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arXiv:1601.07361 · Physical Review A

Three-dimensional visualization of a qutrit

Paweł Kurzyński, Adrian Kołodziejski, Wiesław Laskowski, Marcin Markiewicz

Published Jun 27, 2016• 40 citations
arXivHTMLPDF

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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ 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.

DOI
10.1103/physreva.93.062126

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•Jan 16, 2021•1 citation
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 \ensuremathΓ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ensuremathΓ\ifmmode^\elseˆ\fi​ which satisfy a\ifmmode⋅\else\textperiodcentered\fi\ensuremathΓ\ifmmode^\elseˆ\fi\ifmmode⋅\else\textperiodcentered\fia\ensuremath≤1\mathbf{a}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{a}\ensuremath{\le}1a\ifmmode⋅\else\textperiodcentered\fi\ensuremathΓ\ifmmode^\elseˆ\fi​\ifmmode⋅\else\textperiodcentered\fia\ensuremath≤1. This resembles the well known condition for qubit Bloch vectors in which case \ensuremathΓ\ifmmode^\elseˆ\fi=1\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}=\mathbb{1}\ensuremathΓ\ifmmode^\elseˆ\fi​=1. In our case the vector a\mathbf{a}a corresponds to spin-1 polarization, whereas the tensor \ensuremathΓ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ensuremathΓ\ifmmode^\elseˆ\fi​ is a function of polarization uncertainties. Alternatively, a\mathbf{a}a is a local Bloch vector of a symmetric two-qubit state and \ensuremathΓ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ensuremathΓ\ifmmode^\elseˆ\fi​ is a function of the corresponding correlation tensor.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
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 S7\mathbb{S}^{7}S7 splits into two parts, a four-sphere S4\mathbb{S}^{4}S4 and a two-sphere S2\mathbb{S}^{2}S2. 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.
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.
In the Bloch sphere based representation of qudits with dimensions greater than two, the Heisenberg-Weyl operator basis is not preferred because of presence of complex Bloch vector components. We try to address this issue and parametrize a qutrit using the Heisenberg-Weyl operators by identifying eight real parameters and separate them as four weight and four angular parameters each. The four weight parameters correspond to the weights in front of the four mutually unbiased bases sets formed by the eigenbases of Heisenberg-Weyl observables and they form a four-dimensional unit radius Bloch hypersphere. Inside the four-dimensional hypersphere all points do not correspond to a physical qutrit state but still it has several other features which indicate that it is a natural extension of the qubit Bloch sphere. We study the purity, rank of three level systems, orthogonality and mutual unbiasedness conditions and the distance between two qutrit states inside the hypersphere. We also analyze the two and three-dimensional sections centered at the origin which gives a close structure for physical qutrit states inside the hypersphere. Significantly, we have applied our representation to find mutually unbiased bases(MUBs) and to characterize the unital maps in three dimensions. It should also be possible to extend this idea in higher dimensions.