Quantitative verification of entanglement and fidelities from incomplete measurement data

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.

Keywords:

• entanglement estimation
• fidelity
• robustness
• stabilizer states

Acknowledgements

HW thanks Fernando Brandão, Animesh Datta, Masaki Owari, Alex Retzker and Shashank Virmani for very helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft, the EU Integrated Project “Qubit Applications”, the EPSRC QIP-IRC and the EU STREP project HIP.