  Complete characterization of a noisy multipartite quantum state in terms of
entanglement requires full knowledge of how the entanglement content in the
state is affected by the spatial distribution of noise in the state.
Specifically, we find that if the measurement-basis in the protocol of
computing localizable entanglement and the basis of the Kraus operator
representing the local noisy channel do not commute, the information regarding
the noise is retained in the system even after the qubit is traced out after
measurement. Using this result and the basic properties of entanglement under
noise, we present a set of hierarchies that localizable entanglement over a
specific subsystem in a multiqubit state can obey when local noise acts on the
subparts or on all the qubits of the whole system. In particular, we propose
two types of hierarchies -- one tailored according to the number of noisy
unmeasured qubits, and the other one that depends additionally on the
cardinality of the set of noisy measured qubits, leading to the classification
of quantum states. We report the percentage of states satisfying the proposed
hierarchies in the case of random three- and four-qubit systems and show, using
both analytical methods and numerical simulations, that in almost all the
cases, anticipated hierarchies tend to hold with the variation of the strength
of noise.

Quantum Physics