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Universal Beamsplitters in Boson/Fermion Models: Exploring Complex and Real Sets

This work delves into the classification of beamsplitters within the regimes of bosonic and fermionic models. It defines universal beamsplitters as those that can densely generate SU(m) or SO(m) on m modes. Real beamsplitters have been established to be universal for three or more modes, while the potential universality of complex beamsplitters on the same condition is explored. The text also addresses important open questions regarding the proof of universality, extensions to multi-mode systems, and the implications of phase applications in beamsplitter operations.

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Universal Beamsplitters in Boson/Fermion Models: Exploring Complex and Real Sets

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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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