Constraints on higher-dimensional Bloch vectors

Question

I'm interested in the constraints on the $(4^n-1)$-dimensional generalized Bloch vector (the Bloch vector for $n$ qubits). To the best of my knowledge, these are not analytically characterized for the general case, but what if the vector only has support on a fully commuting or fully anticommuting set of generalized Pauli operators (tensor products of $2\times2$ Pauli matrices)? Does anyone know if the constraints simplify in either of these cases?

Answer 1

Since I've now worked out part of the answer to my question, and apparently (https://stackoverflow.com/help/self-answer) answering your own question is encouraged, I'll post it in case anyone else is interested.

Given a completely anticommuting set $\mathcal{S}$ of Pauli operators on $n$ qubits, we know that for any density operator $\rho$ we may write $$\rho=\frac{1}{2^n}(I+\vec{m}\cdot\vec{\mathcal{S}}+\cdots),$$ where the ellipsis indicates that in general more Paulis will be necessary to specify the whole density matrix (I've written the set $\mathcal{S}$ as a vector $\vec{\mathcal{S}}$ so that I can represent the coefficients simply as the vector $\vec{m}$.) Then the vector $\vec{m}$ satisfies $$|\vec{m}|\le1.$$ This follows fairly straightforwardly from the fact that the observable $\vec{m}\cdot\vec{\mathcal{S}}$ has eigenvalues $\pm|\vec{m}|$ when $\mathcal{S}$ is fully anticommuting.

Furthermore, any (real) vector $\vec{m}$ satisfying $|\vec{m}|\le1$ corresponds to at least one valid density operator with the form given above, namely $$\rho'\equiv\frac{1}{2^n}(I+\vec{m}\cdot\vec{\mathcal{S}}).$$ Since the density operators form a convex set, it is enough to show that this is true for any $|\vec{m}|=1$, which one can do by explicitly constructing the corresponding pure state (which turns out to always exist) and showing that its outer product with itself gives the desired density operator.

Thus, the possible projections of density operators onto to any fully anticommuting set of Pauli operators are parametrized by a unit ball of the appropriate dimension.

As soon as we introduce any commuting operators into the picture, the situation becomes a lot more complicated, and I have not found a simple constraint for the associated Bloch vectors.