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arXiv:0902.2093 · Journal of Modern Optics

Quantitative verification of entanglement and fidelities from incomplete measurement data

Harald Wunderlich, Martin Bodo Plenio

Published Aug 19, 2009• 30 citations
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Abstract

Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.

DOI
10.1080/09500340903184303

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Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.
Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.
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Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper, we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.
ventional full state tomography reaches its limit already for a few qubits and hence novel methods for the verification and benchmarking of quantum devices are called for. We show how the complete reconstruction of density matrices is possible even if one relies only on local information about the state. This results in an experimental effort that is linear in the number of qubits and efficient post-processing – in stark contrast to the exponential scaling of standard tomography. Whenever full tomography is not needed but instead less information required, one would expect that even fewer measurements suffice. Taking entanglement content of solid state samples and bosons in lattices as an example, we show how it may be quantified unconditionally using already routinely performed measurements only.
Entanglement is the crucial resource in the field of quantum information processing and consequently a better understanding of its properties is very desirable. While for bipartite entangled qubit states many tools for characterization exist and also the experimental implementation of the tools is feasible, the situation for multipartite entangled states is different: The available tools have to be suited for a much greater diversity of entanglement properties, e.g. to analyze all the different entanglement classes [1]. A full experimental characterization via state tomography is increasingly difficult as the number of necessary measurement settings grows exponentially with the number of particles. In this contribution we focus on efficient, non-tomographic methods to characterize experimentally observed, multipartite entangled states. First steps towards this goal were demonstrated recently, for instance, for the proof of genuine multipartite entanglement [2] or for the analysis of the persistency of entanglement under loss [3]. Such methods can be further optimized by tailoring them to particular properties of states. We show how three important characteristic quantities, the fidelity, the non-separability and the permutation symmetry, can be evaluated with a significantly lower number of measurement settings than required for a complete tomography. Although the determination of these particular properties provides less information than the reconstruction of the complete density matrix, it allows to judge how well state characteristics are reproduced experimentally. This might be important with respect to an experimental application of states in quantum information. Exemplarily we apply the analysis tools to two genuinely four-qubit entangled states: the Dicke state with two excitations |D 4 〉 [4] and the cluster state |C4 〉 [5, 6]. These states are relevant for particular tasks: The state |D 4 〉 is a resource for telecloning [7], cluster states find an application in one-way quantum computation [6, 8] and are a subgroup of graph states, which play an important role for quantum error correction.
The verification and quantification of experimentally created entanglement by simple measurements, especially between distant particles, is an important basic task in quantum processing. When composite systems are subjected to local measurements the measurement data will exhibit correlations, whether these systems are classical or quantum. Therefore, the observation of correlations in the classical measurement record does not automatically imply the presence of quantum correlations in the system under investigation. In this study, we explore the question of when correlations, or other measurement data, are sufficient to guarantee the existence of a certain amount of quantum correlations in the system and when additional information, such as the degree of purity of the system, is needed to do so. Various measurement settings are discussed, both numerically and analytically. Exact results and lower bounds on the least entanglement consistent with the observations are presented. The approach is suitable both for the bi-partite and the multi-partite setting.