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Revealing Anyonic Statistics with Multiphoton Entanglement

This study explores the criteria for topological order and topological quantum computing, including initialization, cooling, addressability, trapping, measurement, scalability, and low decoherence. It also discusses the Toric Code: ECC and demonstrates the anyonic properties using one plaquette. The experiment showcases state identification, fusion rules, statistics, and potential applications.

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Revealing Anyonic Statistics with Multiphoton Entanglement

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  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]

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