Manuscript Oai283-A@1 · Oai283-A.tex · theorem · 6fb75895

Theorem: Positive-domination certificate

Operational certification of the logical channel › A finite semidefinite upper certificate · lines 838–859 · proof included or linked

Oai283-A@1#thm:SDP

Back to manuscript

Let $H = J (Φ - Ψ)$ . If a Hermitian matrix $Y$ satisfies

Y - H \geq 0, Y + H \geq 0, Tr_{out} Y \leq t I,

then

d_{⋄} (Φ, Ψ) \leq t /2.

Proof. The matrices $(Y + H) /2$ and $(Y - H) /2$ are Choi matrices of CP maps $Λ_{+}$ and $Λ_{-}$ . Their difference is $Φ - Ψ$ , and their sum has Choi matrix $Y$ . For every referenced input state, the two CP outputs are positive. The trace norm of their difference is at most the trace of their sum, which is at most $t$ . Take the supremum and divide by two.

Displayed formulas in this passage

unlabeled · line 841
$Y - H \geq 0, Y + H \geq 0, Tr_{out} Y \leq t I,$
unlabeled · line 846
$d_{⋄} (Φ, Ψ) \leq t /2.$

Nearby in this section

section: A finite semidefinite upper certificate