1. Classifying Beamsplitters Adam Bouland

2. Boson/Fermion Model M modes

3. 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. Let G=

16. Complex Beamsplitters

17. 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= Lemma: If G is discrete, R1,R2,R3 form an irreducible representation of G.

20. Complex Beamsplitters

22. Δ(6n 2 )

23. Complex Beamsplitters Δ(6n 2 ) 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 ?