1. Brownian Entanglement: Entanglement in classical brownian motion Dr. Theo M. Nieuwenhuizen Institute for Theoretical Physics University 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. implies Therefore if, then This holds also for a mixture Thus entanglement is present when for at least one of the cases Quantum entanglement and uncertainty relations

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

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

9. Situations with entanglement In equilibrium, if |g|<a but, any T Particles interact for t 0 Brownian entanglement 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 Quantum entanglement is a purely quantum phenomenon Entanglement can exist in classical physics. Examples also known in laser physics.