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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.

Entanglement of indistinguishable particles

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.

Anti-symmetrised state of two fermions

P. Zanardi, PRA 65 042101 (2002)

Entanglement of indistinguishable particles

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 66052323 (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 indistinguishable particles

ENTANGLEMENT OF IDENTICAL PARTICLES AND THE DETECTION PROCESS

Entanglement 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.

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

Entanglement of indistinguishable particles

Entanglement of indistinguishable particles

Indistinguishable

distinguishable

distinguishable

indistinguishable

Entanglement of indistinguishable particles

MODE ENTANGLEMENT

Entanglement of indistinguishable particles

Mode entanglement

- Another approach to define entanglement of indistingiushable particles is to move into second quantisation formalism.
- Energy modes
- Spatial modes
- Entanglement may exist between modes occupied by particles.

Entanglement of indistinguishable particles

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.

Entanglement of indistinguishable particles

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.

Entanglement of indistinguishable particles

B

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?

Implies creation or destruction of particles:

is forbidden for an isolated system.

Entanglement of indistinguishable particles

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 andD. Jaksch, In preparation.

Entanglement of indistinguishable particles

MODE ENTANGLEMENT OF MASSIVE PARTICLES IS USEFUL FOR QUANTUM COMMUNICATION

Entanglement of indistinguishable particles

Dense coding protocol

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.

- 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.

Entanglement of indistinguishable particles

Dense coding with mode entanglement

- System – double well formed of tightly confined potentials:

- A single particle is initialised in the state:

LH and V. Vedral, arXiv:0907.5404v1

Entanglement of indistinguishable particles

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:

Shared BEC reservoir:

Apply a potential bias to mode A.

Entanglement of indistinguishable particles

Dense coding with mode entanglement

- Exchange of particles between the BEC and mode B.
- Interaction between modes: Drive bosons to the hardcore limit - they behave like Fermions. Allow tunneling so that the particles exchange positions.
- Couple both modes to BEC to rotate to the particle number basis (eliminates the BEC phase).
- Read out: The four outcomes,
- |00>, |01>, |10> and |11>
- correspond to the four Bell states.

- Alice sends her mode to Bob.
- Decoding (Bob performs complete Bell state analysis on both modes):

Entanglement of indistinguishable particles

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.

Entanglement of indistinguishable particles

THANK YOU FOR LISTENINGANY QUESTIONS?

Entanglement of indistinguishable particles

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