Proposition: Exact reference–accessible negativity

For the Bell-input state,

Thus the reference and accessible output are entangled for every $\eta >0$.

Proof. The only coherence of ${\rho }_{QR}$ connects $|0{⟩}_Q|00{⟩}_R$ and $|1{⟩}_Q|11{⟩}_R$, with coefficient $\eta /2$. Under $T_Q$, it enters the block on $|0{⟩}_Q|11{⟩}_R,|1{⟩}_Q|00{⟩}_R$:

Its eigenvalues are

All remaining eigenvalues are nonnegative diagonal entries or zeros. The negative eigenvalue therefore gives (eq:rawneg). A separable state's partial transpose is positive, because transposition preserves positivity of each factor in a separable decomposition. The strictly negative eigenvalue at $\eta >0$ proves entanglement.