Can maximum entanglement occur between states of different dimensions?

Question

If a bipartite state on $H^A \otimes H^B$ is maximally entangled, does the dimension of $H^A$ have to equal to the dimension of $H^B$? If it need not, (in my own very abstract terms), shouldn't the larger Hilbert space have more "structure" in it to form correlations/entanglement with other systems, thus violating the monogamy of (maximum) entanglement?

Answer 1

This boils down to what you mean by "maximal entanglement". Usually, you define maximal entanglement by saying that both reduced density matrices are proportional to the identity. However, if the system consists of two parts with unequal dimension, this is impossible to happen.

Here is why: Given $|\psi\rangle \in \mathcal{H}_A\otimes \mathcal{H}_B$, you can write it as $|\psi\rangle = \sum_{i=1}^k \sqrt{\lambda_i} |i_A\rangle\otimes |i_B\rangle$ where $|i_A\rangle,|i_B\rangle$ form part of an orthonormal basis of $\mathcal{H}_A$ and $\mathcal{H}_B$ respectively (this is known as the Schmidt decomposition).

Now, if you compute the reduced density matrices, you will find that they are given by

$$\rho_A = \sum_i \lambda_i |i_A\rangle\langle i_A|$$

$$\rho_B = \sum_i \lambda_i |i_B\rangle\langle i_B|.$$

Just seeing this, it should be clear that we have a problem if the dimension of $\mathcal{H}_A$ and $\mathcal{H}_B$ do not match: The two matrices have equal rank hence not both of them can have full rank. It is easy to see that the maximal entanglement entropy between the two systems is reached if $\lambda_i=\lambda_j$ for all $j$. The amount is then given by the dimension of the smaller system.

In other words: As you suspect the maximal entanglement of the combined system is dictated by the smaller system and there is "room" in the larger system. The two systems can, however, not be maximally entangled in the usual sense and there is no violation of monogamy.