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
Jan 01, 2016
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.