Error rates of Belavkin weighted quantum measurements and a converse to Holevo’s asymptotic optimality theorem

We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called “pretty good measurement” (PGM) in a number of respects: (1) Holevo’s quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.

Figure 1

$P_{\text{fail}}/P_{\text{fail}}^{\text{opt}}$ for the PGM (upper) and Holevo’s measurement (lower) for binary ensembles with $\left|⟨{\psi }_1,{\psi }_2⟩\right|=\text{cos}\theta$ and $p_1=1-p_2=p$.

Figure 2

(a) $P_{\text{fail}}^{\text{Holevo}}/P_{\text{fail}}^{\text{opt}}$ for binary ensembles and (b) $P_{\text{fail}}^{\text{cubic}}/P_{\text{fail}}^{\text{opt}}$ for binary ensembles.