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.

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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.

  • 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


    A

    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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