Brownian Entanglement: Entanglement in classical brownian motion

1. Brownian Entanglement:Entanglement in classical brownian motion Dr. Theo M. Nieuwenhuizen Institute for Theoretical PhysicsUniversity of Amsterdam Fluctuations, information flow and experimental measurements Paris, 27 Jan 2010

2. Outline “Entanglement is a purely quantum phenomenon” Quantum entanglement Definition of classical entanglement Examples Conclusion

3. Entanglement • Quantum case • Non-entangled pure state • Non-entangled mixed state • In terms of Wigner functions • In classical physics one always has • Only entanglement if is not allowed distribution. • This happens if there are uncertainty relations between x and p

4. Quantum entanglement and uncertainty relations • implies • Therefore if , then • This holds also for a mixture Thus entanglement is present when for at least one of the cases

5. Paul Langevin dynamics and coarse grained velocities Forward Kolmogorov Average coarse grained velocities Departure velocity: overdamped Newtonian Arrival velocity: extra kick Ed Nelson: Osmotic velocity:

6. Ensemble view for N particles • : ensemble of all trajectories through N-dim point x at time t, • embedded with prob. density P(x,t) in ensemble of all configs. • In this sense, x is a random variable • Then also u(x,t) is a random variable • Joint distribution: • Of course:

7. Brownian uncertainty relations and entanglement for N=2 The relation implies Hence uncertainty relation: N=2: Absence of entanglement iff But entanglement occurs if for at least one of the cases

8. Explicit cases for entanglement • Harmonic interaction with |g|<a • Same T; • Distribution remains • Gaussian, if initially • Osmotic velocities • I f , then sufficient condition for entanglement is:

9. Situations with entanglement • In equilibrium, if |g|<a but , any T • Particles interact for t <0, but g=0 for t >0 • Brownianentanglement sudden death:No entanglement for large t • a=0: Entanglement, not present at t=0, can exist in interval

10. Summary • Entanglement due to uncertainty relations on Brownian timescales • No entanglement in Newtonian regime (few collisions of “water molecules” with “tea particle”) • Entanglement occurs for osmotic velocity u defined in terms of ensemble of all (N=2) particles: • It does not exist when each u_j is defined in terms of ensemble of trajectories of particle j alone • Paper: Brownian Entanglement: Allahverdyan, Khrennikov, Nh PRA’05

11. Conclusion Entanglement can exist in classical physics. Examples also known in laser physics. Quantum entanglement is a purely quantum phenomenon