EPR Paradox and Bell’s theorem

EPR Paradox and Bell’s theorem
Kim min-gi Hello everyone, my name is Kim min gi and I’m going to describe about the Einstein-Podolsky-Rogen paradox, simply EPR paradox, Bell’s theorem, and some specific examples.

Index Appearance of Quantum Mechanics
Two different interpretations of ψ Doubts to Copenhagen interpretation Bell’s theorem Contrary to Q.M prediction Bell test experiment Non-locality Index is as follow. I first will explain the background about the EPR paradox, why it came and what they state. And I’ll gonna introduce Bell’s theorem which includes Bell’s inequality related to EPR experiment, and some actual experiments. Finally I’ll talk about the non-locality very briefly.

Appearance of Quantum Mechanics
Before 20th century • Classical Mechanics - Absolute space & time - Matter = particle - Light = wave Young’s double-slit experiment Before 20th century, there was no quantum or relativity theory at all, only exists classical mechanics. People believed the space and time are absolute value, matter is particles, and light is a wave. / of course, there was an argument about the nature of light whether it’s wave or particle, / but through Young’s double-slit experiment, they thought light is just a wave. Isaac Newton Thomas Young Particle Wave

20th century • Einstein’s photoelectric effect experiment(1905) - Duality of light • De Broglie’s matter wave(1924) - Duality of matter • Stern-Gerlach experiment(1922) - Spin • Heisenberg’s uncertainty principle(1927) • Schrodinger’s wave equation(1926) But in 20th century, many researches about quantum phenomena were studied. Albert Einstein proved duality of light with the photoelectric effect in 1905, De Broglie had introduced the concept of matter wave in 1924 so that duality of matter could be interpreted as a true nature of matter. Also in 1922, Stern-Gerlach experiment discovered the presence of spin which had no analogy with previous classical theory, and in 1927 Heisenberg presented the uncertainty principle which has a very important rule in quantum theory. Obviously there could be many other discoveries related to quantum theory, among those discoveries, seemed the most important thing was the constructing quantum wave equation by Schrodinger in 1926.

Schrodinger’s wave equation Great correspondence As all you well know, this is the time-dependent Schrodinger’s wave equation which can describe particle’s position, momentum or energy, etc. When Schrodinger exposes this equation to world, a lot of scientists tried to match this equation with experimental data, and they figured out there was a great correspondence with them. For example, this is the position probability density of electron in atom expected by the equation, and this is well matched with real data.

Two different interpretations of ψ
Copenhagen interpretation • ψ is a wave of ‘probability’ • Named from the place ‘Copenhagen’ where was a middle of argument • Niels Bohr, Max Born, Heisenberg P(r) r 1 Even if there was very few scientists who protest to Schrodinger’s wave equation itself due to the high level of correspondence, there was a certain difference of opinion about the interpretation of wave function psi. As a consequence of this, people who supported quantum theory were roughly divided by two group. One of the group supported the Copenhagen interpretation. This is an interpretation which ψ is a wave of ‘probability’, not one of reality of quanta. Its name was from the place Copenhagen where was a middle of the argument. Representative scientists who supported this view were Niels Bohr, Max Born, and Heisenberg. They said, normally a quanta exits only as a probability like this, but at the time someone detect it, the state is to be fixed at an arbitrary point and the other probability wave collapse to zero. P(r) r 1 • Detecting

Two different interpretations of ψ
Realism • ψ is a wave of quanta itself • Einstein, Schrodinger, De Broglie • Incompleteness of Schrodinger equation - The complete equation will be able to find the exact state of a quanta • “God does not play dice” The other group supported Realism. This is an interpretation which ψ is a wave of quanta itself, not a virtual probability wave. Representative scientists who supported this view were Einstein, Schrodinger, De Broglie. They said the Schrodinger equation, which is only capable of probabilistic prediction, is not a complete equation. So they insist that there will be a discovery of more complete equation in the future. The famous phrase “God does not play dice” by Einstein is the same context of this.

Doubts to Copenhagen interpretation
Schrodinger’s cat thought experiment(1935) • Extreme Copenhagen interpretation : Human’s perception affects detecting results • According to the extreme Copenhagen interpretation, there exists an alive ‘and’ dead cat simultaneously → Exclude the human effect in detecting Those who supported Realism proposed many refutations about Copenhagen interpretation. I would explain one of them briefly. In 1935, Schrodinger proposed the Schrodinger’s cat thought experiment. Simply there is a cat, radioactive substance, it’s detector, and poison gas emission system in an enclosed box. Experimenter set the probability of decay of the substance to 50%. If the radioactive substance decay, the poison gas emission system activate, and the cat would die. But if the substance not decay, emission system deactivate, and cat would leave alive. According to the Extreme Copenhagen interpretation, human’s perception affects the collapse of wave function. Then, before the time human open the box and watch whether the cat die, cat could be alive ‘and’ dead simultaneously, which seemed impossible. But this paradox was solved by dropping extreme Copenhagen interpretation, that is, excluding the human’s effect from detecting. Then, even before human observe the cat’s situation, the wave would already collapse by the detector detecting radioactive decay.

EPR(Einstein-Podolsky-Rosen) Paradox(1935) Alice Bob 𝑆 𝑧 +1 -1 Another refutation aroused in the same year by Einstein, Podolsky, Rogen trio. They pointed out the contrariety of Copenhagen interpretation using a special situation which two particles being entangled each other. In this discussion, I will explain it through the David Bohm’s advanced version of EPR paradox which use electron and positron instead of photon of original version. An electron and a positron, which was originally a spin 0 pion, form a singlet state and take apart each other in opposite direction. When a distance between them is sufficiently far, if Alice measure the spin of one particle, we can know the spin of other particle instantaneously. This state, we can say the two particles are ‘entangled’. In this case, if Alice had measured the spin of Z direction of electron and it came spin up, the spin of Z direction of positron is to be spin down. However, according to Heisenberg’s uncertainty principle, we cannot know the spin of two direction simultaneously, therefore, Bob will not able to measure, for example, the spin of X direction. But if Alice had not measured spin of Z direction, / Bob can measure spin of X direction. What it means is that whether Bob can measure spin of X direction of positron depends on whether Alice did measure spin of Z direction of electron. Then if, / according to Copenhagen interpretation, the spin was undetermined when they separate from pion, the behavior which measuring spin of Z direction by Alice must generate some kind of information to transmit to Bob so that make measuring spin of X direction to be impossible. But because this information goes instantaneously to the other, / it violates the Einstein’s special relativity theory. That’s why they called it ‘paradox’. However, if we use the Realism and suppose the spin of two particles were determined right after they separate, though we wouldn’t know, there is no reason they transmit an information, so that there would be no violation of special relativity. Copenhagen interpretation 𝑆 𝑥 X O 𝑆 𝑥 Violation of Special relativity

Bohm’s Hidden variable theory(1952) P(r) r 1 P(r) r 1 • As an extension of Realism, David Bohm suggested Hidden variable theory in This is a theory that the reason of lack of acknowledge about the detecting position of a quanta is because quantum theory is yet incomplete and if we could find out the hidden variable in quantum theory, we could know the quantum states in all situation regardless of measuring. P(r) r 1 • Hidden variable Detecting

Bell’s theorem Hidden variable setting in EPR experiment
• Hidden variable λ in a probability space Λ • The values observed by Alice(A) or Bob(B) are functions of the detector settings( 𝑎 , 𝑏 , 𝑐 … ∈ 𝑆 2 ) and the λ only 𝐴, 𝐵 : 𝑆 2 ×Λ→{−1, +1} 𝐵( 𝑎 ,λ)=−A( 𝑎 ,λ) In 1964, John Bell assumed Realism is a correct one and assumed the existence of hidden variable to find out what happened if so mathematically. Following are hidden variable setting which Bell did. There is a probability space big lambda and the observed outcomes by both Alice and Bob are resulted by random sampling of the hidden variable lambda. It is a matter of indifference in the following whether we denotes lambda as a single variable, or a set, or even a set of functions, and whether the variables are discrete or continuous. However, we write as if it were a single continuous parameter for convenience. The values observed by Alice or Bob are functions of the local detector settings and the lambda only, where a detector setting is modeled as a location on the unit sphere 𝑆 square. Mathematically it can be represented like this. The outcomes of detector A and B are spin up or spin down. Also, this anti-correlation would hold because of the entanglement. If Alice and Bob use the same direction of detector, they must have negative relationship.

Bell’s theorem Bell’s inequality
• The quantum correlation between A( 𝑎 ,λ) and 𝐵( 𝑏 ,λ), defined as an expectation value of a product of the two components, is C( 𝑎 , 𝑏 )≡ 𝑝 λ A( 𝑎 ,λ)𝐵( 𝑏 ,λ)dλ=− 𝑝(λ) A( 𝑎 ,λ)𝐴( 𝑏 ,λ)dλ (𝑝(λ) : probability density) • If 𝑐 is an another detector setting, C( 𝑎 , 𝑏 )−C( 𝑎 , 𝑐 )=− 𝑝 λ [A( 𝑎 ,λ)𝐴( 𝑏 ,λ)−A( 𝑎 ,λ)A( 𝑐 ,λ)]dλ Next, Bell conducted a certain relationship called Bell’s inequality between the quantum correlation among three detector setting a, b, and c. The quantum correlation between A and B, defined as an expectation value of a product of the two components, is following, where p is a probability density of λ. You can see the applying of anti-correlation in this process. Using this definition, if c is an another detector setting, we can make this form.

Bell’s theorem Bell’s inequality
C( 𝑎 , 𝑏 )−C( 𝑎 , 𝑐 )=− 𝑝 λ A( 𝑎 ,λ)𝐴( 𝑏 ,λ)[1− A( 𝑐 ,λ) A( 𝑏 ,λ) ]dλ = 𝑝 λ A( 𝑎 ,λ)𝐴( 𝑏 ,λ)[A( 𝑏 ,λ)A( 𝑐 ,λ)−1]dλ |C( 𝑎 , 𝑏 )−C( 𝑎 , 𝑐 )|≤ 𝑝 λ [1−A( 𝑏 ,λ)A( 𝑐 ,λ)]dλ 1+𝐶( 𝑏 , 𝑐 )≥|𝐶( 𝑎 , 𝑏 )−𝐶( 𝑎 , 𝑐 )| Using the fact that outcomes of detecting is -1 or +1 only, and through some simple derivation, we can finally get this inequality. And this is the Bell’s inequality using hidden variable theory.

Bell’s theorem Bell’s inequality simple verification
• 8 possible cases of spins electron positron a b c +1 -1 C( 𝑎 , 𝑏 ) C( 𝑏 , 𝑐 ) C( 𝑐 , 𝑎 ) -1 +1 • Calculate C( 𝑎 , 𝑏 ), C( 𝑏 , 𝑐 ), C( 𝑐 , 𝑎 ) in each case We can simply check whether the inequality is correct assuming lambda is just a single constant. Separated electron and positron’s ‘a’, ‘b’, ‘c’ direction of spins, according to hidden variable theory, are determined right after the separation. Thus, there are 8 possible cases of spins as left side table. You can see the positron’s spin directions are opposite compare to electron. / And calculate these three variables in each case of which the result is in right side table. Because we assumed lambda as a constant, we can just simply multiply the two output.

Bell’s theorem Bell’s inequality simple verification
• Calculate LHS and RHS of Bell’s inequality 1+𝐶( 𝑏 , 𝑐 )≥|𝐶( 𝑎 , 𝑏 )−𝐶( 𝑎 , 𝑐 )| • In all cases, the inequality holds LHS RHS 2 1 Using this, we can derive left-hand-side and right-hand-side of the inequality like this table. We can see, in all cases, left-hand-side is same or bigger than right-hand-side, which means Bell’s inequality successively holds.

Contrary to Q.M prediction
Correlation as calculated by Quantum mechanics ( 𝑎 , 𝑏 )=<𝑆 𝜎 2 ∙ 𝑏 𝜎 1 ∙ 𝑎 𝑆> (|𝑆> = (| χ + >| χ − >−| χ − > χ + > ) ( 𝜎 𝑥 = , 𝜎 𝑦 = 0 −𝑖 𝑖 0 , 𝜎 𝑧 = −1 ) 𝐶 𝐶 ( 𝑎 , 𝑏 ) = − 𝑎 ∙ 𝑏 • This doesn’t satisfy Bell’s inequality As for Copenhagen interpretation, we can derive the correlation between ‘a’ and ‘b’ like this formula, where S is the spin singlet state which is expressed in here, and sigma is the Pauli matrices which can be represented by this. Using these formulas, and through a little messy calculation, which you could find the detail in my note, we can obtain this result. However, this doesn’t satisfy Bell’s inequality. And it means, one of them has to be necessarily wrong.

Bell test experiment CHSH inequality
• John Clauser, Michael Horne, Abner Shimonv, Richard Holt(1969) • Advanced version of Bell’s inequality |𝑆|≡| 𝐸 𝑎, 𝑏 − 𝐸 𝑎, 𝑏 ′ + 𝐸 𝑎 ′ , 𝑏 + 𝐸 𝑎 ′ , 𝑏 ′ |≤2 ( 𝐸 𝑎, 𝑏 = 𝑁 +,+ + 𝑁 −,− − 𝑁 +,− − 𝑁 −,+ 𝑁 +,+ + 𝑁 −,− + 𝑁 +,− + 𝑁 −,+ ) ( 𝑁 +,+ : Number of simultaneous occurrences of the outcome +1 on both sides and vice versa • |𝑆| 𝐶 >2 To confirm which interpretation is correct, dozens of experiments have conducted. And they are called ‘Bell test experiments’. Before I show you a few cases, let me explain the other inequality related to hidden variable. In 1969, these scientists improved Bell’s inequality to CHSH inequality. This is an advanced version of Bell’s inequality, and the specific formula is like this. To conduct an experiment and analyze it, we need to define an experimental correlation. The outcomes A and B could each only take one of two values, -1 or +1. It followed that the product, too, could only be -1 or +1, so that the average value of the product would be derived like this, where for example, N++ is the number of simultaneous occurrences of the outcome +1 on both sides of the experiment and vice versa. The reason that I express the denominator not just N total but like that, is that there could be inaccurate detecting data in experiment. In the case of Copenhagen interpretation, if we calculate the CHSH formula, S is bigger than 2, which violates the inequality.

Bell test experiment Freedman and Clauser experiment(1972)
• First actual Bell test • Using Freedman’s inequality Aspect et al(1982) • Using photon polarization • 𝑎 :0°, 𝑎 ′ :22.5°, 𝑏 :45°, 𝑏 ′ :67.5° I want to briefly introduce two of Bell test experiments firstly. In 1972, Freedman and Clauser did the Bell test at first place using their own Freedman’s inequality. Also in 1982, Alain Aspect’s research team conducted the test with the polarization of photon, setting ‘a’ as 0 degree, a’ as 22.5 degree, b as 45 degree and b’ as 67.5 degree.

(η : efficiency of experiment)
Bell test experiment Loopholes in Bell test experiment • Detection efficiency / Fair sampling - Inaccurate measurement by coincidental factors 𝐸 𝑎, 𝑏|𝑐𝑜𝑖𝑛𝑐. − 𝐸 𝑎, 𝑏 ′ |𝑐𝑜𝑖𝑛𝑐. + 𝐸 𝑎 ′ , 𝑏|𝑐𝑜𝑖𝑛𝑐. + 𝐸 𝑎 ′ , 𝑏 ′ |𝑐𝑜𝑖𝑛𝑐. ≤ 4 η −2 (η : efficiency of experiment) - If η is less than 83%, there would be no violation with Q.M prediction - Efficiency of typical optical experiments was around 5~30% But while scientists do the tests, they found some nettlesome problems caused by experimental restrictions. They called it as ‘loopholes in Bell test experiment’. The first one is called ‘detection efficiency problem’, or ‘fair sampling problem’. This loophole is occurred due to the inaccurate measurement by coincidental factors in experiment. If we consider this factor, the CHSH inequality must change the form like this, where eta is an efficiency of experiment. You can see if eta is one, which is 100%, then it becomes the original CHSH inequality. If eta is less than 83%, there would be no violation with quantum mechanical prediction, so that we wouldn’t know which interpretation is correct. But at that time, the efficiency of typical optical experiments was around 5 to 30%

Bell test experiment Loopholes in Bell test experiment
• Detection efficiency / Fair sampling - Fair sampling assumption : Sample of detected pairs is representative of the pairs emitted → Set η as 1 So scientists used a trick to avoid the detection efficiency problem. That is called ‘fair sampling assumption’. It stated that the sample of detected pairs can be representative of the total pairs emitted. And just assume the value of eta as 1.

Bell test experiment Loopholes in Bell test experiment
• Locality / Communication - Prohibit any communication by separating the two sites - Measurement duration must be shorter than the time it would take for any light-speed signal from one site to the other, or indeed, to the source The other loophole is called ‘Locality problem’, or ‘communication problem’. In Bell test experiment, experimenter must prohibit any communication by separating the two sites, and measurement duration must be shorter than the time it would take for any light-speed signal from one site to the other, or indeed, to the source. So this loophole gave scientists technical problems.

Bell test experiment Hensen et al(2015): “loophole-free” Bell test
• Detect two entangled spin of electron which is trapped in nitrogen-vacancy(NV) defect centre in a diamond chip • The diamonds are mounted in closed-cycle cryostats (T=4K) located in laboratories named A and B which distant about 1.3km But very recently, in 2015, professor Hensen and his research team conducted a ‘loophole-free’ Bell test at first place. Basically they detected two entangled spin of electron which is trapped in nitrogen-vacancy defect centre in a diamond chip, and put the data to CHSH inequality. The diamonds are mounted in closed-cycle cryostats at 4 kelvin, located in laboratories named A and B which distant about 1.3km. This is a rough image of trapped electron in a diamond, and this is the inside of Delft university. You can see two laboratories A and B are separated about 1.3 km, and this red dotted line is an actual path of photon. Additional C laboratory will be explained in next slide.

• Constructing entanglement - Event-ready set-up(entanglement swapping) 𝑎, 𝑎 ′ New entanglement 𝑎 𝑏 𝑏 ′ , 𝑏 𝑎 ′ 𝑏 ′ There are various way of constructing entanglement between two particles. Among them, Hensen team chose the ‘event-ready set-up’ method which uses an entanglement swapping. For example, let’s assume there are two pairs of entangled polarization of photon which denoted by a, a’ and b, b’. At first place, there are no relationship between ‘a’, and ‘b’. And then, let’s shoot a’ and b’ to this beam splitter, and measure the polarization directions. But because of the presence of beam splitter, we don’t know which detecting result correspond to a’ or b’. So in this situation, new entanglement between ‘a’ and ‘b’ is generated. It is called the entanglement swapping process. As for the previous topography picture, this source of entangled photon-pair are laboratory A and B, and the beam splitter is in the C laboratory. The prior advantage of this mechanism is two entangled particles are not neccessarily from same source.

• Schematic This is a schematic figure of Hensen’s experiment. Through somewhat complicate mechanism, a photon be entangled with the spin of electron in diamond at box A and B, and transmit the entangled photon to box C which has the beam-splitter. And through the entanglement swapping process, two electrons in diamonds become entangled. At this moment, laboratory A and B detect the spin of electron, and record it to put in the CHSH inequality.

• Space-time analysis of the experiment • Locality - It takes 4.27μs between A and B in speed of light - Measuring duration : 3.7μs < 4.27μs • Detection efficiency - Through 245 trials, result in Figure c - Measuring fidelity A : 97.1±0.2%, B : 96.3±0.3% This is a space-time analysis during the one period of experiment. X axis is distance between A, C and B, and Y axis is time. You can see the two pairs of photon-electron entanglement in A and B which to be created in different time because of the simultaneous arrive at C. And this long green line and blue line correspond to measuring duration of each spin of electron. Between A and B, it takes 4.27 micro seconds with speed of light. But as you can see, the measuring duration is only 3.7 micro seconds which is less than Therefore, this can exclude the locality loophole. Next, let’s consider the detection efficiency problem. Through 245 trials of experiment, they could draw the relation between measuring duration and fidelity of measure with average values. And as for 3.7 micro seconds duration of measure, they could achieve measuring fidelity about 97.1% of A and 96.3% of B. This results indicate far more detection efficiency over 83%, therefore the detection efficiency loophole could be solved either.

Bell test experiment Result
• Substitution of experimental values results in violation of CHSH inequality in all experiments • In Hensen’s experiment, 𝑆=2.42±0.03>2 Copenhagen Interpretation Hidden variable theory In all those various Bell test experiments, substitution of experimental values resulted in violation of CHSH inequality. As for Hensen’s experiment too, S was equal about 2.42 which is bigger than 2. So from these facts, / Hidden variable theory was defeated by Copenhagen interpretation at last.

Non-locality Violation of special relativity in EPR experiment
• Two particles which have an entanglement can interact simultaneously → ‘Non-locality’ quantum characteristic • Many experimental data prove this phenomenon Lastly, I want to explain the non-locality characteristic in quantum theory briefly. Among the possible explanations of EPR experiment, because one based on Realism went down, the only remaining option is an explanation based on Copenhagen interpretation. But it had a problem about violation of special relativity. So scientists regarded two particles which have entanglement as a ‘special’ relation, and assumed they can interact simultaneously. And they called this phenomena ‘non-locality’. In later additional experiments, they confirmed the simultaneous information transmission between two particles, so that they determined the presence of non-locality.

Thank you Thank you for your fine listening.

EPR Paradox and Bell’s theorem

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