Manuscript X-K-ST@1 · X-K-ST.tex · theorem · 93c39c0f

Theorem: Sequential complete-channel telescope

Complete-channel and state-specific error bounds › Matching interfaces are part of the hypothesis · lines 1186–1227 · proof included or linked

X-K-ST@1#thm:error, X-K-ST@1#eq:telescope

Back to manuscript

Let

C = D Γ_{m} \dots Γ_{1} E, C = D Γ_{m} \dots Γ_{1} E

be matched compatible channel compositions. If

d_{⋄} (E, E) \leq e_{in}, d_{⋄} (Γ_{j}, Γ_{j}) \leq e_{j}, d_{⋄} (D, D) \leq e_{out},

then

d_{⋄} (C, C) \leq min {1, e_{in} + j = 1 \sum^{m} e_{j} + e_{out}} .

The same allowance controls every input–reference output state and every retained record marginal.

Proof. Insert intermediate compositions replacing one factor at a time. Each difference has one changed factor, with channels before and after it. Their diamond norms are one, so each contribution is bounded by the corresponding factor difference. Sum, divide by two, and use that half-diamond distance between channels is at most one.

Tensoring with a reference and applying to a state gives the complete trace-distance bound. Partial trace gives the record marginal bound.

Displayed formulas in this passage

X-K-ST@1#eq:telescope · line 1204
$d_{⋄} (C, C) \leq min {1, e_{in} + j = 1 \sum^{m} e_{j} + e_{out}} .$
unlabeled · line 1189
$C = D Γ_{m} \dots Γ_{1} E, C = D Γ_{m} \dots Γ_{1} E$
unlabeled · line 1198
$d_{⋄} (E, E) \leq e_{in}, d_{⋄} (Γ_{j}, Γ_{j}) \leq e_{j}, d_{⋄} (D, D) \leq e_{out},$

Theorem: Sequential complete-channel telescope

Displayed formulas in this passage

Nearby in this section