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Complete characterization of a noisy multipartite quantum state in terms ofentanglement requires full knowledge of how the entanglement content in thestate is affected by the spatial distribution of noise in the state.Specifically, we find that if the measurement-basis in the protocol ofcomputing localizable entanglement and the basis of the Kraus operatorrepresenting the local noisy channel do not commute, the information regardingthe noise is retained in the system even after the qubit is traced out aftermeasurement. Using this result and the basic properties of entanglement undernoise, we present a set of hierarchies that localizable entanglement over aspecific subsystem in a multiqubit state can obey when local noise acts on thesubparts or on all the qubits of the whole system. In particular, we proposetwo types of hierarchies -- one tailored according to the number of noisyunmeasured qubits, and the other one that depends additionally on thecardinality of the set of noisy measured qubits, leading to the classificationof quantum states. We report the percentage of states satisfying the proposedhierarchies in the case of random three- and four-qubit systems and show, usingboth analytical methods and numerical simulations, that in almost all thecases, anticipated hierarchies tend to hold with the variation of the strengthof noise.

Hierarchies of localizable entanglement due to spatial distribution of local noise