Theorem: Spectra and mutual-information balance

For the Bell input,

including zero multiplicities. The residual spectrum is obtained by interchanging $\eta$ and $t$.

With

using $0{\log }_{2}0=0$, one has

Consequently,

and

Proof. For the Bell input the matrices (eq:rhoR) and (eq:rhoE) are diagonal, giving the spectra. The global state $|{\Psi }_{\eta }⟩$ is pure and $S(Q)=1$. For a pure bipartite state the two reduced states have the same nonzero spectrum: expand its coefficient matrix and compare the nonzero eigenvalues of $A{A}^{†}$ and $A^{†}A$. Apply this to $QR|E$ and $QE|R$ to obtain $S(QR)=S(E)$ and $S(QE)=S(R)$. This proves (eq:MIr)–(eq:balance).

At the endpoints the spectra give entropies $0,1$ in opposite orders. At $\eta =t=1/2$, the spectra coincide. Squaring and summing the accessible eigenvalues gives the three displayed purities.