Manuscript Hawk01-A@1 · Hawk01-A.tex · theorem · 11979653

Theorem: Sharp measure-before-query bound

Exact delayed-query scoring and classical nulls · lines 787–837 · proof included or linked

Hawk01-A@1#thm:classical, Hawk01-A@1#eq:EBform, Hawk01-A@1#eq:EBbound

Suppose the complete channel from the trusted qubit is measure-and-prepare:

N (ρ) = y \sum Tr (E_{y} ρ) σ_{y}, E_{y} \geq 0, y \sum E_{y} = I,

with the measurement performed before the fresh query. Then

Q_{opt} (N) \leq \frac{3}{4} .

The bound is sharp, even with a two-outcome measurement and a one-bit classical record. It also holds when the complete pre-query output is an arbitrary classical record generated by a POVM on the trusted qubit.

Proof. Giving the decoder the entire label $y$ can only improve its performance. Write

E_{y} = w_{y} (I + r_{y} \cdot σ), w_{y} \geq 0, ∣ r_{y} ∣ \leq 1, y \sum w_{y} = 1.

For the classical label channel, the requested-bit probability difference is

p (y ∣ x_{j} = 0) - p (y ∣ x_{j} = 1) = \frac{Tr ( E _{y} σ _{j} )}{2},

where $σ_{1} = σ_{x}$ and $σ_{2} = σ_{z}$ . Equation (eq:helstromscore) therefore gives

Q = \frac{1}{2} + \frac{1}{4 2} y \sum w_{y} (∣ r_{y, x} ∣ + ∣ r_{y, z} ∣) \leq \frac{1}{2} + \frac{1}{4} .

The final inequality uses $∣ r_{x} ∣ + ∣ r_{z} ∣ \leq 2 ∣ r ∣ \leq 2$ . Measuring along $(σ_{x} + σ_{z}) / 2$ attains equality. For a continuous outcome space the same proof uses the trace measure of the POVM and its Bloch-vector density.

Displayed formulas in this passage

Hawk01-A@1#eq:EBform · line 791
$N (ρ) = y \sum Tr (E_{y} ρ) σ_{y}, E_{y} \geq 0, y \sum E_{y} = I,$
Hawk01-A@1#eq:EBbound · line 799
$Q_{opt} (N) \leq \frac{3}{4} .$
unlabeled · line 812
$E_{y} = w_{y} (I + r_{y} \cdot σ), w_{y} \geq 0, ∣ r_{y} ∣ \leq 1, y \sum w_{y} = 1.$
unlabeled · line 820
$p (y ∣ x_{j} = 0) - p (y ∣ x_{j} = 1) = \frac{Tr ( E _{y} σ _{j} )}{2},$
unlabeled · line 826
$Q = \frac{1}{2} + \frac{1}{4 2} y \sum w_{y} (∣ r_{y, x} ∣ + ∣ r_{y, z} ∣) \leq \frac{1}{2} + \frac{1}{4} .$

Theorem: Sharp measure-before-query bound

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