Physica Scripta

Phase portrait of the qutrit system

Surajit Sen, Tushar Kanti Dey

Published Sep 1, 2025• 0 citations

Abstract

Abstract We investigate the structure of the Bloch space for the qutrit system by analyzing its properties and phase portraits. Within the framework of the SU(3) representation theory and transition features between various states, the qutrit system is categorized as lambda, vee, and cascade type three-level systems. Each of these systems has its own Bloch equation, resulting in distinct dynamics with unique dynamical trajectories. Expanding upon our earlier work (S. Sen, M. R. Nath, T. K. Dey and G. Gangopadhyay, Ann. Phys. 327 (2012) 224–252), which demonstrated the splitting of the Bloch sphere into two distinct sectors, in this paper we provide their explicit derivation from the dynamical perspective and examine the phase portraits of two sectors separately. These phase portraits, ranging from simple to complex patterns of curves, reveal the characteristic pattern of evolution for each sub-sector of the qutrit Bloch space.

DOI: 10.1088/1402-4896/adfd21

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Physica Scripta

Phase portrait of the qutrit system

Surajit Sen, Tushar Kanti Dey

Published Sep 1, 2025• 0 citations

Publisher

Abstract

DOI: 10.1088/1402-4896/adfd21

Related papers

Bloch Sphere of the Qutrit System

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We present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere $\mathbb{S}^{7}$ splits into two parts, a four-sphere $\mathbb{S}^{4}$ and a two-sphere $\mathbb{S}^{2}$ . The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.We present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere $\mathbb{S}^{7}$ splits into two parts, a four-sphere $\mathbb{S}^{4}$ and a two-sphere $\mathbb{S}^{2}$. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.

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Geometry of the generalized Bloch sphere for qutrits

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Paweł Kurzyński, Adrian Kołodziejski, Wiesław Laskowski +1 more

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.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.

arXiv:1601.07361Physical Review AJun 27, 201640 citations

\mathbb{S}^{7}

splits into two parts, a four-sphere

\mathbb{S}^{4}

and a two-sphere

\mathbb{S}^{2}

. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.

\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.