Trace-distance-preserving binary measurement
= Trace-distance-preserving binary measurement
For two <density operators> $\rho,\sigma$, let $P_+$ project onto the positive spectral subspace of $\rho-\sigma$. The binary <POVM> $\{P_+,I-P_+\}$ has output probability difference $(D,-D)$, where $D=D(\rho,\sigma)$. Its <measurement channel> therefore preserves this pair's <trace distance> exactly. The optimizing <quantum measurement> depends on the pair; a single fixed measurement need not preserve every pair's distance.