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”

Quantum entanglement

Definition of classical entanglement

Examples

Conclusion

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

• 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

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 variable

• Then also u(x,t) is a random variable

• Joint distribution:

• Of course:

Brownian uncertainty relations and entanglement for N=2

The 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|<a

• Same T;

• Distribution remains

Gaussian, if initially

• Osmotic velocities

• I f , then sufficient condition for entanglement is:

Explicit cases for entanglement

Situations with entanglement

• In equilibrium, if |g|<a but , any T

• Particles interact for t <0, but g=0 for t >0

• Brownian entanglement sudden death: No entanglement for large t

• a=0: Entanglement, not present at t=0, can exist in interval

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

Conclusion

Quantum entanglement is a purely quantum phenomenon

Entanglement can exist in classical physics. Examples also known in laser physics.