On Bell s Joint Probability Distribution Assumption and Proposed Experiments of Quantum Measurement

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1 On Bell s Joint Probability Distribution Assumption and Proposed Experiments of Quantum Measurement Hai-Long Zhao 7th branch post office 15th P.O.BOX 11#, Lanzhou, 73750, China Abstract: In the derivation of Bell s inequalities, there is an additional assumption of joint probability distribution besides the assumptions of locality and realism. It has been shown that if this assumption does not hold, then Bell s inequalities fail. Our further analysis shows that Bell s joint probability distribution holds only for two independent events but not for the joint measurements of EPR pairs. We point out that polarization entangled photon pair is actually a pair of circularly polarized photons or a circularly polarized photon and a linearly polarized photon whose hidden variables are correlated. This hypothesis can explain the experimental results of EPR pairs. Several experiments are proposed to test the relevant problems in quantum measurement. The first use delayed measurement on one photon of EPR pair to demonstrate directly whether measurement on the other could have any non-local influence on it. The other two are used to reveal the constituents of polarization Bell states. The last one verifies the deduction of determinism that two particles with the same hidden variable will behave the same under the same measuring condition. PACS: Ta; Ud; 4.50.Xa 1. Introduction Quantum theory gives only probabilistic predictions for individual events based on the probabilistic interpretation of wave function, which leads to the suspicion of the completeness of quantum mechanics and the puzzle of the non-locality of the measurement of EPR pairs [1]. Indeed, if hidden variable theory is not introduced into quantum measurement, we can hardly understand the distant correlation of EPR pairs, e.g. quantum teleportation and quantum swapping [,3]. Bell pointed out that any theory that was based on the joint assumptions of locality and realism conflicted with the quantum mechanical expectation [4]. Since then, various local and non-local hidden variable models against Bell s inequalities have been proposed (see, for example, [5-1]), of which the most attractive one is the time-related and setting-dependent model suggested by Hess and Philipp [11,1], but was criticized by Gill et al and Myrvold for being non-local [13,14]. As a matter of fact, there is an additional assumption of joint probability distribution besides the assumptions of locality and realism [15-17]. Bell supposed that joint probability distribution is only a function of hidden variable and is irrelevant to measuring condition. However, the validity of this assumption is dubious. As pointed out by de la Peňa et al and Nagasawa that if this assumption does not hold, then Bell s inequalities fail [16,17]. On the other hand, it has been shown that even if non-locality is taken into account, Bell s inequalities may also be violated [18,19]. So we focus on Bell s joint probability distribution assumption and discuss its validity. We point out that this assumption is equivalent to the measurement of two independent events and is inapplicable to the joint measurement of EPR pairs. In the meanwhile, we suggest polarization uncertainty as the hidden variable for polarization degree of freedom. In terms of quantum entanglement, the spin (polarization) of a pair of EPR particles is 1

2 indefinite and dependent on each other. By analysis of existing experiments [0-31], we show that polarization Bell states (maximally entangled states) can be composed of two circularly polarized photons or a circularly polarized photon and a linearly polarized photon (given their hidden variables are correlated). The experimental results of EPR pairs are explained based on this assumption. If hidden variable does exist, then the quantum state of one of the EPR pair will not change when measurement is made on the other, and the outcomes of a pair of particles with the same hidden variable will be the same under the same circumstance. We propose three types of experiments to test above hypotheses. The experiments are easy to realize for the experimental setups are very simple.. Discussion on Bell s joint probability distribution assumption and suggested hidden variable In local hidden variable theory Bell s inequalities play an important role. Bell regarded that his correlation function was founded on the vital assumption of Einstein that the result of B does not depend on the setting of measuring device a, nor A on b [4]. P ( = A( B(, (1) where A ( = ± 1, B ( = ± 1, ρ ( satisfies normalized condition = 1. Bell regarded joint probability distribution as only a function of λ. de la Peňa et al suggested that joint probability distribution may depend on measuring condition [16]. Nagasawa later made a detailed analysis based on strict mathematical definition [17]. We now discuss this problem in a simple way. Eq. (1) includes four joint probabilities, which are P ( = 1, B = 1), ( A P ( = P P P P. Since P( actually denotes joint probabilities, A P = 1, B = 1), P ( A = 1, B = 1), P ( A = 1, B = 1), respectively. Then we have ρ must be the joint probability density function with respect to the measurement outcomes A and B, i.e. ρ = A = ± 1, B = ± 1). As the results of A and B depend on the settings of measuring devices and hidden variables of the pair, we have ρ =. If it does not vary with measuring condition, then it becomes the case that Bell considered. For a pair of EPR particles it s easy to understand that they share a same hidden variable. But there is no prior reason that joint probability distribution is irrelevant to measuring condition. Although it satisfies the normalized condition, its expression may vary with the settings of measuring devices. Two possible curves are plotted in Fig.1 representing the different measuring conditions a, b and a, b, respectively. a, b, 0 1 λ Fig. 1. Possible joint probability distributions under different measuring conditions. One might think that ρ = conflicts with the locality. In fact, locality assumption has already been included in the expressions of A = A( and B = B(.

3 Since joint measurement outcomes are related to a, b and λ, it s natural that joint probability distribution is a function of a, b and λ, which is unconcerned with non-locality. Now we discuss joint probability in another way. As A = A(, B = B(, we have P ( a) = A(, P ( = A(, i.e. the probability spaces of the two events are different. In order to calculate the joint probability we must carry it out in the same probability space. For convenience we calculate the joint probability in the probability space of A. Then Eq. (1) is modified as P ( = A( B( λ A) = A( B( a,, () where B ( does not mean that the setting of measuring device a could have any non-local influence on the result of B. It denotes the result of B conditioned to the settings of measuring device that is related to a and b. To be specific consider the case that a pair EPR photons is incident upon a pair of polarizers. B( a, represents the measurement outcome of B under the condition that the orientation of polarizer is set to the direction a ± b and the result of A is already known. It can be seen that Eq. () is just the joint probability expressed with conditional probability, which applies to the calculation of joint probability of two dependent events. If the two events are independent, then Eq. () is turned into (1). Similarly, we have P ( c) = A( B( c,, (3) P ( c) = A( B( c,. (4) Substituting b, in Eq. (4) with a,, we have P ( c) = A( B( c,. (5) With the Eqs. (), (3) and (5), Bell s inequalities cannot be obtained. We do not discuss the detailed derivation process. Most of the researchers think that the measurements of EPR pairs are stochastically independent events, and they attribute this independence to locality [13-15]. Then Eq. (1) is valid. Such assumption is unreasonable. Although the experimental results of A and B do not depend on the setting of measuring device on the other side, this does not necessarily imply that the two events are independent. In fact, dependent events that have the feature of locality exist widely in macroscopic and microscopic worlds. For example, suppose there are ten holes, of which a little cat can pass through six and a big cat can pass through four. Then the probability that both cats can pass through a hole is not =0.4 but 0.4. This is because if the big cat can pass through, then the little cat can pass through with certainty. For a pair of particles in singlet state considered by Bell, perfect anticorrelation exists in the case of a = b, i.e. A( = B(. So we cannot conclude independence from locality, just as we cannot think that the joint probability distribution is unrelated to measuring condition. From above analysis we see that the problem with Bell s inequalities is not the assumptions of locality and realism but the assumption of joint probability distribution (or the independence of the measurement outcomes of EPR pairs), which is the fundamental reason that Bell s inequalities conflict with the prediction of quantum mechanics. For a pair of EPR particles whose hidden variables are correlated, their measurement outcomes are correlated. In 3

4 the derivation of Bell s inequalities, only the perfect anticorrelation is considered when the settings of measuring devices on the two sides are the same. In other cases this correlation is not considered. So Eq. (1) actually represents the correlation of two independent events. In order to indicate the intrinsic correlation of two dependent events, one way is to suppose that joint probability density function varies with measuring condition, the other is the expression with conditional probability. In both cases we cannot derive Bell s inequalities. In the following we discuss the problem of quantum measurement based on the assumption that local hidden variable exists. We first explore the physical meaning of hidden variable. Due to wave-particle duality and the fact that the fluctuation of a quantum state is in three-dimensional space, the internal fluctuation states of two particles may be different even if their external motion states are the same. We take the spin (polarization) of a particle as an example. In classical theory angular momentum is a vector, whose magnitude and the projections in three directions are all well-defined. In quantum mechanics, the magnitude of angular momentum is well-defined, and we can only determine its projection l in one direction. The angular position φ and the other two projections lx and l are all indefinite. φ and lz satisfy the uncertainty relation φ l z h /. Both φ and l z indicate the fluctuation state of the spin (polarization) of a particle in the projective direction. So the two parameters may be used as the hidden variables. In classical mechanics and quantum field theory, we have principle of least action. We now try to introduce this principle into quantum measurement. We define φ l z as the action for spin (polarization) of a particle. When a photon is incident upon a polarizer, it has two choices. Consequently, there are two possible collapsed polarization directions. We suppose photon always chooses the direction with a less action. For a linearly polarized photon, its polarization direction may be regarded as the direction with the least action, namely in this direction we have φ l = h /. Circularly polarized light may be thought of z being composed of two orthogonal polarized components, so circularly polarized photon has two directions with a least action. Similarly, we define the product of the uncertainties of position and momentum as the action for the motion of center of mass of a photon. As spin (polarization) is a relativistic quantum effect, it s likely that the corresponding hidden variables are irrelevant to time. We will test this hypothesis in the experiment below. In general cases, when measurement is made on a particle, its quantum state will collapse into another one, and the collapsing process is nonlinear and irreversible. A small change of external circumstance or internal fluctuation state may lead to a different result, i.e. the measurement outcome is sensitive to external circumstance and internal fluctuation state of a particle. So the collapse of quantum state is chaotic in essence. From this point of view, the evolutions of microcosm and macrocosm, and even the universe are chaotic in essence. 3. Interpretation of EPR-type experiments The experiment used to test Bell s inequalities with polarization state of photon pairs is shown in Fig.. A pair of EPR photons is incident from opposite directions upon a pair of polarization analyzers a and b with different orientations. We denote the transmitted channel and reflected channel with and, respectively. The results for quantum mechanics are [5] y φ z state in 4

5 D1 D3 S D4 D a b Fig.. Experimental test of Bell s inequalities. P ( a) = P ( a) = 1/, (6) P ( P ( ) = 1/, (7) = b 1 P ( = P ( = cos ( a, (8) 1 P ( = P ( = sin ( a. (9) In terms of quantum entanglement the polarization of a pair of EPR photons is indefinite. If hidden variable exists, the polarization of each photon should be well-defined. Consider the experiment of photon pairs emitted by the J = 0 J = 1 J = 0 cascade atomic calcium [0,1]. According to classical theory, the two photons are circularly polarized. For the experiment of J = 1 J = 1 J = 0 cascade atomic mercury [], one photon is linearly polarized and the other circularly polarized. In the down-conversation of nonlinear crystal [3-31], the wave functions of two orthogonally polarized photons overlap at crystal or beam splitter. If a π / phase difference exists between the two photons, they will form two circularly polarized photons. The combination of half-wave plate and quarter-wave plate can transform a Bell state into other Bell states [5]. From these facts, we may think that Bell state can be composed of a pair of circularly polarized photons or a circularly polarized photon and a linearly polarized photon (two linearly polarized photons cannot form a Bell state, which we will discuss below). For the twin photons generated in cascade radiation or down-conversation, we may think that their hidden variables are correlated. Additionally, when the wave functions of two photons with nearly equal wavelengths overlap at crystal or beam splitter, the hidden variables of the two photons will become correlated owing to the exchange of their spin (polarization) angular momentums (exchange effect will be weak for large wavelength difference [31]). We first consider Bell state composed of a pair of circularly polarized photons. For a circularly polarized photon, the probabilities of being transmitted and reflected are both 1/ no matter how we orientate the polarizer. Thus for single probabilities we get the results of Eqs. (6) and (7). We make use of projective geometry to calculate the joint probabilities of a pair of dependent events. We suppose P ( to be the square of the projection of the modulus of the eigenstate in direction a onto direction b, i.e. 1 P ( = ( cos( a ) = cos ( a. As the moduli of the eigenstates in directions a 5

6 and b are equal, we get the same joint probability by projecting b onto a. The other three joint probabilities can be obtained with the same method. They all agree with the expectations of quantum mechanics. If we use conditional probability and Malus law, we will get the same results. But note that one must not think that measurement on one side can influence the result on the other side. For example, do not think that measurement on the left side will lead to the collapse of the quantum state of the photon on the right side and turns it into linearly polarized photon. Then on the basis of Malus law Eqs. (8) and (9) are obtained. We should understand in this way. If the photon on the left side can pass through a polarizer with the orientation of a, then the photon on the right side can certainly pass through a polarizer with the same orientation. In the case that the photon on the left side can pass through a polarizer with the orientation of a, the probability that the photon on the right side can pass through a polarizer with the orientation of b is cos ( a. Note that only for a pair of circularly polarized photons with correlated hidden variables can we use this projective method. For a pair of uncorrelated photons, we have P ( = ( a) P ( = 1/ 4. P As for the Bell state composed of a circularly polarized photon and a linearly polarized photon, we suppose the former is incident upon polarizer a and the latter upon polarizer b. We first project a onto b. Since P ( a) = 1/ and the angle between the orientations of the 1 two polarizers is a b, we have P ( a, = cos ( a. We then project b onto a. Let the angle between the polarization direction of photon and the orientation of polarizer b is x. Then according to Malus law, we have ( = cos ( b x) cos ( a dx = P ( = cos 1 cos ( b x). Suppose the polarization directions of all the photons distribute uniformly in space, the joint probability is P 1 π π 0 ( a. (10) If the polarization directions of photons distribute only in two orthogonal directions, we have P ( = cos x cos ( a sin x cos ( a = cos ( a, (11) which also agrees with the expectation of quantum mechanics. But the projective relation of two linearly polarized photons is much complicated and we cannot obtain the above result, which we will discuss in the following. 4. Proposed experiments of quantum measurement 4.1. Experimental test of the locality of the measurements of Bell states In terms of quantum entanglement, measurement on one particle of the EPR pair will lead to the collapse of the quantum state of the other. While according to hidden variable theory, this is not the case. For example, suppose ϕ state is composed of two circularly polarized photons. When we measure one photon with polarizer and turn it into linearly polarized photon, the other will instantaneously collapsed into linear polarization according to quantum entanglement. In terms of hidden variable theory, the other will remain circular polarization. Does this violate the conservation of angular momentum? If we only consider the system composed of a pair of photons, the angular momentum of the system is certainly not 6

7 conserved. In the measuring process, a third component the measuring device is involved. If the measuring device is included, the momentum and angular momentum of the system are still conserved. In order to discriminate between the two hypotheses, we must seek a material which can exhibit different effects when circularly and linearly polarized photons pass through it respectively. Here we make use of roto-optic effect (or Faraday effect) to distinguish between circularly and linearly polarized photons. This is because a linearly polarized photon can be regarded as the combination of left-handed and right-handed circularly polarized components. When it passes through a roto-material, the velocities of the two components are different according to Fresnel s roto-optic theory. Then there exists a phase shift between the two components. The polarization plane of the photon will rotate and the quantum state will change. As a circularly polarized photon passes through the roto-material, only a neglectable overall phase is added to the quantum state. The experimental setup is shown in Fig. 3, where I and II are a pair of polarizers with the same orientation, and Ro is a roto-material which rotates the polarization plane of linearly polarized photon by π /. Circularized polarized φ state photon pairs are generated from source of SPDC. Let the distance between source S and Ro be larger than that between S and polarizer I (L>L1). Then the leftwards-traveling photon will first be analyzed. Co is an optical path length compensator which is used to guarantee the simultaneous detection of a pair of photons within the coincidence window of counters D1 and D. If roto-material is a Faraday rotator, then the compensator can be used with another same one that is power-off. D1 Co I L1 S L Ro II D Fig. 3. Experimental test of the locality of the collapse of polarization Bell state. We now see the expectations of the two theories. According to quantum entanglement, when the leftwards-traveling photon passes through polarizer I, the polarization direction of the rightwards-traveling photon will instantaneously collapse to the orientation of polarizer I. Its polarization plane is then rotated by π / when it passes through Ro. Thus it will be reflected by polarizer II. If the leftwards-traveling photon is reflected by polarizer I, the coincidence rate is zero whatever the rightwards-traveling photon is transmitted or reflected. So the expectation of coincidence rate is zero in terms of quantum entanglement. According to hidden variable theory, measurement on one photon does not influence the other. On the other hand, roto-material does not change the quantum state of circularly polarized photon. So the coincidence counting rate will remain unchanged and is always 1/. If ϕ state is composed of a circularly and a linearly polarized photons, and suppose that they are emitted uniformly into both sides, the coincidence rate will be 1/4. If hidden variable varies with time, as suggested by Hess and Philipp [11,1], the coincidence rate will vary with the position of the polarizer on the right side. Thus we can distinguish whether a pair of φ state photons can influence each other. Similar experiments can be made for the other three Bell states. 7

8 4.. Experimental test of the constituents of Bell states We have supposed in the above that Bell states can be composed of circularly or linearly polarized photon pairs. To verify this assumption, we use a pair of linearly polarized photons generated from type-i (II) non-collinear SPDC source. Since the two photons are generated from a same photon, the hidden variables of the two daughter photons are correlated. So polarization Bell state is easy to obtain by converting one (or two) photon into circularly polarized state, which can be realized by inserting one (or two) quarter-wave plate (QWP) into one (or two) optical path. For example, ϕ state can be generated by inserting two quarter-wave plates into the optical paths of type I non-collinear SPDC source, as in Fig. 4. If type-ii collinear SPDC source is used, a polarizing beam splitter (PBS) may be adopted to separate the two orthogonal polarized photons. Then ψ state can be obtained with one (or two) QWP preceding the polarizer. Pump laser NC QWP Pol IF D1 Coinc counter Type-I Fig. 4. Generation of φ QWP Pol IF D state from Type-I down-conversation. Although Bennett et al have suggested quantum nonlocality without entanglement and Pryde et al have demonstrated this phenomenon by a specific parameter estimation experiment [3,33], the above experiment will explicitly show that entangled state is not mysterious, it is only quantum state with correlated hidden variables. The following experiment uses the exchange effect of a pair of orthogonally polarized photons at beam splitter to obtain ψ state. The experimental setup is shown in Fig. 5. A beam of linearly polarized laser enters Mach-Zehnder interferometer (MZI), and a half-wave plate (HWP) is inserted into one arm of MZI to rotate the polarization plane by π /. BS is 50/50 beam splitter. A pair of linearly polarized photons exchanges angular momentum at the output port. If the relative phase of the two photons in the two arms is correctly chosen, the output states exiting from the two output ports are circularly polarized states. Additionally, the hidden variables of the two photons will be correlated due to the exchange of angular momentum. In order to obtain state, a glass plate may be inserted into the other arm or we can scan one of the mirrors of MZI to change the relative phase of the two photons in the two arms. The advantages of the experiment are that Bell state can be obtained without SPDC and interference filters while the intensity of photon pairs can be arbitrary bright. If we replace the first BS with a PBS, the HWP can be removed, and the polarization of pump laser should be set to ± 45. o ± ψ 8

9 M Laser BS HWP BS D M 1 Pol Pol Fig. 5. ψ state obtained by the exchange effect of a pair of photons in beam splitter Experimental test of determinism in quantum measurement If quantum measurement is deterministic, then experimental result is determined by measuring condition and hidden variable, and there are no random disturbances during the measuring process. We may further infer that the collapsed quantum states of a pair of particles with a same quantum state must be the same under the same measuring condition. We now verify this assumption by experiment in Fig. 6. We add another pair of polarizers in Fig.. The orientations of polarizers I and I are the same, and this also applies to the polarizers II and II. The source generated circularly polarized φ state photon pairs. According to Eq. (7), as the photon pairs are detected by the first pair of polarizers, we have P = 1/. Half of the photon pairs will pass through the first pair of polarizers and reach the second pair of polarizers. When they are analyzed by the second pair of polarizers, their behaviors will still be the same, i.e. if one is transmitted, the other will also be transmitted. Thus for the second pair of polarizers, we have P = cos θ, P = sin θ, P = 0, D1 = P where θ is the angle between the orientations of the two pairs of polarizers. According to quantum mechanics, the two photons are no longer entangled after the first measurement. Since their subsequent experimental results are mutually independent, or if the result of quantum measurement is random, the probabilities of and will not be zero. Thus P P there exists conceptual difficulty for quantum mechanics to explain the total correlation of a pair of particles without entanglement, which can be readily understood in deterministic hidden variable theory. D1 D3 D4 S D II I I II Fig. 6. Two successive polarization measurements on EPR photon pairs. 9

10 As the collapsed quantum states of a pair of photons after the first measurement are the same, they can be restored into state by inserting two quarter-wave plates into the optical paths between the two pairs of polarizers. If only one quarter-wave plate is inserted, φ state will be obtained. φ We now see the coincidence counting result when the orientations of the second pair of polarizers are different. Suppose the orientation of the first pair of polarizers is in the x axis, and the orientations of the second pair of polarizers in the directions of a and b, respectively. Let a, b and x lie in one plane. a and b are the directions perpendicular to a and b, respectively, as shown in Fig. 7. a b a θ b x Fig. 7. Orientations of the two pairs of polarizers. As the collapsed states of a pair of photons are the same after the first measurement, their subsequent outcomes are still correlated and we can use projective method as stated above but with a bit of difference. In the case of a pair of circularly polarized photons, the single probabilities P (a) and P ( are equal, we can get the same joint probability whether by projecting a onto b or by projecting b onto a. In the case of a pair of linearly polarized photons, the single probabilities that the two photons pass through the second pair of polarizers respectively are not equal. Then different projective sequences will lead to different results. If we project a onto b, we get P ( a, = cos acos P ( θ, where θ = a b. If we project b onto a, we obtain P ( = cos bcos θ. As joint probability cannot be greater than single probabilities, and the latter may not satisfy this requirement, we choose P ( a,b = cos acos ) θ for the moment. We now consider the expression of P (. According to the rule of projecting from one channel with a smaller probability onto the other with a larger probability, we obtain P ( = cos asin θ in the case of cos a sin b and P ( = sin bsin θ for cos a sin b. As the requirement of P ( a, P ( a, = P ( a) = cos a must be satisfied, and considering the smooth joining of the expression of probability, we take cos a cos θ cos a sin b P ( =. (1) cos a sin b sin θ cos a sin b It can be verified that in addition to satisfying the projective relation in the instances of θ = 0 and θ = π /, Eq. (1) also meets the expectations of P ( = cos a for b = 0 and P ( = 0 for a = π /. So it is a reasonable probability formula. With the expression of P ( we can calculate the other three joint probabilities P (, P ( and P ( using the relations of P ( P ( = cos a, P ( P ( = cos b 10

11 and P ( P ( = sin b. As the single probabilities that the two photons pass through the second pair of polarizers respectively are not equal, a and b are in the asymmetric situations. So there may exist another projective relation. In the case of Fig. 7, when b rotates between 0 and a, the joint probability ( may remain unchanged and is always equal to cos a, i.e. joint P P P P P P probability takes the smaller one of the two single probabilities. This implies that for two dependent events under certain conditions (for example, a and b both lie in the same quadrants), if one event with a smaller probability occurs, then another event with a larger probability will occur with certainty, just like the instance of cats passing through holes we have cited in the above. The four joint probability expressions can be written as i.e. ( = cos ( = 0 ( = cos ( = sin a b cos ( P ( a) = P ( = cos b =. (13) a It can be seen that in the instance of θ = 0 we get the same joint probability as Eq. (1), a. In other cases, we cannot determine whether Eq. (1) or (13) is correct, which can only be verified by experiment. But for a deterministic measurement theory, the requirement that joint probability equals the single probabilities must be satisfied in the case of θ = 0. If we suppose the polarization directions (the x axis in Fig. 7) of photon pairs distribute uniformly in space, and then average over polarization direction to get joint probability, we find that whether using Eq. (1) or (13) the result will not agree with the expectation of quantum mechanics. We do not present the detailed calculation process. So a pair of linearly polarized photons cannot form a Bell state. 5. Conclusion Compared with the assumptions of locality and realism, the reliability of Bell s joint probability distribution assumption is much weak. There is no prior reason to support this assumption. Our further analysis shows that this assumption holds only for two independent events but not for dependent events, such as the joint measurement of a pair of EPR particles. This is the reason that Bell s inequalities conflicts with quantum mechanics. The results of Bell-type experiments can be explained with the projective relation of the quantum states composed of circularly or linearly polarized photons. So far there is no experiment suggested to distinguish between the locality and non-locality assumptions. Our first experiment is aimed for this purpose, which uses roto-optic effect to distinguish between circularly and linearly polarized photons, and delayed measurement is made on one photon of EPR pair to test whether measurement on the other could influence it. The second and third experiments are used to verify the hypothesis that two circularly polarized photons or a circularly polarized and a linearly polarized photons can make up of a polarization Bell state. The last experiment tests determinism, which shows that measurement outcome is determined by the intrinsic property of the particle and measuring condition, and random disturbances do not exist in quantum measurement. Whatever the 11

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