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Geometric Constructions over \({\mathbb {C}}\) and \({\mathbb {F}}_2\) for Quantum Information

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Quantum Physics and Geometry

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 25))

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Abstract

In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers \({\mathbb {C}}\) and the other one is over the two elements field \(\mathbb {F}_2\). Both constructions have been employed in the past 15 years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems (three qubits, three fermions, three bosonic qubits…) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized N-Pauli group.

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Notes

  1. 1.

    This concept of contextuality also appears in Bell’s paper [9, 59].

  2. 2.

    In quantum physics, the outcomes of a measurement are encoded in an hermitian operator, called an observable. The eigenvalues of the observable correspond to the possible outcomes of the measurement and the eigenvectors correspond to the possible projections of the state after measurement.

  3. 3.

    Physically one may imagine that each part of the system is in a different location and experimentalists only apply local quantum transformations, i.e. some unitaries defined by local Hamiltonians.

  4. 4.

    In this paper an algebraic variety will always be the zero locus of a collection of homogeneous polynomials.

  5. 5.

    In the four-qubit case, the ring of SLOCC invariant polynomials is generated by four polynomials denoted by H, L, M and D in [57]. One way of defining the quotient map is to consider \(\Phi :\mathcal {H}\to {\mathbb {C}}^4\) defined by \(\Phi (\hat {x})=(H(\hat {x}),L(\hat {x}),N(\hat {x}),D(\hat {x}))\), see [37].

  6. 6.

    The discriminant of the miniversal deformation of a singularity parametrizes all singular deformations of the singularity [3].

  7. 7.

    The spin group Spin(2N) corresponds to the simply connected double cover of SO(2N) [27].

  8. 8.

    A point-line incidence structure is called a generalized quadrangle of type (s, t), and denoted by GQ(s, t) iff it is an incidence structure such that every point is on t + 1 lines and every line contains s + 1 points such that if p∉L, ∃!q ∈ L such that p and q are collinear.

  9. 9.

    Up to a transformation of coordinates, this is a set of points \(x\in \mathbb {P}^{2N-1}_2\) satisfying the standard equation x 1 x 2 + x 3 x 4 + ⋯ + x 2N−1 x 2N = 0.

  10. 10.

    Up to a transformation of coordinates this is defined as points \(x\in \mathbb {P}^{2N-1}_2\) such that f(x 1, x 1) + x 2 x 3 + …x 2N−1 x 2N = 0.

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Acknowledgements

The first part of this review paper was presented at the international workshop “Quantum Physics and Geometry” organized at Levico Terme in July 2017. I would like to thank the organizers for inviting me to present an overview on my research and to contribute to this UMI Lecture Notes. The research presented in this review has been done collectively; I would like to warmly thank my co-authors Jean-Gabriel Luque, Jean-Yves Thibon, Michel Planat, Metod Saniga, Péter Lévay and Hamza Jaffali for our rich collaboration over the past 6 years. This work was partially supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03).

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  1. Laboratoire Interdisciplinaire Carnot de Bourgogne, University Bourgogne Franche-Comté, Belfort, France

    Frédéric Holweck

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  1. Frédéric Holweck
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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Trento, Italy

    Edoardo Ballico

  2. Dipartimento di Matematica, Università di Trento, Trento, Italy

    Alessandra Bernardi

  3. BEC Center, INO-CNR, Trento, Italy

    Iacopo Carusotto

  4. Dipartimento di Matematica, Università di Trento, Trento, Italy

    Sonia Mazzucchi

  5. Dipartimento di Matematica, Università di Trento, Trento, Italy

    Valter Moretti

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Holweck, F. (2019). Geometric Constructions over \({\mathbb {C}}\) and \({\mathbb {F}}_2\) for Quantum Information. In: Ballico, E., Bernardi, A., Carusotto, I., Mazzucchi, S., Moretti, V. (eds) Quantum Physics and Geometry. Lecture Notes of the Unione Matematica Italiana, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-06122-7_5

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