Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Operational meaning and stability of the witness

Environmental dynamics and the two-budget principle › A separately specified coherence witness · lines 1020–1077 · proof included or linked

Psy02-A@1#thm:witness, Psy02-A@1#eq:witness-stability

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For the witness (eq:coherence-witness),

D_{s} = 0 \leq M \leq 1 sup \abs Tr (M Λ_{s} (ρ)) - Tr (M Λ_{s} (Δ ρ)) .

Further deterministic processing cannot increase $D_{s}$ .

Suppose

d_{⋄} (Λ_{s}, Λ_{s}^{0}) \leq e_{s}, d_{tr} (ρ, ρ_{0}) \leq p,

with the same $Δ$ . Then

\abs D_{s} (ρ; Λ, Δ) - D_{s} (ρ_{0}; Λ^{0}, Δ) \leq 2 e_{s} + 2 p .

Thus a survival certificate is

D_{s} (ρ_{0}; Λ^{0}, Δ) - 2 e_{s} - 2 p \geq d_{m i n} > 0.

Proof. The output difference is Hermitian and traceless. Its positive and negative parts have equal trace, each equal to half its trace norm. The positive spectral projector attains the effect supremum.

Trace distance contracts under a channel. For channel perturbations, each of the two normalized input states contributes trace-norm error at most $2 e_{s}$ . After multiplying their sum by $1/2$ , the allowance is $2 e_{s}$ . For preparation changes, contractivity of $Λ_{s}$ and $Δ$ gives

\frac{1}{2} \norm Λ_{s} [(ρ - ρ_{0}) - Δ (ρ - ρ_{0})]_{1} \leq \norm ρ - ρ_{0}_{1} = 2 p .

The reverse triangle inequality proves (eq:witness-stability).

Displayed formulas in this passage

Psy02-A@1#eq:witness-stability · line 1043
$\abs D_{s} (ρ; Λ, Δ) - D_{s} (ρ_{0}; Λ^{0}, Δ) \leq 2 e_{s} + 2 p .$
unlabeled · line 1023
$D_{s} = 0 \leq M \leq 1 sup \abs Tr (M Λ_{s} (ρ)) - Tr (M Λ_{s} (Δ ρ)) .$
unlabeled · line 1036
$d_{⋄} (Λ_{s}, Λ_{s}^{0}) \leq e_{s}, d_{tr} (ρ, ρ_{0}) \leq p,$
unlabeled · line 1050
$D_{s} (ρ_{0}; Λ^{0}, Δ) - 2 e_{s} - 2 p \geq d_{m i n} > 0.$
unlabeled · line 1069
$\frac{1}{2} \norm Λ_{s} [(ρ - ρ_{0}) - Δ (ρ - ρ_{0})]_{1} \leq \norm ρ - ρ_{0}_{1} = 2 p .$

Theorem: Operational meaning and stability of the witness

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