Post on 01-Jan-2016
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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”
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.