quantum mysteries again!

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

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! EPR paradox (1935) or quantumnon-locality? “strange action at a distance” or common state?

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

4. θ 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

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

6. θ θ 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)

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

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

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

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

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

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

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

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

15. 1 2 3 pedestrian’s set up! 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) e particle source three settings:1,2,3 and two results:Red or Green

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

17. 1 2 3 1 2 3 quantum correlations entangled particles have prob=cos2 (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 the quantum prob=1/4 issmallerthan1/3 violating classical statistics!

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