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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) 1 singlet state (EPR pair) sin g 2 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 quantum non-locality? “strange action at a distance” or common state? 2 quantum vs. classical correlations what are correlations due to? fast communication (via exchange of messages) or common preparation (via hidden variables) 3 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 θ n-direction with angle θ c( n ) n c( n ) n cos sin 0 (n) 2 c(n) 2 c i S 1 i S z (1) 1 e sin e cos z (1) 1 2 2 S 1 S ( 2) 1 ( 2) 2 2 c( n ) sin , c( n ) cos P sin , P cos 2 2 2 2 4 quantum correlation function measure the spins in two directions with angle θ θ C ( ) S1( z ) S 2 ( ) P sin 2 P cos P (1)( 1) P (1)( 1) P (1)( 1) P (1)( 1) P P P P P P P P P P P P 2 2 P cos 2 P sin 2 2 2 2 2 cos 2 sin 2 2 2 cos 2 sin 2 cos 2 2 2 S z (1) 1 S ( 2 ) 1 e.g. C (0) S1( z ) S 2( z ) 1 (1)( 1) (1)( 1) 2 1 cos 0 remember, the mean value of SzSθ is taken on the singlet (entangled) state 5 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) S 1 z (1) P θ θ P 1 P 1 P Cclas ( ) S1( z ) S 2( ) S ( 2 ) 1 clas P P P P P P P P 1 1 2 2 1 1 6 quantum vs. classical Cqua ( ) S1( z ) S 2( ) 1 qua cos Cclas ( ) S1( z ) S 2 ( ) -1 2 clas 1 quantum correlations are stronger than classical (Bell showed QM can go out of mathematical 7 limits!!!) measure one spin • measure the spin in various directions (settings) with results 1 (in units of 2 ) a 1 a 1 in z-dir in n-dir (θ) 8 …measure both spins (in a singlet) • measure the spin in various directions (settings) with results 1 (in units of 2 ) at locations A and B location A a 1 a 1 in z-dir in n-dir (θ) location B b 1 b 1 in z-dir in n-dir (-θ) consider now the linear combination of correlations g ab ab' a' b a' b' a(b b' ) a' (b b' ) how many possible results? 16 what are they?g= +2 or -2 9 Bell-CHSH inequality λ: hidden variable mean correlations ab a b d S g ab ab' a' b a' b' 2 10 quantum correlation function violates it! ab sin g a b sin g a b cos Bell- CHSH inequality: ab ab' a' b a' b' 2 cos 0 cos cos cos2 2 1 2 cos cos2 2 violation of the inequality at π/3: |1+2(1/2)-(-1/2)|=2.5>2! 11 violation of Bell’s inequality S 1 2 cos( ) cos(2 ) 2 S 2 0 π/3 π/2 maximum violation at π/3! π θ 12 remember! Bell’s inequality is only maths! physics (QM) often violates it! 13 quantum mysteries for everybody! D. Mermin, Am. J. Phys. 49, 940 (1981) 14 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) 1 sin g 1 2 e particle source 2 3 2 3 three settings: 1,2,3 and two results: Red or Green 15 classical correlations hidden variables: particles carry identical instruction sets (eight possibilities) RRR, RGG, GRG,GGR, GRR, RGR, RRG, GGG SAME e.g. if RRR then: for 12 RR, for 23 RR, for 13 RR the same are 100% of the time (TWO) DIFFR e.g. if RGG then: for 12 RG, for 23 GG, for 13 RG (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 16 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 1 2 3 1 2 3 1 1 4 1 4 1 4 1 1 4 1 4 1 4 1 the quantum prob=1/4 is smaller than 1/3 violating classical statistics! 17 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) A B superposition & entanglement A1 A2 B1B2 A1 A2 A3 B1B2 B3 18 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 non-locality was verified in experiments via violation of Bell’s inequalities 19