entanglement of indistinguishable particles
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Entanglement of indistinguishable particles. Libby Heaney Paraty Workshop, 2009. Particle entanglement. Entanglement usually considered between degrees of freedom of two or more well separated quantum systems. Hilbert space has a tensor product structure. - PowerPoint PPT PresentationTRANSCRIPT
Atom and Laser Physics Group,Oxford Physics
September 8th, 2009
Entanglement of indistinguishable particles
Libby Heaney
Paraty Workshop, 2009
September 8th, 2009Entanglement of indistinguishable particles
Particle entanglement
Entanglement usually considered between degrees of freedom of two or more well separated quantum systems.
Hilbert space has a tensor product structure.
Entanglement is assigned to the state alone.
Indistinguishable particles
Identical particles whose wavefunctions overlap in space.
Hilbert space no longer has the tensor product structure required to correctly define entanglement.
Cannot assign to either particle a specific set of degrees of freedom.
September 8th, 2009Entanglement of indistinguishable particles
P. Zanardi, PRA 65 042101 (2002)
Anti-symmetrised state of two fermions
Entanglement of indistinguishable particles
Two methods for defining entanglement of indistinguishable particles:
Include detection process in the definition of particle entanglement.Tichy, et al. arXiv:0902.1684v3.
Use the formalism of second quantisation and consider entanglement of modes.
e.g. P. Zanardi, PRA 65 042101 (2002), Ch. Simon, PRA 66 052323 (2002), … J. Goold, et al. PRA 80 022338 (2009).
Is mode entanglement as genuine as particle entanglement? LH and V. Vedral, arXiv:0907.5404v1.
September 8th, 2009Entanglement of indistinguishable particles
ENTANGLEMENT OF IDENTICAL PARTICLES AND THE DETECTION PROCESS
September 8th, 2009Entanglement of indistinguishable particles
Entanglement of identical particles
Assign identity to particles by including the detection process. A priori entanglement of the state is the distinguishable particle entanglement.
Physical entanglement – apply an entanglement measure to the above state.
Note for indistinguishable particles there is a non-zero probability of detecting
both particles in the same region of space.
September 8th, 2009Entanglement of indistinguishable particles
Tichy, de Melo, Kus, Mintert and Buchleitner, arXiv:0902.1684v3
Entanglement of indistinguishable particles
September 8th, 2009Entanglement of indistinguishable particles
distinguishable
indistinguishable
distinguishableIndistinguishable
MODE ENTANGLEMENT
September 8th, 2009Entanglement of indistinguishable particles
Mode entanglement
September 8th, 2009Entanglement of indistinguishable particles
Simple example of mode entanglement
Entanglement between two spatial modes occupied by a single particles.
In second quantisation:
September 8th, 2009Entanglement of indistinguishable particles
1st quantisation:Superposition of A and B.
2nd quantisation:Entanglement of A and B.
Is mode entanglement genuine entanglement? For photons it is generally accepted that mode
entanglement is as genuine as particle entanglement.
» Tan et al PRL 66 252 (1991).
» Hessmo et al, PRL 92 180401 (2004).
» Van Enk, PRA 72 064306 (2006).
No experiments have tested mode entanglement of massive particles.
Disputed whether mode entanglement of massive particles is genuine, due to a particle number superselection rule.
September 8th, 2009Entanglement of indistinguishable particles
Since the correlations of entanglement are basis independent, to verify entanglement requires measurements in at least two bases.
For mode entanglement, one measurement setting could be the particle
number basis, but what about another measurement setting?
Particle number superselection rule
September 8th, 2009Entanglement of indistinguishable particles
A B
Implies creation or destruction of particles:is forbidden for an isolated system.
Overcoming the particle number superselection rule Locally overcome the particle number superselection rule by exchanging
particles with a particle reservoir. Eg. Dowling et al. Phys. Rev. A, 74, 052113 (2006), see also Bartlett, et al., Rev. Mod.
Phys. 79 555 (2007).
LH and J. Anders, PRA 80 032104 (2009), S.-W. Lee, LH and D. Jaksch, In preparation.
September 8th, 2009Entanglement of indistinguishable particles
MODE ENTANGLEMENT OF MASSIVE PARTICLES IS USEFUL FOR QUANTUM COMMUNICATION
September 8th, 2009Entanglement of indistinguishable particles
Dense coding protocol
System: Maximally entangled Bell state.
Encoding: Alice acts on her qubit to encode one of four messages.
Alice sends her qubit to Bob.
Decoding: Bob performs Bell state analysis to recover which of the four messages Bob transmitted.
September 8th, 2009Entanglement of indistinguishable particles
Classically, i.e. with bits, one can send 2 messages per use of the channel, C=1.
Quantum mechanically, i.e. with qubits (and by utilizing entanglement), one can send 4 messages per use of the channel, C=2.
Dense coding with mode entanglement
System – double well formed of tightly confined potentials:
September 8th, 2009Entanglement of indistinguishable particles
LH and V. Vedral, arXiv:0907.5404v1
A single particle is initialised in the state:
Dense coding with mode entanglement Encoding (X and Z operations on mode A): Here no coupling between modes (J=0) – Alice acts solely on her mode.
September 8th, 2009Entanglement of indistinguishable particles
Z operation: X operation:
Apply a potential bias to mode A.
Shared BEC reservoir:
Dense coding with mode entanglement
September 8th, 2009Entanglement of indistinguishable particles
(1) Exchange of particles between the BEC
and mode B.
(1) Interaction between modes: Drive
bosons to the hardcore limit - they
behave like Fermions. Allow tunneling
so that the particles exchange
positions.
(1) Couple both modes to BEC to rotate to
the particle number basis (eliminates
the BEC phase).
(2) Read out: The four outcomes,
|00>, |01>, |10> and |11>
correspond to the four Bell states.
Summary
Entanglement between the degrees of freedoms of indistinguishable particles is meaningful if one takes the detection process into account.
Indistinguishability can even generate entanglement between particles that have no a priori entanglement.
A tensor product Hilbert space is recovered by considering entanglement of modes occupied by particles.
The particle number superselection rule can be locally overcome by coupling to a reservoir Bose-Einstein condensate.
Mode entanglement of massive particles can, in principle, be used as a resource for quantum communication.
September 8th, 2009Entanglement of indistinguishable particles
THANK YOU FOR LISTENING
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September 8th, 2009Entanglement of indistinguishable particles