Three-dimensional visualization of a qutrit

Abstract

We present a surprisingly simple three-dimensional Bloch sphere representation of a qutrit, i.e., a single three-level quantum system. We start with a symmetric state of a two-qubit system and relate it to the spin-1 representation. Using this representation we associate each qutrit state with a three-dimensional vector $\mathbf{a}$ and a metric tensor $\mathbf{\hat\Gamma}$ which satisfy $\mathbf{a}\cdot \mathbf{\hat\Gamma} \cdot \mathbf{a}\leq 1$. This resembles the well known condition for qubit Bloch vectors in which case $\mathbf{\hat\Gamma}=\hat{I}$. In our case the vector $\mathbf{a}$ corresponds to spin-1 polarisation, whereas the tensor $\mathbf{\hat\Gamma}$ is a function of polarisation uncertainties. Alternatively, $\mathbf{a}$ is a local Bloch vector of a symmetric two-qubit state and $\mathbf{\hat\Gamma}$ is a function of the corresponding correlation tensor. We also discuss how a purity of states and state-dynamics is depicted in our representation.