Proposition: Reference–accessible negativity

For the Bell-input accessible state,

It is entangled for every $\eta >0$.

Proof. The coherence connects $|0,00⟩$ and $|1,11⟩$, with coefficient $\eta /2$. After partial transpose it enters the block on $|0,11⟩,|1,00⟩$:

The eigenvalues are

All other eigenvalues are nonnegative. The absolute value of the negative eigenvalue gives (eq:negativity). A separable state has positive partial transpose, so strict negativity proves entanglement.