Consider a system composed of two subsystems $A$ and $B$ living in $\mathcal{H}=\mathcal{H}_A \otimes \mathcal{H}_B$. The density matrix of the system $AB$ is defined to be $\rho$. The entanglement negativity of $\rho$, defined as $$\mathcal{N}_A(\rho) := \frac12(\|\rho^{T_A}\|_1 -1),$$ where $\rho^{T_A}$ is the partial transposition of $\rho$ and $\|\cdot\|_1$ is the trace norm, measures by how much $\rho^{T_A}$ fails to be positive semidefinite. This is useful since would $\rho$ be separable, $\rho^{T_A}$ would be positive semidefinite, hence $\mathcal{N}_A(\rho)=0$. This, along with some other nice properties makes $\mathcal{N}$ a nice entanglement measure.

What I don’t understand is why one can specifically interpret $\mathcal{N}_A(\rho)$ as the entanglement between $A$ and $B$.

Why is this true?