EPR paradox and Bell inequality

Anton´ın Cernochˇ

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

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 1 / 12 Quantum physics

Milestones Planc - black-body radiation Einstein - photoelectric effect Heisenberg - 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 principles randomness superposition principle

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 2 / 12 Copenhagen interpretation Niels Bohr and Werner Heisenberg Wavefunction physical systems have not definite properties before they are measured quantum mechanics only predict the probabilities of measurements outputs wavefunction is non-local measurement changes the system collapse of wavefunction →

Other theories many-worlds interpretation the De Broglie-Bohm (pilot-wave) interpretation quantum decoherence theorie

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 3 / 12 EPR paradox

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

wave function does not provide a complete description of physical reality ”element of reality” determines the measurement results these hidden variables are local ”God does not play dice with the universe”

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 4 / 12 Bell inequality

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

local realism cannot reproduce all the predictions of quantum mechanical theory

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

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 5 / 12 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-perfectly correlated (anti-correlated) pair 2 C(a, b) C(a, b0) + C(a0, b) + C(a0, b0) 2 a, b − ≤ − ≤ ∀ N++ N+− N−+ + N−− C = − − N++ + N+− + N−+ + N−− 1 1 2 − +1 +1

1. photon 2. photon

1 PBS PBS − A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 6 / 12 Proof

four random independent variables a, a0, b, b0 with possible values 1 ± B = ab + ab0 + a0b a0b0 −

a +1 +1 +1 +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 −1 −1 a0 +1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 −1 −1 −1 b +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 b0 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 B +2 +2 −2 −2 +2 −2 +2 −2 −2 +2 −2 +2 −2 −2 +2 +2

B = 2 ± statistical outcomes ⇒ 2 B +2 − ≤ h i ≤

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 7 / 12 Previous measurements

quantum limit

2√2 2.8284271 ≈ Poh et al., Singapore

2.82759 0.00051 ± my yesterdays value

2.599 0.015 ±

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 8 / 12 Standard deviation σ

longer accumulation of data higher precision (µ/σ) → A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 9 / 12 Bell states

Maximally entangled states

1 Φ+ = √1 ( HH + VV ) Ψ+ = ( HV + VH ) | i 2 | i | i | i √2 | i | i 1 Φ− = √1 ( HH VV ) Ψ− = ( HV VH ) | i 2 | i − | i | i √2 | i − | i

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 measurements ⇒ if you change measurement basis by HWP – rotate it by half angle!

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 10 / 12 Table of HWP settings

C(0◦, 22.5◦) C(0◦, 67.5◦)

ϑ1 ϑ2 ϑ1 ϑ2 ◦ ◦ ◦ ◦ N++ 0 11.25 N++ 0 33.75 ◦ ◦ ◦ ◦ N+− 0 56.25 N+− 0 78.75 ◦ ◦ ◦ ◦ N−+ 45 11.25 N−+ 45 33.75 ◦ ◦ ◦ ◦ N−− 45 56.25 N−− 45 78.75

C(45◦, 22.5◦) C(45◦, 67.5◦)

ϑ1 ϑ2 ϑ1 ϑ2 ◦ ◦ ◦ ◦ N++ 22.5 11.25 N++ 22.5 33.75 ◦ ◦ ◦ ◦ N+− 22.5 56.25 N+− 22.5 78.75 ◦ ◦ ◦ ◦ N−+ 67.5 11.25 N−+ 67.5 33.75 ◦ ◦ ◦ ◦ N−− 67.5 56.25 N−− 67.5 78.75

A. Cernochˇ (RCPTM/JLO) EPR & Bell inequality 31.8.2016 11 / 12 Bell factor calculation

Bmax σ(Bmax )

2 2 X 2 Nu = N++ N+− N−+ + N−− σ (N ) = σ (N ) = σ (N ) − − u l ±± Nl = N++ + N+− + N−+ + N−− N C = u N 1 l σ2( ) = ( 2 + 2) σ2( ) C 4 Nu Nl Nu Nl Bmax = C + C + C + C − 1 2 3 4 4 2 X 2 σ (Bmax ) = σ (Cn) n=1

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