1. 1 quantum mysteries again! quantum mysteries again! classical vs. quantum correlations ‘ quantum mechanics is weird” N. Bohr Bell’s inequality? QM VIOLATES IT! D. Mermin, Am. J. Phys. 49, 940 (1981)

2. 2 singlet state (EPR pair) take two spins and move them apart (no common preparation or exchange of signals between them) and measure them in various directions (settings). What are the results? always opposite! quantum non-locality EPR paradox (1935) or quantum non-locality? common state? “strange action at a distance” or common state?

3. 3 quantum vs. classical correlations fast communication (via exchange of messages) fast communication (via exchange of messages)or common preparation (via hidden variables) common preparation (via hidden variables) what are correlations due to?

4. 4 2 spins in the singlet state if spin 1 is up & spin 2 is down in the z-dir if spin 1 is up in the z-dir the spin 2 is down in the n-direction with angle θ θ

5. 5 quantum correlation function measure the spins in two directions with angle θ θ remember, the mean value of S z S θ is taken on the singlet (entangled) state

6. 6 classical correlation function suppose spins have definite (if unknown) values, then the orientation of spin is random (of course the spins are opposite to each other) θ θ

7. 7 quantum vs. classical 1 quantum correlations are stronger than classical (Bell showed QM can go out of mathematical limits!!!)

8. 8 measure one spin measure the spin in various directions (settings) with results (in units of ) in z-dir in n-dir (θ)

9. 9 …measure both spins (in a singlet) measure the spin in various directions (settings) with results (in units of ) at locations A and B in z-dir in n-dir (θ)   location A   location B in z-dir in n-dir (-θ) consider now the linear combination of correlations how many possible results? what are they? 16 g= +2 or -2

10. 10 Bell-CHSH inequality mean correlations λ: hidden variable λ: hidden variable

11. 11 quantum correlation function violates it! Bell- CHSH inequality: violation of the inequality at π/3: |1+2(1/2)-(-1/2)|=2.5>2!

12. 12 violation of Bell’s inequality maximum violation at π/3! 0 2 π π/2 π/3 θ

13. 13 remember!Bell’s inequality remember! Bell’s inequality is only maths! physics (QM) often violates it!

14. 14 quantum mysteries for everybody! D. Mermin, Am. J. Phys. 49, 940 (1981)

15. 15 pedestrian’s set up! e particle source e particle source 32 1 entangled particle source and A & B detectors: entangled particle source and A & B detectors: public language: three settings (1,2,3) & two flash Red or Green our language: dirs of measurement (0, -π/3,+π/3) & up or down) settings:two results: three settings: 1,2,3 and two results: Red or Green

16. 16 classical correlations identical hidden variables: particles carry identical instruction sets (eight possibilities) RRR, RGG, GRG,GGR, GRR, RGR, RRG, GGG e.g. if RGG then: for 12 RG, for 23 GG, for 13 RG e.g. if RGG then: for 12 RG, for 23 GG, for 13 RG the same are 100% of the time the same are 100% of the time prob to be the same =1/3 (prob no smaller than 1/3) e.g. if RRR then: for 12 RR, for 23 RR, for 13 RR e.g. if RRR then: for 12 RR, for 23 RR, for 13 RR prob to flash same colour prob to flash same colour can never be smaller than 1/3 SAME (TWO) DIFFR(SIX) this is Bell’s inequality

17. 17 quantum correlations entangled entangled particles have prob=cos 2 ( θ/2) to flash the same colour (why?), for θ=0, -120, 120 we have prob=1, ¼, ¼ to flash the same colour, respectively 1 2 3 1 2 31 2 3 the quantum prob=1/4 is smaller than 1/3 violating classical statistics!

18. 18 our world is non-local! Einstein: quantum physics is incomplete (EPR paradox) Bell: quantum physics violates mathematical inequalities (Bell’s inequalities) experiment showed: Bell is right! (non-local quantum correlations exist) superposition & entanglement

19. 19 end of lecture quantum mysteries revisited: entanglement is the key! non-locality quantum correlations: violate Bell’s inequalities (neither fast communication nor common preparation) superposition of distant states was verified in experiments via violation of Bell’s inequalities quantum world: neither deterministic nor local!