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Introduction and Theoretical Background

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Quantum Entanglement of Complex Structures of Photons

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Abstract

The nature of light and its main properties, either seen classically as an electromagnetic wave or quantum mechanically as a photon, is not only the focus of much ongoing research, but has been covered by many excellent textbooks. Thus, only a short introduction to the theory will be conducted in the following chapter mainly to establish the employed nomenclature.

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Notes

  1. 1.

    The derivation from the Maxwell equations can be found for example in the textbook [4] or the thesis of A. Mair [5].

  2. 2.

    A detailed derivation can be found e.g. in [3, 17].

  3. 3.

    A more detailed description can be found here [18].

  4. 4.

    This effect is similar to the rotation of the intensity mentioned earlier, if two LG modes with different Gouy phase are superposed.

  5. 5.

    Although the square shaped modes might not be a simple solution to the PWE, they should be describable with a complex superposition of many “simple” modes (maybe an infinite sum of modes). This stems from the fact that all earlier discussed modes form a complete set of spatial modes. Thus, they are able to describe any paraxial light field. A more detailed discussion however is outside of the scope of this thesis.

  6. 6.

    I abstain from discussing in detail the more realistic description of physical states in terms of density matrices for mixed states and only refer the reader to the corresponding literature [52, 54]. Nevertheless, the simplifying assumption of pure states is a very good approximation to all real physical systems in the presented results. Thus, all drawn conclusions hold.

  7. 7.

    Note that even single higher-dimensional systems—so-called qudit states—could also be used to justify the statement (for example it was shown for single qutrits in [57]). However, they are not in the scope of this thesis, where qubit systems are the focus of all investigations.

  8. 8.

    Although multi-partite systems of more than two parties can form very interesting in quantum states, e.g. GHZ states [58], they are not in the scope of this thesis.

  9. 9.

    Without explaining the concept of local realism in more detail, I use the modern, standard way of defining locality, i.e. two space-like separated systems cannot influence each other faster than the speed of light, and realism, i.e. measurable properties do not exist before and independent of the measurements.

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  1. Vienna Center for Quantum Science and Technology, University of Vienna/IQOQI Vienna, Vienna, Austria

    Robert Fickler

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Fickler, R. (2016). Introduction and Theoretical Background. In: Quantum Entanglement of Complex Structures of Photons. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-22231-8_2

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