Brownian Entanglement: Entanglement in classical brownian motionDr. Theo M. Nieuwenhuizen Institute for Theoretical Physics University of AmsterdamFluctuations, information flow and experimental measurementsParis, 27 Jan 2010

OutlineEntanglement is a purely quantum phenomenon

Quantum entanglement

Definition of classical entanglement

Examples

Conclusion

EntanglementQuantum caseNon-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

implies

Therefore if , then

This holds also for a mixture Thus entanglement is present when

for at least one of the casesQuantum entanglement and uncertainty relations

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

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 variableThen also u(x,t) is a random variable

Joint distribution:

Of course:

Brownian uncertainty relations and entanglement for N=2The relation

implies

Hence uncertainty relation:

N=2: Absence of entanglement iff

But entanglement occurs iffor at least one of the cases

Harmonic interaction with |g|

Situations with entanglementIn equilibrium, if |g|

Summary Entanglement due to uncertainty relations on Brownian timescalesNo 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 PRA05

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