Perspective Chapter: On the Contradiction between Special Relativity and Quantum Entanglement

Abstract

Demonstrations of quantum entanglement (QE), which confirm the violation of Bell’s inequality, indicate that under certain conditions action at a distance is possible. This consequence seems to contradict the relativistic principle of causality, which asserts that an effect never precedes its cause, in any reference frame. By analyzing a numerical example of Bell’s experiment with entangled pairs of photons, we show how observers in two inertial reference frames can disagree about the causality relation between two events. One observer claims that event 1 is the cause of event 2, while the other claims that event 1 is the result of event 2. The solution we suggest to the paradox is that in entangled systems, one can find pairs of “entangled events” which have symmetrical causality relations. Each of the events can serve as a cause or as an effect, depending on the frame of reference in which they are observed.

1. Introduction

Recently, several independent “loophole-free Bell violation” experiments were reported (e.g. [1, 2, 3]). These works, as well as previous demonstrations of quantum entanglement (QE) since the pioneering work of Freedman and Clauser [4], have shown that under certain conditions, action at a distance between entangled systems is possible. A straightforward interpretation of the results is that faster-than-light (FTL) or superluminal communication is feasible. This consequence may seem to contradict an established tenet of Special Relativity (SR) that claims that FTL transfer of information violates causality and is therefore impossible. The purpose of this chapter was to examine in detail where the conflict arises and to propose a possible solution to the paradox.

In fact, the paradox was already hidden in the Einstein-Podosky-Rosen (EPR) thought experiment [5]. If two identical particles, A and B, move in opposite directions after a brief interaction which ensured that v_B = −v_A, and if at a specific moment, t₀, when A and B are far apart, the position of A and the momentum of B are measured, one can know both the position and the momentum of A at t₀, since p_A = −p_B. The authors claimed that this thought experiment contradicted Heisenberg’s uncertainty principle, and consequently that quantum mechanics (QM) is incomplete.

In addition to this explicit criticism against QM, another claim can be based on the EPR thought experiment, concerning an inconsistency between QM and SR. Suppose we perform a very accurate measurement of the magnitude of the momentum of A and find that it is p_A. According to the uncertainty principle, the measurement changes the state of A so that the uncertainty in its location becomes infinite, or very large (since Δp×Δx ≈ ħ). But since A and B are entangled, the measurement should change the state of B as well. If we now measure p_B we should certainly get p_B = − p_A while the uncertainty in the location of B becomes infinite, or very large. This is a substantial change in the wave function Ψ_B, which prior to the measurement on A, could be presented by a wave packet in which both Δx_B and Δp_B were finite.

The question is, how is it possible that an operation made on A instantly influences the state of B which, in principle, could be thousands of kilometers away from A. The effect of the measurement on A on the state of B is immediate, since the time interval Δt between the measurements on A and on B can be arbitrarily short. It seems to contradict SR which claims that no interaction can travel faster than the speed of light.

2. The EPR paradox in the spin version

An alternative version of the EPR paradox is based on measuring the spin direction, instead of momentum and location [6]. In this version, A and B are particles with spin 1/2, which were created with opposite spins (e.g., by the decay of a particle with spin 0). They arrive at two detectors D_A and D_B where their spin directions are measured with respect to an arbitrary z-axis. Due to the conservation of angular momentum, if the spin of A is found to be positive (|+1/2⟩), then the spin of B must be negative (|–1/2⟩), and vice versa. This is truly unrelated to the distance that separates the particles at the moment of measurement.

If we repeat the measurement but measure both spins along the x-axis (perpendicularly to the z-axis), the result will be the same. If the spin of A is found to be +1/2, the measured spin of B will be –1/2, and vice versa. The results can be explained in two different ways.

1. The particles are entangled in such a way that when one spin is measured, the other spin becomes its opposite. The quantum state of the system, which could be a combination of |+1/2⟩ and |–1/2⟩ for both particles prior to the measurement, collapses into |+1/2⟩ for particle A and |–1/2⟩ for particle B (or vice versa) because of the measurement. This is the QM or the Copenhagen explanation.

2. The two particles were created with definite (opposite) spins around any axis in space we may choose. This is “the hidden variable” explanation.

Both explanations seem to be problematic. In order to illustrate the problem with the first explanation, let us assume that particle A arrives at the detector D_A and after a short time interval, Δt, particle B arrives at the detector D_B. The information about the measured spin of A should travel from D_A to D_B faster than the speed of light, since the distance between them can be many kilometers, while Δt can be arbitrarily small. This seems to violate SR.

According to the second explanation, no information is transferred from one detector to the other. However, we have to assume that each of the two particles is created with definite eigenvalues of both S_z and S_x. But, according to QM, two perpendicular components of the angular momentum cannot be simultaneously in defined states. If a particle has a definite spin direction relative to the z-axis, its spin direction on the x-axis should be a superposition of |+1/2⟩ and |–1/2⟩ so that a measurement can give each state with equal probabilities. If the information about the results of potential measurements of the spin along any arbitrary axis exists prior to the measurement, then that information must somehow be concealed. Hence, this model was dubbed “hidden variables.”

3. Bell’s experimental setup for photons

Until the article of John Stewart Bell [7], no experiment was made in order to find out which of the two explanations is true, since they were thought to be indistinguishable in terms of experimental results. Bell pointed out that the two explanations predict the same results if D_A and D_B are oriented in the same direction in space. However, the results predicted by the two explanations may differ, if the detectors measure the spins in different directions. Bell adopted the spin example advocated by Bohm and Aharonov [6], but since most experiments were made with photons (where the polarization of the photon substitutes the spin component), it will be more convenient in the ensuing discussion to present Bell’s thought experiment with photons. The following scheme, depicted in Figure 1, is a simplified version of the two-channel experimental setup first employed by Aspect et al. [8].

Figure 1.

The setup for Bell’s experiment with photons.

A source (S) emits two photons, A and B, which travel in opposite directions and have the same polarization state. The photons reach two detectors (e.g., sensitive photomultipliers), D_A and D_B, which are able to detect single photons. In front of each detector, there is a polarizer (P_A and P_B). If the photon passes the polarizer and arrives at the detector, the event is registered as 1. If the photon is stopped by the polarizer and does not reach the detector, the event is recorded as 0.

Let us assume first that the axes of both polarizers are oriented in the z direction, as shown in Figure 1. Since the two photons were emitted with the same polarization, we can assume that there will be perfect correspondence between the results recorded by the two detectors: if one detector records the series: 1, 0, 0, 1, 0, 1..., the second detector will record exactly the same series. On the other hand, if polarizer P_A is oriented in the z direction while polarizer P_B is oriented in the x direction, there will be a perfect mismatch between the results: if D_A records 1 then D_B will record 0, and vice versa.

Suppose that there is an angle θ between the two polarizers. We define a “matching function” F(θ), which is the ratio of the number of matches between the two detectors to the total number of readings, in a long series of measurements. We also define a “mismatch function” E(θ) as the percentage of mismatches between the two detectors. It’s easy to see that:

$$E\left(\theta \right)=1–F\left(\theta \right)\left(\text{for}\mathrm{any}\theta \right)$$E1

$$F\left(\theta =0^0\right)=1;E\left(\theta =0^0\right)=0$$E2

$$F\left(\theta ={90}^0\right)=0;E\left(\theta ={90}^0\right)=1$$E3

Let us consider the case of θ = 0⁰. The correlation between D_A and D_B (Eq. (2)) comes as no surprise if the initial polarization of the pair of photons is parallel to the z-axis (|z⟩) or to the x-axis (|x⟩). However, we expect to get the same correlation, even when the initial polarization of the photons forms an arbitrary angle ϕ (0⁰ < ϕ < 90⁰) with the z-axis. In this case, the initial polarization prior to the measurement can be considered as a superposition of |z⟩ and |x⟩, and the measurement actually causes the collapse of the wave function of each photon to one of the eigenstates, |z⟩ or |x⟩. As mentioned above, the fact that in two remote locations, the collapse is to the same eigenstate can be explained in two alternative ways: the QM interpretation (which involves an immediate action at a distance) or the hidden-variable model.

4. Bell’s inequality and Bell’s theorem

Let us explore the following four-stage thought experiment, which will lead us to Bell’s Inequality.

Stage 1: P_A and P_B both point in the z direction (θ = 0), as shown in Figure 2-1. Two entangled photons are emitted from the source. Photon A arrives at P_A while photon B arrives at P_B. The readings of the two detectors will be the same all the time, therefore E(θ) = 0.

Figure 2.

A four-stage thought experiment that leads to Bell’s inequality.

Stage 2: P_B is rotated counterclockwise at an angle θ (0⁰ < θ < 45⁰, e.g., θ = 15⁰), as shown in Figure 2-2. Once more, two photons are emitted from the source. Let us assume that photon A passes through P_A and reaches D_A. The probability that photon B will reach D_B is no longer 100%; sometimes it will arrive at D_B and sometimes it will not. In some cases, photon A will be blocked while photon B arrives at D_B. Therefore, E(θ), which corresponds to the average mismatch between D_A and D_B, is no longer 0.

Stage 3: P_B is restored to its previous position (parallel to the z-axis) while P_A is rotated clockwise at the same angle θ, as shown in Figure 2-3. The situation is symmetrical to Figure 2-2, and we expect the average mismatch to be E(θ) as in Stage 2.

Stage 4: We leave P_A as in Stage 3, i.e., skewed at an angle θ clockwise. We rotate P_B anticlockwise at the same angle θ. The angle between the two polarizers is now 2θ, as shown in Figure 2-4. We repeat the series of measurements that were performed in the previous stages. The mismatch between D_A and D_B is now E(2θ).

We shall now see that the two models, which explain the correlation between the readings of D_A and the readings of D_B, provide different predictions about the relationship between E(2θ) and E(θ). Therefore, by measuring E(θ) and E(2θ) for various values of θ, it can be determined as to which of the two models is correct.

According to the hidden-variable model—that rejects action at a distance—there is no connection between the two polarizers. There is no way that polarizer P_A could sense the state of polarizer P_B. Therefore, the rotation of polarizer P_B between Stage 3 and Stage 4 cannot affect polarizer P_A. Consequently, at Stage 4, polarizer P_A will continue to pass or block photons with the same rate of mismatch which we would get if P_B stayed upright. In other words, at Stage 4, the mismatch between the series a (which shows the readings of D_A) and a hypothetical series c (which represents the results we would get from D_B at Stage 4, if P_B remained upright) is still E(θ).

By the same token, the mismatch between c and b (the actual readings of D_B at Stage 4) will also be E(θ). One might conclude that the mismatch between a and b will be the sum of the mismatch between a and c and the mismatch between b and c, namely, that E(2θ) = 2E(θ). However, this conclusion is too hasty. In some cases, an element in a will be the opposite of the corresponding element in c, and the same element in b will also be the opposite of the element in c. In this case, the elements in a and in b will be identical. Therefore, the average mismatch between a and b at Stage 4, namely E(2θ), can be 2E(θ) but can also be smaller than 2E(θ). We can write:

$$E\left(2\theta \right)\le 2E\left(\theta \right)\left[{\text{Bell}}^{’}\mathrm{s}\text{inequality}\right]$$E4

We can demonstrate the relation between E(2θ) and 2E(θ) by the following example. Let us assume that.

$$c=0,0,1,0,1,0,1,1,0,1,1,0$$E5

In order to create series a, we duplicate c but randomly change 4 of the 12 digits. To create series b, we once more duplicate c and randomly change 4 of the 12 digits. The mismatch between c and a as well as between c and b will be 1/3.

$$E\left(c,a\right)=E\left(c,b\right)=1/3$$E6

If the elements of c, which were changed to create b, were different from the elements of c which were changed to create a, the mismatch between a and b will be 2/3. However, if the same elements were changed in both cases, the mismatch between a and b will be 0. In the general case, we can write:

$$E\left(a,b\right)\le 2/3$$E7

In correspondence with Eq. (4). Eq. (4) is Bell’s inequality for the specific experimental procedure described above. According to Bell’s theorem, Eq. (4) would be verified experimentally if the hidden-variable model is true. On the other hand, if Bell’s inequality is violated, then the QM model is true.

5. E(θ) according to quantum mechanics

We can go on and evaluate E(θ) according to the traditional interpretation of QM. Let us assume that in Stage 2 of the thought experiment described in Section 4, photon A passes through P_A and reaches D_A. This means that the polarization of photon A is parallel to the z-axis. Since the two photons are entangled, this is also the polarization of photon B. The angle between the axes of P_A and P_B is θ. Therefore, the polarization of photon B is skewed relatively to P_B at an angle θ.

According to Malus’s law, when a polarized beam of light hits a perfect polarizer, the intensity of the light that passes through the polarizer is given by:

$$I=I_0{cos}^2\theta$$E8

where I₀ is the initial intensity and θ is the angle between the light’s direction of polarization and the axis of the polarizer.

When we regard the beam as a stream of photons, we can ascribe to each photon a defined direction of polarization which creates an angle θ with P_B. According to Eq. (8), out of N photons, approximately Ncos²θ will pass P_B and about Nsin²θ will be blocked. The probability that a single photon will reach D_A or D_B, while its companion will be blocked at the other detector is therefore sin²θ, and this will be the average mismatch rate in a long series of measurements¹. Thus, according to QM, at Stages 2 and 3:

$$E\left(\theta \right)={sin}^2\theta \left(\text{Stages}2\text{and}3\right)$$E9

In Stage 4, the angle between the polarizers is 2θ and, according to QM, the mismatch rate will be:

$$E\left(2\theta \right)={sin}^{2}2\theta$$E10

If, for example, θ = 30⁰, we shall get in Stages 2 and 3 a mismatch rate of:

$$E\left({30}^0\right)={sin}^2\left({30}^0\right)={0.5}^2=0.25$$E11

while in Stage 4 the mismatch rate will be:

$$E\left({60}^0\right)={sin}^2\left({60}^0\right)=0.75$$E12

In this case, E(2θ) > 2E(θ) in contrast to Bell’s inequality (Eq. (4)). Indeed, it’s easy to prove that E(2θ) > 2E(θ) for any θ in the range: 0⁰ < θ < 45⁰.

To prove this, we notice that:

$${sin}^2\left(2\theta \right)=4{sin}^2\left(\theta \right)\times {cos}^2\left(\theta \right)$$E13

In the range 0⁰ ≤ θ ≤ 45⁰, the function cos²(θ) is a monotonically descending function, which has a minimum at θ = 45⁰, where cos²(θ) = 0.5. At that point:

$${sin}^2\left(2\theta \right)={\text{4sin}}^2\left(\theta \right)\times 0.5=2{sin}^2\left(\theta \right)\left(\theta =45°\right)$$E14

namely, for θ = 45°, E(2θ) = 2E(θ). If 0⁰ < θ < 45° then cos(2θ) > 0.5 and sin²(2θ) > 2sin²(θ), which means that E(2θ) > 2E(θ), in contrast with Bell’s inequality.

Figure 3 depicts E(θ) according to the QM description (E(θ) = sin²θ) in the range 0⁰ ≤ θ ≤ 90⁰. On the straight line (described by E(θ) = θ/90⁰), the relation between E(θ) and E(2θ) is: E(2θ) = 2E(θ). One can see that in the range 0⁰ < θ < 45°, the function E(θ) = sin²θ (which represents QM’s results) lies below the straight line, while Bell’s inequality Eq. (4) is valid only for points that are above the straight line. Thus, by performing the experiment described in Section 4, one can find out whether the QM model or the hidden-variable model is correct.

Figure 3.

The points depict E(θ) according to quantum mechanics (QM) (E(θ) = sin²θ). On the straight line (described by E(θ) = θ/90°), E(2θ) = 2E(θ).

In a pioneering work of Freedman and Clauser [4], an experiment similar to the thought experiment described above was performed. They measured the polarization correlation of two entangled photons emitted in an atomic cascade of calcium. The wavelengths of the photons were 551.3 nm and 422.7 nm. The measurements were made at nine different angles in the range 0⁰ ≤ θ ≤ 90°. The results were in agreement with quantum mechanics and violated Bell’s inequality to a high statistical accuracy. Actually, the curve describing the results (Figure 3 in [4]) is similar to E(θ) = sin²θ in Figure 3 above, except for the values on the vertical axis which reflect the fact that the efficiency of the detectors was less than 100%. The authors considered the results as strong evidence against local hidden-variable theories.

Over the years, additional experiments demonstrated clearly the violation of Bell’s inequality. It was established that in experiments such as those described in the EPR and Bell’s papers, a measurement performed on one particle does affect the other particle which can be far away. The effect is immediate, indicating that a single unified wave function continues to describe the two particles even when they are far apart. Thus, a measurement performed on one particle causes the collapse of the wave function in the other one as well. In order to resolve the contradiction between this action at a distance and SR, we have to see why the contradiction arises in the first place.

6. Special relativity and action at a distance

Let S and S′ denote two inertial reference frames. S′ is moving with respect to S at a constant velocity υ along the x-axis. At time t = 0, the spatial axes and clocks of S and S′ coincide. Suppose that an event that occurred in S, at point x₁ in time t₁ creates a signal that travels at a velocity u. The signal arrives at point x₂ in time t₂ and creates a second event there (e.g., turning on a lamp). The coordinates and times of the two events in S′ are (x′₁, t′₁) and (x′₂, t′₂). According to the Lorentz transformation:

$$t_1^{\prime }=\gamma \left(t_1-\frac{\upsilon x_1}{c^2}\right);t_2^{\prime }=\gamma \left(t_2-\frac{\upsilon x_2}{c^2}\right)$$E15

Where γ = 1/(1–υ²/c²)^1/2. From (15), we get:

$$\mathrm{\Delta }{t}^{\prime }=\gamma \left(\mathrm{\Delta }t–\upsilon \mathrm{\Delta }x/c^2\right)$$E16

Where Δt′ = t′₂ – t′₁; Δt = t₂ – t₁; Δx = x₂ – x₁.

Since Δx = uΔt, we can write:

$$\mathrm{\Delta }{t}^{\prime }=\gamma \mathrm{\Delta }t\left(1–\mathit{u\upsilon }/c^2\right)$$E17

If the signal velocity u is greater than the speed of light (u > c), we can define a reference frame S′ that moves at a velocity υ which is smaller than c but close enough to c so that uυ /c² > 1 (in order for that to happen, υ should meet the condition c > v > c²/u). The expression (1 – uυ/c²) in Eq. (17) will be negative, and therefore if Δt is positive, Δt′ should be negative and vice versa. Thus, in S′, event 1 will occur after event 2 and an observer in S′ would see the effect precede its cause. For example, let u = 1.1c and Δx = c × 1 s while υ = 0.98c (γ = 5.025). With these values:

Δt = Δx/u = 0.9091 s.

Δt′ = 5.025 × 0.9091(1–0.98 × 1.1) = − 0.356 s.

The minus sign indicates that the order of events in S and S′ is reversed. According to an observer in S′, event 1 (which is the cause of event 2) occurred 0.356 s after event 2. The assumption that a signal can travel faster than the speed of light leads to a violation of the relativistic principle of causality, which asserts that an effect never precedes its cause, in any reference frame. This is why SR forbids instantaneous action at a distance as well as traveling of matter, energy, or information at speeds greater than the speed of light.

In order to demonstrate the inconsistency of Bell-like experiments with SR, let us return to Stage 1 in the experiment described in Section 4. Both polarizers P_A and P_B are oriented in the z direction (θ = 0), as shown in Figure 2-1. Thus, the readings of the two detectors are the same all the time and E(θ) = 0. Let us assume that D_A, D_B, and the source are on the x-axis of a rest frame S. The source is at x₀ = 0 and the coordinates of D_A and D_B are:

$$x_1=x\left(D_A\right)=15\mathrm{m};x_2=x\left(D_B\right)=-15.3\mathrm{m}$$E18

In order to facilitate the calculations, we redefine our unit of length so that c = 3 × 10⁸ m/s exactly. We denote by t₁ and t₂ the times of two events. Event 1: “Photon A reached P_A and then D_A registered 0 or 1,” Event 2: “Photon B reached P_B and then D_B registered 0 or 1.” It’s easy to see that:

$$t_1=50\mathrm{ns};t_2=51\mathrm{ns}$$E19

Before checking the times in another frame, S′, let us discuss the following question: Can an observer in S consider event 2 as the result of event 1? I claim that she can. The two events can be considered a combination of cause and effect for the following reasons.

1. In frame S, event 1 precedes event 2 by 1 ns.

2. Prior to the occurrence of event 1, the reading of D_B could be either 0 or 1 in equal probabilities. After event 1 takes place, the reading of D_B is definitely determined.

3. If D_A was removed after the photons left the source, and before they reached the polarizers, the reading of D_B could be either 0 or 1 in equal probabilities. The fact that D_A operated and registered the arriving photon influenced the reading of D_B.

Therefore, event 2 can be considered the result of event 1. An argument against this claim is that the two events are separated by a space-like interval. According to SR, only if two events are separated by a time-like or light-like interval can one event influence the other. However, it can be claimed that the concept of causality is metaphysically prior to the relativistic restrictions. Actually, it was argued that many standard philosophical theories would treat the relationship between such two events as causal, despite the contradiction with SR [9].

Let us assume a second frame S′ which is moving with respect to S in the negative direction of the x-axis at a velocity υ = −0.6c. Let us assume that when the two photons are ejected from the source, the clocks at S and S′ show t = t′ = 0, and the origins coincide. By using the Lorentz transformation (Eq. 15), we find:

$$t_1^{\prime }=100\mathrm{ns};t_2^{\prime }=25.5\mathrm{ns}$$E20

Thus, in S′ event 2 occurs before event 1, although we defined event 1 as the cause of event 2. Actually, since the velocity of the signal which carries the information between D_A and D_B is infinite, the paradox appears for υ as small as |υ| ≈ 0.01c for the numerical values of the example above. In general, it occurs whenever:

$$-\upsilon \ge \frac{t_2-t_1}{x_1-x_2}{c}^2$$E21

7. Solutions to the paradox

Several solutions can be offered for the contradiction, which was demonstrated in the previous section between SR and Bell-like experiments. Ballentine and Jarrett [10] suggested a distinction between a “strong” locality principle and a “weak” one that is needed to satisfy the demands of relativity. They claimed that QM satisfies the latter and therefore there is no contradiction between QM and SR. Instead, one can argue that we do not have two separate events here but only one spatially extended but indivisible event which is “the collapse of the wave function which represents the polarization of the two photons” ([9], p. 41). Another alternative is to formulate a theory of causation which requires some conditions which two events need to fulfill in order to represent a cause-and-effect relationship, and then show that these conditions are not realized here ([9], p. 42).

The solution, which I suggest to the paradox, is based on the following principles:

1. Event 1 and event 2, in the example discussed in Section 6, are two distinct and separate events that occur at different points in space–time.

2. There is a cause-and-effect connection between the two events.

3. According to an observer in S, event 2 is a result of event 1.

4. According to an observer in S′, on the other hand, event 1 is a result of event 2.

5. The disagreement between the two observers does not violate the causality principle of SR, since in this particular case the cause-and-effect relationship between the two events is symmetrical: each of them can be regarded as a result of the other, depending on the frame of reference in which they are observed.

Usually, when there is a causal relationship between two remote events, they are physically different. That’s why the effect cannot precede its cause in any reference frame. For example, if event 1 is the ejection of a signal from point (x₁, y₁, z₁, t₁) and event 2 is the arrival of the signal to point (x₂, y₂, z₂, t₂) where it turns on a lamp, we demand that, in any reference frame, t₂ > t₁. This demand is fulfilled only if the velocity of the signal does not exceed the speed of light, as shown in Section 6.

However, in Bell-like experiments, like the one described in Section 4, there is no physical difference between the two events: they are totally symmetrical. Each of them can serve as a cause or as an effect, depending on the frame of reference in which they are observed. If event 1 is observed before event 2, event 1 is the cause and event 2 is the result. If the order of times is reversed, then event 2 is the cause and event 1 is the result.

It is customary to think that the causal relation between two events can be one of the four types:

1. Event 2 is a result of event 1.

2. Event 1 is a result of event 2.

3. Both events have a common cause.

4. There is no causal relation between the two events.

The analysis of the ostensible contradiction between Bell’s theorem and SR indicates that there is an additional possibility. In entangled systems, one can find pairs of “entangled events” which have symmetrical cause-and-effect relations. Each of them can appear to be the cause of the other, depending on the frame of reference in which they are observed. This fifth possibility solves the paradox which the action at a distance creates. Experimental results (e.g., [11]) can be interpreted as supporting this suggestion.