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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
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