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Physical Review A

Fidelity estimation and entanglement verification for experimentally produced four-qubit cluster states

Yuuki Tokunaga, Takashi Yamamoto, Masato Koashi, Nobuyuki Imoto

Published Aug 23, 2006• 33 citations
Publisher

Abstract

We propose methods of fidelity estimation and entanglement verification for experimentally produced four-qubit cluster states. We show that we can obtain a high lower bound of the fidelity using only four local projective measurement settings. The lower bound is close to the exact fidelity, which is determined only by at least nine local projective measurement settings. We also present witness operators for distinguishing entanglement around a four-qubit cluster state from specific classes of genuine four-qubit entanglement, e.g., a class including GHZ and $W$ types of entanglement.

DOI
10.1103/physreva.74.020301

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We propose methods of fidelity estimation and entanglement verification for experimentally produced four-qubit cluster states. We show that we can obtain a high lower bound of the fidelity using only four local projective measurement settings. The lower bound is close to the exact fidelity, which is determined only by at least nine local projective measurement settings. We also present witness operators for distinguishing entanglement around a four-qubit cluster state from specific classes of genuine four-qubit entanglement, e.g., a class including GHZ and WWW types of entanglement.
We experimentally demonstrate a simple scheme for generating a four-photon entangled cluster state with fidelity over 0.860+/-0.015. We show that the fidelity is high enough to guarantee that the produced state is distinguished from Greenberger-Horne-Zeilinger, W, and Dicke types of genuine four-qubit entanglement. We also demonstrate basic operations of one-way quantum computing using the produced state and show that the output state fidelities surpass classical bounds, which indicates that the entanglement in the produced state essentially contributes to the quantum operation.
The size of the Hilbert space for a multiqubit state scales exponentially with the number of constituent qubits. Often this leads to a similar exponential scaling of the experimental resources required to characterize the state. Contrary to this, we propose a physically motivated method for experimentally assessing the fidelity of an important class of entangled states known as cluster states. The proposed method always yields a lower bound of the fidelity with a number of measurement settings scaling only linearly with the system size, and is tailored to correctly account for the errors most likely to occur in experiments. For one-dimensional cluster states, the constructed fidelity measure is tight to lowest order in the error probability for experimentally realistic noise sources and thus closely matches the true fidelity. Furthermore, it is tight for the majority of higher-order errors, except for a small subset of certain nonlocal multiqubit errors irrelevant in typical experimental situations. The scheme also performs very well for higher-dimensional cluster states, assessing correctly the majority of experimentally relevant errors.
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.