Three-dimensional visualization of a qutrit
Paweł Kurzyński, Adrian Kołodziejski, Wiesław Laskowski, Marcin Markiewicz · Physical Review A · 2016
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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ which satisfy $\mathbf{a}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{a}\ensuremath{\le}1$. This resembles the well known condition for qubit Bloch vectors in which case $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}=\mathbb{1}$. In our case the vector $\mathbf{a}$ corresponds to spin-1 polarization, whereas the tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of polarization uncertainties. Alternatively, $\mathbf{a}$ is a local Bloch vector of a symmetric two-qubit state and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of the corresponding correlation tensor.