Estimating localizable entanglement from witnesses

Abstract

Computing localizable entanglement for noisy many-particle quantum states is\ndifficult due to the optimization over all possible sets of local projection\nmeasurements. Therefore, it is crucial to develop lower bounds, which can\nprovide useful information about the behaviour of localizable entanglement, and\nwhich can be determined by measuring a limited number of operators, or by\nperforming least number of measurements on the state, preferably without\nperforming a full state tomography. In this paper, we adopt two different yet\nrelated approaches to obtain a witness-based, and a measurement-based lower\nbounds for localizable entanglement. The former is determined by the minimal\namount of entanglement that can be present in a subsystem of the multipartite\nquantum state, which is consistent with the expectation value of an\nentanglement witness. Determining this bound does not require any information\nabout the state beyond the expectation value of the witness operator, which\nrenders this approach highly practical in experiments. The latter bound of\nlocalizable entanglement is computed by restricting the local projection\nmeasurements over the qubits outside the subsystem of interest to a suitably\nchosen basis. We discuss the behaviour of both lower bounds against local\nphysical noise on the qubits, and discuss their dependence on noise strength\nand system size. We also analytically determine the measurement-based lower\nbound in the case of graph states under local uncorrelated Pauli noise.\n