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For $|J,J⟩$, tracing out subsystem 2 gives the reduced density matrixwhich is pure, so its entanglement entropy is zero.

For $|J,J-1⟩$, the two subsystem-2 factors in part (iii) are orthogonal. The partial trace therefore removes the cross terms and givesIts nonzero eigenvalues are $j_1/J$ and $j_2/J$, so the Von Neumann entropy isWhen $j_1=j_2$, both eigenvalues are $1/2$ and $S=\log 2$, the maximal entropy for a rank-two reduced state.

Apply the assumed data-processing inequality for quantum relative entropy to the normalized partial trace over $C$, with the two input statesThe channel sends them to ${\rho }_{AB}\otimes I_C/d_C$ and $I_A/d_A\otimes {\rho }_B\otimes I_C/d_C$. Additivity over the common maximally mixed factor reduces data processing toExpanding the Umegaki relative entropy in terms of Von Neumann entropy givesAfter cancelling $\log d_A$, this is precisely the Strong subadditivity of Von Neumann entropy

For a density operator $\rho$ on a finite-dimensional Hilbert space, the Von Neumann entropy iswith $0\log 0=0$. Its concavity says that, for $0\le p\le 1$,

Tracing out $B$ gives ${\rho }_A=I/2$. Its Von Neumann entropy is $-\mathrm{tr}{\rho }_A{\log }_2{\rho }_A=1$, and $\mathrm{tr}{\rho }_A^2=1/2$, so it is mixed.

A bipartite density operator is a product state when it factors as ${\rho }_A\otimes {\rho }_B$. A pure product state factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$; its reduced density matrices have rank one and zero entanglement entropy. Mixed product states can have nonzero marginal Von Neumann entropy.

For any tripartite density operator, expanding the three quantum mutual information terms in Von Neumann entropy gives $I(X:BD)=I(X:B)+I(XB:D)-I(B:D)$. This algebraic identity allows separate bounds on two information contributions, while nonnegativity of quantum relative entropy allows the last term to be dropped in an upper bound.

In an orthonormal basis, rank-one dephasing is the quantum channel $\Delta (\rho )={\sum }_i|{\psi }_i⟩⟨{\psi }_i|\rho |{\psi }_i⟩⟨{\psi }_i|$. It removes off-diagonal entries while preserving the basis probabilities. Its output Von Neumann entropy is their Shannon entropy, and the relative-entropy identity for rank-one dephasing quantifies the entropy increase.

Schumacher compression is quantum source coding into the typical subspace of ${\rho }^{\otimes n}$. Asymptotically reliable compression is possible at every qubit rate above the Von Neumann entropy $S(\rho )$.