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

2. Overview In condensed matter anyons appear as ground or excited states of two dimensional systems: –Superconducting electrons in a strong magnetic field (Fractional Quantum Hall Effect) –Lattice systems (Kitaev’s toric code/hexagonal lattice, Wen’s models, Ioffe’s model, Freedman-Nayak- Shtengel model…) Energy gap protects anyons for applications Relatively large systems to have many anyons and move them in arbitrary paths

3. Overview

4. Anyons Anyons have non-trivial statistics. Bosons Fermions Anyons 3D 2D Consider as composite particles of fluxes and charges. Then phase is like the Aharonov-Bohm effect.

5. Criteria for Topo Order and TQC (CM) 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) [G. Brennen and J.K.P., Proc. Roy. Soc. A, 2007] [I. Cirac, S. Simon, private communication]

6. Criteria for Topo Order and TQC (CM) 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) [G. Brennen and J.K.P., Proc. Roy. Soc. A, 2007]

7. Criteria for Topo Order and TQC (CM) 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) Ocneanu rigidity: algebraic structure is rigid to small perturbations of the Hamiltonian [G. Brennen and J.K.P., Proc. Roy. Soc. A, 2007]

8. Toric Code: ECC Consider the lattice Hamiltonian p p p p s s s s s X1X1 X4X4 X3X3 X2X2 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. p

9. Toric Code: ECC Consider the lattice Hamiltonian p p p p s s s s Good quantum numbers:  eigenvalues of XXXX and ZZZZ terms are 1 and -1 Also Hamiltonian exactly solvable: s X1X1 X4X4 X3X3 X2X2 p

10. Toric Code: ECC Consider the lattice Hamiltonian p p p p p s s s s Indeed, the ground state is: 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. [F. Verstraete, et al., PRL, 96, 220601 (2006)] s X1X1 X4X4 X3X3 X2X2

11. Toric Code: ECC p p p p p s s s s 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. Z X s X1X1 X4X4 X3X3 X2X2

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

13. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 Energy of ground state 1 Z1Z1 Energy of excited state s

14. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 Note that the second anyon from the Z rotation is outside the system. Move an X anyon around the Z one. What we really want is the path that it traces and this can be spanned on the spins 1,2,3,4. 1 Z1Z1 s

15. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette. 1 Z1Z1 s

16. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 1 X1X1 Z1Z1 s Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette.

17. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 1 X1X1 X2X2 Z1Z1 s Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette.

18. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 1 X1X1 X2X2 X3X3 Z1Z1 s Assume there is an X anyon outside the system. With successive X rotations it can be transported around the plaquette.

19. One Plaquette One can demonstrate the anyonic statistics with only this plaquette. First create excitation with Z rotation at one spin: 4 2 3 1 X1X1 X2X2 X3X3 X4X4 Z1Z1 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: s

20. One Plaquette After a complete rotation of an X anyon around a Z anyon (two successive exchanges) the resulting state gets a phase (a minus sign): hence ANYONS with statistical angle A property we used is that A crucial point is that these properties can be demonstrated without the Hamiltonian!!! An interference experiment can reveal the presence of the phase factor. Anyonic statistics

21. Extended System From the formula 4 2 3 1 X1X1 X2X2 X3X3 X4X4 Z1Z1 s 5 6 7 8 9 the extended state is given by The two X, Z anyonic state is given by

22. 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

23. Interference Process That we obtained the state is due to the minus sign produced from the anyonic statistics. If it wasn’t there then we would have returned to the vacuum state. Distinguish between states: has even number of 1’s. has odd number of 1’s.

24. Experiment

25. 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]

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

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

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

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

30. Experiment

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