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Entanglement of indistinguishable particles

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Entanglement of indistinguishable particles

Libby Heaney

Paraty Workshop, 2009

- 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

- 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

- 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).

- Include detection process in the definition of particle entanglement.
- Is mode entanglement as genuine as particle entanglement?
LH and V. Vedral, arXiv:0907.5404v1.

Entanglement of indistinguishable particles

Entanglement of indistinguishable 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

Indistinguishable

distinguishable

distinguishable

indistinguishable

Entanglement of indistinguishable particles

Entanglement of indistinguishable particles

- Another approach to define entanglement of indistingiushable particles is to move into second quantisation formalism.
- Energy modes
- Spatial modes

Entanglement of indistinguishable particles

- 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

- 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).

Entanglement of indistinguishable particles

A

B

- 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

- 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).

Entanglement of indistinguishable particles

Entanglement 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.

- 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

- 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

- 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

- 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

- 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.

Entanglement of indistinguishable particles

Entanglement of indistinguishable particles