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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)
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singlet state (EPR pair)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 quantumquantum non-localitynon-locality? “strange action at a distance” or common statecommon state??
2sin
g
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quantum vs. classical quantum vs. classical correlationscorrelations
fast communicationfast communication (via exchange of messages) (via exchange of messages)oror
common preparationcommon preparation (via hidden variables) (via hidden variables)
what are correlations due to?what are correlations due to?
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2 spins in the singlet state2 spins in the singlet state
if spin 1 is up & spin 2 is down in the z-dir
1
1
)2(
)1(
S
S z
2cos,
2sin
2cos
2sin
2sin
2cos
1
0
)()(
)()(
)()(
nn
i
n
i
n
n
n
n
n
cc
ec
ec
cc
if spin 1 is up in the z-dir the spin 2 is down in the n-direction with angle θ
θ
2
cos,2
sin 22 PP
1
1
)2(
)1(
S
S z
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quantum correlation functionquantum correlation function
measure the spins in two directions with angle θ
1
1
)2(
)1(
S
S z
2sin
2cos
2cos
2sin
2
2
2
2
P
P
P
P
cos
2cos
2sin2
2sin
2cos2
)1)(1()1)(1()1)(1()1)(1(
)(
22
22
)(2)(1
PPPP
PPPP
PPPP
PPPP
SSC z
θ
0cos1
)1)(1()1)(1(2
1
)0(.. )(2)(1
zz SSCge
remember, the mean value of remember, the mean value of SSzzSSθθ is is
taken on the singlet (entangled) statetaken on the singlet (entangled) state
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classical correlation functionclassical 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)
1
1
)2(
)1(
S
S z
P
P
P
P
1
1
12
112
1
)( )(2)(1
PPPP
PPPP
SSCclaszclas
θ
θ
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quantum vs. classical quantum vs. classical
12
)( )(2)(1
claszclas SSC
cos
)( )(2)(1
quazqua SSC
11
-1-1
quantum correlations are stronger than classical quantum correlations are stronger than classical (Bell showed QM can go out of mathematical (Bell showed QM can go out of mathematical
limits!!!)limits!!!)
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measure one spinmeasure one spin
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• measure the spin in various directions (settings) with results (in units of )1
1
1
a
a in z-dirin z-dirin n-dir in n-dir (θ)(θ)
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……measure both spins (in a singlet)measure both spins (in a singlet)
2
• measure the spin in various directions (settings) with results (in units of ) at locations A and B1
1
1
b
b
1
1
a
a in z-dirin z-dirin n-dir in n-dir (θ)(θ)
location A location B
in z-dirin z-dirin n-dir in n-dir ((--θ)θ)
consider now the linear combination of correlationsconsider now the linear combination of correlations
)'(')'(
''''
bbabba
babaababg
how many possible results? what are they? how many possible results? what are they? 16 g= +2 or -2 16 g= +2 or -2
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Bell-CHSH inequalityBell-CHSH inequality
mean correlationsmean correlations
λλ: hidden variable: hidden variable
2'''' babaababgS
dbaab
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quantum correlation function quantum correlation function violates it!violates it!
cossinsin babaab gg
Bell- CHSH inequality:Bell- CHSH inequality:
22coscos21
22coscoscos0cos
2''''
babaabab
violation of the inequalityviolation of the inequality at at π/3π/3: : |1+2(1/2)-(-1/2)|=2.5>2!|1+2(1/2)-(-1/2)|=2.5>2!
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violationviolation of of Bell’s inequalityBell’s inequality
2S
maximum violation at maximum violation at π/3π/3!!
00
22
πππ/π/22π/3π/3θθ
)2cos()cos(21 S
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remember!remember! Bell’s inequalityBell’s inequalityis only maths!is only maths!
physics (QM) often violates it!
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quantum mysteries for everybody!
D. Mermin, Am. J. Phys. 49, 940 (1981)
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pedestrian’s set up!pedestrian’s set up!
e particle sourcee particle source
32
1
32
12sin
g
entangled particle sourceentangled particle source and A & B detectors: and A & B detectors: public language: three settings (1,2,3) & two flash Red or Greenour language: dirs of measurement (0, -π/3,+π/3) & up or down)
three settingssettings:: 1,2,3 and two two resultsresults:: Red or Green
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classical correlationsclassical correlations hidden variables: particles carry identicalidentical instruction sets (eight possibilities)
RRR, RRRR, RGGGG, , GGRRGG,,GGGGR, R, GGRR, RRR, RGGR, RRR, RRGG, , GGGGGG
e.g. if e.g. if RRGG GG then: for 12then: for 12 RRG, G, for 23for 23 GG, GG, for 13for 13 RRGG
the same are 100% of the timethe same are 100% of the time
prob to be the same =1/3 (prob to be the same =1/3 (prob no smaller than 1/3)prob no smaller than 1/3)
e.g. if e.g. if RRRRRR then: for 12then: for 12 RRRR, , for 23for 23 RR,RR, for 13for 13 RRRR
prob to flash same colour prob to flash same colour can never be smaller than 1/3
SAME SAME (TWO)(TWO)
DIFFRDIFFR(SIX)(SIX)
this is Bell’s inequalitythis is Bell’s inequality
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quantum correlationsquantum correlations entangledentangled particles have prob=cos2 ( θ/2) to flash the same colour (why?), for θ=0, -120, 120 we haveprob=1, ¼, ¼ to flash the same colour, respectively
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1
4
14
11
4
14
1
4
11
1 2 31 2 3
1 1 2 2 3 3
the quantum prob=1/4 the quantum prob=1/4 isis smallersmaller thanthan 1/31/3 violating classical statistics!violating classical statistics!
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our world is non-local!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)
BA
2121 BBAA
321321 BBBAAA
superposition & entanglement
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end of lectureend of lecturequantum mysteries revisited:quantum mysteries revisited:
entanglement is the key!
non-localitynon-locality
quantum correlations: violate Bell’s inequalities (neither fast communication nor common preparation)
superposition of distant states
was verified in experimentswas verified in experimentsvia violation of Bell’s inequalitiesvia violation of Bell’s inequalities
quantum world: neither deterministic nor local!
