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 $\widehat{\mathrm{\Gamma }}$ which satisfy $\mathbf{a}·\widehat{\mathrm{\Gamma }}·\mathbf{a}\le 1$. This resembles the well known condition for qubit Bloch vectors in which case $\widehat{\mathrm{\Gamma }}=\mathbb{1}$. In our case the vector $\mathbf{a}$ corresponds to spin-1 polarization, whereas the tensor $\widehat{\mathrm{\Gamma }}$ is a function of polarization uncertainties. Alternatively, $\mathbf{a}$ is a local Bloch vector of a symmetric two-qubit state and $\widehat{\mathrm{\Gamma }}$ is a function of the corresponding correlation tensor.

• Received 13 February 2016
• Revised 16 May 2016

DOI:https://doi.org/10.1103/PhysRevA.93.062126