I How Does Quantum Negativity Vary with Partial Transposes in Bipartite Systems?

[Jufa]

TL;DR

I am struggling with this concept mainly for two reasons: it is non-symmetric and I find it difficult to encounter a proper definition for mixed states.

Let as consider a system $H = A\otimes B$

I've been said that quantum negativity, i.e. taking the partial transpose w.r.t A or B and summing the magnitude of the negative eigenvalues obtained, is a measure of how entangled are the parties A and B.
First question:
Why is it that we do not always obtain the same negativity regardless of the system from which we take the partial transpose? After all the negativity tells how entangled is the bipartite system, so intuitively one can expect something like$N(\rho^A)=N(\rho^B)$. Nevertheless it is not difficult to fins some examples where this equality does not hold
Second question:
How do we define the negativity for mixed states? As other entanglement measures, I understand that the negativity of a bipartite state is the lower that can be found out of any of the possible collectivities may produce our mixed state but, again, from which system do we take the partial trace?

Thanks in advance

 

[azntoon]

.For the first question, it is important to note that the partial transpose of a bipartite state with respect to one of the systems (say system A) may not be the same as the partial transpose of the same state with respect to the other system (say system B). This is because the partial transpose operation is not commutative; in other words, the partial transpose of a state $\rho$ w.r.t. system A is not necessarily equal to the partial transpose of the same state w.r.t. system B. For the second question, the negativity of a mixed state is defined as the sum of the absolute values of the negative eigenvalues of the partial transpose with respect to either system A or B (whichever yields the lowest value). This is because the partial transpose operation is not commutative, so it is possible that taking the partial transpose of a mixed state with respect to one system (say system A) may yield different results than taking the partial transpose of the same state with respect to the other system (say system B). In such cases, the lower value should be used to calculate the negativity.