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Pauli matrices, together with the identity matrix can generate any $2\times 2$ matrix. By adding the condition that the matrices must be hermitian and with trace 1, we can represent density matrices for spin-$\frac{1}{2}$ systems as $$ \rho=\frac{1}{2}(\mathbb{I} + \mathbf{P}\cdot\boldsymbol{\sigma}), $$ where we can determine the polarization $\mathbf{P}$ with the ensemble averages by $[S_i]\propto [\sigma_i]=P_i$.

For spin-$1$ systems, we can use the Gell-Mann matrices ($\boldsymbol{\lambda})$ instead of the Pauli matrices to represent $3\times 3$ density matrices: $$ \rho = \frac{1}{3}(\mathbb{I}+\boldsymbol{\Lambda}\cdot\boldsymbol{\lambda}), $$ where $\boldsymbol{\Lambda}$ is a vector of length $8$. However, the Gell-Mann matrices are not directly related to the spin components like Pauli matrices are. So we can represent these density matrices another way (I can't find any literature on this): $$ \rho=\frac{1}{3}(\mathbb{I} + \mathbf{P}\cdot\boldsymbol{\sigma}+\mathbf{W}\cdot\mathbf{T}), $$ where $\mathbf{W}$ is a vector of length 5, and $\mathbf{T}$ consists of 5 matrices where $$ T_{ij}=\frac{1}{2}(J_iJ_j+J_jJ_i)-\frac{2}{3}\delta_{ij}, \hspace{3mm}i,j \in \{1,2,3\}, \hspace{3mm} i\leq j. $$ ($J_i$ are the angular momentum operators for spin $1$.) Then, I suppose $\mathbf{P}$ is related to $[\mathbf{J}]$ the same way as before, and that the elements of $\mathbf{W}$ are related to $[J_iJ_j]$.

How are these $T_{ij}$ matrices called, and where can I read about this representation of the density matrix for spin-$1$ systems (the one without the Gell-Mann matrices)? Are the last two statements true? These are things that are mentioned in my QM notes but I can't find anything similar anywhere.

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  • $\begingroup$ I had a similar problem looking at the polarisation of light (photons are spin-1 their polarisation maps to their spin). I never solved it but this paper was very helpful: citeseerx.ist.psu.edu/viewdoc/… (Although it seems you already have its information yourself) $\endgroup$ – Dast Sep 25 '19 at 18:38
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You might well be giving the Gell-Mann matrices a bad rap. They are all traceless hermitian, but they are real, except for the three imaginary ones, $\lambda_2,\lambda_5,\lambda_7$ which are imaginary antisymmetric, so, multiplied by i , comprise the three antisymmetric generators of SO(3) in the triplet (spin 1) representation. Behold.

So your three σ are the antisymmetric ones above (with indices 2,5,7), and your five symmetric ones (with indices 1,3,4,6,8) are the rest, T, suitably normalized. They are, together with the identity, a complete set for hermitean 3×3 matrices, and orthonormal, $$\operatorname {Tr} (\lambda_a \lambda_b)=2\delta_{ab}, $$ so tracing with σ for its expectation value nets you your result.

I don't think there is a popular name for the antisymmetric/symmetric split, but if you have ever understood their (G-M matrices’) structure constants, you rely on this very divide to appreciate why they are so sparse.

But the same argument holds for the sparseness of the symmetric d coefficients for the anticommutators.

That is to say, from the evident symmetry of the matrix $$\{\lambda_a,\lambda_b\}=\frac{4}{3}\delta_{ab} +2d_{abc}\lambda_c $$ where $d_{abc}$ are the completely symmetric coefficient constants, only the symmetric ones from the T set enter on the right hand side when both matrices in the anticommutator on the left hand side are antisymmetric, as in your question: The ds vanish if the number of indices from the set {2,5,7} is odd!

In conclusion, the anticommutator of any two σ is a linear combination of Ts, so, inverting these 6 equations (constrained by tracelessness) defines the 5 Ts in terms of the 3 σs.

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It may be that what you're looking for is the $\mathfrak{so}(3)$ basis of $\mathfrak{su}(3)$ spanned by the three angular momentum and the five quadrupole operators. As real matrices the angular momenta are antisymmetric matrices, and the quadrupole are traceless symmetric matrices. As hermitian operators there are $i$'s here and there.

These matrices were quite popular in the days of the nuclear $\mathfrak{su}(3)$ model for rotational bands in deformed nuclei. Google does not provide obviously accessible references but the two canonical references are

Harvey, Malcolm. "The Nuclear SU 3 Model." In Advances in nuclear physics, pp. 67-182. Springer, Boston, MA, 1968, and the original papers by Elliott: Elliot, J. P., Proc. Roy. Soc., A 245: 128 and 562, 1958.

It took the following matrices from

Rowe, D.J., Le Blanc, R. and Repka, J., 1989. A rotor expansion of thesu (3) Lie algebra. Journal of Physics A: Mathematical and General, 22(8), p.L309.

With $C_{ij}$ denoting the $3\times 3$ matrix with $1$ in row $i$ and column $j$, and $0$ elsewhere, we have: \begin{align} L_0&=-i(C_{23}-C_{32})\, ,\quad L_{\pm 1}= -i (C_{31}-C_{13}\pm (C_{12}-C_{21})\, ,\\ Q_{20}&= 2C_{11}-C_{22}-C_{33}\, ,\quad Q_{2\pm 1}= \mp \sqrt{\frac{3}{2}}(C_{12}+C_{21}\pm i C_{13}\pm iC_{31}\\ Q_{2\pm 2}&=\sqrt{\frac{3}{2}}(C_{22}-C_{33}\pm i C_{23}\pm i C_{32}) \end{align} The quadrupole moments are elements of an $L=2$ tensor operator.

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