= Invariant-pure-state discrimination against a maximally mixed reset
{title2=$P_{\rm succ}=1-1/(2d)$}
Suppose one equally likely action returns a known normalized <pure state> $|s\rangle$ unchanged, while another returns the <maximally mixed state> $I/d$. A <basic measurement> in any <orthonormal basis> containing $|s\rangle$ 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 $|s\rangle\langle s|-I/d$ 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 $1/2+\|\rho_0-\rho_1\|_1/4$. 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.<The Theory of Quantum Information by John Watrous>[§3.1.1, Theorem 3.4 and equation (3.20)]{ref}
Back to article page
