Entanglement of indistinguishable particles

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.


P. Zanardi, PRA 65 042101 (2002)

Anti-symmetrised state of two fermions


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.


ENTANGLEMENT OF IDENTICAL PARTICLES AND THE DETECTION PROCESS


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.


Tichy, de Melo, Kus, Mintert and Buchleitner, arXiv:0902.1684v3



distinguishable

indistinguishable

distinguishableIndistinguishable

MODE ENTANGLEMENT


Mode entanglement


Simple example of mode entanglement

Entanglement between two spatial modes occupied by a single particles.

In second quantisation:


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.


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


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.


MODE ENTANGLEMENT OF MASSIVE PARTICLES IS USEFUL FOR QUANTUM COMMUNICATION


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.


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:


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.


Z operation: X operation:

Apply a potential bias to mode A.

Shared BEC reservoir:

Dense coding with mode entanglement


(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.


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Entanglement of indistinguishable particles

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