bibkey: “godley2023absorber” authors: “Alfred Godley and Mădălin Guţă” year: 2023 title: “Adaptive measurement filter: efficient strategy for optimal estimation of quantum Markov chains” doi: “10.22331/q-2023-04-06-973” url: “https://arxiv.org/abs/2204.08964v3” claim: “Lemma 4.1 constructs a coherent absorber whose fixed interaction preserves a shared stationary purification and returns each noise unit to its input state.” strata_touched: [] license: “citation-only” triage: “anchor”

Shared stationary initialization and coherent absorption

The primary source is Godley–Guţă, Quantum 7, 973 (2023), Lemma 4.1 and its proof. The stated lemma assumes a primitive finite-dimensional quantum Markov chain. It purifies the source’s stationary state with an absorber of the same dimension. After the source interacts with a fresh pure noise unit, the source marginal remains stationary. The two resulting purifications are therefore connected by a unitary on absorber and noise unit. This single unitary restores the shared purification and the original pure noise state at every step when the initial joint state is that purification.

The proof’s purification step uses stationarity and an orthonormal Schmidt family; it does not supply an independently prepared absorber for arbitrary source–reference inputs. Starting from a different joint state is a different initialization problem. Stationary or asymptotic output statements must not be read as exact blank output from every independent initial state.

The same construction is recalled with explicit Kraus formulas in Girotti–Godley–Guţă, Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator, Quantum 9, 1835 (2025), §3.1, DOI:10.22331/q-2025-08-27-1835. That paper expressly refers back to Lemma 4.1 for the construction.

In §8 of the phase-boundary volume, this established construction is an intermediate comparison in Corollary 8.4. The displayed rank-two stationary state and its purification specify the comparison source. The new repository derivation is Theorem 8.3: under independent pure receiver initialization, all source–reference inputs, one fixed local unitary and exact blank output at every step, the specified source needs receiver dimension two for one step and 2N−1 for N steps when N is at least two. The lower bound and matching construction use actual coefficient domains and their orthogonal isometric images. The cited absorber lemma does not optimize this independent-initialization problem.

This source comparison attributes established ingredients and their initialization conditions; it is not a global originality certificate.