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Classifying Beamsplitters Adam Bouland

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Boson/Fermion Model M modes

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Boson/Fermion Model

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

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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?

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Beamsplitters Obviously not universal:

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Beamsplitters Obviously not universal: Not obvious:

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

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

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

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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.

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Complex Beamsplitters Determinant ±1

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Complex Beamsplitters

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Let G=

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Complex Beamsplitters

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

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

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

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Complex Beamsplitters

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Δ(6n 2 )

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Complex Beamsplitters Δ(6n 2 ) Algebraic Number Theory

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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?

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Questions ?

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11.2 Geometric Representations of Complex Numbers.

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