Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ with dimension $d$. Consider a subspace $S \subset V^{\otimes n}$ representing the code subspace of a quantum error correcting code. We can define the entanglement spectrum of $S$ with respect to a bipartition $A \cup B = \{1, 2, ..., n\}$ as the eigenvalues of the reduced density matrix $\rho_A = \text{Tr}_B(\rho)$, where $\rho$ is the maximally mixed state on $S$.

Now, suppose we impose a specific combinatorial structure on the entanglement spectrum. For instance, let's require that the multiplicities of the eigenvalues of $\rho_A$ correspond to the coefficients of a specific polynomial $P(x)$ with integer coefficients, i.e., if $\lambda_1, ..., \lambda_k$ are the distinct eigenvalues of $\rho_A$ with multiplicities $m_1, ..., m_k$, then $P(x) = \sum_{i=1}^k m_i x^{\lambda_i}$.

Given a polynomial $P(x)$ with integer coefficients and a dimension $d$, can we characterize the set of integers $n$ for which there exists a subspace $S \subset V^{\otimes n}$ with an entanglement spectrum whose eigenvalue multiplicities are given by the coefficients of $P(x)$?

Furthermore, can we establish a connection between the properties of $P(x)$ (e.g., its roots, degree, coefficients) and the properties of the corresponding quantum error correcting code $S$ (e.g., its distance, code rate)?

I know that the multiplicities of the eigenvalues in the entanglement spectrum are directly related to the dimensions of certain subspaces associated with the bipartition $A \cup B$. Specifically, consider the Schmidt decomposition of a state $|\psi\rangle \in S$:

$|\psi\rangle = \sum_i \sqrt{\lambda_i} |i_A\rangle \otimes |i_B\rangle$

where $\lambda_i$ are the eigenvalues of the reduced density matrix $\rho_A$, and $|i_A\rangle$ and $|i_B\rangle$ are orthonormal bases for the subsystems $A$ and $B$, respectively. The multiplicity $m_i$ of an eigenvalue $\lambda_i$ corresponds to the dimension of the subspace spanned by the vectors $|i_A\rangle$ associated with that eigenvalue.

Maybe we can try to count the number of states in $S$ that contribute to each eigenvalue multiplicity. This might involve analyzing the structure of the code subspace and how it relates to the bipartition. For example, if the code has certain symmetries, we can exploit them to simplify the counting process.