Download presentation

Presentation is loading. Please wait.

Published byKimberly Egan Modified over 2 years ago

1
Classifying Beamsplitters Adam Bouland

2
Boson/Fermion Model M modes

3
Boson/Fermion Model

4

5
Beamsplitters Def: A set of beamsplitters is universal if it densely generates SU(m) or SO(m) on m modes.

6
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?

7
Beamsplitters Obviously not universal:

8
Beamsplitters Obviously not universal: Not obvious:

9
Real Beamsplitters Thm: [B. Aaronson 12] Any real nontrivial beamsplitter is universal on 3 modes.

10
Real Beamsplitters Thm: [B. Aaronson 12] Any real nontrivial beamsplitter is universal on 3 modes. What about complex beamsplitters?

11
Complex Beamsplitters Goal: Any non-trivial (complex) beamsplitter is universal on 3 modes.

12
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.

13
Complex Beamsplitters Determinant ±1

14
Complex Beamsplitters

15
Let G=

16
Complex Beamsplitters

17
Subgroups of SU(3): 6 infinite families 12 exceptional groups

18
Complex Beamsplitters Subgroups of SU(3): 6 infinite families 12 exceptional groups

19
Complex Beamsplitters Let G= Lemma: If G is discrete, R1,R2,R3 form an irreducible representation of G.

20
Complex Beamsplitters

21

22
Δ(6n 2 )

23
Complex Beamsplitters Δ(6n 2 ) Algebraic Number Theory

24
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?

25
Questions ?

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google