1. Entanglement of indistinguishable particles
Libby Heaney
Paraty Workshop, 2009
September 8th, 2009

2. 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.
>> Define entanglement.
>> Give examples of when particle entanglement is a useful concept.
>> Here measurement of one particle can be unambiguously assigned to either region of space A or region of space B. Here the spatial regions A and B label the particle.
>> So even though particles are identical they are effectively distinguishable since there is only infinitesimal probability of finding them both in same region of space.
>> The Hilbert space is a tensor product of the Hilbert spaces of the subsystems. The Hilbert spaces are spanned by a basis set for the individual subsystems.
Entanglement of indistinguishable particles
September 8th, 2009

3. 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.
>> Define what we mean by indistinguishable particles. Identical particles whose de Broglie wavelengths are overlapping in space. Such as atoms in a BEC
>> Overlap occurs when the de Broglie wavelengths become more extended as the temperature drops or the particles may move closer together.
>>Identical particles have the same set of quantum numbers. May have different internal states, which could in principle be used to label the particles, but if detector only resolves spatial region, would still consider the particles to be indistinguishable.
>> Hilbert space is a projection on to the symmetric or antisymmetric subspaces of the hilbert space for the same two distinguishable particles.
>> Asym Hilbert space is spanned by all the states that acquire a phase shift pi upon exchange of two particles.
>> Sym subspace is spanned by all the states that acquire no phase upon exchange of the two particles.
Instance of a superselection rule, i.e. a fundamental limitation on the possibility to prepare the system in a given state.
>> Cannot identify a single particle with a specific factor space of the underlying Hilbert space, so that we cannot assign to the particle, even in principle, any set of degrees of freedom.
Anti-symmetrised state of two fermions
P. Zanardi, PRA (2002)
Entanglement of indistinguishable particles
September 8th, 2009

4. 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: v3.
Use the formalism of second quantisation and consider entanglement of modes.
e.g. P. Zanardi, PRA (2002), Ch. Simon, PRA (2002), … J. Goold, et al. PRA (2009).
Is mode entanglement as genuine as particle entanglement?
LH and V. Vedral, arXiv: v1.
Entanglement of indistinguishable particles
September 8th, 2009

5. ENTANGLEMENT OF IDENTICAL PARTICLES AND THE DETECTION PROCESS
Entanglement of indistinguishable particles
September 8th, 2009

6. 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.
>> Include detectors that unambiguously detect the internal degrees of freedom of particles on the left and right side of the system.
>> Sometimes one cannot unambiguously assign specific detectors to a particular particle.
Tichy, de Melo, Kus, Mintert and Buchleitner, arXiv: v3
Entanglement of indistinguishable particles
September 8th, 2009

7. Entanglement of indistinguishable particles
>> Take this paradigmatic a priori entanglement of internal degrees of freedom controlled by the parameter epsilon. Each particle is prepared in an external state denoted by A and B.
>>Since A and B are orthogonal, they could in principle be used to distinguish the particles but to do that the detectors would have to address the entire system and not just the left and right hand sides.
>> When the particles are distinguishable, the dotted curve, i.e. one particle is on the left and one is always on the right, we can see the concurrence just follows the a priori entanglement. For instance when epsilon is zero or pi/2 the concurrence is zero.
>> Indistinguishable particles the behaviour is quite different – detectors can still register entanglement even if the particles were prepared with no entanglement between their internal degrees of freedom to begin with. This is exciting because could generate entanglement by preparing one particle in a spin up and the other in spin down and allow their wavefunctions to overlap. But this only works probabilistically of course.
Entanglement of indistinguishable particles
September 8th, 2009

8. MODE ENTANGLEMENT Entanglement of indistinguishable particles
September 8th, 2009

9. 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.
>> consider bosons only here – massive or massless
>> Any confining volume has a set of single particle states – labeled by k, of energy Ek. Create a particle in one of the energy modes by applying the creation operator to the vacuum. The creation/annihilation operator satisfy the ccr.
>> Recover a tensor product structure in Hilbert space
>> Can also define a complete set of spatial modes. Transform the energy mode creation operator into position using the energy eigenfunctions. Psi(x) is the field operator for point x.
>> Populating one energy mode is equivalent to populating all points in space in a superposed manner.
Entanglement of indistinguishable particles
September 8th, 2009

10. 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
September 8th, 2009

11. Is mode entanglement genuine entanglement?
For photons it is generally accepted that mode entanglement is as genuine as particle entanglement.
Tan et al PRL (1991).
Hessmo et al, PRL (2004).
Van Enk, PRA (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.
>> The non-locality of a single photon was first predicted by Tan Walls and Collett, further modified by Lucien Hardy and subsequently by Dunningham and Vedral.
>> Experimentally verified by a swedish group in 2004.
>> A simple argument for why the entanglement of a single photon can be considered genuine was given by van Enk, describe picture.
Entanglement of indistinguishable particles
September 8th, 2009

12. 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? A B
Implies creation or destruction of particles:
is forbidden for an isolated system.
Entanglement of indistinguishable particles
September 8th, 2009

13. 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, (2006), see also Bartlett, et al., Rev. Mod. Phys (2007).
LH and J. Anders, PRA  (2009), S.-W. Lee, LH and D. Jaksch, In preparation.
Entanglement of indistinguishable particles
September 8th, 2009

14. MODE ENTANGLEMENT OF MASSIVE PARTICLES IS USEFUL FOR QUANTUM COMMUNICATION
Entanglement of indistinguishable particles
September 8th, 2009

15. 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
September 8th, 2009

16. 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: v1
Entanglement of indistinguishable particles
September 8th, 2009

17. 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.
>>Particles in the BEC trapped by a different internal state to the particles in mode A. Couple the two by using a Raman laser set up
Entanglement of indistinguishable particles
September 8th, 2009

18. 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):
Clear communincation – differentiate between Alice and Bob.
Entanglement of indistinguishable particles
September 8th, 2009

19. 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
September 8th, 2009

20. THANK YOU FOR LISTENING ANY QUESTIONS?
Entanglement of indistinguishable particles
September 8th, 2009