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.
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