Operators in the addition of $N$ spin 1/2 particles

Question

Set-up: We are given $N$ spin $\frac{1}{2}$ particles, with associated Pauli operators $\{X_i, Y_i, Z_i\}_{i=1}^N$. We can quickly verify that the operators $\bar{X}=X_1+X_2+\ldots X_n$, $\bar{Y}=Y_1+Y_2+\ldots Y_n$, $\bar{Z}=Z_1+Z_2+\ldots Z_n$ satisfy the commutation relations

$$[\bar{X},\bar{Y}]=2i \bar{Z}, [\bar{Y},\bar{Z}]=2i \bar{X}, [\bar{Z},\bar{X}]=2i \bar{Y}.$$

Thus, $\frac{1}{2}\bar{X}, \frac{1}{2}\bar{Y}, \frac{1}{2}\bar{Z}$ satisfy the angular momentum commutation relation (with $\hbar$ set to $1$ for convenience).

The decomposition of $\frac{1}{2}\bar{X}, \frac{1}{2}\bar{Y}, \frac{1}{2}\bar{Z}$ in terms of irreducible representations is also well understood. For example, answers in this and this stack-exchange question discuss the number of times a spin $\frac{k}{2}$ irreducible representation appears in the decomposition. Let us formally write the decomposition as:

$$\frac{1}{2}\bar{X}=\bigoplus_{\ell} J_x^{(\ell)}, \frac{1}{2}\bar{Y} = \bigoplus_{\ell} J_y^{(\ell)}, \frac{1}{2}\bar{Z}=\bigoplus_{\ell} J_z^{(\ell)}$$

where $\ell$ labels each irreducible decomposition and $J_x^{(\ell)},J_y^{(\ell)}, J_z^{(\ell)}$ are the matrices in the $\ell-$th decomposition.

Question: Is there an explicit expression for $J_x^{(\ell)},J_y^{(\ell)}, J_z^{(\ell)}$ (for each $\ell$) in terms of Pauli operators $\{X_i, Y_i, Z_i\}_{i=1}^N$? There is clearly some expression, since Pauli operators span the space of matrices on $N$ spin $\frac{1}{2}$ particles. But I am looking for an explicit one, which hopefully would be a nice multi-variate polynomial in the Paulis.

Answer 1

Each matrix $J_i^{(l)}$ is an angular momentum matrix in some irreducible representation. As mentioned, each decomposition $l$ corresponds to some (nonunique) spin-$k/2$. All that is left is writing the angular momentum matrices for a spin-$k/2$ system, which is a common exercise.

Your question is now whether a given spin matrix in a given dimension can easily be written in terms of Pauli matrices. The answer is actually no, because tensor products of Pauli matrices always correspond to $2^n$-dimensional matrices for some positive integer $n$, while spin matrices must have dimension $k+1$. For example, a spin-$1$ matrix may look like $$J_z=\frac{1}{2}\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}.$$ There's no way of writing this in terms of Pauli matrices; in fact, since $3$ is a prime number, one cannot decompose this matrix in terms of sums of tensor products of matrices of smaller sizes (other than trivial things like scalars multiplied by $3\times 3$ matrices). It is thus impossible in general to write the angular momentum matrices in terms of Pauli matrices alone.

The generalizations of Pauli matrices that you are looking for are probably these spin matrices themselves. Alternatively, you can look at Gell-Mann matrices or clock and shift matrices as described in this answer for generalizing Pauli matrices.

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The updated question says we are not directly interested in the matrices $J_i^{(l)}$ but actually in representing the matrices acting on the full system, like $0\oplus 0\oplus J_i^{(l)}\oplus 0\cdots$. Of course these can be achieved, but the results shouldn't be expected to be illuminating...

To make the connection, we need to express each single-element matrix $E_{jk}$ as a linear combination of tensor products of Pauli matrices. This can be done by expanding the locations of the nonzero elements in binary with $N$ digits, like $j=0101000\equiv\mathbf{j}$, to establish the connection $E_{jk}=|\mathbf{j}\rangle\langle \mathbf{k}|$. In this fashion, we have $E_{jk}=|j_1\rangle\langle k_1|\otimes\cdot\otimes |j_N\rangle \langle k_N|$. Then, we use Pauli matrices as a basis for each $|j_i\rangle\langle k_i|$ to write $E_{jk}$ as a linear combination of tensor products of Pauli matrices. Finally, we express each $J_i^{(l)}$ as a linear combination of the single-element matrices $E_{jk}$, which is the same as expanding them in the computational basis, so the matrix elements follow directly from the standard definitions of the angular momentum matrices (like $J_z$ in dimension $3$ quoted above). There are lots of indices flying around, from the location of the $l$th block to the location within the $l$th block, as well as the bit strings vs the locations corresponding to those bit strings, but it is all doable if one insists.