epr paradox and bell inequality -...

EPR paradox and Bell inequality

Antonın Cernoch

RCPTM, Joint Laboratory of Optics of Palacky University and Physical Institute of Academyof Sciences of the Czech Republic

A. Cernoch (RCPTM/JLO) EPR & Bell inequality 31.8.2016 1 / 12

Quantum physics

MilestonesPlanc - black-body radiation

Einstein - photoelectric effectHeisenberg - uncertainty principle

Compton, Raman, Zeeman, Millikan, Bohr, Moseley, Debye,Sommerfeld, de Broglie, Schrodinger, Born, von Neumann, Dirac,Fermi, Pauli, von Laue, Dyson, Hilbert, Wien, Bose, Sommerfeld, . . .

Two main principlesrandomnesssuperposition principle


Copenhagen interpretationNiels Bohr and Werner Heisenberg

Wavefunctionphysical systems have not definite propertiesbefore they are measuredquantum mechanics only predict the probabilitiesof measurements outputswavefunction is non-localmeasurement changes the system→ collapse ofwavefunction

Other theoriesmany-worlds interpretationthe De Broglie-Bohm (pilot-wave) interpretationquantum decoherence theorie


EPR paradox

Albert Einstein, Boris Podolsky and Nathan RosenCan quantum-mechanical description of physical reality beconsidered complete? Phys. Rev. 47, 777 (1935)

wave function does not provide a completedescription of physical reality”element of reality” determines the measurementresultsthese hidden variables are local

”God does not play dice withthe universe”


Bell inequality

John Stewart Bell ”On the Einstein Podolsky Rosen Paradox”.Physics 1, 195–200 (1964)

local realism cannot reproduce all the predictionsof quantum mechanical theory

If EPR are rightdifferent measurements (a, b, c) on two distant particles (A and B)measurement results on A is not affected by setting ofmeasurement on B and vice versameasurement result ±1statistics over many realizations→ correlation Cthen C(a, c)− C(b,a)− C(b, c) ≤ 1


CHSH inequality

John Clauser, Michael Horne, Abner Shimony and Richard A. Holt”Proposed experiment to test local hidden-variable theories”. Phys. Rev. Lett. 23,880–4 (1969)

generalization of Bell inequality – works also for not-perfectlycorrelated (anti-correlated) pair

−2 ≤ C(a,b)− C(a,b′) + C(a′,b) + C(a′,b′) ≤ 2 ∀a,b

C =N++ − N+− − N−+ + N−−N++ + N+− + N−+ + N−−

1 2

1. photon 2. photon

PBS PBS

+1

−1

+1

−1


Proof

four random independent variables a,a′,b,b′

with possible values ±1B = ab + ab′ + a′b − a′b′

a +1 +1 +1 +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 −1 −1a′ +1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 −1 −1 −1b +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1b′ +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1B +2 +2 −2 −2 +2 −2 +2 −2 −2 +2 −2 +2 −2 −2 +2 +2

B = ±2

statistical outcomes⇒

−2 ≤ 〈B〉 ≤ +2


Previous measurements

quantum limit

2√

2 ≈ 2.8284271

Poh et al., Singapore

2.82759± 0.00051

my yesterdays value

2.599± 0.015


Standard deviation σ

longer accumulation of data→ higher precision (µ/σ)


Bell states

Maximally entangled states

|Φ+〉 = 1√2

(|HH〉+ |VV 〉) |Ψ+〉 =1√2

(|HV 〉+ |VH〉)

|Φ−〉 = 1√2

(|HH〉 − |VV 〉) |Ψ−〉 =1√2

(|HV 〉 − |VH〉)

Angles which maximize Bell inequality violation→ Bmax

−C(0◦,22.5◦) + C(0◦,67.5◦) + C(45◦,22.5◦) + C(45◦,67.5◦)4 coincidence measurement for each correlation⇒ 16 measurementsif you change measurement basis by HWP – rotate it by half angle!


Table of HWP settings

C(0◦,22.5◦)

ϑ1 ϑ2N++ 0◦ 11.25◦

N+− 0◦ 56.25◦

N−+ 45◦ 11.25◦

N−− 45◦ 56.25◦

C(45◦,22.5◦)

ϑ1 ϑ2N++ 22.5◦ 11.25◦

N+− 22.5◦ 56.25◦

N−+ 67.5◦ 11.25◦

N−− 67.5◦ 56.25◦

C(0◦,67.5◦)

ϑ1 ϑ2N++ 0◦ 33.75◦

N+− 0◦ 78.75◦

N−+ 45◦ 33.75◦

N−− 45◦ 78.75◦

C(45◦,67.5◦)

ϑ1 ϑ2N++ 22.5◦ 33.75◦

N+− 22.5◦ 78.75◦

N−+ 67.5◦ 33.75◦

N−− 67.5◦ 78.75◦


Bell factor calculation

Bmax

Nu = N++ − N+− − N−+ + N−−Nl = N++ + N+− + N−+ + N−−

C =Nu

Nl

Bmax = −C1 + C2 + C3 + C4

σ(Bmax )

σ2(Nu) = σ2(Nl) =∑

σ2(N±±)

σ2(C) =1

N4l

(N2u + N2

l )σ2(Nu)

σ2(Bmax ) =4∑

n=1

σ2(Cn)


epr paradox and bell inequality -...

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