graphs in Fig.4 show the samenessof the polarization interference fringes for different val-ues of path delays corresponding to different degrees ofmixedness for the momentum degree of freedom. For afurther exploration of the quantum feature of the dualitywe prepare also a mixed state of photons by polarizationvariable. The entanglement dualism has so far not beenprobed for a mixed state, which can arise because of de-coherence. To this end, we consider a density matrix ofthe polarization state: ρ = (1 − p ) | Ψ pure ih Ψ pure | + p | H V ih H V | + | V H ih V H | )(11)where | Ψ pure i = √ ( | H i | V i − | V i | H i ) (12)is an initial pure maximal entangled polarization stateand p is a coefficient defining the degree of mixture.In order to prepare a mixed polarization state and donot destroy the momentum entanglement it is necessaryto introduce a quantum distinguishability in a control-lable manner. We complement one of the output port ofthe interferometer by a set of quartz plates ( see Fig.1). Athick quartz plate with a vertically oriented optical axisintroduced delay between vertically and horizontally po-larized photons that led to their distinguishability, andhence, to the emergence of a mixture.Thus, we destroy only polarization entanglement with-out affect on the arriving time of photons on the BS. Itshould be noted that there is no direct interaction on theBS between pairs of photons, because they have differ-ent polarizations (there is no direct two-photon interac-tion). That is why the thick quarts plate QP could beincluded in the ”preparation” part of the setup, whilePBSs, half-,quater- waveplates and detectors form the”measurement” part.We have prepared three states of (11) with various de-gree of mixture: p = 1; 0 .
3; 0 ( p = 1 corresponds to totalymixed state, p = 0 . p = 1 we ob-served a flat interference fringe curve, for p = 0 . IV. DISCUSSIONS
We have verified a predicted dualism in the entangledstate of two identical particles using the modification ofa Bell-state apparatus. Our apparatus is very differentfrom the one in the theoretical proposal [17], as well asthe one reported in a recent test [19]. Having an al-ternative method is always of benefit in the sense thatone would probably intend to extend, in the future, suchtests to entangled states of also material particles. Inthis sense they are sufficiently weakly interacting then the method proposed here is more advisable due to a lesschallenging phase stabilisation between different paths,with respect to the setup [17, 19]. Furthermore, we havealso verified a form of robustness of the states displayingthe dualism – even when decoherence or distinguisha-bility deteriorates the entanglement of one of the tworelevant degrees of freedom, the other may remain unaf-fected (maximally entangled). Thus, one has essentiallytwo ways of exploiting the same resource state. This isbest exemplified in Fig.1. If polarization entanglement isfound to be persistently degraded by decoherence or aneavesdropper, then communicating parties Alice an Bobcan use arrangement (b) to exploit the momentum entan-glement in the state for quantum communications. Thisinteresting property of the decoherence in one of the de-grees of freedom not affecting the entanglement in termsof the other dual degree of freedom needs to be analyzedin depth in order to explore its further ramifications.
Acknowledgments
This work was funded by NATO SPS Project984397. E.V.Moreva acknowledges the support of Rus-sian Foundation for Basic Research (project 13-02-01170-a). SB acknowledges support of the EPSRC Grant No.EP/J014664/1. [1] M.A. Nielsen and I.L. Chuang,
Quantum Computationand Quantum Information , (Cambridge University Press,Cambridge,2000); ”Quantum Information, Computationand Cryptography”,Ed. F.Benatti et al., (Springer Ver-lag, Berlin, 2010).[2] G. Chen, D.A. Church, B.-G. Englert, C. Henkel, B.Rohwedder, M.O. Scully, and M.S. Zubairy