Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 60 / 4

Nielsen's pure-state conversion theorem gives the exact deterministic LOCC criterion. Let the Schmidt decompositions be $|\psi ⟩={\sum }_j\sqrt{{\lambda }_j}|a_j{b}_j⟩$ and $|\varphi ⟩={\sum }_j\sqrt{{\mu }_j}|a_j^{\prime }{b}_j^{\prime }⟩$. The vectors $\lambda ,\mu$ consist of squared Schmidt coefficients, equivalently the eigenvalues of either reduced density operator. Order each in decreasing order and pad with zeros to a common length $d$.

Then deterministic conversion is possible if and only ifThis is majorization, with the input vector majorized by the output vector. The direction matters: a maximally entangled state has a uniform vector, which is majorized by a product state's vector $(1,0,\dots )$, so entanglement can be discarded by LOCC. The criterion is for certain exact conversion, without catalysts or postselection on a successful branch.

Let $r$ be the input Schmidt rank, so ${\lambda }_j=0$ for $j>r$ and ${\sum }_{j=1}^r{\lambda }_j=1$. If $r=d$, the output rank is already at most $r$. Otherwise, Nielsen's pure-state conversion theorem and majorization at $k=r$ giveThus all output coefficients beyond $r$ vanish, provingThis is monotonicity of Schmidt rank under LOCC. It also holds separately in any nonzero postselected branch: represent the input amplitudes by a matrix $C$; a local branch maps it to $AC{B}^T$, whose rank cannot exceed the rank of $C$. The deterministic result requested here follows already from the majorization criterion.

For any tripartite density operator, Strong subadditivity of Von Neumann entropy isUse quantum relative entropy $D(\rho \Vert \sigma )=\mathrm{Tr}\rho ({\log }_2\rho -{\log }_2\sigma )$, with the usual support condition. The equivalent relative-entropy comparison isIndeed the two sides expand respectively as $S(A)+S(BC)-S(ABC)$ and $S(A)+S(B)-S(AB)$. Subtracting cancels $S(A)$ and leaves exactly the strong-subadditivity gap. Marginal-product supports contain the support of the joint state, so these expressions are finite; singular marginals can also be handled by full-rank regularization and a limit.

Finally, tracing out $C$ sends the numerator and denominator of the first relative entropy to those of the second. The data-processing inequality for quantum relative entropy therefore proves the comparison. This is strong subadditivity from relative-entropy monotonicity, rather than an assumption that classical entropy proofs automatically apply to quantum states.

Take a Stinespring dilation $V:B⟶B^{\prime }E$ of the local quantum channel and define ${\tau }_{A{B}^{\prime }E}=(I_A\otimes V){\rho }_{AB}(I_A\otimes V^{†})$. Tracing out $E$ gives the prescribed output, with $A^{\prime }=A$. An isometry preserves the nonzero eigenvalues of a density operator; therefore $S(B^{\prime }E{)}_{\tau }=S(B{)}_{\rho }$, $S(A{B}^{\prime }E{)}_{\tau }=S(AB{)}_{\rho }$, and $S(A{)}_{\tau }=S(A{)}_{\rho }$.

Consequently $I(A:B^{\prime }E{)}_{\tau }=I(A:B{)}_{\rho }$. The loss of quantum mutual information isThe last inequality is Strong subadditivity of Von Neumann entropy, or nonnegativity of quantum conditional mutual information. HenceThis mutual-information loss as conditional mutual information shows exactly which correlations are discarded into the environment. No purity assumption on the original state is needed.