EPR Paradox and Bell s Inequality

Transcription

1 EPR Paradox and Bell s Inequality James Cross

2 1 Introduction The field of quantum mechanics is practically synonymous with modern physics. The basics of quantum theory are taught in every introductory level physics class, and its applications can be seen in the fields of chemistry, mathematics, and even biology. Quantum theory has redefined how we look at the universe around, and it has caused us to view every day phenomena with new understanding. While quantum theory and its applications are accepted by any practicing scientist during the 21st century, that was not always the case. In the first half of the 20th century, there was an intense debate among the physics community on this new and bizarre quantum theory that redefined our understanding of how the universe works. In this paper we will outline the thought experiment made by physicists Einstein, Podolsky, and Rosen, where in their paper Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? We will then examine how experimental data and Bell s Inequality were used to prove that quantum theory is, in fact, complete. In this paper, the terms quantum, quantum theory, and quantum mechanics will be used interchangeably. 2 EPR Paradox The EPR Paradox is the name given to the conclusion of the thought experiment given in the 1935 paper posted by physicists Einstein, Podolsky, and Rose (denoted EPR for the rest of this paper) in their paper Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? In this paper, EPR argued that quantum mechanics, though a successful description of nature, could not represent a final, complete theory. They asserted that in a complete theory, there is an element corresponding to each element of reality (cite). EPR gave the following condition of completeness: A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. To illustrate this, they presented a thought experiment to show that quantum mechanics cannot be a complete description of reality. 2.1 EPR Thought Experiment The following is an outline of their thought experiment. Suppose some event occurs on the origin of the x axis of some arbitrary number line (Figure 1) and emits two identical particles I 1 and I 2. I 1 travels in the positive x-direction with momentum p 1 while I 2 travels in the negative x-direction with momentum p 2 (Figure 1). Figure 1: 1

3 2.2 Uncertainty If the total momentum of the system was initially p = 0, then by the conservation of momentum we get that (1) p 1 = p 2, and since the particles have identical mass, it follows that (2) x 1 = x 2 (except at t = 0). From the uncertainty principle of quantum mechanics, we have the following equation (3) x 1 p 1 h where h is Planc s constant Within quantum theory, it is asserted that measuring an object collapses its wave function, driving the uncertainty of the property to zero. Thus, measurement of an object directly affects the qualities of the object. By measuring either the position or momenta of the particle, the other quantity is changed. 2.3 Measurements As stated previously, within quantum mechanics, measurements disturb the object being measured. If position is measured on a particle, then a measurement of its momentum would yield a difference result than if momentum were measured first. Within the proposed experiment, it is possible that measuring the position of I 2 could cause I 2 to send out information to also collapse the wavefunction of I 1 so that they remain identical particles. However, we can make these particles arbitrarily far apart before we measure them. Since information can only travel as fast as the speed of light, it is possible to out them in a positioon where both measurements can be made before I 1 can collapse the wave function of I 2. Suppose a measurement was made on I 2 to determine its position x 2. Then due to equation (2), we can predict x 1 with certainty without having disturbed I 1, so x 1 becomes real. The same argument follows for predicting p 1 from p 2. In quantum, measurements are represented by applying operators. By uncertainty, the position and momentum measurements are not communitive, and thus equal some non-zero number: 1 ]=i h, 2 ]=i h However, the operators for the position of one particle and the momentum of the other do commute, so their operator is zero. 2 ]=0 1 ]=0 Let ˆx = xˆ 1 xˆ 2 be the difference of the position operators and let ˆp = pˆ 1 pˆ 2 be the sum of the momentum operators. We can work out the commutator of ˆx and ˆp to find that it is zero. [ ˆx, ˆp]= 1 ]- 2 ]+ 2 ]- 1 ]=i h-i h+0-0=0 [5] 2

4 2.4 Outcome Since the commutator zero, then both the position and momentum can be determined. This, however, is a contradiction with the uncertainty principle of quantum mechanics. EPR concludes that either quantum theory is incomplete, or that that I 1 can instantaneously send information to I2, which would require a complete overhaul of the understanding of the universe. This is the EPR Paradox. 3 Bell s Inequality For the following thirty years after the EPR Paradox was published, quantum theory stood as a useful descriptor of the universe, but one that heavily debated on its completeness. This changed in 1964 when John Bell published a paper titled On the Einstein Podolsky Rosen Paradox.The essence of Bells theorem is that quantum mechanical probabilities cannot arise from the ignorance of local pre-existing variables. In other words, if we want to assign pre-existing (but hidden) properties to explain probabilities in quantum measurements, these properties must be non-local. This non-locality is of the worst possible kind: an agent with access to the non-local variables would be able to transmit information instantly to a distant location, thus violating relativistic causality, or in Einstein s words, spooky action at a distance. 3.1 Locality and Reality Let us define a local theory as a one where the outcomes of an experiment on a system are independent of the actions performed on a different system which has no causal connection with the first. For example, the temperature of this room is independent on whether I choose to wear purple socks today. Einsteins relativity provides a stringent condition for causal connections: if two events are outside their respective light cones, there cannot be any causal connection among them. [2] Let us define a real theory as one whose experiments uncover properties that are preexisting. In other words, in a real theory it is meaningful to assign a property to a system (e.g. the position of a particle) independently of whether the measurement of such property 2 is carried out. [2] Modern formulations of quantum mechanics must incorporate Bells result at their core: either they refuse the idea that measurements uncover preexisting properties, or they must make use of non-local properties. In the latter case, they must also introduce some censorship mechanism to prevent the use of hidden variables to transmit information. An example of the first formulation is the conventional Copenhagen interpretation of quantum mechanics, which states that the properties arise from the interaction between the quantum system and the measurement apparatus, they are not pre-existing, i.e. unperformed experiments have no results. An example of the second formulation is the de Broglie-Bohm interpretation of quantum mechanics that assumes that particle trajectories are hidden variables (they exist independently of position measurements). 3

5 a b c a b b c a c Table 1: Possible outcomes In its simplest form, Bell s theorem states that quantum mechanics is either non-local or non-real, it cannot be both. To show this, Bell derived a basic inequality that satisfied all local and real theories. He then showed that all experiments in quantum mechanics violate this inequality, and therefore quantum cannot be both. Below we will show Bell s inequality and then how it leads to this conclusion. 3.2 Bell s Inequality Suppose we take a particle in state X, and subject it to an experiment with two possible outcomes: pass and fail (call it test a). We also have a second test b, and a third test c (Figure 2). Three identical particles (with some state X)are then put through the three tests, with the first particle going to test a, the second the test b, and the third to test c. There are 8 possible outcomes of these three tests. Using 0 and 1 to represent failure and pass respectively, we can make Table 1 to show all possible outcomes. Under lo- Figure 2: cality, we assume that a given state X corresponds to one and only one row in this table. From the three tests, we now look for three specific outcomes: pass a and fail b (a b),pass b and fail c (b c), and pass a and fail c (a c). These outcomes are denoted in Table 1 by the + symbol. Notice that whenever a c is true, either a b or b c must also be true. However, there are scenarios in which both b c and a b can occur independently. From this observation, we derive the following formula: 4

6 (3) P(a b)+p(b c) P(a c)[2] That is, the probability of a c occuring is always less than or equal to the probability of a b occuring plus the probablity of b c occuring. This is Bell s Inequality, which we will use to prove Bell s Theorum. 3.3 Bell s Theorum To prove Bell s Theorum, we now provide a quantum system which violates the above inequality. Suppose that some event hapens at the orgin and two particles (α 1 and α 2 ) in a singlet state (entangled) were sent out in opposite directions (much like in Figure 1). Due to their entanglement, α 1 will always have opposite spin of α 2. Let us assume that the spin of these particles is due to some inherient charactaristc or state of the particle, that is, each particle has difinite spin before it is detected. Now set up two detectors so that the spin of both particles can be measured. Depending on the orientation of the particles spin relative to a detector it will be deflected up or down. Its paired twin will have opposite spin and be deflected in the opposite direction. The particles are spin up or spin down. Let us consider the following three scenerios: 1. Particles detectable with spin up at 0 but not at Particles detectable with spin up at 45 but not at Particles detectable with spin up at 0 but not at 90 One can determine if a particle is not detectable with spin up at a given angle by looking at the particle it is paired with. If a particle is detectable with spin up at some angle then the particle it is paired with cannot be because they have opposite spin. Count the detections of paired particles α 1 and α 2 that fall into the following three categories in which all detections are with spin up. 1. α 1 detectable at 0 and its pair α 2 at 45 (a b) 2. α 1 detectable at 45 and its pair α 2 at 90 (b c) 3. α 1 detectable at 0 and its pair α 2 at 90 (a c) These three categories fit the constraints of (3). From the literature (cite), it has been determined that quantum can predict the probability that two particles will be detected at spin up with an angle φ between the detectors as follows. [3] (4) 1 2 (sin(φ 2 ))2 5

7 Substituting in (3), we get Which evaluates as 1 2 (sin(45 2 )) (sin(45 2 ))2 1 2 (sin(90 2 )) which violates Bell s Inequality. 4 Conclusions This proves Bells theorem: all local real theories must satisfy Bell s inequality (3) which is violated by quantum mechanics. Then, quantum mechanics cannot be a local counterfactual theory: it must either be nonreal (as in the Copenhagen interpretation) or non-local (as in the de Broglie-Bohm interpretation). References [1] A. Einstein, B. Podolsky, N. Rosen. Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev, 47, 777, [2] J. S. Bell. On the Einstein Podolsky Rosen Paradox,. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys, 38, 447, [3] A. Peres. Unperformed experiments have no results. Am. J. Phys, 46, 745, [4] A. Aspect, P. Grangier, G. Rodger. Experimental Realization of EinsteinPodolsky- Rosen-Bohm Gedankenexperiment: A New Violation of Bells Inequalities. Phys. Rev. Lett, 49, 91, [5] G. Ghirardi. On a recent proof of nonlocality without inequalities. Found. Phys, 41, 1309,