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