Manuscript Psy03-C@1 · Psy03-C-v1.tex · section · d3380b90

An exact delayed phase-label pilot

An exact delayed phase-label pilot · lines 1904–1932

Psy03-C@1#eq:qpilot

On a qubit, define

Δ_{λ} (ρ) = \frac{1 + λ}{2} ρ + \frac{1 - λ}{2} Z ρZ, 0 \leq λ \leq 1,

P_{η} (ρ) = η ρ + (1 - η) \frac{I}{2}, 0 \leq η \leq 1,

and

U_{x} = e^{- i θ (x) Z /2}, Λ_{x, s} = Ad_{U_{x}} P_{η_{s}} Δ_{λ_{s}} .

These are CPTP maps by their explicit mixture representations.

An early label is encoded as

ρ_{0}^{\pm} = \frac{I \pm X}{2} .

All later input symbols are independent of that label. No external register retaining the label is supplied to the readout.

Displayed formulas in this passage

unlabeled · line 1907
$Δ_{λ} (ρ) = \frac{1 + λ}{2} ρ + \frac{1 - λ}{2} Z ρZ, 0 \leq λ \leq 1,$
unlabeled · line 1912
$P_{η} (ρ) = η ρ + (1 - η) \frac{I}{2}, 0 \leq η \leq 1,$
Psy03-C@1#eq:qpilot · line 1917
$U_{x} = e^{- i θ (x) Z /2}, Λ_{x, s} = Ad_{U_{x}} P_{η_{s}} Δ_{λ_{s}} .$
unlabeled · line 1926
$ρ_{0}^{\pm} = \frac{I \pm X}{2} .$