Manuscript Psy04-B@1 · Psy04-B-v1.tex · theorem · 62c2b6f0

Theorem: Optimal one-qubit score

Sequential qubit routing and explicit Enter operations › The sharp one-qubit upper benchmark · lines 1143–1195 · proof included or linked

Psy04-B@1#thm:qubitoptimal

Under the one-qubit cut accounting of Assumption ass:capacity, arbitrary qubit encodings and binary POVMs satisfy

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

Input-independent shared classical randomness does not increase this optimum.

Proof. Condition on shared randomness. Write

ρ_{s, t} = \frac{I + r _{s, t} \cdot σ}{2}, \norm r_{s, t}_{2} \leq 1,

where $s = (- 1)^{x_{1}}$ , $t = (- 1)^{x_{2}}$ .

For binary measurement $j$ , let

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

The condition $- I \leq A_{j} \leq I$ implies

∣ a_{j 0} ∣ + \norm a_{j}_{2} \leq 1.

The scalar terms cancel in the uniform input average:

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{\norm a _{1} + a _{2} _{2} + \norm a _{1} - a _{2} _{2}}{8} .

By Cauchy–Schwarz and the parallelogram identity, the numerator is at most

2 \norm a_{1}_{2}^{2} + \norm a_{2}_{2}^{2} \leq 22 .

The construction above attains equality. Average over the shared randomness.

Displayed formulas in this passage

unlabeled · line 1148
$Q \leq \frac{1}{2} + \frac{1}{2 2} .$
unlabeled · line 1158
$ρ_{s, t} = \frac{I + r _{s, t} \cdot σ}{2}, \norm r_{s, t}_{2} \leq 1,$
unlabeled · line 1165
$A_{j} = E_{0 ∣ j} - E_{1 ∣ j} = a_{j 0} I + a_{j} \cdot σ .$
unlabeled · line 1170
$∣ a_{j 0} ∣ + \norm a_{j}_{2} \leq 1.$
unlabeled · line 1174
$Q = \frac{1}{2} + \frac{1}{16} s, t = \pm 1 \sum r_{s, t} \cdot (s a_{1} + t a_{2}) .$
unlabeled · line 1181
$Q \leq \frac{1}{2} + \frac{\norm a _{1} + a _{2} _{2} + \norm a _{1} - a _{2} _{2}}{8} .$
unlabeled · line 1189
$2 \norm a_{1}_{2}^{2} + \norm a_{2}_{2}^{2} \leq 22 .$