Entanglement Negativity and Reduced Density Matrices

Question

The negativity of a density matrix is defined as $$ \mathcal{N}(\rho_{AB}) = \frac{1}{2} \bigl(\|\rho_{AB}^{T_A}\|_1 - 1\bigr), $$

where $\rho_{AB}$ is a density matrix on the bipartite system $A \otimes B$, $T_A$ denotes the partial transpose with respect to subsystem $A$, and $\|\cdot\|_1$ is the trace norm (sum of absolute values of the eigenvalues).

Question: What is known about the relationship between the entanglement negativity $\mathcal{N}(\rho_{AB})$ and the spectrum of the reduced density matrix $\rho_B = \mathrm{Tr}_A(\rho_{AB}^{T_A})$? Specifically, if the partial transpose $\rho_{AB}^{T_A}$ has negative eigenvalues, does that imply anything about the possibility of negative eigenvalues for $\rho_B$? (Note that $\rho_A$ remains positive semi-definite under transposition since the tranpose on subsytem $A$ is a positive map.)

Answer 1

As Norbert pointed out in his comment, $\rho_B={\rm Tr}_A(\rho_{AB}^{T_A})$ will never have negative eigenvalues, and one way to see this is via the Kraus form of the partial trace: $${\rm Tr}_A(\rho_{AB}^{T_A})=\sum_j\big(\langle j|\otimes {\bf1}\big)\rho_{AB}^{T_A}\big(|j\rangle\otimes {\bf1}\big)=\sum_j\big(\langle j|\otimes {\bf1}\big)\rho_{AB}\big(|j\rangle\otimes {\bf1}\big)={\rm Tr}_A(\rho_{AB})\geq 0$$ So as a result of the Kraus operators of the partial trace being local, its power to probe non-local properties is limited. Intuitively, because the partial transpose is not physical and doesn’t correspond to any local measurement process, any entanglement measure derived from it necessarily encodes global information about the state—information that is inaccessible through local operations alone.