Experimentally testing Bell's theorem based on Hardy's nonlocal

SCIENCE CHINA Physics, Mechanics & Astronomy

. Article .

February 2015 Vol. 58 No. 2: 024201 doi: 10.1007/s11433-014-5495-0

Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs GUO WeiJie1 , FAN DaiHe1 & WEI LianFu1,2* 2

1 Quantum Optoelectronics Laboratory, Southwest Jiaotong University, Chengdu 610031, China; State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China

Received December 28, 2013; accepted April 14, 2014; published online September 22, 2014

Bell’s theorem argues the existence of quantum nonlocality which goes basically against the hidden variable theory (HVT). Many experiments have been done via testing the violations of Bell’s inequalities to statistically verify the Bell’s theorem. Alternatively, by testing the Hardy’s ladder proofs we experimentally demonstrate the deterministic violation of HVT and thus confirm the quantum nonlocality. Our tests are implemented with non-maximal entangled photon pairs generated by spontaneous parametric down conversions (SPDCs). We show that the degree freedom of photon entanglement could be significantly enhanced by using interference filters. As a consequence, the Hardy’s ladder proofs could be tested and Bell’s theorem is verified robustly. The probability of violating the locality reach to 41.9%, which is close to the expectably ideal value 46.4% for the photon pairs with degree of entanglement ε = 0.93. The higher violating probability is possible by further optimizing the experimental parameters. quantum nonlocality, entanglement production, Hardy’s proof PACS number(s): 03.65.Ud, 03.67.Mn, 42.50.Xa Citation:

Guo W J, Fan D H, Wei L F. Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs. Sci China-Phys Mech Astron, 2015, 58: 024201, doi: 10.1007/s11433-014-5495-0

Entanglement, that is, quantum nonlocal correlations, is a critical characteristic in quantum mechanics. Einstein et al. [1] regarded it as a “spooky” feature of quantum mechanics. The key of Bell’s theorem is, quantum nonlocality cannot be convincingly explained by any local hidden variable theory (HVT). This property of quantum systems has been applied to quantum computation [2], quantum teleportation [3] as well as quantum cryptography [4]. Currently, experimental verifications of quantum nonlocality via testing the violations of Bell inequalities [5,6] have been demonstrated with various bipartite entanglements, such as photons [7,8], trapped ions [9] and superconducting circuits [10,11]. Alternatively, Greenberger et al. [12] showed that quantum nonlocality could be tested deterministically without statisti*Corresponding author (email: [email protected])

c Science China Press and Springer-Verlag Berlin Heidelberg 2014 

cal inequality. However, the proof as suggested by Greenburger et al. [12] requires minimum of three entangled particles. To remove such a entanglement which seems difficulty to be achieved at that time, Hardy [13] proposed another proof of nonlocality without inequality by using only two particles. The Hardy’s nonlocal proof (HNLP) is based on such a contradiction that, certain results delivered by the exact logics in local HVT could be violated with certain probabilities in quantum mechanics. It has been argued that such a proof is the best approach to implement the loophole-free experimental test of Bell’s theorem without inequality [14]. The original HNLP, termed the Hardy’s proof (HP) for simplicity, involves only one ladder and the maximally-available probability of violating the HVT is determined as 9%. This argument had been verified experimentally in 1995 [15]. Boschi et al. [16] generalized the HP to the logic containing K (K > 1) ladphys.scichina.com

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ders and proposed the so-called Hardy’s ladder proofs (HLPs) for HNLP. The probability of violating the local HVT by the HLPs can be significantly enhanced and the maximal value could reach 50% for the sufficiently many ladders (that is, K → ∞). This improves essentially the feasibility of the experimental test of Bell’s theorem by the HNLP. Consequently, the HNLPs with K = 2 were experimentally tested with the energy-time correlation of single photons [17] and orbital angular momentum entanglements of twisted photons [18], respectively. The maximum probabilities of obtaining the violation of the HVT reach 17% and 13.9%, respectively. With the traditional polarization entanglements of two photons, Giorgi et al. [19] implemented the HNLP tests up to 20 ladders, and the relevant probability of violating the HVT reaches 40%. Consequently, herein we report we report our experiments for testing the HNLP with K > 1000 using also the polarization-entangled photon pairs. The results verify more robustly the predictions of the HNLP and thus support the arguments from Bell’s theorem. Herein we briefly review the basic logic of the HNLP and then discuss how to perform the relevant experiments for the tests. Our experimental tests of the HNLPs by using the polarization-entangled photon pairs are then reported.

1 Hardy’s nonlocal proof and the experimental testability Hardy’s proofs on nonlocality, that is, Hardy’s nonlocal proofs, are based on the logical contradictions between the deterministic classical deductions and the probabilistic quantum mechanics sequiturs. One of these logic contradictions can be summarized as the follows. Suppose we have two observers, Alice and Bob. Alice measures two variables: A0 and A1 , and Bob measures the other variables: B0 and B1 . For simplicity we assume that Ak (k = 0, 1) have the outcomes Ak = ±1, and also the similarity for the Bob. Let us define P(Ai , A j ) as the joint probability for the measured results Ai = B j = 1, and P(A¯i , B j ) the joint probability of obtaining the outcomes Ai = −1 and B j = 1. Here, A¯i = −Ai . Seemingly, for any local HVT, if, ⎧ P(A , B ) = 0, ⎪ ⎪ ⎨ ¯0 0 P(A , B ) = 0, (1) ⎪ ⎪ ⎩ P(A0 , B¯1) = 0, 1 0

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The above one ladder logic contradiction between the locality in HVT and the nonlocality in quantum mechanics can be easily generalized to the case with K ladders. For this, we consider K + 1 dichotomic observables Ak and Bk (k = 0, 1, . . . , K). Assume that the following joint probability condition as thus: ⎧ P(A0 , B0 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ P( A¯0 , B1 ) = 0, ⎪ ⎪ ⎪ ⎪ ¯0 ) = 0, ⎪ P(A 1, B ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎨ ¯ P(Ak−1 , Bk ) = 0, (3) ⎪ ⎪ ⎪ ⎪ ¯k−1) = 0, ⎪ P(A , B k ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎪ P(AK , B¯K−1 ) = 0, ⎪ ⎪ ⎩ P(A¯ , B ) = 0, K−1 K are satisfied, then P(AK , BK ) = 0 is derived definitely in the local HVT. But, in quantum mechanics the argument, P(AK , BK )  0

could be implemented probably (that is, with certain probability). This is the so-called K-ladder HNLP, which is shown schematically in Figure 2. Physically, the above HNLPs could be tested by properly setting the observables A and B, as well as performing the relevant orthogonal measurements (that is, |Ak , |A⊥k  and |Bk , |B⊥k , respectively). For example, the horizontal polarization H and the vertical polarization V can serve as the desirable orthogonal measurements of a photon. Therefore, using a

Figure 1

Logic contradiction between the locality in HVT and the nonlo-

cality in quantum mechanics for one ladder.

are obtained simultaneously, then, P(A1 , B1) = 0

(2)

is derived definitely. However, in quantum mechanics it is possible to find certain quantum states in which the outcomes of the observables A0 , A1 , B0 , B1 satisfying the arguments eq. (1) but P(A1 , B1)  0, that is, violating the HVT prediction eq. (2) with certain probability. This is nothing but the logic in the original Hardy’s proof on nonlocality. Such a logic contradiction between locality and quantum mechanics is shown schematically in Figure 1.

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Figure 2 Logic of a K-ladder HNLP.

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pair of polarization-entangled photons [13,16,20] can be described by the wave function: |Ψ = α|HA |HB − β|VA |VB , |α|2 + |β|2 = 1,

(5)

the HNLP can be tested as the follows. We select a series of projective measurement basis {|Ak , |A⊥k  and |Bk , |B⊥k }. They are related to the horizontalvertical as such: ⎧ ⎪ |HA = ck |Ak  + c⊥k |A⊥k , ⎪ ⎪ ⎪ ⎪ ⎨ |VA = (c⊥k )∗ |Ak  − (ck )∗ |A⊥k , (6) ⎪ ⎪ |HB = ck |Bk  + c⊥ |B⊥ , ⎪ ⎪ ⎪ ⎩ |VB = (c⊥ )∗ |Bk  −k (ckk )∗ |B⊥ . k k Now, in order to deliver a contradiction between the locality in HVT and the nonlocality in quantum mechanics, we need to look for certain quantum states in which |Ak  and |Bk  satisfy the following conditions: ⎧ ⎪ |A |B |Ψ|2 = 0, ⎪ ⎪ ⎪  0 0⊥ 2 ⎨ Ak |Bk−1|Ψ = 0, (7) ⎪ ⎪   ⎪ 2 ⎪ ⊥ ⎩ Ak−1 |Bk |Ψ = 0. For simplicity, the parameters α and β in eq. (5) are taken to be real and positive and ck and c⊥k are selected also to be real. Thus, from eq. (6) we have ⎧ ⎪ |Ak  = ck |HA + c⊥k |VA , ⎪ ⎪ ⎪ ⎪ ⎨ |A⊥k  = c⊥k |HA − ck |VA , (8) ⎪ ⎪ |Bk  = ck |HB + c⊥k |VB , ⎪ ⎪ ⎪ ⎩ |B⊥  = c⊥ |HB − ck |VB . k k Substituting eq. (8) into eq. (7), we have thus ⎧ 1 ⎪ ⎪ c0  β  2 ⎪ ⎪ ⎪ = , ⎪ ⎪ α ⎨ c⊥0 ⎪ ⎪ ⎪ ck β ck−1 ⎪ ⎪ ⎪ ⎪ ⎩ c⊥ = − α · c⊥ . k k−1

(9)

Finally, the probability PK of violating the local argument is given by PK = |AK |BK |Ψ|2

2 2K+1 − βα2K+1 αβ = . β2K+1 + α2K+1

Figure 3

(Color online) Probabilities Pk of violating the locality for differ-

ent ladders K and the quantum states with different degree of entanglement ε = β/α.

2 Experimentally testing the NLHPs with entangled-photon pairs Nonmaximally entangled states utilized to test the NLHPs are produced in the process of spontaneous parametric down conversion (SPDC) by pumping a pair of type-I beta-barium borate (BBO) crystals. Generally, such an entangled-photon pairs can be described by the following wave function: √ |Ψ = (|HH + εeiφ |VV)/ 1 + ε2 . (11) For the present case, we have ε = β/α and φ = π, and thus, √ |Ψ = (|HH − ε|VV)/ 1 + ε2 . (12) Here, the parameter ε is controllable. Eq. (10) implies that, for a given K a proper value of ε can always be set for obtaining the maximum value of PK . Also, for a given ε, the value of PK tends theoretically to a maximum with K increasing. Therefore, for the given ε and K, the measurement settings for the observers A and B could be set as: |A(θ) = sin(θA )|HA  + cos(θA )|VA ,

(13)

|B(θ) = sin(θB )|HB  + cos(θB )|VB ,

(14)

and (10)

It is readily apparent that when PK = 0 for the maximally entangled states no matter what K is. Also, for K = 1 the maximum probability of violating the locality is P1 = 9% (when α/β = 0.46) [13]. In particular, one can readily show that, as K tends to infinity, α tends to β (but not equal), the maximum value of PK tends to 50%. This feature can be shown schematically in Figure 3. Certainly, for the locality in HVT the value of PK equals always to 0. This implies that, once a nonzero value of Pk is measured ideally, then the locality is violated and consequently nonlocality predicted by quantum mechanics can be verified experimentally.

respectively. Here, the angle θA, B respects to the vertical axis and should be set properly (according to eq. (9)) as thus:  1 (15) θkA = θkB = arctan (−1)k (ε)k+ 2 . The experimental system used here for testing the NLHPs is shown in Figure 4. By rotating the 1#HWP behind lens to change the polarization of the pump laser, non-maximally entangled states with controllable degree of entanglement ε = tan(2χ) can be generated. Here, χ is the angle of the HWP with respect to the vertical. Theoretically, the probability of coincident detection depending on the choose of measurement setting (not using the 2# QWP) can be read as: P(θA , θB ) = |θA |θB |Ψ|2

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2  = sin θA sin θB + εeiφ cos θA cos θB  /(1 + ε2 ).

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This indicates that the phase φ in eq. (11) could be controlled as π, in principle, by rotating the 1#QWP to minimize the probability of coincident detection. In experiment, this can be achieved by fixing the ε for proper value of the θ parameters, e.g., θA = θB = 45◦ for convenience. Conversely, by using the usual quantum tomography technique [21,22] via adding 2#QWP the density matrix of the entangled-photon pairs for different degrees of entanglement can be reconstructed. Thus, the values of the parameters ε and φ can be determined. For simplicity, we first test the NLHP with one ladder, that is, the original Hardy’s proof on nonlocality. Figure 5 show the measured probability of violating the locality versus the degree of entanglement ε. The solid line represents the theoretical predictions and the points with error bars are the experimental results. Seemingly, the experimental data agrees well with the theoretical predictions based on the NLHP with K = 1. This indicates that the experimental results support the nonlocality predicted by quantum mechanics. The numerical simulations in Figure 3 show that, for a sufficiently large ladders, that is, K → ∞, nonlocaltiy indicates that the probability of violating the locality (that is, the probability of coincident counts in the two detectors in the experiments) could tend to 50% when ε → 1. We performed the relevant experiments to verify such an argument. The experimental results are shown schematically in Figure 6, wherein ε = 0.93. It is seen that the experimental data is entirely consistent with the theoretical predictions in quantum mechanics. Note that most of experimental data match the theoretical predictions, although there still exists certain deviations, particularly for Figure 6 where the deviation between the experimental limit value 41.9% and the theoretical one 46.4%. One reason to explain such a deviation is the is the experimental accuracy. Specifically, the alignment angles of the wave plates that measure photon polarizations are the most

Figure 4 Schematic illustration of the experimental test of NLHPs. entangled-photon pairs are produced by two type-I BBO crystals, pumped by a 405 nm laser with vertical polarization. Lens are used to focus the light beams. PBS is the polarization-beam-splitter, HWP represents the half-wave plate and QWP for the quarter-wave plate. Ape (aperture) and IF (Interference filters whose center wavelength is 810 nm, bandwidth is 10 nm) are used to filter out the stray photons. Photons are detected by the APDs and their correlations are measured by the correlator.

Figure 5 (Color online) Probability PK (K = 1) versus the degree of entanglement ε. Here, solid line is the predictions from quantum mechanics, and the points with error bars are the experimental data.

Figure 6 (Color online) Probability of violating the locality versus the ladder number K for ε = 0.93. Here, the ladder number K is taken from 1 to 1000. It is shown that the experimental data (with error bars) is phenomenally consistent with the theoretical predictions, except a small systematic deviation (i.e., the limit value obtained by the experiments is 41.9%, but the theoretical one is 46.4%.

important source of errors. Thus, the misalignment of the 2#HWP affects the value of PK directly; the accuracy of the HWP in our experiment is 2◦ . For the data shown in Figure 6, we can calculate the maximum deviation is ±0.9%. If we take the calibration of HWP into account, then the value of PK can be 43%. Furthermore, the value of ε is dependent of the angle of 1#HWP, i.e., δε ≈ 2δχ/ cos(2χ)2 . Here δε and δχ represent the deviations of ε and χ, respectively. When ε is near to 1, we have δε ≈ 4δχ. Finally, the uncertainty of φ brings errors too. It is emphasized that eq. (7) could not be satisfied strictly in the practical experiments. In order to make sure these deviations do not wash out the Hardy’s logical contradiction, we need to introduce a quantity [16] as thus: S K ≡ P(AK , BK ) −

K

[P(Ak , B⊥k−1) + P(A⊥k−1 , Bk )]

k=1

− P(A0 , B0 ),

(17)

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and the National Fundamental Research Program of China (Grant No. 2010CB923104).

Figure 7 (Color online) Violation degree (probability) of the locality versus the degree of entanglement. Note that the NLHP test should not contain the point of the maximal entanglement, i.e., ε = 1.

which represents that, if S K > 0, then the local theory has been violated. For example, in one of our experiments we have ε = 0.364, and the measured value of PK (K = 1) is 8.1%. In this case the value of S K is calculated as 5.8%, which apparently violate the locality (even if the eq. (7) is not satisfied strictly).

3 Discussion Given that many experiments have been done to verify the Bell’s theorem by testing the violations of various Bell’s inequalities, in this paper we have reported another optical experiment to verify again the Bell’s theorem by testing the NLHPs without inequality. In our experiment, the degree of entanglement of the entangled-photon pairs generated by pumping the BBO crystals are effectively adjustable. In particular, the ladder number K can be set arbitrarily. Our testing experiments include two types, one is to test the NLHP with K = 1 for different degree of entanglement, and the other is to test the NLHP with a fixed ε for different ladder number K. All these experiments support the predictions of quantum mechanics and violating the locality. Similar to the test of the Bell’s inequality, we note also that violation degree of locality increases with the degree of entanglement (see Figure 7) Note that, at the point ε = 1 the Bell inequality could be violated maximally. However, at this point the testing of NLHP fails completely. This indicates that, the larger degree of entanglement (except the maximal value 1) is more advantageous to test for the NLHP for sufficiently much ladders, that is, K → ∞, and the maximal probability of violating the locality can tend to 50% ideally.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61308008, 91321104, U1330201 and 11174373),

1 Einstein A, Podolsky B, Rosen N. Can quantum mechanical description of physical reality be considered complete. Phys Rev, 1935, 47: 777–780 ˇ Zukowski ˙ 2 Brukner C, M, Pan J W, et al. Bell’s inequalities and quantum communication complexity. Phys Rev Lett, 2004, 92: 127901 3 Bennett C H, Brassard G, Crepeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys Rev Lett, 1993, 70: 1895–1899 4 Ekert A K. Quantum cryptography based on Bell’s theorem. Phys Rev Lett, 1991, 67: 661–663 5 Bell J S. On the einstein-podolsky-rosen paradox. Physics, 1964, 1: 195–200 6 Clauser J F, Horne M A, Shimony A, et al. Proposed experiment to test local hidden-variable theories. Phys Rev Lett, 1969, 23: 880–884 7 Weihs G, Jennewein T, Simon C, et al. Violation of Bell’s inequality under strict Einstein locality conditions. Phys Rev Lett, 1998, 81: 5039–5043 8 Aspect A, Dalibard J, Roger G. Experimental test of Bell’s inequalities using time-varying analyzers. Phys Rev Lett, 1982, 49: 1804–1807 9 Rowe M A, Kiepinski D, Meyer V, et al. Experimental violation of a Bell’s inequality with efficient detection. Nature, 2001, 409: 791–794 10 Wei L F, Liu Y X, Nori F. Testing Bell’s inequality in a constantly coupled Josephson circuit by effective single-qubit operations. Phys Rev B, 2005, 72: 104516 11 Wei L F, Liu Y X, Storcz M J, et al. Macroscopic Einstein-PodolskyRosen pairs in superconducting circuits. Phys Rev A, 2006, 73: 052307 12 Greenberger D M, Horne M A, Shimony A, et al. Bell’s theorem without inequalities. Am J Phys, 1990, 58: 1131–1143 13 Hardy L. Nonlocality for two particles without inequalities for almost all entangled states. Phys Rev Lett, 1993, 71: 1665–1668 14 Eberhard P H. Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment. Phys Rev A, 1993, 47: 747–750 15 Torgerson J R, Branning D, Monken C H, et al. Experimental demonstration of the violation of local realism without Bell inequalities. Phys Lett A, 1995, 204: 323–328 16 Boschi D, Branca S, Martini F D, et al. Ladder proof of nonlocality without inequalities: Theoretical and experimental results. Phys Rev Lett, 1997, 79: 2755–2758 17 Vallone G, Gianani I, Enrique B, et al. Testing Hardys nonlocality proof with genuine energy-time entanglement. Phys Rev A, 2011, 83: 042105 18 Chen L X, Romero J. Hardy’s nonlocality proof using twisted photons. Opt Express, 2012, 20: 21687 19 Giorgi G, DiNepi G, Mataloni P, et al. A high brightness parametric source of entangled photon states. Laser Phys, 2003, 13: 350–354 20 White A G, James D F V, Eberhard P H, et al. Nonmaximally entangled states: Production, characterization, and utilization. Phys Rev Lett, 1999, 83: 3103–3107 21 Fan D H, Guo W J, Wei L F. Experimentally testing the Bell inequality violation by optimizing the measurement settings. J Opt Soc Am B, 2012, 29: 3429–3433 22 James D F V, Kwait P G, Munro W J, et al. Measurement of qubits. Phys Rev A, 2001, 64: 052312