The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Christopher Eltschka1, Marcus Huber2,3, Simon Morelli2,3, and Jens Siewert4,5

1Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
2Atominstitut, Technische Universität Wien, 1020 Vienna, Austria
3Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
4Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain
5IKERBASQUE Basque Foundation for Science, E-48009 Bilbao, Spain

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Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.

The quantum state space exhibits a much richer structure than its classical counterpart. This is already true for the qubit, the simplest quantum system, represented by the Bloch ball. Contrary to what the regular structure of the classical probability space and of the Bloch ball might suggest, the quantum state space for higher dimensional systems is an irregular object whose description can be a challenging task already for low dimensions.
In this work we present (two versions of) a three dimensional model of the eight dimensional state space of a qutrit, capturing its essential geometric and algebraic characteristics. Besides providing an intuition for three-level systems, our model also gives insight to higher dimensional state spaces that feature properties not present in the Bloch ball. Finally we show the usefulness of the model in various applications, such as the representation of the mixture of two states, the unitary transformation of a state and the behavior of the state space under the action of quantum channels.

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► References

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[2] Karol Życzkowski, Trends in Mathematics 105 (2023) ISBN:978-3-031-30283-1.

[3] Paul M. Alsing and Carlo Cafaro, "From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians", International Journal of Geometric Methods in Modern Physics 21 08, 2450152 (2024).

[4] José J. Gil, Andreas Norrman, Ari T. Friberg, and Tero Setälä, "Descriptors of dimensionality for n × n density matrices", The European Physical Journal Plus 138 5, 476 (2023).

[5] Paul M. Alsing and Carlo Cafaro, "From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians", International Journal of Geometric Methods in Modern Physics 21 08, 2450151 (2024).

[6] Gautam Sharma, Sibasish Ghosh, and Sk Sazim, "Bloch sphere analog of qudits using Heisenberg-Weyl Operators", Physica Scripta 99 4, 045105 (2024).

[7] Michael J. Grabowecky, Christopher A. J. Pollack, Andrew R. Cameron, Robert W. Spekkens, and Kevin J. Resch, "Experimentally bounding deviations from quantum theory for a photonic three-level system using theory-agnostic tomography", Physical Review A 105 3, 032204 (2022).

[8] Asmae Benhemou, Toonyawat Angkhanawin, Charles S. Adams, Dan E. Browne, and Jiannis K. Pachos, "Universality of Z3 parafermions via edge-mode interaction and quantum simulation of topological space evolution with Rydberg atoms", Physical Review Research 5 2, 023076 (2023).

[9] Simon Morelli, Christopher Eltschka, Marcus Huber, and Jens Siewert, "Correlation constraints and the Bloch geometry of two qubits", Physical Review A 109 1, 012423 (2024).

[10] A. R. P. Rau, "Symmetries and Geometries of Qubits, and Their Uses", Symmetry 13 9, 1732 (2021).

[11] Shravan Shravan, Simon Morelli, Otfried Gühne, and Satoya Imai, "Geometry of two-body correlations in three-qubit states", arXiv:2309.09549, (2023).

[12] Lu Wei, Zhian Jia, Dagomir Kaszlikowski, and Sheng Tan, "Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems", arXiv:2202.10989, (2022).

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