ISSN 1054-660X, Laser Physics, 2007, Vol. 17, No. 7, pp. 993–1000.

QUANTUM OPTICS, LASER PHYSICS, AND SPECTROSCOPY

© MAIK “Nauka /Interperiodica” (Russia), 2007. Original Text © Astro, Ltd., 2007.

Experiments of Quantum Nonlocality with Polarization-Momentum Entangled Photon Pairs G. Vallonea, *, P. Matalonia, F. De Martinia, and M. Barbierib a Dipartimento

di Fisica dell’Universitá “La Sapienza” and Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Roma, 00185 Italy b The University of Queensland, Department of Physics, 4072, Brisbane, QLD, Australia Webpage: http://quantumoptics.phys.uniromal.it/ *e-mail: [email protected] Received January 16, 2007

Abstract—We present the results of some experimental tests of quantum nonlocality performed by two-photon states, entangled both in polarization and momentum, namely hyperentangled states and two-photon four-qubit linear cluster states. These states, which double the number of available qubits with respect to the standard twophoton entangled states, are engineered by a simple experimental method, which adopts linear optics and a single type I nonlinear crystal. The tests of local realism performed with these states represent a generalization of the Greenberger, Home, and Zeilinger (GHZ) theorem to the case of two entangled particles. PACS numbers: 42.50.Dv, 42.50.Xa, 42.50.-p DOI: 10.1134/S1054660X07070146

1. INTRODUCTION Quantum nonlocality arises from the impossibility of obtaining the predictions of quantum mechanics by a local realistic model requiring the adoption of ancillary variables. Since these parameters remain hidden in quantum description, they are usually referred to as local hidden variables (LHV). The need for introducing LHV to overcome an implicit incompleteness of quantum mechanics was pointed out by Einstein, Podolsky, and Rosen (EPR) in their seminal paper in 1935 [1], where they introduced three important concepts: Locality: A measurement on system A is independent of what is done on system B, which is spatially separated from A; Realism: if, without disturbing in any way a system, we can predict with certainty (i.e., with a probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity; Completeness: each element of physical reality must have a counterpart in physical theory. EPR showed that quantum mechanics cannot be simultaneously local, real, and complete, and were convinced that quantum theory should satisfy the reasonable assumption of locality and reality. Therefore, they concluded that quantum mechanics is incomplete. In 1964, John Bell discovered his famous inequality ruling out the possibility to introduce LHV [2]. In his proof, he demonstrated that any LHV model cannot explain the statistical correlations present in two-qubit (i.e., a quantum two-level system) entangled states.

The huge amount of experimental data obtained so far by Bell’s nonlocality tests, in particular with photons, confirms the quantum mechanical predictions and leads to the common belief that quantum mechanics cannot be simultaneously local and real (apart from the problems related to the so-called loopholes, concerning the necessity to adopt auxiliary assumptions). Indeed Bell’s inequality is nowadays exploited as a tool to detect entanglement in quantum cryptography and in quantum computation [3]. As an alternative approach, Greenberger, Home, and Zeilinger (GHZ) proposed using the perfect correlations in multiparticle entangled states to illustrate a deeper contradiction between quantum mechanics and LHV theories. Indeed, they demonstrated that local realism is unable to account for statistical correlations, but it is even logically inconsistent with perfect correlations. Furthermore, when using the GHZ approach, the elements of reality are properly defined by the correlation themselves. As a consequence, we don’t need to assume that the measured quantities satisfy the EPR criteria, since it can actually be checked. For years, it was thought that no GHZ-type proof could be found in the case of two particles. Recently, it was shown that such a task is, indeed, feasible, provided that more than one qubit is encoded on each particle [4]. These states can be engineered in a feasible way by photonic techniques. In fact, the use of linear and nonlinear optics allow us to generate the so-called hyperentangled states, i.e., photon pairs entangled in more than one degree of freedom, for instance, polarization and linear momentum. In this way, the available

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Hilbert space is expanded with respect to the usual schemes based on two-qubit entanglement. This experimental approach presents several advantages with respect to its multiparticle counterpart, such as higher generation rates and more resistance to decoherence. Moreover, these schemes could represent a promising resource to implement a loophole-free test of nonlocality. Recently, other GHZ-type proofs involving multiqubit cluster states, which represent the fundamental tool for one-way quantum computation [5], were introduced [6, 7]. The paper is organized as follow: in Section 2, we discuss the concept of simultaneous elements of reality; in Section 3, we give a detailed description of the source of hyperentangled photon states adopted to perform nonlocality tests. Section 4 concerns the description of the “all-versus-nothing” (AVN) experiment, a generalization of the GHZ test with hyperentangled states [4, 8]. Finally, Section 5 is aimed at showing the realization of a more powerful nonlocality experiment by using the so-called two-photon four-qubit linear cluster states. 2. SIMULTANEOUS ELEMENTS OF REALITY Consider two particles entangled in one degree of freedom. All of the inequalities that allow as to test quantum nonlocality are, in general, expressed in terms of the expectation values of some operators n

〈 S〉 =

∑∑

(1)

i=1j=1

where ci, j are properly chosen coefficients and Ai, Bj represent the observables measured by Alice and Bob, respectively. Note that, in each term of the sum of (1), only one operator for Alice and one for Bob are considered. When two particles are entangled in more than one degree of freedom, more interesting inequalities can be derived. In fact, in this case, there are compatible local observables, which can be viewed as simultaneous elements of reality [9]. For instance, Alice can measure two commuting local operators A1, A2 and Bob can measure two observables B1, B2. These are chosen in order to show a perfect correlation in the quantum state |φ〉 〈φ| A 1 B 1 |φ〉 = 1, 〈φ| A 2 B 2 |φ〉 = 1.

〈 A i … A j B k …B l〉 ,

(3)

where all of the operators are EPR elements of reality, and the operators appearing in the same average are compatible observables. With hyperentangled photons, it is straightforward to imagine that compatible observables refer to different degrees of freedom of the particles. The problem of performing a test of this kind mainly resides in the difficulty of achieving perfect correlations in real life. The present technology allows us to generate entangled photon pairs with high visibilities, nearly satisfying correlations such as (2). One possible strategy to account for the imperfections consists of relaxing the EPR criterion and defining as elements of reality those that can be almost perfectly predicted [10]. 3. THE SOURCE

n

c i, j 〈 A i ⊗ B j〉 ,

theless, one can argue that, since the value of A1 can be predicted from a distant measurement on the subsystem 2, in order not to violate locality, this outcome cannot be influenced by the apparatus on system 1. Therefore, EPR criteria themselves require that a local realistic model must be noncontextual as well: the same value must be assigned to A1, regardless of whether it is measured alone, with A2, or with any other observable. By virtue of this property, the inequality may contain terms such as 〈A1A2B1〉, and a generic term can be written as

(2)

Since the outcomes of A1 and A2 can be predicted with certainty from spacelike separated measurements without introducing any disturbance, according to the EPR criterion, they are, at the same time, local elements of reality. The fact that A1 and A2 are commuting operators does not imply that they can be thought of as physically compatible; in fact, it could happen that the measurement of one observable may disturb the other. Never-

The spontaneous parametric down conversion (SPDC) source used in our experiments allows for the efficient generation of the polarization (π) entangled 1 states |Φθ〉π = ------- (|H〉A|H〉B + eiθ |V〉A |V〉B) (with |H〉 and 2 |V〉 corresponding to the horizontal and vertical polarization, respectively) over the entire emission cone of a type I, 0.5-mm-thick, β-BaB2O4 (BBO) crystal. A and B stand for Alice’s and Bob’s photons, respectively (see Fig. 1). Entanglement arises from the superposition of the degenerate emission cones of the crystal, excited in two opposite directions kp and –kp by a V-polarized laser beam. Phase setting θ = 0, π is performed by fine translation of the spherical mirror M shown in Fig. 1a, allowing for the efficient generation of the polarization Bell states 1 ± |Φ 〉 π = ------- ( |H〉 A |H〉 B ± |V 〉 A |V 〉 B ). 2

(4)

For the experimental details concerning the generation of π entanglement by this source, we refer to [11, 12]. Momentum (path) entanglement is created by the same source by selecting two pairs of symmetric modes, lA – rB and rA – lB, within the conical emission of the crystal [13] by adopting a four-hole screen (cf. Fig. 1a). In this way, the momentum (k) entangled state LASER PHYSICS

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995 (b)

Glass M

λ/4

HW Alice (A)

BBO

rA lA lB rB

θ Hyperentangled

Bob (B) HW*

Cluster

Fig. 1. Source of polarization-momentum eutangled photons. (a) Generation of hypereutangled states |Bell〉π ⊗ |ψ±〉k. With or without the HW* oriented at 45° and intercepting modes rA and rB we generate |Ψ±〉π ⊗ |ψ±〉k or |Φ±〉π ⊗ |ψ±〉k respectively. (b) Generation of the linear Cluster state by the insertion of the half-wave plate (HW) with vertical optical axis on mode rA (see Section 5).

1 |ψϕ〉k = ------- (|l〉A |r〉B + eiϕ |r〉A |l〉B) is generated for either 2 one of the two SPDC emission cones. The phase ϕ can be set by suitably tilting a thin glass plate intercepting mode rA (see Fig. 1a) and allows us to generate the Bell momentum states 1 ± |ψ 〉 k = ------- ( |l〉 A |r〉 B ± |r〉 A |l〉 B ). 2

(5)

Then, by combining polarization and momentum entanglement, the four hyperentangled states |Φ±〉π ⊗ |ψ±〉k can be generated. By insertion of a zero-order λ/2 waveplate (wp) oriented at 45° (HW* in Fig. 1a) in one output side, say the r side, it is possible to generate the states |Ψ±〉π ⊗ |ψ±〉k, with |Ψ±〉π corresponding to the other two polarization Bell states: 1 ± |Ψ 〉 π = ------- ( |H〉 A |V 〉 B ± |V 〉 A |H〉 B ). 2

(6)

In fact, the HW* exchanges the polarizations |H〉 |V〉 of the photon in the modes |r〉A and |r〉B, and, then, acts in the following way: ±

HW*

+ |Ψ 〉 π ⊗ |l〉 A |r〉 B ,

±

HW*

± |Ψ 〉 π ⊗ |r〉 A |l〉 B .

|Φ 〉 π ⊗ |l〉 A |r〉 B |Φ 〉 π ⊗ |r〉 A |l〉 B

±

±

(7)

By using the previous equations, it’s easy to show that HW* transforms the state |Φ+〉π ⊗ |ψ±〉k into the state |Ψ+〉π ⊗ |ψ±〉k, while the state |Φ–〉π ⊗ |ψ±〉k is trans− +

formed into |Ψ–〉π ⊗ | ψ 〉k (i.e., the polarization transformation is accompanied by a π shift in the phase ϕ of the momentum state). As a consequence, the whole set of the hyperentangled states |Π〉 ≡ |Bell〉π ⊗ |ψ±〉k can be generated, with |Bell〉π representing one of the four πentangled Bell states. In this way, four qubits are available with only two photons. These states represent the LASER PHYSICS

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starting point for performing powerful nonlocality tests of quantum mechanics. Concerning the measurement apparatus, the two generated photons are spatially and temporally combined onto a beam splitter (BS) through the interferometric apparatus shown in Fig. 2. These are further analyzed half waveplates and PBSs. Spatial and spectral selection are achieved by using a 1-mm-diameter diaphragm and 6-nm bandwidth interference filter on each output mode. As we will show in subsection 4.1, this apparatus represents the core of all measurements we performed in our tests. In these conditions, nearly 1500 photon coincidences per second between the Alice’s and Bob’s sites could be measured. Typical values of polarization and momentum visibility are 0.95 and 0.90, respectively [14]. 4. ALL-VERSUS-NOTHING NONLOCALITY TEST The nonlocal character of hyperentangled states can be tested by exploiting them as a whole system spanning the (4 × 4) Hilbert space and performing the so-called “all-versus-nothing” (AVN) proof of nonlocality [4, 8, 15]. It is based on the fact that assigning to certain Pauli observables a value which preexists the measurement leads to a logical inconsistency. It was previously demonstrated that this theorem represents a generalization of the GHZ argument to the case of two observers [4]. The proof is based on the introduction of the following Pauli π and k observables: Z j = σ z j = |H〉 j 〈H| – |V 〉 j 〈V |, X j = σ x j = |H〉 j 〈V | + |V 〉 j 〈H|, z j = σ 'z j = |l〉 j 〈l| – |r〉 j 〈r|, x j = σ 'x j = |l〉 j 〈r| + |r〉 j 〈l|, ( j = A, B ),

(8)

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(b) l'A

rA HWP rB

l'B

HWP PBS lA

r'A

lB

r'B

∆x

Fig. 2. Measurement apparatus. ∆x represents the optical path delay that can be simultaneously changed for both the lA and lB paths. The beam splitter (BS) and/or the half waveplates on the r modes before the BS can be used or not depending on the measured observables.

which are easily achievable by standard linear optics. The first two equations refer to polarization observables, while the last two equation correspond to momentum observables. It can be verified that Pauli operators (8) allow state transformations: ––

––

Z A ⋅ Z B |Ξ 〉 = – |Ξ 〉, ––

––

z A ⋅ z B |Ξ 〉 = – |Ξ 〉, ––

––

X A ⋅ X B |Ξ 〉 = – |Ξ 〉, ––

––

x A ⋅ x B |Ξ 〉 = – |Ξ 〉, ––

––

––

––

––

––

––

––

Z A z A ⋅ Z B ⋅ z B |Ξ 〉 = |Ξ 〉,

(9)

X A x A ⋅ X B ⋅ x B |Ξ 〉 = |Ξ 〉, Z A ⋅ x A ⋅ Z B x B |Ξ 〉 = |Ξ 〉, X A ⋅ z A ⋅ X B z B |Ξ 〉 = |Ξ 〉, ––

––

Z A z A ⋅ X A x A ⋅ Z B x B ⋅ X B z B |Ξ 〉 = – |Ξ 〉 for the double-singlet hyperentangled state |Ξ––〉 = |Ψ−〉π ⊗ |ψ–〉k. The above equations indicate that the observables ZA, zA, XA, xA, ZAza, and XAxA for Alice and ZB, zB, XB, xB, ZBxB, and XBzB for Bob are elements of reality. Notice that the operators separated by dots

should be thought of as single operators; for instance, ZAzA should be considered as a single operator, while the measurement of ZBzB is given by the product of the results obtained measuring ZB and zB separately. Of course, this difference has no relevance in quantum mechanics (QM) but becomes important with LHV theories, where we must distinguish between different elements of physical reality. In order to explain correlations (9) with a LHV model, we must introduce the response function for each element of reality, which assigns a λ-determined measurement outcome and formalizes the hypothesis of reality. For instance,
a(λ)(b(λ)) gives the outcome of the measured observable a (b) at the Alice (Bob) side corresponding to the hidden variable value equal to λ.1 The locality properties are expressed by the condition that
a(λ) cannot depend on the observables b chosen by Bob and, symmetrically, b(λ) cannot depend on the observables a chosen by Alice, since the apparatus which measures the spin at one site cannot be influenced by the setting of the other one. This LHV model is constrained by correlation (9). For instance, by using the perfect correlation given in the first equation of (9), we obtain the following relation between the response 1 Note

that both a than b must be elements of reality. LASER PHYSICS

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functions:
Z A(λ)  Z B(λ) = –1. In this way, starting from (9), we can write nine relations between the response function. If we multiply eight of those relations, we find a contradiction with the last one. Then, the LHV models can correctly predict only eight of the nine correlations. In order to account for the absence of perfect correlations in a real experiment, inequalities need to be used. The quantum correlation functions are expressed in terms of LHV as LHV

〈 a ⊗ b〉 =

∫
( λ )  ( λ )ρ ( λ ) dλ. a

b

(10)

The LHV are distributed with a probability density ρ(λ) whose normalization writes

∫ ρ ( λ ) dλ = 1.

(11)

Indeed, by properly combining the predictions of (9), we can estimate the expectation value of the operator

 = –Z A ⋅ Z B – z A ⋅ z B – X A ⋅ X B – x A ⋅ x B + Z A z A ⋅ Z B ⋅ z B + X A x A ⋅ X B ⋅ x B + Z A ⋅ x A ⋅ Z B x B (12) + X A ⋅ zA ⋅ X BzB + Z AzA ⋅ X A x A ⋅ Z B xB ⋅ X BzB. In the case of the state |Ξ– –〉, the quantum mechanics expectation value of this operator is 〈〉 = 9, while the limit imposed by local realism is 〈〉 ≤ 7 [4]. Hence, we expect the quantum behavior of the state |Ξ– –〉 in the region 7 ≤ 〈〉 ≤ 9. In a previous AVN experiment [8], we obtained 〈〉 = 8.114 ± 0.011, corresponding to a violation by 101 standard deviations with respect to the local realism limit 〈〉 = 7. However, in that case, it was implicitly assumed that the numerical results of the measurement of some operators, namely ZAzA, XAxA, zBxB, and XBzB, are equal to the product of the measured values of ZA and zA (or XA and xA, zB and xB, XB and zB) obtained by different apparata. Hence, we repeated the AVN experiment by using different settings in order to avoid any supplementary assumption. In the new measurement procedure, three different experimental setups in Alice’s site correspond to other three settings at Bob’s site (see Fig. 3). Indeed, by referring to the last three equations of (9), it comes out that Alice must use three different settings: the first one contextually measures the two operators ZA and xA, besides their product ZA · xA. The second apparatus allows us to measure XA and zA (and, then, XA · zA). Finally, by the third one, the contextual measurement of (ZAzA) and (XAxA) is performed. In the same way, Bob needs three other measurement settings: the first one for the contextual measurement of the two operators XB and xB (and, then, XB · xB), the second which measures ZB and zB (and, then, ZB · zB), and the third apparatus for the contextual measurement of (ZBxB) and (XBzB). The experiLASER PHYSICS

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mental values of the nine operators appearing in (9) are then obtained by adopting all possible combinations between Alice’s and Bob’s measurement settings. 4.1. The Measurement Let’s now analyze the measurement schemes. It is quite easy to explain the first two measurement apparatus of Alice (A1, A2) and Bob (B1, B2). They must measure two operators, one in the momentum space and the other in the polarization space. Then, they could perform the measurement in two step. In the first step, they distinguish between the two momentum eigenstates: for instance, at Alice’s site, the states |l〉A and |r〉A are eigenstates of zA, while, when using the beam splitter (BS), the two BS output are eigenstates of xA. In the second step, Alice and Bob discriminate the polarization eigenstates with the help of one HWP, whose optical axis is oriented at an angle of α/2 with respect to the vertical direction, followed by the PBS. The HWP + PBS system is equivalent to a single polarizing beam splitter (called PBS α' ) whose vertical axis is oriented at α with respect to the vertical direction. In the following, we will refer to the HWP +PBS system as PBS α' . The other two measurement apparata (A3 and B3) require more attention. Consider Alice’s apparatus A3, which contextually measures (ZAzA) and (XAxA). Since these two operators commute, it’s possible to find a common eigenvectors basis. Explicitly, we have ±

±

Z A z A |σ 〉 A = |σ 〉 A , ±

±

X A x A |σ 〉 A = ± |σ 〉 A , ±

±

±

±

Z A z A |τ 〉 A = – |τ 〉 A ,

(13)

X A x A |τ 〉 A = ± |τ 〉 A , where we have introduced the single-photon Bell basis 1 ± |σ 〉 A = ------- ( |H〉 A |l〉 A ± |V 〉 A |r〉 A ), 2 1 ± |τ 〉 A = ------- ( |V 〉 A |l〉 A ± |H〉 A |r〉 A ). 2

(14)

If we are able to distinguish between |σ+〉A, |σ –〉A, |τ+〉A, and |τ –〉A, we can contextually measure the two operators (ZAzA) and (XAxA). The apparatus A3, also used in [16] and discriminating the four single-photon Bell states in (14), is shown in Fig. 3: the HWP is at 45° on

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XA · zA

PBS

ZAzA · XAxA

PBS

PBS HWP (45°)

BS

HWP (0°/45°)

BS

HWP (22.5°/67.5°)

A1 B1

A2

A3

B2

B3

PBS

PBS

HWP (0°/45°)

PBS HWP (0°)

BS

HWP (22.5°/67.5°)

XB · xB

BS

HWP (0°/45°) ZB · xB

HWP (22.5°/67.5°)

ZBxB · XBzB

Fig. 3. Experimental apparatus allowing the measurement of the expectation values of the nine operators appearing in the operator . Upper side: the three different settings adopted to perform Alice’s measurements. Lower side: the three different settings adopted to perform Bob’s measurements. HWP: λ/2 waveplates set for analysis at the angles indicated in parentheses.

the lA arm of the interferometer and, next, the BS transform the four single-photon Bell states into ±

|σ 〉 A

HWP

1 ------- |V 〉 A ( |l〉 A ± |r〉 A ) 2

1 ± |τ 〉 A HWP ------- |H〉 A ( |l〉 A ± |r〉 A ) 2

BS

BS

⎧ |V 〉 A |l' 〉 A ⎨ ⎩ |V 〉 A |r'〉 A , ⎧ |H〉 A |l' 〉 A ⎨ ⎩ |H〉 A |r'〉 A .

±

– |t 〉 B

HWP(0°) + BS

HWP(0°) + BS

(15)

⎧ |–〉 B |r'〉 B ⎨ ⎩ |–〉 B |l'〉 B ,

(17)

⎧ |+〉 B |r'〉 B ⎨ ⎩ |+〉 B |l'〉 B ,

' . which can be discriminated by two PBS 45°

By two PBS 0' (one on mode l A' and one on mode r A' ), we easily discriminate those four states. A similar argument holds for the last Bob apparatus. In order to contextually measure the two commuting operators (ZBxB) and (XBzB), we only need to completely discriminate the common eigenvectors basis, namely 1 ± |s 〉 B = ------- ( |+〉 B |l〉 B ± |–〉 |r〉 B ), 2 1 ± |t 〉 B = ------- ( |–〉 B |l〉 B ± |+〉 |r〉 B ), 2

±

– |s 〉 B

(16)

1 where the states |±〉B = ------- (|H〉B ± |V〉B) correspond to 2 linear polarized states in the direction ±45°. The HWP(0°) together with the BS transform them into

In the actual experiment, we slightly modify the measurement apparatus for three operators between the nine needed for • XAxA · XB · xB: we use A3 and B1 with HWP (45°) on both lA and lB. This is possible because the insertion of HWP (45°) on the lB side doesn’t change Bob’s apparatus B1; in fact, this HWP simply permutates the states 1 ------- |±〉B ⊗ (|l〉B ± |r〉B) that will be discriminated by the 2 ' . BS and the PBS 45° • ZA · xA · ZBxB: we use A1 and B3 with HWP (0°) on both lA and rB. In fact, the insertion of HWP (0°) on the lA side doesn’t change Alice’s apparatus A1; in fact, this 1 HWP simply permutates the states ------- |H〉A ⊗ (|l〉A ± 2 LASER PHYSICS

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1 |r〉A), ------- |V〉A ⊗ (|l〉A ± |r〉A) that will be discriminated by 2 the BS and the PBS 0' . • ZAzA · XAxA · ZBxB · XBzB: we use A3 and B3 with a HWP (45°) followed by a HWP(0°) on both lA and rB. Indeed, the insertion of a HWP(0°) (after the HWP (45°) in A3 Fig. 3) on the a1 side doesn’t change Alice’s apparatus A3; this HWP simply permutates the states on the right-hand side of (15) that must be successively discriminated. Similarly, the insertion of HWP (45°) (before the HWP (0°) in B3 Fig. 3 on the lB side doesn’t perform any change, because it simply permutates the states (16). Using these setups, we measured the following values: 〈ZA · ZB〉 = –0.9348 ± 0.0037, 〈XA · XB〉 = –0.9059 ± 0.0034, 〈zA · zB〉 = –0.9893 ± 0.0031, 〈xA · xB〉 = –0.8665 ± 0.0021, 〈ZAzA · ZB · zB〉 = 0.9218 ± 0.0037,

(18)

〈XAxA · XB · xB〉 = 0.8027 ± 0.0024, 〈Za · xA · ZBxB〉 = 0.8048 ± 0.0027, 〈XA · zA · XBzB〉 = 0.8839 ± 0.0031, 〈ZAzA · XAxA · ZBxB · XBzB〉 = –0.8258 ± 0.0042, which give an expectation value 〈〉 = 7.9354 ± 0.0097, violating local realism by 96 standard deviations. Coincidence measurements were performed lasting an average time of 30 s. It is worth noting that the correlation measurements are affected by the nonperfect symmetry of the beam splitter (transmission T = 0.55, reflection R = 0.45). Moreover, it has been observed that the BS introduces a phase shift in the momentum state, which depends on the photon polarization. A detailed discussion concerning how nonperfect polarization and momentum correlations affect the results given in [17]. 5. NONLOCALITY WITH TWO-PHOTON LINEAR CLUSTER STATES Another AVN proof based on only two observers, which provides a stronger contradiction between quantum and classical predictions, was recently introduced by Cabello [7]. It requires two photons entangled in polarization and linear momentum and prepared in the so-called linear cluster state 1 |C 4〉 = --- ( |Hr〉 A |Hl〉 B + |Vr〉 A |Vl〉 B 2 + |Hl〉 A |Hr〉 B – |Vl〉 A |Vr〉 B ) 1 + – = ------- ( |Φ 〉 |r〉 A |l〉 B + |Φ 〉 |l〉 A |r〉 B ). 2 LASER PHYSICS

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The cluster state was created from the hyperentangled state |Ξ〉 = |Φ–〉 ⊗ |ψ+〉, by inserting in the rA mode a zero-order half waveplate (HW) with the optical axis oriented along the vertical direction (cf. Fig. 1b). The HW leaves the state |Φ–〉|l〉A|r〉B unchanged, while the |Φ+〉|r〉A|l〉B also transtransformation |Φ–〉|r〉A|l〉B forms |Ξ〉 into |C4〉. The operation represented by the insertion of HW acts “locally” on photon A only, but, because of the polarization and momentum qubits associated with photon A, the HW operates nonlocally as a controlled phase (CP) between the target qubit and the control qubit (i.e., the polarization and the momentum degree of freedom of photon A), thus, entangling the four qubits together. Cluster states, which have been proposed for the realization of the linear optics one-way quantum computation [5, 18], also represent a powerful tool for performing nonlocality tests [19, 20]. It is well known that the ratio between the quantum and classical prediction, in fact, grows exponentially with the number of qubit systems prepared in the state. However, it is difficult to produce GHZ states with more than n = 4 qubits. Moreover, in a real experiment n-particle GHZ states suffer a decoherence that also grows with the number of particles. The two-photon cluster generated by our technique state represents a genuine four-qubit entangled state [21]. Compared to multipartite cluster states, it has the advantage of a high purity and a relatively high generation rate. Hence, it represents an alternative solution to perform more advanced nonlocality tests and, possibly, for the realization of one-way quantum computation. The AVN proof we used to test the nonlocal behavior of linear cluster states requires only two observers and combines the advantages of a small number of quantum predictions (just four) and of a larger amount of evidence against local realism. It is based on the following eigenvalue equations: X A z A X B |C 4〉 = – |C 4〉,

(20a)

z A z B |C 4〉 = – |C 4〉,

(20b)

x A Z B x B |C 4〉 = + |C 4〉,

(20c)

Z A y A y B |C 4〉 = + |C 4〉,

(20d)

Y A z A Y B |C 4〉 = + |C 4〉,

(20e)

X A X B z B |C 4〉 = + |C 4〉,

(20f)

Y A Y B z B |C 4〉 = – |C 4〉,

(20g)

X A x A Y B y B |C 4〉 = + |C 4〉,

(20h)

Y A x A X B y B |C 4〉 = + |C 4〉,

(20i)

1000

VALLONE et al.

where we used

posed, which gives a stronger discrepancy between quantum and classical predictions.

Y j = σ y j = i |V 〉 j 〈H| – i |H〉 j 〈V |, y j = σ 'x j = i |r〉 j 〈l| – i |l〉 j 〈r|

(21)

( j = A, B ), and the operators defined in Eq. (8). The first seven equalities of (20) demonstrate that the local observables XA, YA, xA, XB, YB, yB, and zB are elements of reality in the EPR sense [1]. The last four equalities are used in the AVN proof through the following quantum mechanical expectation value of the cluster state (19): 〈 S〉 = 〈C 4| X A X B z B – Y A Y B z B + X A x A Y A y B + Y A x A X A y B |C 4〉 = 4.

REFERENCES (22)

Note that, following the local realistic theory based on the previously defined elements of reality, we expect S ≤ 2. The experimental results obtained by using the experimental apparatus in Fig. 2 give, as a final result, Tr [ Sρ exp ] = 3.4145 ± 0.0095.

ACKNOWLEDGMENTS This work was supported by FIRB 2001 (Realization of Quantum Teleportation and Quantum Cloning by the Optical Parametric Squeezing Process) and PRIN 2005 (New Perspectives in Entanglement and Hyperentanglement Generation and Manipulation) of MIUR (Italy).

(23)

This result corresponds to a violation of the classical bound by 148 standard deviations. Note that this result provides an enhanced discrepancy between the quantum versus classical predictions (4 versus 2) with respect to the standard CHSH inequality ( 2 2 versus 2) [14]. 6. CONCLUSIONS The results of two all-versus-nothing experiments, of the GHZ type, performed by two-photon states entangled in polarization and momentum, have been presented in this paper. These states, spanning the 4 × 4 Hilbert space, permit us to operate with four qubits by using only two photons. They make possible tasks which cannot be achieved by 2 × 2 entangled photon pairs; for instance, they provide stronger contradictions between quantum and classical predictions and are more robust against noise. The possibility offered by these states is expected to be, in the near future, a useful and far reaching resource of QI technology. A further improvement of this technique consists in the realization of the novel two-photon four-qubit linear cluster states. We have tested the nonlocal character of these states by performing a more advanced two-observer AVN experiment of quantum nonlocality, recently pro-

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LASER PHYSICS

Vol. 17

No. 7

2007