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Linear Canonical Transforms on Quantum States of Light

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Linear Canonical Transforms

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 198))

Abstract

Many quantum information and quantum computation protocols exploit high-dimensional Hilbert spaces. Photons, which constitute the main carrier of information between nodes of quantum networks, can store high-dimensional quantum bits in their spatial degrees of freedom. These degrees of freedom can be tailored by resorting to the symplectic invariant approach based on lossless linear canonical transformations. These transformations enable one to manipulate the transverse structure of a single photon prepared in superpositions of paraxial modes. We present a basic introduction of these transformations acting on photons and discuss some of their applications for elementary quantum information processing.

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Notes

  1. 1.

    We use a slightly different symplectic metric matrix Ω to that used in Chap. 2. For every \(\mathbf{T} \in Sp(4, \mathbb{R})\), the symplecticity condition reads as T Ω T T = Ω.

  2. 2.

    The metaplectic group is a double cover of the symplectic group.

  3. 3.

    For simplicity, we focus on the transverse profile of a single-photon, considering a certain polarization \(\sigma\) and a frequency ω, see Eq. (15.25). Also, without loss of generality, we consider all \(\mathcal{O}_{\ell,p}\) having ℓ ≥ 0. Spheres with ℓ < 0 exhibit a state configuration identical to those with ℓ > 0, after inversion with respect to their centers.

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Acknowledgements

This work was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Science, under contract # DE-AC02-06CH11357. G.F.C. gratefully acknowledges the University of Castilla-La Mancha for financial support. G.F.C and A.P. acknowledge fruitful discussions with T. Alieva, J.A. Rodrigo, M.L. Calvo, M. VanValkenburgh, and S. Walborn.

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Authors and Affiliations

  1. Department of Mathematics and IMACI-Institute of Applied Mathematics in Science and Engineering, UCLM-Universidad de Castilla-La Mancha, Ciudad Real, 13071, Spain

    Gabriel F. Calvo & Antonio Picón

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  1. Gabriel F. Calvo
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  2. Antonio Picón
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Correspondence to Antonio Picón .

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Editors and Affiliations

  1. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland

    John J. Healy

  2. The Scientific and Technological Research Council of Turkey, Ankara, Ankara, Turkey

    M. Alper Kutay

  3. Department of Electrical Engineering, Bilkent University, Ankara, Ankara, Turkey

    Haldun M. Ozaktas

  4. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Dublin, Ireland

    John T. Sheridan

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Calvo, G.F., Picón, A. (2016). Linear Canonical Transforms on Quantum States of Light. In: Healy, J., Alper Kutay, M., Ozaktas, H., Sheridan, J. (eds) Linear Canonical Transforms. Springer Series in Optical Sciences, vol 198. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3028-9_15

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