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Classifying Beamsplitters

Classifying Beamsplitters. Adam Bouland. Boson/Fermion Model. M modes. Boson/Fermion Model. Boson/Fermion Model. Beamsplitters. Def: A set of beamsplitters is universal if it densely generates SU(m) or SO(m) on m modes. Beamsplitters.

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Classifying Beamsplitters

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  1. Classifying Beamsplitters Adam Bouland

  2. Boson/Fermion Model M modes

  3. Boson/Fermion Model

  4. Boson/Fermion Model

  5. Beamsplitters • Def: A set of beamsplitters is universal if it densely generates SU(m) or SO(m) on m modes.

  6. Beamsplitters • Def: A set of beamsplitters is universal if it densely generates SU(m) or SO(m) on m modes. Q: Which sets of beamsplitters are universal?

  7. Beamsplitters • Obviously not universal:

  8. Beamsplitters • Obviously not universal: • Not obvious:

  9. Real Beamsplitters Thm: [B. Aaronson ’12] Any real nontrivial beamsplitter is universal on ≥3 modes.

  10. Real Beamsplitters Thm: [B. Aaronson ’12] Any real nontrivial beamsplitter is universal on ≥3 modes. What about complex beamsplitters?

  11. Complex Beamsplitters Goal: Any non-trivial (complex) beamsplitter is universal on ≥3 modes.

  12. Complex Beamsplitters Goal: Any non-trivial (complex) beamsplitter is universal on ≥3 modes. Can show: Any non-trivial beamsplitter generates a continuous group on ≥3 modes.

  13. Complex Beamsplitters Determinant ±1

  14. Complex Beamsplitters

  15. Complex Beamsplitters Let G=<R1,R2,R3>

  16. Complex Beamsplitters

  17. Complex Beamsplitters Subgroups of SU(3): 6 infinite families 12 exceptional groups

  18. Complex Beamsplitters Subgroups of SU(3): 6 infinite families 12 exceptional groups

  19. Complex Beamsplitters Let G=<R1,R2,R3> Lemma: If G is discrete, R1,R2,R3 form an irreducible representation of G.

  20. Complex Beamsplitters

  21. Complex Beamsplitters

  22. Complex Beamsplitters Δ(6n2)

  23. Complex Beamsplitters Δ(6n2) Algebraic Number Theory

  24. Open questions • Can we complete the proof to show any beamsplitter is universal? • Can we extend this to multi-mode beamsplitters? • What if the beamsplitter applies a phase as well?

  25. Questions ?

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