Quantifying high dimensional entanglement with two mutually unbiased bases

Paul Erker1,2,3, Mario Krenn4,5, and Marcus Huber1,5,6,7

1Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
2Faculty of Informatics, Università della Svizzera italiana, Via G. Buffi 13, 6900 Lugano, Switzerland
3Facoltà indipendente di Gandria, Lunga scala, 6978 Gandria, Switzerland
4Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
5Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
6Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
7ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels, Barcelona, Spain

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We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.

High dimensional entanglement naturally created in down conversion processes carries great potential for improving quantum communication tasks. A central challenge is developing feasible methods for its experimental certification that scale well in system dimension. Here, we make progress on this task by developing a general method for quantifying high-dimensional entanglement using correlations in only two local unbiased bases. Surprisingly, in the ideal case all entanglement can be certified with only two measurements, even if detailed phase relations between the measurements are not known. This enables an experimental proposal using only a camera and a lens as measurement devices, which we show could realistically compete with the currently highest certified entanglement in more complicated setups.

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