Brownian Entanglement: Entanglement in classical brownian motion

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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 caseNonentangled pure state
Nonentangled 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 Ndim 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.