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

Entanglement of indistinguishable particles

Libby Heaney

Paraty Workshop, 2009


Particle entanglement

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

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 particles1

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 IDENTICAL PARTICLES AND THE DETECTION PROCESS

Entanglement of indistinguishable particles


Entanglement of identical 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 particles2

Entanglement of indistinguishable particles

Indistinguishable

distinguishable

distinguishable

indistinguishable

Entanglement of indistinguishable particles


Mode entanglement

MODE ENTANGLEMENT

Entanglement of indistinguishable particles


Mode entanglement1

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

    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

    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


    Particle number superselection rule

    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

    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

    MODE ENTANGLEMENT OF MASSIVE PARTICLES IS USEFUL FOR QUANTUM COMMUNICATION

    Entanglement of indistinguishable particles


    Dense coding protocol

    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

    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 entanglement1

    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 entanglement2

    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

    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 listening any questions

    THANK YOU FOR LISTENINGANY QUESTIONS?

    Entanglement of indistinguishable particles


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