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arXiv:2601.06240

The Qutrit Bloch Sphere

Vinod K Mishra

quant-phSubmitted Jan 9, 2026• Announced Jan 12, 2026• 0 citations
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Abstract

It is very important to understand if a qutrit can be visualized in a 3-dimensional Bloch sphere. In this work, a mathematical model for performing this operation is presented.

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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 a\mathbf{a}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 a\mathbf{a}a corresponds to spin-1 polarization, whereas the tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of polarization uncertainties. Alternatively, a\mathbf{a}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.
Visual methods are of great utility in understanding and interpreting quantum mechanics at all levels of understanding. The Bloch sphere, for example, is an invaluable and widely used tool for visualising quantum dynamics of a two level qubit system. In this work we present an `octant' visualisation method for qutrits bearing similarity to the Bloch sphere, that encompasses all eight degrees of freedom necessary to fully describe a three level state whilst remaining intuitive to interpret. Using this framework, a set of typical three level processes are modelled, described and displayed.