# Classifying Beamsplitters Adam Bouland. Boson/Fermion Model M modes.

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Boson/Fermion Model M modes

Boson/Fermion Model

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

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?

Beamsplitters Obviously not universal:

Beamsplitters Obviously not universal: Not obvious:

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

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

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

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.

Complex Beamsplitters Determinant ±1

Complex Beamsplitters

Let G=

Complex Beamsplitters

Subgroups of SU(3): 6 infinite families 12 exceptional groups

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

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

Complex Beamsplitters

Δ(6n 2 )

Complex Beamsplitters Δ(6n 2 ) Algebraic Number Theory

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?

Questions ?