# What are the range of values of the negativity in quantum mechanics?

The negativity of a quantum state, given by

$N(\rho)=\frac{||\rho^{\Gamma_A}||-1}{2}$

where $\rho^{\Gamma_A}$ is the partial transpose with respect to subsystem $A$ of the density matrix of a quantum state and $||.||$ is the trace norm. The measures the degree of entanglement between quantum states. I can plot this for a given quantum state but would like to check my answer is correct. I can't find anywhere what the maximum and minimum values of the negativity can be, and which refers to maximum entanglement. Does anyone have this information?

On the other hand, negativity is zero on separable states. For the same reason, this is the lowest value. (More directly, the partial transpose is trace preserving, so the eigenvalues of $\rho^{\Gamma_A}$ add up to $1$. Thus, the sum of their absolute values must be $\ge 1$.)