There are two procedures that I know of for finding the von Neumann entanglement entropy of a bipartite system split between $A$ and $B$ - a bad way and a better way. I have in mind computationally calculating the entanglement entropy for a spin chain of even length $L$.
The bad way is to take the outer product $\rho = |\psi \rangle \langle \psi |$, trace out the $B$ subsystem to get $\rho_A = Tr_B[\rho]$, and diagonalize $\rho_A$ to get the entanglement spectrum $\{r_i\}$. One can rapidly plug in the entanglement spectrum to get the von Neumann entanglement entropy. Though this procedure is how entanglement entropy is defined, there's a better operational way to compute it below.
The better way is to reshape $|\psi\rangle$ according to the following isomorphism: $|\psi\rangle = \sum_{i,j} c_{ij} |A_i\rangle |B_j \rangle \cong \sum_{i,j} c_{ij} |A_i\rangle \langle B_j |$. This allows a singular value decomposition to get the Schmidt coefficients $\{\lambda_i\}$. The entanglement spectrum obeys $r_i = |\lambda_i|^2$, and so we can get the von Neumann entanglement entropy as above.
The better way was better because it avoided large memory usage. For a spin-1/2 chain, the bad way required constructing the $2^L$ by $2^L$ matrix $\rho$ (a total of $4^L$ entries). On the other hand, vector-matrix isomorphism above only has a total of $2^L$ entries - much better. This also means the better way is much faster to calculate the entanglement spectrum, since it's for a smaller matrix.
The above example is to motivate my question. I recently learned about the concept of logarithmic negativity which is defined on density matrices. That is, given a density matrix $\rho_{AB}$ describing subsystems $A$ and $B$, the procedure is to take $\log( | \rho_{AB}^{T_B} |)$ where $T_B$ is the partial transpose on subsystem $B$.
In particular, for my problem, I'm considering a pure state $|\psi\rangle$ with corresponding density matrix $\rho$ that I write in a tripartite form involving partitions $A$, $B$, $C$. I use the vector-matrix isomorphism above to get $M = \sum_i c_i |A_i\rangle |B_i\rangle \langle C_i|$ and construct $\rho_{AB} = Tr_C[\rho]$ via an appropriate matrix multiplication of $M$ with $M^\dagger$. I then take the partial transpose to get $\rho_{AB}^{T_B}$. I then find the spectrum of $\rho_{AB}^{T_B}$ and compute the logarithmic negativity.
My trouble is that if the subsystem $C$ is quite small, $\rho_{AB}$ is a massive matrix.
For example, if there's only one qubit in $C$, then $\rho_{AB}$ is a $2^{L-1}$ by $2^{L-1}$ matrix with $4^{L-1}$ entries. This is hard to store and especially hard to find the spectrum.
Is there any way to compute the entanglement negativity without constructing and diagonalizing $\rho_{AB}^T$, similarly to how one could calculate the entanglement entropy of a pure state slightly obliquely in "the better way" above?
