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Determining the quantum expectation value by measuring a single photon


One description provides only probabilities for obtaining various eigenvalues of a quantum variable. The eigenvalues and the corresponding probabilities specify the expectation value of a physical observable, which is known to be a statistical property of an ensemble of quantum systems. In contrast to this paradigm, here we demonstrate a method for measuring the expectation value of a physical variable on a single particle, namely, the polarization of a single protected photon. This realization of quantum protective measurements could find applications in the foundations of quantum mechanics and quantum-enhanced measurements.

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Figure 1: Estimation of the polarization expectation value 〈P〉 by means of a single protective measurement.
Figure 2: Experimental setup.
Figure 3: Illustrative drawing showing the measurement of unprotected and protected photons.
Figure 4: Results obtained for the input state |ψ17π/60〉 = 0.629|H〉 + 0.777|V〉.
Figure 5: Comparison between the uncertainty on P with the PM approach (u(P)) and the one given by projective measurement (uPBS(P)).


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This work has received funding from the European Union’s Horizon 2020 and the EMPIR Participating States in the context of the project EMPIR-14IND05 ‘MIQC2’, and from the INRIM ‘Seed’ project ‘GeQuM’. E.C. was supported by ERC AdG NLST. L.V. acknowledges support of the Israel Science Foundation Grant No. 1311/14 and the German-Israeli Foundation for Scientific Research and Development Grant No. I-1275-303.14. We wish to thank Y. Aharonov, S. Popescu and M. G. A. Paris for helpful discussion.

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I.P.D., M.Gramegna, F.P., A.A., M.Genovese (responsible for the laboratories), E.C. and L.V. (both responsible for the theoretical framework) planned the experiment. The experimental realization was achieved (supervised by I.P.D., G.B., M.Gramegna and M.Genovese) by F.P. (leading role), A.A. and E.R. The SPAD camera was developed and optimized for this experiment by R.L., F.V. and A.T. The manuscript was prepared with inputs by all the authors. They also had a fruitful systematic discussion on the progress of the work.

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Correspondence to Fabrizio Piacentini.

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The authors declare no competing financial interests.

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Piacentini, F., Avella, A., Rebufello, E. et al. Determining the quantum expectation value by measuring a single photon. Nature Phys 13, 1191–1194 (2017).

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