Optical experiment to test negative probability in context of quantum-measurement selection

Negative probability values have been widely employed as an indicator of the nonclassicality of quantum systems. Known as a quasiprobability distribution, they are regarded as a useful tool that provides significant insight into the underlying fundamentals of quantum theory when compared to the classical statistics. However, in this approach, an operational interpretation of these negative values with respect to the definition of probability—the relative frequency of occurred event—is missing. An alternative approach is therefore considered where the quasiprobability operationally reveals the negativity of measured quantities. We here present an experimental realization of the operational quasiprobability, which consists of sequential measurements in time. To this end, we implement two sets of polarization measurements of single photons. We find that the measured negativity can be interpreted in the context of selecting measurements, and it reflects the nonclassical nature of photons. Our results suggest a new operational way to unravel the nonclassicality of photons in the context of measurement selection.

i (error of each component i ) 2 × (times used). Here, we assumed that there is no correlation between the errors of different components. Note that the given error values (error bars in the figures) are maximally estimated. The statistical fluctuation over measurements at many time intervals can be inferred from the distribution of the experimental values (red circles) in Fig. 2a in the main text, which is much smaller than the given error bar.

II. SECOND ORDER CORRELATION FUNCTION OF THE RESOURCES
We examined the anti-bunching characteristics of the heralded single photons and single photons from a single molecule by measuring the second-order correlation function g (2) (τ ). In the case of the SPDC source, as shown in Fig. S1a, photon pair is initially separated using a * rjhui82@gmail.com † kglee@hanyang.ac.kr ‡ hyoung@hanyang.ac.kr PBS into the signal and the idler paths. Both paths are then sub-divided by the 50:50 non-polarizing beam splitter (NPBS) into two branches. By combining one branch of the signal and one of the idler using an AND gate, it becomes possible to mimic the coincidence of the heralded photon. Therefore, the Hanbury-Brown and Twiss (HBT) measurement of the outputs of two AND gates implies the g (2) (τ ) of the heralded photons. Two digital chips (PO74G08) are used for the AND gates. The HBT measurement was performed using the start-stop mode of a TCSPC device (PicoQuant, PicoHarp 300). The resulting experimental curve is shown together with a theoretical calculation in Fig. S1b. In the calculation, we considered the following system parameters; the timing jitter of the APDs (= 0.61 ns) and the coincidence time window of the AND gates (= 5.5 ns). The average value of the dip in the time delay range of -3 ns ∼ 3 ns is only 0.036. The near zero value of g (2) (0) ensures that the heralded photons are the most similar to the single photons. For the case of the single molecule, the emitted photons are divided by a NPBS into two branches and are directly used as the inputs of the start-stop measurements. The results are shown in Fig. S1c.

III. ADDITIONAL EXPERIMENT BY POST-SELECTED WEAK FIELD
We here discuss the negativity of the operational quasiprobability using a weak-field. Given that such light does not exhibit the anti-bunching characteristic, the weak-field can be regarded as classical light. However, we can detect the negativity with a post-selection process. This indicates that our method provides an operational way to detect the nonclassicality of optical fields within the context of selecting measurement procedures.
The input source was prepared as follows; we used picosecond pulses from a mode-locked Ti:sapphire laser (Mira 900). The centre wavelength is set to 800 nm and the pulse repetition rate is reduced down to 3.8 MHz with a pulse picker (Coherent 9200). Then, using neutral density filters, the intensity of the beam is attenuated so that the average number of photons range from 10 −3 to 10 −1 per pulse. We implemented the same measurement setups as shown in Fig. 1 in the main text. We postselected the raw data to evaluate the negativity in a way that only single APD clicks were sampled and the rest  (2) dip is 0.036. c, The g (2) (τ ) function of single photons emitted from a single terrylene molecule. The result reads g (2) (0) = 0.14. of events, e.g., more than two clicks simultaneously were neglected.
Experimental results together with the theoretical predictions are presented in Fig. S2a for the negativity and in Fig. S2b-e for each W(a 1 , a 2 ). We started with the average photon number 6 × 10 −3 per pulse (red open and filled circles in Fig. S2a). The maximum negativity was obtained as 0.078 without the correction of the dark count of the APDs (red filled circle). This maximum in-creased to 0.099 after correcting for the dark count (red open circles); we measured the dark count of each APD and subtracted this value from the total measured counts. For a higher average photon number (10 −1 per pulse), the maximum negativity was 0.097 even without the correction for dark-count (see green circle). This is because for a higher detection count, the contribution of the dark count of the APD (∼ 10 3 counts per second) becomes smaller. Note that all maxima are obtained for θ = 44 • and φ = 0 • . We followed the error analysis in Sec. I.