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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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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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Complex BeamsplittersLet G=<R1,R2,R3>
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Complex Beamsplitters
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Complex Beamsplitters
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 BeamsplittersLet G=<R1,R2,R3>
Lemma: If G is discrete, R1,R2,R3 form an irreducible representation of G.
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Complex Beamsplitters
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Complex Beamsplitters
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Complex Beamsplitters
Δ(6n2)
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Complex Beamsplitters
Δ(6n2)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
?