Bloch–Gell–Mann Representation in Qudit Systems

• The Bloch–Gell–Mann representation is a formalism that expands density matrices using traceless Hermitian generators to provide a complete geometric and algebraic characterization of d-level quantum states.
• It employs generalized Gell–Mann matrices to decompose quantum states, facilitating precise state reconstruction, quantum dynamics analysis, and applications in quantum machine learning.
• This unified approach integrates algebraic structure, positivity constraints, and dynamical evolution across dimensions, extending from qubits to multi-qudit and higher spin systems.

The Bloch–Gell–Mann representation is a generalization of the well-known Bloch sphere formalism for qubits that provides a complete geometric and algebraic characterization of finite-dimensional quantum states (qudits). By expanding density matrices and observables in the orthonormal basis of traceless Hermitian generators of $\mathrm{SU}(d)$—the generalized Gell–Mann matrices—any $d$-level system can be parametrized by its so-called Bloch–Gell–Mann vector. This approach enables a unified description of state space geometry, operator expansions, dynamical evolution, and operational applications, and has broad utility in quantum information science, quantum foundations, and quantum machine learning [(Valtinos et al., 2023); (Loubenets, 2019); (Loubenets et al., 2020); (Chew et al., 2020); (Giraud et al., 2014)].

1. Definition of Generalized Gell–Mann Matrices

Let $d$ be the Hilbert space dimension. The $d^2-1$ traceless Hermitian generators $\{\lambda_i\}$ of $\mathfrak{su}(d)$ are constructed as follows (Loubenets, 2019, Loubenets et al., 2020):

• Symmetric off-diagonal generators: $$\lambda^{(s)}_{m k} = |m\rangle\langle k| + |k\rangle\langle m|$$ for $1\leq m<k\leq d$;
• Antisymmetric off-diagonal generators: $$\lambda^{(a)}_{m k} = -i|m\rangle\langle k| + i|k\rangle\langle m|$$;
• Diagonal generators: For $l=1,\dots,d-1$,

$$\lambda^{(d)}_l = \sqrt{\frac{2}{l(l+1)}}\left(\sum_{j=1}^{l} |j\rangle\langle j| - l|l+1\rangle\langle l+1|\right).$$

These matrices are orthonormal under the Hilbert–Schmidt inner product: $$\mathrm{Tr}(\lambda_i \lambda_j) = 2 \delta_{ij},\qquad \lambda_i^\dagger = \lambda_i,\qquad \mathrm{Tr}(\lambda_i) = 0.$$ The case $d=2$ recovers the usual Pauli matrices; $d=3$ yields the Gell–Mann matrices, explicitly: $$\lambda_1 = \begin{pmatrix} 0 & 1 & 0\ 1 & 0 & 0\ 0 & 0 & 0 \end{pmatrix}, \quad \lambda_2 = \begin{pmatrix} 0 & -i & 0\ i & 0 & 0\ 0 & 0 & 0 \end{pmatrix}, \ldots, \lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & -2 \end{pmatrix}$$ (Valtinos et al., 2023, Loubenets et al., 2020).

2. Expansion of Quantum States: Bloch–Gell–Mann Parametrization

Any Hermitian, trace-one operator (density matrix) $\rho$ on $\mathbb{C}^d$ admits the expansion (Loubenets, 2019, Loubenets et al., 2020): $$\rho = \frac{1}{d} I + \frac{1}{2} \sum_{i=1}^{d^2-1} r_i \lambda_i,$$ where $r_i = \mathrm{Tr}(\rho \lambda_i)$ are real and form the $(d^2-1)$-dimensional Bloch–Gell–Mann vector. The normalization ensures $\mathrm{Tr}(\rho) = 1$ and Hermiticity follows from that of the $\lambda_i$. The collection $$\{I/\sqrt{d},\lambda_1/\sqrt{2},\ldots,\lambda_{d^2-1}/\sqrt{2}\}$$ forms an orthonormal basis for the Hilbert–Schmidt space of Hermitian matrices (Loubenets, 2019).

For $d=3$ (qutrits), for instance,

$$\rho = \frac{1}{3} I_3 + \frac{1}{2} \sum_{i=1}^{8} r_i \lambda_i,\qquad r_i = \mathrm{Tr}(\rho\lambda_i) \in \mathbb{R}^8$$

(Valtinos et al., 2023).

3. Structure Constants and Algebraic Properties

The Gell–Mann matrices close under commutation and anticommutation, with structure constants $f_{ijk}$ (totally antisymmetric) and $d_{ijk}$ (totally symmetric) (Loubenets, 2019, Loubenets et al., 2020): $$[\lambda_i,\lambda_j] = 2i f_{ijk} \lambda_k, \qquad \{\lambda_i,\lambda_j\} = \frac{4}{d}\delta_{ij} I + 2 d_{ijk} \lambda_k,$$ where

$$f_{ijk} = \frac{1}{4i}\mathrm{Tr}([\lambda_i, \lambda_j]\lambda_k),\quad d_{ijk} = \frac{1}{4}\mathrm{Tr}(\{\lambda_i, \lambda_j\}\lambda_k).$$

These constants enter both dynamical equations and the closure relations for the product of generators, central to the structure of $\mathfrak{su}(d)$ (Loubenets et al., 2020).

4. Geometry and Constraints of the Generalized Bloch Body

The set of physically valid Bloch–Gell–Mann vectors forms a convex subset of $\mathbb{R}^{d^2-1}$, defined by positivity constraints, Hermiticity, and normalization (Valtinos et al., 2023, Loubenets, 2019):

• Purity bound: $$\mathrm{Tr}(\rho^2)\leq1 \Longrightarrow \|\mathbf{r}\|^2\leq 4(d-1)/d$$.
• Positivity: All eigenvalues of $\rho$ are $\geq0$. For $d=2$, the positivity region is the unit ball; for $d>2$, it is a proper convex subset of the corresponding ball. For example, for qutrits ($d=3$) the “Bloch body” is inside a ball of radius $\sqrt{4/3}$ but its boundary is a high-dimensional, algebraically complicated manifold defined by higher-order invariants such as $\det\rho\geq 0$ (Valtinos et al., 2023, Loubenets, 2019).

For three-qubit (su(8)) systems, the physical region is a convex compact subset of the $\sqrt{7/4}$ ball in $\mathbb{R}^{63}$ (Chew et al., 2020). In general, boundary points corresponding to pure states form a complex projective manifold $\mathbb{C}P^{d-1}$.

5. Extension to Tensors and Higher Spins

The Bloch–Gell–Mann formalism admits generalization to arbitrary spin-$j$ systems using tensor expansions (Weinberg matrices) (Giraud et al., 2014). For spin-$j$ ($d=2j+1$), any density matrix $\rho$ may be expanded as

$$\rho = \frac{1}{2^N} \sum_{(\mu_1,\ldots,\mu_N)=0}^3 x_{\mu_1\ldots\mu_N} S_{\mu_1\ldots\mu_N},$$

where the $S_{\mu_1\ldots\mu_N}$ form a symmetric, Hermitian tensor basis and the coefficients are real. This construction recovers the standard Gell–Mann generators as a subset for $j>1/2$, while for $j=1/2$ it reduces to the Pauli matrices. This tensor formalism naturally encodes transformation properties under $\mathrm{SU}(2)$ and partial trace operations, and is fundamental for the characterization of phenomena such as quantum polarization and anticoherence (Giraud et al., 2014).

6. Applications: Quantum Information and Quantum Machine Learning

The Bloch–Gell–Mann representation is essential in several areas:

• Quantum tomography: Extraction of Bloch vector components allows full state reconstruction from expectation values of the generalized Gell–Mann operators (Loubenets, 2019, Chew et al., 2020).
• Quantum dynamics: The von Neumann equation for a general qudit reduces to a differential equation for the Bloch vector, governed by the structure constants $f_{ijk}$:

$\dot{r}_i = 2\sum_{j,k} f_{ijk} h_j r_k,$

for a Hamiltonian $H = h_0 I + \frac{1}{2} \sum h_i \lambda_i$ (Loubenets et al., 2020).

• Quantum machine learning: The Gell–Mann feature map encodes $8$-dimensional data directly into a qutrit using the exponential map $\exp[-i \sum_a w_a \lambda_a]$. This yields richer high-dimensional embeddings and more expressive circuit ansätze for classification, support vector machines, and variational quantum algorithms compared to qubit-based feature maps (Valtinos et al., 2023).

7. Specific Implementations and Coordinate Mapping

For composite and higher-dimensional systems, the generalized Gell–Mann matrices provide a bridge to the Pauli-tensor basis, particularly for multi-qubit systems ($d=2^n$). For example, $\mathfrak{su}(8)$ generators can be explicitly expanded as linear combinations of three-qubit Pauli operators, with the mapping coefficients determined by change-of-basis matrices (Chew et al., 2020). Measuring all $2^n$ Pauli correlators enables reconstruction of the Bloch–Gell–Mann vector for states of $n$ qubits. This basis change is instrumental in experimental tomography, Hamiltonian learning, and quantum simulation.

---

References:

• (Valtinos et al., 2023)
• (Loubenets, 2019)
• (Loubenets et al., 2020)
• (Chew et al., 2020)
• (Giraud et al., 2014)