Trace distance and state discrimination

A density operator is positive and has unit trace. For two candidate states define the Hermitian difference $\Delta=\rho-\sigma$. Its eigenvalues sum to zero, so the sum of its positive eigenvalues equals the absolute sum of its negative eigenvalues. The trace norm adds the absolute eigenvalues of a Hermitian operator. Including the factor one half makes the resulting distance range from zero for identical states to one for states with orthogonal supports.

A binary measurement has effects $M$ and $I-M$, with $0\le M\le I$. If outcome $M$ means guessing rho, equal priors give a success probability equal to one half plus one half of $\operatorname{Tr}(M\Delta)$. In the eigenbasis of the difference, this quantity is maximized by accepting every positive eigenspace and rejecting every negative one. Zero eigenspaces can be assigned either way without affecting success. This derivation explains the optimal measurement rather than merely naming a norm.

If both matrices are diagonal in a common basis, measuring that basis produces classical distributions. Trace distance then becomes total variation distance: sum the absolute differences of probabilities and divide by two. No quantum measurement improves on reading this common eigenbasis. For noncommuting matrices, however, the computational basis can hide their difference. The pure states with opposite x-directed Bloch vectors have identical computational-basis probabilities but are perfectly distinguished by an x-basis measurement.

Unequal priors require a weighted difference $p\rho-(1-p)\sigma$. Its trace is generally nonzero, and the optimum success probability is one half of one plus its trace norm. Substituting the equal-prior distance formula into that task is incorrect. Similarly, repeated independent copies require comparing tensor powers rather than multiplying a single-copy success probability. A receiver can sometimes improve discrimination further by a collective measurement across the entire block.

A quantum channel is a linear, completely positive, trace-preserving map. Applying the same channel to either hypothesis cannot increase their trace distance. Any output measurement could instead be implemented as a measurement on the input followed by that channel, so it cannot outperform the unrestricted input optimum. Partial trace is one example: losing a tensor factor can erase distinguishing correlations even if each retained subsystem remains perfectly well defined.

This conclusion concerns deterministic processing with all outcomes included. Conditioning on a rare success flag can increase the distance between normalized conditional states, but its probability must enter the overall task. It also does not say every particular measurement attains the distance. A detector that measures only population can miss coherence entirely. Distinguish a fundamental impossibility from a restriction imposed by the available apparatus, especially when comparing a theoretical error bound with an experimental readout.