Revealing anyonic statistics by multiphoton entanglement

1. Revealing anyonic statistics by multiphoton entanglement Jiannis K. Pachos Witlef Wieczorek Christian Schmid Nikolai Kiesel Reinhold Pohlner Harald Weinfurter arXiv:0710.0895 QEC07, USC, December 2007

2. Criteria for Topo Order and TQC (CM) [G. Brennen and J.K.P., Proc. Roy. Soc. A, 2007] • Initialization (creation of highly entangled state with TO) • Dynamically, adiabatically, cooling, (possibly H) • Addressability (anyon generation, manipulation) • Trapping, adiabatic transport, pair creation & annihilation. • Measurement (Topological entropy of ground state, interference of anyons) • Scalability (Large system, many anyons, desired braiding) • Low decoherence (Temperature, impurities, anyon identification) [I. Cirac, S. Simon, private communication]

3. Toric Code: ECC Consider the lattice Hamiltonian p s s p p p Spins on the vertices. Two different types of plaquettes, p and s, which support ZZZZ or XXXX interactions respectively. The four spin interactions involve spins at the same plaquette. s s p X1 s X4 X2 X3

4. Toric Code: ECC Consider the lattice Hamiltonian p s s Indeed, the ground state is: p p p s s p The |00…0> state is a ground state of the ZZZZ term and the (I+XXXX) term projects that state to the ground state of the XXXX term. X1 s X4 X2 [F. Verstraete, et al., PRL, 96, 220601 (2006)] X3

5. Toric Code: ECC • Excitations are produced by Z or X rotations of one spin. • These rotations anticommute • with the X- or Z-part of the • Hamiltonian, respectively. • Z excitations on s plaquettes. • X excitations on p plaquettes. • X and Z excitations behave as anyons with respect to each other. p Z s s p p p s s X p X1 s X4 X2 X3

6. One Plaquette It is possible to demonstrate the anyonic properties with onesplaquette only. The Hamiltonian: 1 The ground state: 2 s 4 3 GHZ state!

7. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: Z1 1 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. 2 s 4 3

8. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: Z1 X1 1 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. 2 s 4 3

9. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: Z1 X1 1 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. X2 2 s 4 3

10. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: Z1 X1 1 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. X2 2 s 4 3 X3

11. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: Z1 X1 1 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. The final state is given by: X2 2 s X4 4 3 X3

12. Interference Process Create state With half Z rotation on spin 1, , one can create the superposition between a Z anyon and the vacuum: for . Then the X anyon is rotated around it: Then we make the inverse half Z rotation

13. Experiment

14. Experiment Qubit states 0 and 1 are encoded in the polarization, V and H, of four photonic modes. [J.K.P., W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, H. Weinfurter, arXiv:0710.0895]

15. Vacuum Counts Anyon Counts Experiment Qubit states 0 and 1 are encoded in the polarization, V and H, of four photonic modes. The states that come from this setup are of the form:  Measurements and manipulations are repeated over all modes.

16. Vacuum Correlations x-displacement: -7° and +2° Anyon Correlations y-displacement ~ EPRxEPR Experiment: State identification Consider correlations: Visibility > 64% Fidelity: Witness for genuine 4-qubit GHZ entanglement:

17. Experiment: Fusion Rules Fusion rules Generation of anyon: Fusion exe=1: (invariance of vacuum state) Fusion ex1=e: (Invariance of anyon state)

18. Experiment: Statistics Interference - loop around empty plaquette: - loop around occupied plaquette: - interference of the two processes:

19. 4 qubit GHZ stabilizers Properties and Applications • Invariance of vacuum w.r.t. to closed paths: • Useful for: • quantum error correction, • topological quantum memory, • quantum anonymous broadcasting • Implement Hamiltonian, larger systems… A C B [arXiv:0710.0895; J.K.P., Annals of Physics 2007, IJQI 2007]