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Certification of a non-projective qudit measurement using multiport beamsplitters


The most common form of measurement in quantum mechanics projects a wavefunction onto orthogonal states that correspond to definite outcomes. However, generalized quantum measurements that do not fully project quantum states are possible and have an important role in quantum information tasks. Unfortunately, it is difficult to certify that an experiment harvests the advantages made possible by generalized measurements, especially beyond the simplest two-level qubit system. Here we show that multiport beamsplitters allow for the robust realization of high-quality generalized measurements in higher-dimensional systems with more than two levels. Using multicore optical fibre technology, we implement a seven-outcome generalized measurement in a four-dimensional Hilbert space with a fidelity of 99.7%. We present a practical quantum communication task and demonstrate a success rate that cannot be simulated in any conceivable quantum protocol based on standard projective measurements on quantum messages of the same dimension. Our approach, which is compatible with modern photonic platforms, showcases an avenue for faithful and high-quality implementation of genuinely non-projective quantum measurements beyond qubit systems.

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Fig. 1: Generalized measurment scheme using a MBS.
Fig. 2: Communication task for certifying a generalized measurement.
Fig. 3: Experimental setup for certification of generalized measurements.
Fig. 4: Examples of experimental results for the certification scheme.
Fig. 5: A diagram of the path-encoding scheme for the multicore fibre in dimension D = 7.

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Data availability

Source data are provided with this paper. Any additional data related to the findings of this paper are available upon reasonable request.

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Computer codes related to the findings of this paper are available upon request.


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This work was supported by ANID – Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) (grant nos. 11201348, 1180558, 1190901, 1200266, 1200859 and 3200779) and ANID – Millennium Science Initiative Program ICN17_012. J.C. acknowledges financial support from ANID/REC/PAI77190088. L.P. was supported by ANID-PFCHA/DOCTORADO-BECAS-CHILE/2019-72200275, the Spanish Project No. PGC2018-094792-B-I00 (MCIU/AEI/FEDER, UE) and CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM). A.T. is supported by the Wenner-Gren Foundation.

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D.M., E.S.G. and J.C. performed the experiment and analyzed the data under the supervision of S.P.W. and G.L. A.D. and L.P. theoretically modelled the quantum device. A.T. developed the theory of the protocol. All authors contributed to the writing of the manuscript.

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Correspondence to Armin Tavakoli.

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Martínez, D., Gómez, E.S., Cariñe, J. et al. Certification of a non-projective qudit measurement using multiport beamsplitters. Nat. Phys. 19, 190–195 (2023).

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