Abstract
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 the 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 multipartite 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.
Highlights
Over the last two decades, quantum entanglement [1] has emerged as a crucial resource in a plethora of quantum information processing tasks, including quantum teleportation [1,2,3], quantum dense coding [4,5,6], quantum cryptography [7, 8], and measurement-based quantum computation [9,10,11]
One of the approaches is based on local witnesses, whose expectation values can be used to obtain a lower bound of the localizable entanglement
Using graph states for demonstration, we show that in the case of graph states exposed to noise, the measurement-based lower bounds (MLBs) is greater or equal to the witness-based lower bound (WLB)
Summary
Over the last two decades, quantum entanglement [1] has emerged as a crucial resource in a plethora of quantum information processing tasks, including quantum teleportation [1,2,3], quantum dense coding [4,5,6], quantum cryptography [7, 8], and measurement-based quantum computation [9,10,11]. An additional complication arises from the fact that one needs to deal with experimental N-qubit states which due to noise necessarily deviate to some degree from ideal, often pure target states In such cases, determination of the localizable entanglement becomes difficult due to the limited number of computable measures of entanglement in multipartite mixed states [43], if one is interested in localizable entanglement in sets involving more than two qubits. Determination of the localizable entanglement becomes difficult due to the limited number of computable measures of entanglement in multipartite mixed states [43], if one is interested in localizable entanglement in sets involving more than two qubits In this situation, a promising approach towards understanding the behaviour of localizable entanglement under noise for large stabilizer states is to develop non-trivial as well as computable lower bounds of the actual quantity.
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