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

Proposition: Number-preserving complete dephasing

Environmental dynamics and the two-budget principle › Counterexamples to occupation-only and marginal-only inference · lines 1202–1224 · proof included or linked

Psy02-A@1#prop:dephasing

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In one mode,

n (ρ_{+}) = n (Δ ρ_{+}) = \frac{1}{2},

and both inputs satisfy the same number-based concentration bound. Yet

D (ρ_{+}; id, Δ) = \frac{1}{2}, D (ρ_{+}; Δ, Δ) = 0.

Thus finite number and small scaled observable spread do not imply preservation of this quantum witness.

Proof. Both states have number populations $1/2, 1/2$ . Their difference has eigenvalues $\pm 1/2$ . The dephasing channel annihilates that difference.

Displayed formulas in this passage

unlabeled · line 1205
$n (ρ_{+}) = n (Δ ρ_{+}) = \frac{1}{2},$
unlabeled · line 1211
$D (ρ_{+}; id, Δ) = \frac{1}{2}, D (ρ_{+}; Δ, Δ) = 0.$

Nearby in this section

proposition: Proposition: One accessible marginal does not determine recovery
section: Counterexamples to occupation-only and marginal-only inference