Bell s Theorem...What?! Entanglement and Other Puzzles

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1 Bell s Theorem...What?! Entanglement and Other Puzzles Kyle Knoepfel 27 February 2008 University of Notre Dame Bell s Theorem p.1/49

2 Some Quotes about Quantum Mechanics Erwin Schrödinger: I do not like it, and I am sorry I ever had anything to do with it. Max von Laue (regarding de Broglie s theory of electrons having wave properties): If that turns out to be true, I ll quit physics. Niels Bohr: Anyone who is not shocked by quantum theory has not understood a single word... Richard Feynman: I think it is safe to say that no one understands quantum mechanics. Carlo Rubbia (when asked why quarks behave the way they do): Nobody has a ******* idea why. That s the way it goes. Golden rule number one: never ask those questions. Bell s Theorem p.2/49

3 Outline Quantum Mechanics (QM) Introduction Different Formalisms,Pictures & Interpretations Wavefunction Evolution Conceptual Struggles with QM Einstein-Podolsky-Rosen (EPR) Paradox John S. Bell The Inequalities The Experiments The Theorem Should we expect any of this? References Bell s Theorem p.3/49

4 Quantum Mechanics Review Which of the following statements about QM are false? Bell s Theorem p.4/49

5 Quantum Mechanics Review Which of the following statements about QM are false? 1. If O = O, then the operator O is self-adjoint. 2. x p x /2 is always true. 3. The N-body Schrödinger equation is local in physical R The Schrödinger equation was discovered before the Klein-Gordon equation. Bell s Theorem p.5/49

6 Quantum Mechanics Review Which of the following statements about QM are false? 1. If O = O, then the operator O is self-adjoint. 2. x p x /2 is always true. 3. The N-body Schrödinger equation is local in physical R The Schrödinger equation was discovered before the Klein-Gordon equation. All statements are false! Bell s Theorem p.6/49

7 Quantum Mechanics Formulations There are at least 9 different formulations of QM. 1. Matrix Formulation (Heisenberg) 2. Wavefunction Formulation (Schrödinger) 3. Path Integral Formulation (Feynman) 4. Second Quantization Formulation Bell s Theorem p.7/49

8 Quantum Mechanics Formulations There are at least 9 different formulations of QM. 1. Matrix Formulation (Heisenberg) 2. Wavefunction Formulation (Schrödinger) 3. Path Integral Formulation (Feynman) 4. Second Quantization Formulation 5. Phase Space Formulation (Wigner) 6. Density Matrix Formulation 7. Variational Formulation (Schwinger) 8. The Pilot Wave Formulation (de Broglie Bohm) 9. The Hamilton-Jacobi Formulation Bell s Theorem p.8/49

9 QM Pictures & Interpretations There are infinitely many QM pictures placement of time-dependence. 1. Schrödinger Picture: time-dependent wavefunctions 2. Heisenberg Picture: time-dependent operators 3. Interaction Picture: time-dependence split Bell s Theorem p.9/49

10 QM Pictures & Interpretations There are infinitely many QM pictures placement of time-dependence. 1. Schrödinger Picture: time-dependent wavefunctions 2. Heisenberg Picture: time-dependent operators 3. Interaction Picture: time-dependence split There are many different interpretations of QM. VERY Difficult to Characterize. 1. Realistic Interpretation (wavefunction describes reality) 2. Positivistic Interpretation (wavefunction contains only the information about reality) 3. Copenhagen Interpretation Orthodox Interpretation 4. Causal Interpretation (Bohm) 5. The Many-worlds Interpretation (Everett) 6. Ithaca Interpretation (Mermin) Bell s Theorem p.10/49

11 Wavefunction Evolution Two Postulates Postulate 1: Schrödinger Equation i 2 ψ(r, t) = t 2m 2 ψ(r, t) + V (r, t)ψ(r, t) Natural Evolution Entirely deterministic! Bell s Theorem p.11/49

12 Wavefunction Evolution Two Postulates Postulate 1: Schrödinger Equation i 2 ψ(r, t) = t 2m 2 ψ(r, t) + V (r, t)ψ(r, t) Natural Evolution Entirely deterministic! Postulate 2: Wavefunction Collapse ψ θ = N θ P θ ψ where P θ = θ θ projects ψ into the θ eigenstate. Measurement Entirely random! Bell s Theorem p.12/49

13 QM Puzzles I 1. Quantum Mechanics Classical Mechanics NOT today s topic. The transition from QM to Classical Physics is not understood. When and how does it happen? Is it the limit where 0? Is it the limit where the commutation relation becomes the Poisson bracket? [X, P ] QM = i [x, p] CM = 1 Is it the limit where operators merely multiply wavefunctions by their corresponding values? Schrödinger Equation E = p2 2m + V (r, t) Is it the Correspondence Principle? All good ideas, but is that all there is to it? Nobody knows! Your chance to become famous! Bell s Theorem p.13/49

14 QM Puzzles II 2. Measurement Problem NOT today s topic. Act of measurement is just introducing a time-dependent potential: V (r, t) V (r, t) + V M (r, t) Shouldn t the wavefunction be: Ψ = ψ P hys. [ ] ψ App. not ψ θ = N θ P θ ψ What collapses the wavefunction? (Von Neumann Chain,Infinite Regress) Schrödinger s Cat Dead & alive until measurement made Ψ = α + dead + β alive Reductio ad Absurdum. Bell s Theorem p.14/49

15 QM Puzzles II 2. Measurement Problem NOT today s topic. Act of measurement is just introducing a time-dependent potential: V (r, t) V (r, t) + V M (r, t) Shouldn t the wavefunction be: Ψ = ψ P hys. [ ] ψ App. not ψ θ = N θ P θ ψ What collapses the wavefunction? (Von Neumann Chain,Infinite Regress) Schrödinger s Cat Dead & alive until measurement made Ψ = α + dead + β alive Reductio ad Absurdum. 3. Non-locality. Welcome to entanglement, let s begin with EPR. Bell s Theorem p.15/49

16 Einstein-Podolsky-Rosen (EPR) Paradox Albert Einstein Boris Podolsky Nathan Rosen Bell s Theorem p.16/49

17 EPR Paradox Assume entangled state Ψ such that: L 1 + L 2 = 0 S 1 + S 2 = 0 J 1 + J 2 = 0 Assume locality: measurements on one system cannot affect another system if they are separated by a space-like interval! Bell s Theorem p.17/49

18 EPR Paradox Assume entangled state Ψ such that: L 1 + L 2 = 0 S 1 + S 2 = 0 J 1 + J 2 = 0 Assume locality: measurements on one system cannot affect another system if they are separated by a space-like interval! Measure Particle 1: Infer Particle 2: S 1x L 1y S 1z S 2x L 2y S 2z J 1x At any given time, Particle 2 does not know what property of Particle 1 will be measured. J 2x Bell s Theorem p.18/49

19 EPR Paradox Assume entangled state Ψ such that: L 1 + L 2 = 0 S 1 + S 2 = 0 J 1 + J 2 = 0 Assume locality: measurements on one system cannot affect another system if they are separated by a space-like interval! Measure Particle 1: Infer Particle 2: S 1x L 1y S 1z S 2x L 2y S 2z J 1x At any given time, Particle 2 does not know what property of Particle 1 will be measured. Particle 2 MUST have a value corresponding to any measurement made on Particle 1! CONTRADICTION! Not all variables commute i.e. not possible to have simultaneous values! J 2x Bell s Theorem p.19/49

20 EPR Conclusions One is thus led to conclude that the description of reality as given by a wave function is not complete (EPR). Assumptions that lead to this conclusion: 1. Logic holds at the QM scale. 2. QM objects have intrinsic reality independent of measurements. 3. Locality is preserved in quantum mechanics. Bell s Theorem p.20/49

21 EPR Conclusions One is thus led to conclude that the description of reality as given by a wave function is not complete (EPR). Assumptions that lead to this conclusion: 1. Logic holds at the QM scale. 2. QM objects have intrinsic reality independent of measurements. 3. Locality is preserved in quantum mechanics. Given above assumptions, EPR argument cannot be disproved! Bell s Theorem p.21/49

22 EPR Conclusions One is thus led to conclude that the description of reality as given by a wave function is not complete (EPR). Assumptions that lead to this conclusion: 1. Logic holds at the QM scale. 2. QM objects have intrinsic reality independent of measurements. 3. Locality is preserved in quantum mechanics. Given above assumptions, EPR argument cannot be disproved! [S]ince either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality ot P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this (EPR). Bell s Theorem p.22/49

23 From Einstein to Bell While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible (EPR). Bell s Theorem p.23/49

From Einstein to Bell While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description

24 From Einstein to Bell While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible (EPR). Copenhagen (Bohr) - Einstein debate raged on during this time....their arguments do not justify their conclusion that the quantum description turns out to be essentially incomplete...this description can be characterized as a rational use of the possibilities of an unambiguous interpretation of the process of measurement compatible with the finite and uncontrollable interaction between the object and the instrument of measurement in the context of quantum theory. (Bohr). Bohr and Einstein, 1925 WHAT? John Bell is first person in 30 years to propose additional variables theory to solve EPR. Bell s Theorem p.24/49

25 John S. Bell B: 1928 Belfast, North Ireland D: 1990 Geneva, Switzerland Family was not well-off (financially); 3 siblings, only John continued with education Entered physics department at Queen s as a technician Graduated from Queen s with first-class honors in 1948 in experimental physics Married Mary Ross, physicist/mathematician from Glasgow; both worked in accelerator physics Derived CPT Theorem in 1951 Lüders and Pauli just beat him! Several important contributions to QFT Bell s Theorem p.25/49

Bell - The 1964 Paper (I) Based on idea by Bohm and Aharonov where one measures

Measure spins of both particles, but introduce parameter that accounts for the

2 b If hidden variable theory is reasonable, then the expectation value using λ

26 Bell - The 1964 Paper (I) Based on idea by Bohm and Aharonov where one measures spin projections of entangled particles. Measure spins of both particles, but introduce parameter that accounts for the hidden variables: λ A (a, λ) = Measurement of σ 1 a B (b, λ) = Measurement of σ 2 b If hidden variable theory is reasonable, then the expectation value using λ P (a, b) = dλρ (λ) A(a, λ)b(b, λ) where dλρ (λ) = 1 should equal the QM result σ 1 a σ 2 b = a b Bell s Theorem p.26/49

27 Bell - The 1964 Paper (II) Bell s Original Inequality: 1 + P (b, c) P (a, b) P (a, c) For b c, expand right-hand side in Taylor series to first order P (b, c b) b c c P (a, c = b) 1 Does this equal the QM value of σ 1 b σ 2 b = 1? Bell s Theorem p.27/49

28 Bell - The 1964 Paper (II) Bell s Original Inequality: 1 + P (b, c) P (a, b) P (a, c) For b c, expand right-hand side in Taylor series to first order P (b, c b) b c c P (a, c = b) 1 Does this equal the QM value of σ 1 b σ 2 b = 1? No, but it s subtle. Best you can do: P (b, c b) 1 Unless P is constant, c P (a, c = b) is not defined. Bell continues the paper with δ ε proof that hidden variables theory cannot reproduce quantum mechanics. 4 (ε + δ) 2 1 Additional variables theory inconsistent with QM. Bell s Theorem p.28/49

29 Bell Inequalities Definition: Any inequality derived from assuming the existence of additional variables. Original Inequality 1 + P (b, c) P (a, b) P (a, c) Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) Version More Common P (a, b) + P ( a, b ) + P ( a, b ) P ( a, b ) 2 Greenberger-Horne-Zeilinger (GHZ) Inequality 3-body states Ψ = 1 2 ( +, +, +,, ) σ 1x σ 2x σ 3x = 1 EP R Hardy s Impossibilities Bell-Kochen-Specker (Contextuality) Bell Experiment: Any experiment which seeks to prove/disprove the Bell inequalities. Bell s Theorem p.29/49

30 Bell Experiments Aspect, et al ( ) Alain Aspect Series of experiments testing generalized Bell-CHSH inequality correlated photons. Measured photon polarizations by looking at calcium transition 4p 2 (J = 0) 4s 2 (J = 0) transitions. Bell Inequality Prediction: -1 S 0 where S is some combination of P s. Bell s Theorem p.30/49

31 Bell Experiments Aspect, et al ( ) Alain Aspect Series of experiments testing generalized Bell-CHSH inequality correlated photons. Measured photon polarizations by looking at calcium transition 4p 2 (J = 0) 4s 2 (J = 0) transitions. Bell Inequality Prediction: -1 S 0 where S is some combination of P s. Measured quantity: S exp. = ± Violates inequality by 9 standard deviations! Also measured violation over space-like interval of over 10 m. Bell s Theorem p.31/49

32 Bell Experiments Belle Collaboration (2007) One can study QM entanglement with Υ(4S) BB Decays. B 0 B 0 system is correlated since both particles evolving exactly in phase with each other. At any given time: Ψ = 1 2 ( B 0 1 B 0 2 B 0 1 B 0 2 ) Flavor measurement of one particle uniquely determines flavor of other. Bell s Theorem p.32/49

33 Bell Experiments Belle Collaboration (2007) One can study QM entanglement with Υ(4S) BB Decays. B 0 B 0 system is correlated since both particles evolving exactly in phase with each other. At any given time: Ψ = 1 2 ( B 0 1 B 0 2 B 0 1 B 0 2 ) Flavor measurement of one particle uniquely determines flavor of other. Why is this interesting? Test QM at high energy scale. Bell Test cannot be done for BB system; but one can still test QM by measuring time-dependent decay rate. Quantum Mechanics A(SF, OF ) = 1 ± cos m t Asym. = A SF A OF A SF + A OF = cos m t Bell s Theorem p.33/49

34 Bell Experiments Belle Collaboration (2007) 1. Quantum Mechanics Asym. = cos m t 2. Spontaneous Disentanglement Asym. = 1 2 [cos m (t 1 + t 2 ) + cos m t] 3. Local Realism: Pompili-Selleri Asym. max P S = 1 {1 cos( m t)} cos( t min ) + sin( m t) sin( m t min ) Asym. min P S = 1 min(2 + Ψ, 2 Ψ) where Ψ = {1 + cos( m t)} cos( t min ) sin( m t) sin( m t min ) Bell s Theorem p.34/49

35 Bell Experiments Belle Collaboration (2007) Flavor determined by B 0 D l + ν decays. Quantum Mechanics: χ 2 /ndof = 5.2/11 PREFERRED! SD: χ 2 /ndof = 174/11 PS: χ 2 /ndof = 31.3/11 Bell s Theorem p.35/49

36 Bell s Theorem There have been many Bell tests, all favoring QM. What was wrong with EPR? What is wrong, if we believe QM, is to attribute well-defined values to each emitted pair [of entangled particles]. (Laloë) Unperformed experiments have no result. (Peres) We often discussed his notions on objective reality. I recall that during on walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it. (A. Pais) Bell s Theorem p.36/49

37 Bell s Theorem There have been many Bell tests, all favoring QM. What was wrong with EPR? What is wrong, if we believe QM, is to attribute well-defined values to each emitted pair [of entangled particles]. (Laloë) Unperformed experiments have no result. (Peres) We often discussed his notions on objective reality. I recall that during on walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it. (A. Pais) Theorem (Logical): The following system of three assumptions is self-contradictory 1. The validity of the EPR notion of elements of reality, 2. Locality, 3. The predictions of quantum mechanics are always correct. Theorem (Conventional): No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Bell s Theorem p.37/49

38 Nonlocality in QM Should non-locality in QM really surprise us? I argue No; it shouldn t. i 2 ψ(r, t) = t 2m 2 ψ + V (r, t)ψ(r, t) Not local in physical space. Consider adding another particle. i t ψ(r 1, r 2, t) = 2 2m 1 2 1ψ + 2 2m 2 2 2ψ + V (r 1, r 2, t)ψ(r 1, r 2, t) Bell s Theorem p.38/49

39 Nonlocality in QM Should non-locality in QM really surprise us? I argue No; it shouldn t. i 2 ψ(r, t) = t 2m 2 ψ + V (r, t)ψ(r, t) Not local in physical space. Consider adding another particle. i t ψ(r 1, r 2, t) = 2 2m 1 2 1ψ + 2 2m 2 2 2ψ + V (r 1, r 2, t)ψ(r 1, r 2, t) Variable space is now 6+1 dimensional, not 3+1. Furthermore, Schrödinger Equation is not Lorentz invariant non-relativistic, so c is reasonable. Bell s Theorem p.39/49

40 Nonlocality in QM Should non-locality in QM really surprise us? I argue No; it shouldn t. i 2 ψ(r, t) = t 2m 2 ψ + V (r, t)ψ(r, t) Not local in physical space. Consider adding another particle. i t ψ(r 1, r 2, t) = 2 2m 1 2 1ψ + 2 2m 2 2 2ψ + V (r 1, r 2, t)ψ(r 1, r 2, t) Variable space is now 6+1 dimensional, not 3+1. Furthermore, Schrödinger Equation is not Lorentz invariant non-relativistic, so c is reasonable. But both experiments I presented are relativistic! Enter Quantum Field Theory. Bell s Theorem p.40/49

41 (Non)locality in QFT Locality in QFT Equations of motion are local: Klein-Gordon, Dirac, and Proca equations. Exchange of bosons occurs at a point interaction: W γ b u,c,t A21 s Bell s Theorem p.41/49

42 (Non)locality in QFT Locality in QFT Equations of motion are local: Klein-Gordon, Dirac, and Proca equations. Exchange of bosons occurs at a point interaction: W γ b u,c,t A21 s Nonlocality in QFT Relevant observable is the cross-section, dependent on the S-matrix: S = n=0 i n n! d 4 x 1 d 4 x n T [L I (x 1 ) L I (x n )] Bell s Theorem p.42/49

43 (Non)locality in QFT Locality in QFT Equations of motion are local: Klein-Gordon, Dirac, and Proca equations. Exchange of bosons occurs at a point interaction: W γ b u,c,t A21 s Nonlocality in QFT Relevant observable is the cross-section, dependent on the S-matrix: S = n=0 i n n! d 4 x 1 d 4 x n T [L I (x 1 ) L I (x n )] Is that all there is to it? What happens if you add Lorentz Invariance, Spin Statistics, and Locality? Bell s Theorem p.43/49

44 (Non)locality in QFT Locality in QFT Equations of motion are local: Klein-Gordon, Dirac, and Proca equations. Exchange of bosons occurs at a point interaction: W γ b u,c,t A21 s Nonlocality in QFT Relevant observable is the cross-section, dependent on the S-matrix: S = n=0 i n n! d 4 x 1 d 4 x n T [L I (x 1 ) L I (x n )] Is that all there is to it? What happens if you add Lorentz Invariance, Spin Statistics, and Locality? CPT! What do we do now...??? For next time. Bell s Theorem p.44/49

45 What didn t I cover? Hidden variables associated with the measuring apparatus Von Neumann Impossibility Theorem QM Loopholes/Conspiracies Decoherence No-communication Theorem Quantum Cryptography Quantum Computing Quantum Histories PLUS A WHOLE LOT MORE! Bell s Theorem p.45/49

46 What didn t I cover? Hidden variables associated with the measuring apparatus Von Neumann Impossibility Theorem QM Loopholes/Conspiracies Decoherence No-communication Theorem Quantum Cryptography Quantum Computing Quantum Histories PLUS A WHOLE LOT MORE! Very interesting theory - it makes no sense at all. (Groucho Marx) Bell s Theorem p.46/49

47 References Aspect, et al. Experimental Tests of Realistic Local Theories via Bell s Theorem. PRL (1981) Bell, J.S. On the Einstein Podolsky Rosen Paradox. Physics, 1 (195), 1964 Belle Collaboration, Measurement of Einstein-Podolsky-Rosen-Type Flavor Entanglement in Υ(4S) BB Decays. PRL (2007) Einstein,et al. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev 47 (777), May 1935 Gasiorowitz, Quantum Physics, Ed: 3. Wiley International: Hoboken, NJ (2003) Laloë. Do we really understand quantum mechanics? Am. J. Phys. 69 (6), June 2001 Mermin. Is the moon there when nobody looks? Phys. Today, Apr (38) Wikipedia article/discussion on Bells Theorem. Bell s Theorem p.47/49

48 BACKUP SLIDES Bell s Theorem p.48/49

49 Locality Gauss s Law (EM) Global Integral Equation: Equality holds by integration over entire surface area. E da = Q encl. ɛ 0 Use Gauss s Law to transform surface integral to volume integral: E da = E d 3 x Use integral expression of Q encl. to obtain: E d 3 x = 1 ρ d 3 x or ɛ 0 ( E ρɛ0 ) d 3 x = 0 In order for equality to hold for arbitrary integration region, integrand must be zero for each point (x, y, z) included in the integral. We then get the Local Differential Equation: E = ρ ɛ 0 Bell s Theorem p.49/49

Einstein-Podolsky-Rosen paradox and Bell s inequalities

Einstein-Podolsky-Rosen paradox and Bell s inequalities Jan Schütz November 27, 2005 Abstract Considering the Gedankenexperiment of Einstein, Podolsky, and Rosen as example the nonlocal character of quantum