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.
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.
 M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
 J. Martin, O. Giraud, P.A. Braun, D. Braun, and T. Bastin, Multiqubit symmetric states with high geometric entanglement, Physical Review A 81, 062347 (2010).
 R.P. Feynman, F.L. Vernon, and R.W. Hellwarth, Geometrical Representation of the Schrödinger Equation for Solving Maser Problems, J. Appl. Phys. 28, 49 (1957).
 M.S. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation, Phys. Rev.A 68, 062322 (2003).
 G. Kimura and A. Kossakowski, The Bloch-vector space for $N$-level systems: the spherical-coordinate point of view, Open Syst. Inf. Dyn. 12, 207 (2005).
 I.P. Mendaš, The classification of three-parameter density matrices for a qutrit, J. Phys. A: Math. .Gen. 39, 11313 (2006).
 S. Vinjanampathy and A.R.P. Rau, Bloch sphere like construction of SU(3) Hamiltonians using Unitary Integration, J. Phys. A: Math. Theor. 42, 425303 (2009).
 C.F. Dunkl, P. Gawron, J.A. Holbrook, J.A. Miszczak, Z. Puchała, and K. Życzkowski, Numerical shadow and geometry of quantum states, J. Phys. A: Math. Theor. 44, 335301 (2011).
 G. Sarbicki and I. Bengtsson, Dissecting the qutrit, J. Phys. A: Math. Theor. 46, 035306 (2012).
 G.N.M. Tabia and D.M. Appleby, Exploring the geometry of qutrit state space using symmetric informationally complete probabilities, Phys. Rev. A 88, 012131 (2013).
 S. Goyal, B.N. Simon, R. Singh, and S. Simon, Geometry of the generalized Bloch sphere for qutrits, J. Phys. A: Math. Theor. 49, 165203 (2016).
 K. Szymański, S. Weis, and K. Życzkowski, Classification of joint numerical ranges of three hermitian matrices of size three, Lin. Alg. Appl. 545, 148 (2018).
 J. Xie, A. Zhang, N. Cao, H. Xu, K. Zheng, Y.T. Poon, N.S. Sze, P. Xu, B. Zeng, and L. Zhang, Observing Geometry of Quantum States in a Three-Level System, Phys. Rev. Lett. 125, 150401 (2020).
 S. Dogra, A. Vepsäläinen, and G.S. Paraoanu, Majorana representation of adiabatic and superadiabatic processes in three-level systems, Phys. Rev. Research 2, 043079 (2020).
 R.A. Bertlmann and P. Krammer, Bloch vectors for qudits, Journal of Physics A: Mathematical and Theoretical 41, 235303 (2008).
 Note that we use a different normalization compared to that in Ref. Bengtsson_2012 because of a different choice of prefactor for the Hilbert-Schmidt norm.
 T. Baumgratz, M. Cramer, and M.B. Plenio, Quantifying Coherence, Phys. Rev. Lett. 113, 140401 (2014).
 P. Rungta, V. Bužek, C.M. Caves, M. Hillery, and G.J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
 M. Grassl, L. Kong, Z. Wei, Z. Yin, and B. Zeng, Quantum Error-Correcting Codes for Qudit Amplitude Damping, IEEE Transactions on Information Theory 64, 4674 (2018).
 A. R. P. Rau, "Symmetries and Geometries of Qubits, and their Uses", arXiv:2103.14105.
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