Estimating localizable entanglement from witnesses. (arXiv:1803.02753v1 [quant-ph])

Computing localizable entanglement for noisy many-particle quantum states is
difficult due to the optimization over all possible sets of local projection
measurements. Therefore, it is crucial to develop lower bounds, which can
provide useful information about the behaviour of localizable entanglement, and
which can be determined by measuring a limited number of operators, or by
performing least number of measurements on the state, preferably without
performing a full state tomography. In this paper, we adopt two different yet
related approaches to obtain a witness-based, and a measurement-based lower
bounds for localizable entanglement. The former is determined by the minimal
amount of entanglement that can be present in a subsystem of the multi-partite
quantum state, which is consistent with the expectation value of an
entanglement witness. Determining this bound does not require any information
about the state beyond the expectation value of the witness operator, which
renders this approach highly practical in experiments. The latter bound of
localizable entanglement is computed by restricting the local projection
measurements over the qubits outside the subsystem of interest to a suitably
chosen basis. We discuss the behaviour of both lower bounds against local
physical noise on the qubits, and discuss their dependence on noise strength
and system size. We also analytically determine the measurement-based lower
bound in the case of graph states under local uncorrelated Pauli noise.

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