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EPR Paradox, Bell Inequalities

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EPR Paradox, Bell Inequalities EPR paradox Bell inequalities 1 Can quantum-mechanical description of physical reality be considered complete? • A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 - 780 (1935) In a complete theory there is an element corresponding to each element of reality.A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non- commuting operators, the knowledge of one precludes the knowledge of the other.Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete. Abstract © American Physical Society. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. 2 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 3 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 4 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 5 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 6 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 7 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 8 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 9 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 10 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 11 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 12 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 13 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 14 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 15 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 16 Entangled Pair • Prepare the state =( 01 + 10 ) / p 2 of two identical particles (spins)| i | i | i • Particles move to Alice and Bob, that measure their angular momentum Sz, obtaining either +1 or -1 A B • Experiment repeated many times: perfect anti- correlation 17 Traveler pair • Anti-correlation also in “classical” experiment • Two travelers with balls inside two luggages Images by MIT OpenCourseWare. • Alice and Bob check the luggages: perfect anti- correlation 18 Traveler pair • Anti-correlation also in “classical” experiment • Two travelers with balls inside two luggages Images by MIT OpenCourseWare. • Alice and Bob check the luggages: perfect anti- correlation 19 Two properties of balls • Now assume that the balls can be red or green and matte or shiny. • Anti-correlation also for the property of gloss 20 Two axes • In QM, the 2 properties are the spin along 2 axes: (x σz • We can rewrite the state in ( x basis, =(01 + 10 )/p2=( + + + )/p2 | i | i | i | -i | - i • thus measuring (x Alice and Bob obtain same anti-correlation 21 Classical hypothesis Realism & Locality 22 Realism • At preparation, particles a and b possess both the properties (color and gloss for the classical balls σx, σz, with σx,z = ±1 for the quantum particles) 23 Locality • When I measure particle a, I cannot modify instantaneously the result of measuring particle b. There is no action at distance (faster than light) 24 EPR Paradox • Any complete description of the world must respect local realism • Local realism is violated by quantum mechanics ➜ Quantum mechanism is not a complete description of the world 25 Bell Inequalities Quantitative measure of violation of local realism 26 Correlation for any axes • Assume two spins in Bell State ✓ =(01 + 10 )/p2 | i | i | i Alice measure a A obtaining a, while Bob measure z • B B B a b = cos ✓az +sin✓a x getting b ∈ {+1,-1}. • What is the correlation ab = aAaB ? h i h z b i 27 Some calculations... 1 aAaB = ( 01 aAaB 01 + 01 aAaB 10 + h z b i 2 h | z b | i h | z b | i + 10 aAaB 01 + 10 aAaB 10 ) h | z b | i h | z b | i 1 A B A B = 0 az 0 1 ab 1 + 0 az 1 1 ab 0 + 2 h | | ih | | i h | | ih | | i + 1 aA 1 0 aB 0 + 1 aA 0 0 aB 1 h | z | ih | b | i h | z | ih | b | i 1 = 1 aB 1 0 aB 0 = cos ✓ 2 h | b | i-h | b | i - 28 Bell experiment • Alice measures along either a or a’ • Bob measures along either b or b’ a a’ ✓ b ab = cos ✓ aa0 = cos ¢ b’ b c bb0 = cos ¢ a0b0 = cos ✓ c d 29 Correlations • The correlations among the measurements are then: ab = a0 b0 = cos ✓ h i h i - a0 b = cos(✓ ¢) ab0 = cos(✓ + ¢) h i - - h i - • We want to calculate 〈S〉 S = ab + a0 b0 + ab0 a0 b h i h i h i h i-h i 30 Locality + Realism • At each measurement, I should be able to calculate the value of the operator: A B A B A B A B Sk =(σ σ )k +(σ σ )k +(σ σ )k (σ σ )k a b a0 b0 a b0 − a0 b 1 The expectation value is then S =lim Sk • h i N N !1 Xk 31 Realism • Even if I measure the spin along a, the property spin along a’ is real (that is, it has a definite value) 32 Calculate outcome of Sk • Rewrite as A B B A B B Sk = ( (( + ( )k ( (( ( )k a b b0 - a0 b - b0 B B Outcomes of a a are 0, +2, 2 • b ± b 0 { - } B B B B If a b + a b is 2 , a a is 0 and vice-versa • 0 ± b - b 0 Then S = 2 a or S = 2a0 • k ± a k ± a 33 Locality • The fact of measuring b or b’ does not change the value of a or a’ (that have outcomes +/-1). Then: S = 2 k ± • The expectation value of S is then 2 < S < +2 − h i 34 Bell inequality • If S > 2 at least one of the two hypothesis (locality|h i| or realism) is not true 35 Bell inequality • Choose a=z; a’=x; b=-x+z; b’=x+z; a ab = a0 b0 = cos ✓ab = 1/p2 b h i h i - - b’ p ab0 = cos ✓ab0 = 1/ 2 a’ h i - - a0 b = cos ✓ =+1/p2 h i - a0b • We obtain 4 S = ab + a0 b0 + ab0 a0 b = = 2p2 < 2 h i h i h i h i-h i -p2 - - 36 References • J.
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