Does a basis of maximally entangled states exist for two-qubit or two-qutrit system so that the density matrices of the basis states don't commute?

Question

I want to find a basis of maximally entangled states $|\Psi_i\rangle$, for $\mathcal{H}^{2} \otimes \mathcal{H}^{2}$ and, $\mathcal{H}^{3} \otimes \mathcal{H}^{3}$ such that the density matrices of those states don't commute with each other.

If $\rho_{i} = |\Psi_i\rangle \langle\Psi_i|$

I need the following condition to be met: $[\rho_{i},\rho_{j}] \ne 0$ for at least one pair of indices $i, j$.

I have tried using quasi Bell states, and Bell states in $|+\rangle, |-\rangle$ basis etc. but haven't succeeded yet.

If one or more such bases exist, how should I go about constructing one? If such a basis doesn't exist, what would be the next best basis in terms of entanglement, given that non-commutativity and entangled states are my priority. The orthogonality requirement can be discarded if necessary.

Answer 1

No such (orthonormal) basis can exist. An orthonormal basis $\{|\psi_i\rangle\}$ requires $\langle \psi_i | \psi_j \rangle = 0$ for $i\neq j$, and so clearly

\begin{align} [\rho_i, \rho_j] &= |\psi_i\rangle \langle \psi_i | \psi_j\rangle \langle \psi_j | - | \psi_j\rangle \langle \psi_j |\psi_i\rangle \langle \psi_i | \\ &= 0 \end{align}

So to get a basis whose elements don't commute you have to sacrifice orthogonality. In that case its hard to recommend what the "next best basis" would be without knowing what your goal is. You could start with a Bell basis and apply a local rotation to just one of the states, for example replace the state $ \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ with $(I\otimes R_z(\theta)) \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, which is still maximally entangled but no longer commutes with $\frac{1}{\sqrt{2}}(|00\rangle - |11 \rangle)$.

Also note that if you choose to relax the orthogonality constraint you can make your search a bit easier. Assume you've found any $\rho_i, \rho_j$ that do not commute. Then their commutator is still nonzero under an arbitrary unitary $U$: \begin{align} [\rho_i, \rho_j] &= A \\\rightarrow U[\rho_i, \rho_j]U^\dagger &= U\rho_i U^\dagger U \rho_j U^\dagger - U\rho_j U^\dagger U \rho_i U^\dagger \\&= [U\rho_i U^\dagger, U \rho_j U^\dagger] \\&= UAU^\dagger \end{align}

So regardless of how entangled the states that you found were, you can apply a transformation so that at least one of them becomes maximally entangled.