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
Quantum entanglement and uncertainty relations
•
implies
• Therefore if
, then
• This holds also for a mixture
Thus entanglement is present when
for at least one of the cases
Paul Langevin dynamics and coarse grained velocities
Forward Kolmogorov
Average coarse grained velocities
Departure velocity: overdamped Newtonian
Arrival velocity: extra kick
Ed Nelson:
Osmotic velocity:
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 if
for at least one of the cases
Explicit cases for entanglement
• Harmonic interaction
with |g|<a
• Same T;
• Distribution remains
Gaussian, if initially
• Osmotic
velocities
• I f
, then sufficient condition for entanglement is:
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
Entanglement can exist in classical physics.
Examples also known in laser physics.
Quantum entanglement is a purely quantum phenomenon