Manuscript Hawk02-A@1 · Hawk02-A.tex · theorem · 7fbae745

Theorem: Reference-preserving sequential error

Stage 2: the backreaction interface › Sequential comparison requires complete lifts · lines 686–715 · proof included or linked

Hawk02-A@1#thm:sequential, Hawk02-A@1#eq:sequential

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Under Assumption ass:stepwise,

d_{⋄} (Φ_{N} \circ \dots \circ Φ_{1}, Φ_{N} \circ \dots \circ Φ_{1}) \leq min {1, j = 1 \sum^{N} η_{j}} .

Additional CPTP preparation, cutoff, or endpoint comparisons contribute their own diamond distances by the same rule.

Proof. Insert one replacement at a time:

Φ_{N} \dots Φ_{1} - Φ_{N} \dots Φ_{1} = j = 1 \sum^{N} Φ_{N} \dots Φ_{j + 1} (Φ_{j} - Φ_{j}) Φ_{j - 1} \dots Φ_{1} .

Pre- and postcomposition by channels do not increase the diamond norm. Apply the triangle inequality and divide by two. Channel distance is at most one. The argument already includes arbitrary references.

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Hawk02-A@1#eq:sequential · line 689
$d_{⋄} (Φ_{N} \circ \dots \circ Φ_{1}, Φ_{N} \circ \dots \circ Φ_{1}) \leq min {1, j = 1 \sum^{N} η_{j}} .$
unlabeled · line 703
$Φ_{N} \dots Φ_{1} - Φ_{N} \dots Φ_{1} = j = 1 \sum^{N} Φ_{N} \dots Φ_{j + 1} (Φ_{j} - Φ_{j}) Φ_{j - 1} \dots Φ_{1} .$

Theorem: Reference-preserving sequential error

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