Invariant-pure-state discrimination against a maximally mixed reset ($P_{\mathrm{succ}}=1-1/(2d)$)

Suppose one equally likely action returns a known normalized pure state $|s⟩$ unchanged, while another returns the maximally mixed state $I/d$. A basic measurement in any orthonormal basis containing $|s⟩$ distinguishes them with success $1-1/(2d)$: guess the pure-state action on its basis label and the reset action on the remaining $d-1$ labels. The first action succeeds with probability one and the second with probability $(d-1)/d$.

The Holevo–Helstrom theorem also proves optimality for these returned states. The Hermitian difference $|s⟩⟨s|-I/d$ has eigenvalues $1-1/d$ and $-1/d$ with multiplicity $d-1$, hence trace norm $2(1-1/d)$. Its optimal equal-prior discrimination success is $1/2+\Vert {\rho }_0-{\rho }_1{\Vert }_1/4$. Random computational-basis permutations fix the uniform superposition, so they supply an example of the invariant action. This compares the returned states in the stated unassisted protocol; it makes no assertion about a more general channel-discrimination problem with an entangled ancillary probe.^[ref]