Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 4

Let $\Delta =\rho -\sigma ={\Delta }_{+}-{\Delta }_{-}$ be its positive-negative decomposition. Since $\mathrm{Tr}\Delta =0$, the positive and negative parts have equal trace, namely $D(\rho ,\sigma )$. For every positive contraction $M$,Here both products have nonnegative trace, and $M\le I$ bounds the first term. Equality is attained by the orthogonal projection $P_{+}$ onto the positive spectral subspace of $\Delta$, since $P_{+}{\Delta }_{+}={\Delta }_{+}$ and $P_{+}{\Delta }_{-}=0$. Thus the variational characterization of trace distance is

Choose an orthonormal basis $\{|j⟩{\}}_{j=1}^d$ of the input space and define operators from the input to the $k$-dimensional outcome register byThese give a Kraus representation:The POVM completeness relation givesTherefore $\Phi$ is a completely positive map and is trace preserving: it is a quantum channel representing a deterministic measurement channel.

Let $P_{+}$ be the maximizing projection in the variational characterization of trace distance. Use the binary POVM $\{P_{+},I-P_{+}\}$ and its measurement channel. The difference between its two output probability vectors iswhere $D=D(\rho ,\sigma )$ and the second equality uses $\mathrm{Tr}(\rho -\sigma )=0$. Since trace distance between diagonal density operators is half the L1 norm of their probability-vector difference,Thus a binary quantum measurement can preserve the distinguishability of this particular pair exactly. This is the trace-distance-preserving binary measurement, which may depend on the two states.

Choose the trace-distance-preserving binary measurement from the previous part. Let $p,q$ be its output probability distributions, so $\Vert p-q{\Vert }_1=2D(\rho ,\sigma )$. The data-processing inequality for quantum relative entropy givesFor diagonal density operators, the quantum expression is the classical Kullback-Leibler divergence. Applying the supplied Pinsker's inequality givesConsequently the quantum Pinsker inequality isThe $\ln 2$ factor converts natural-logarithm relative entropy to bits. If the relative entropy is infinite because its support condition fails, the inequality holds automatically.