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, 9th September 2007

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

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

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.htmlhttp://www.uni-potsdam.de/u/ostermeyer/web/contents/quantcrypto/nav_qc.html

Application:Quantum teleportation2

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Quantum state/secret sharing1

Entanglement swapping3 Testing Bell inequality4

Finding the output state:

Employ covariance matrix5 if only Gaussian states used.

Wigner function2 has 2n degrees of freedom for n modes:

Apply transformation directly to canonical variables but

requires O(n2) parameters.

W q,p dnq

2 nq q '/ 2 q q '/ 2 eiq 'p

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.

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

Mathematical Foundation for Lie Group Theory:

Cartan and Casimir operators

Irreducible representations

Coherent states: Perelomov'sdefinition

Lie group

Lie algebra

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

Beam Splitter and Two-Mode Squeezer

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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

Motivation:Quantum teleportation2

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Quantum state/secret sharing1

Entanglement swapping3 Testing Bell inequality4

Multipartite Squeezing and SU(1,1) Symmetry:

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Multipartite Squeezing… Cont’d...

Three mode: k, 1

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a

a=0

k+ 1

a,k a 1, k

,

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.

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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

Complicated Scenarios: Concatenation

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Complicated Scenarios: Bloch-Messiah Theorem

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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

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(n2) 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.

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)