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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. Outline. “Entanglement is a purely quantum phenomenon” - PowerPoint PPT Presentation

### Transcript of Brownian Entanglement: Entanglement in classical brownian motion

• 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.