Classifying Beamsplitters

Adam Bouland

Boson/Fermion Model

M modes

Boson/Fermion Model

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

Complex BeamsplittersLet G=<R1,R2,R3>

Complex Beamsplitters

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 BeamsplittersLet G=<R1,R2,R3>

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

Complex Beamsplitters

Complex Beamsplitters

Complex Beamsplitters

Δ(6n2)

Complex Beamsplitters

Δ(6n2)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

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