Measurement-Cost Triage for Non-Markovian Entanglement Diagnostics in Hyperentangled Photonic Qubits

We formulate a layer-resolved diagnostic protocol for separating different kinds of quantuminformation
content in a hyperentangled four-qubit state undergoing local non-Markovian
pure dephasing. The protocol uses Bell-sector coherence flow as a low-cost entrance test for
candidate backflow windows, but does not identify this signal with entanglement recovery.
Informative windows are then promoted to local complementarity, pair-concurrence, and
simultaneous all-cut negative-partial-transpose (NPT) diagnostics. For the benchmark family
studied here, the pair concurrence CAB(t), the local complementarity defect DA(t), and the
white-noise visibility threshold vc(t) of the all-cut NPT layer have closed-form expressions.
These relations show explicitly that coherence backflow, bipartite entanglement recovery,
local budget redistribution, and all-cut NPT activation are distinct diagnostic statements.
Robustness checks under controlled-phase, visibility, reservoir-parameter, and small stateperturbation
variations support the use of the hierarchy as a quantum-information decision
protocol rather than as a single state measure. The result is not a new entanglement or
non-Markovianity monotone; it is an operational way to decide which information layer is
certified by a given measurement record and when richer reconstructions are justified.