Theorem: Bell spectra and information balance

For this Bell input,

including zero multiplicities. The residual spectrum follows by interchanging $\eta ,t$.

With $0{\log }_{2}0=0$ and $I(X:Y)=S(X)+S(Y)-S(XY)$,

In particular,

These spectra and numerical values are input-specific.

Proof. The Bell-input marginal matrices are diagonal. The joint state is pure and $S(Q)=1$. The two marginals of a pure bipartite state have the same nonzero spectrum, as follows by comparing the nonzero eigenvalues of its coefficient matrices $A{A}^{†}$ and $A^{†}A$. Thus $S(QR)=S(E)$ and $S(QE)=S(R)$. This proves the information identities; the displayed values follow from the spectra.