Manuscript Psy04-A@1 · Psy04-A-v1.tex · theorem · d5800e2a

Theorem: Sharp qubit upper benchmark

A sequential one-qubit construction › Optimality for arbitrary binary qubit measurements · lines 620–672 · proof included or linked

Psy04-A@1#thm:quantumoptimal

Under Assumption ass:qubit, arbitrary qubit preparations and binary POVMs satisfy

Q \leq \frac{1}{2} + \frac{1}{2 2} .

Theorem thm:quantumattainment attains equality.

Proof. Condition on the shared classical randomness. Write

ρ_{x} = \frac{I + r _{x} \cdot σ}{2}, \abs r_{x} \leq 1.

For the two binary measurements, their difference observables have the form

A_{j} = E_{0 ∣ j} - E_{1 ∣ j} = a_{j 0} I + a_{j} \cdot σ .

Since $- I \leq A_{j} \leq I$ ,

\abs a_{j 0} + \abs a_{j} \leq 1.

Set $s = (- 1)^{x_{1}}$ , $t = (- 1)^{x_{2}}$ . The scalar terms cancel in the uniform input average, leaving

Q = \frac{1}{2} + \frac{1}{16} s, t = \pm 1 \sum r_{s, t} \cdot (s a_{1} + t a_{2}) .

Therefore

Q \leq \frac{1}{2} + \frac{\abs a _{1} + a _{2} + \abs a _{1} - a _{2}}{8} .

By Cauchy–Schwarz and the parallelogram identity,

\abs a_{1} + a_{2} + \abs a_{1} - a_{2} \leq 2 \abs a_{1}^{2} + \abs a_{2}^{2} \leq 22 .

This proves the bound. Average over the shared randomness.

Displayed formulas in this passage

unlabeled · line 624
$Q \leq \frac{1}{2} + \frac{1}{2 2} .$
unlabeled · line 633
$ρ_{x} = \frac{I + r _{x} \cdot σ}{2}, \abs r_{x} \leq 1.$
unlabeled · line 639
$A_{j} = E_{0 ∣ j} - E_{1 ∣ j} = a_{j 0} I + a_{j} \cdot σ .$
unlabeled · line 644
$\abs a_{j 0} + \abs a_{j} \leq 1.$
unlabeled · line 649
$Q = \frac{1}{2} + \frac{1}{16} s, t = \pm 1 \sum r_{s, t} \cdot (s a_{1} + t a_{2}) .$
unlabeled · line 656
$Q \leq \frac{1}{2} + \frac{\abs a _{1} + a _{2} + \abs a _{1} - a _{2}}{8} .$
unlabeled · line 663
$\abs a_{1} + a_{2} + \abs a_{1} - a_{2} \leq 2 \abs a_{1}^{2} + \abs a_{2}^{2} \leq 22 .$