# What does it mean geometrically (in terms of vectors in the Bloch sphere) if the commutator of two density matrices $ρ_1$ and $ρ_2$ vanishes?

When the commutator of two operators vanishes then we can measure one without affecting the other. I'm not sure how this translates in the case of density matrices.

If the density matrices are representing pure states then the density matrices would represent projection operators onto the subspace spanned by the given state. So I think a vanishing commutator on two density matrices could mean that either the subspace spanned by the two states are orthogonal or that they are the same. Is this correct?

If two operators commute, they have the same eigenvectors. For the density matrix of a qubit, the eigenvectors can be visualised as being along a particular axis of the Bloch sphere, corresponding to the direction of the Bloch vector. So, two density matrices commute if their Bloch vectors ($$\vec{n}$$ and $$\vec{m}$$) are parallel, i.e. there exists a real number $$\alpha$$ such that $$\vec{n}=\alpha\vec{m}$$ (note the $$\alpha$$ may be negative. I'm also ignoring the trivial case of $$\vec{m}=0$$).
• @Rammus That is already taken care of. For qubits, this can only happen for pure states where $\vec{n}=-\vec{m}$. – DaftWullie May 21 at 7:53
• @Kay45 No, my answer covers that (pure is just the special case $\alpha=\pm 1$). – DaftWullie May 21 at 11:08