1. Multi-Partite Squeezing and SU (1,1) Symmetry Zahra Shaterzadeh Yazdi Institute for Quantum Information Science, University of Calgary with Peter S. Turner and Dr. Barry C. Sanders IICQI 2007 Conference, Kish University, Kish Island, Iran Sunday, 9 th September 2007

2. Outline: Significance of squeezing for QIP An efficient Method to characterize two-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open question Conclusion

3. Outline: Significance of squeezing for QIP An efficient Method to characterize multi-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open questions Conclusion

4. Introduction: Squeezed light is the key source of entanglement for optical quantum information processing tasks. Such light can only be produced in a single mode and two modes by means of squeezers. More general QIP tasks require the squeezed light to be distributed amongst multiple modes by passive optical elements such as beam splitters and phase shifters and squeezers.  http://www.pi4.uni-stuttgart.de/NeueSeite/index.html?research/homepage_ultrafast/ultrafast_propagation.html  http://www.uni-potsdam.de/u/ostermeyer/web/contents/quantcrypto/nav_qc.html

5. Application: Quantum teleportation 2 S bc MPMPMPMP MXMXMXMX B ab c b a S bc M M B ab B cd a d c b S bc B ab B cd b a d c B ab S bc S ac c b a Quantum state/secret sharing 1 Entanglement swapping 3 Testing Bell inequality 4

6. Finding the output state: Employ covariance matrix 5 if only Gaussian states used. Wigner function 2 has 2n degrees of freedom for n modes: Apply transformation directly to canonical variables but requires O(n 2 ) parameters.

7. Goals:  Employ Lie group theory to describe the mathematical transformation for active multi-mode interferometers with few squeezers.  This approach provides an elegant and efficient characterization of a large class of output states generated by such networks for any input states.  This simplification arises by identifying appropriate symmetries through making use of the available group representations.

8. Outline: Significance of squeezed states for QIP An efficient Method to characterize two-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open questions Conclusion

9. Mathematical Foundation for Lie Group Theory: Cartan and Casimir operators Irreducible representations Coherent states: Perelomov's definition Lie group Lie algebra

10. SU(2) Symmetry and SU(1,1) Symmetry

11. Beam Splitter and Two-Mode Squeezer S ab b a a a BabBabBabBab b

12. Outline: Significance of squeezing for QIP An efficient Method to characterize multi-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open questions Conclusion

13. Motivation: Quantum teleportation 2 S bc MPMPMPMP MXMXMXMX B ab c b a S bc M M B ab B cd a d c b S bc B ab B cd b a d c B ab S bc S ac c b a Quantum state/secret sharing 1 Entanglement swapping 3 Testing Bell inequality 4

14. Multipartite Squeezing and SU(1,1) Symmetry: arar a4a4 a3a3 a2a2 a1a1 b1b1 bsbs b2b2 b3b3 b4b4 arar a4a4 a3a3 a2a2 a1a1 b1b1 bsbs b2b2 b3b3 b4b4 S AB S ab B B B B B B B

15. Multipartite Squeezing… Cont’d...,

16. What is nice about our approach?!!! It enables us to use a variety of mathematical properties that have already been established for this group, greatly facilitating calculations. Examples: The SU(1,1) Clebsch-Gordan coefficients are useful if we want to concatenate some of these typical networks ‘in parallel’. The output states of such networks can be described by the coherent states of SU(1,1). The significance of our result is that, in contrast to existing methods, it allows for arbitrary input states. This method can therefore be used for a large class of output states. S S

17. Outline: Significance of squeezing for QIP An efficient Method to characterize two-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open question Conclusion

18. Complicated Scenarios: Concatenation S AB S CD S EF S GH S S S S B B B B B B B

19. Complicated Scenarios: Bloch-Messiah Theorem SCDSCD S AB S EF S CD S GH U V † S AB S CD S EF S GH S S S S B B B B B B B B B B B B B B

20. Outline: Significance of squeezing for QIP An efficient Method to characterize multi-mode squeezing Multi-mode squeezing and SU(1,1) symmetry Open questions Conclusion

21. Conclusions: Characterized typical multi-mode optical networks as SU(1,1) transformations: Multi-mode squeezed states generated in such networks are the SU(1,1) coherent states. Simplifies calculations from O(n 2 ) to constant number of parameters. Identified the symmetries based on the group representations. This approach is independent of input states (such as assuming covariance matrix or Wigner functions), because SU(1,1) weight states are equivalent to pseudo Fock states.

22. References: 1. T. Tyc and B. C. Sanders, PRA 65, 042310 (2002) 2. S. L. Braunstein and H. J. Kimble, PRL 80, 869 (1998). 3. O. Glock et al., PRA 68, 1 (2001) 4. S. D. Bartlett et al., PRA 63, 042310 (2001) 5.J. Eisert and M. B. Plenio, PRL 89, 097901 (2002) 6.S. L. Braunstein, PRA 71, 055801 (2005)