Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Exact column reduction

Assisted stability and finite Choi certificates › A classical-input diamond norm · lines 1563–1593 · proof included or linked

VII-G@1#prop:diamond-column, VII-G@1#eq:column-diamond

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The assisted hybrid norm satisfies

∥ F - F ∥_{⋄} = g max ∥ D_{g} ∥_{⋄} .

Moreover,

∥ D_{g} ∥_{⋄} \leq h \sum ∥ D_{h ∣ g} ∥_{⋄} .

Proof. For an amplified block input $X = (X_{g})_{g}$ ,

∥[(F - F) \otimes id] (X) ∥_{1} \leq g \sum ∥(D_{g} \otimes id) (X_{g}) ∥_{1} \leq g max ∥ D_{g} ∥_{⋄} g \sum ∥ X_{g} ∥_{1} .

For the full-matrix extension, pinching is a trace-norm contraction also after amplification. This follows either from duality with a unital completely positive adjoint or by averaging diagonal phase unitaries. Hence the same upper bound holds there.

An input supported in one maximizing block gives the reverse inequality. The branch bound is the triangle inequality in the output direct sum.

Displayed formulas in this passage

VII-G@1#eq:column-diamond · line 1566
$∥ F - F ∥_{⋄} = g max ∥ D_{g} ∥_{⋄} .$
unlabeled · line 1572
$∥ D_{g} ∥_{⋄} \leq h \sum ∥ D_{h ∣ g} ∥_{⋄} .$
unlabeled · line 1580
$∥[(F - F) \otimes id] (X) ∥_{1} \leq g \sum ∥(D_{g} \otimes id) (X_{g}) ∥_{1} \leq g max ∥ D_{g} ∥_{⋄} g \sum ∥ X_{g} ∥_{1} .$

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section: A classical-input diamond norm