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 specificallyI have read that if $\mathcal{N}_A(\rho)\neq 0$ then one can claim $A$ is entangled with $B$. This is what I don’t understand. By definition, $\mathcal{N}_A(\rho)$ measures by how much $\rho^{T_A}$ fails to be positive semidefinite, an essential property of a separable and hence a non-entangled system. Great, we know whether $\rho$ is entangled or not. However, just because we are told $\rho$ is entangled it doesn’t necessarily mean that the degrees of freedom in $A$ are entangled with those in $B$ right? I guess my problem could steem from the fact that I don’t understand the physical consequences of taking a partial transpose of $\rho$ w.r.t. some subsystem (i.e. what is the physical significance of $\rho^{T_A}$?). interpret $\mathcal{N}_A(\rho)$ as the entanglement between $A$ and $B$.
Why is this true?Edit: First of all for your all your comments and generous patience. I edited the question to better address my last issue with understanding entanglement negativity.
