Experimental Self-Characterization of Quantum Measurements

The accurate and reliable description of measurement devices is a central problem in both observing uniquely nonclassical behaviors and realizing quantum technologies from powerful computing to precision metrology. To date quantum tomography is the prevalent tool to characterize quantum detectors. However, such a characterization relies on accurately characterized probe states, rendering reliability of the characterization lost in circular argument. Here we report a self-characterization method of quantum measurements based on reconstructing the response range—the entirety of attainable measurement outcomes, eliminating the reliance on known states. We characterize two representative measurements implemented with photonic setups and obtain fidelities above 99.99% with the conventional tomographic reconstructions. This initiates range-based techniques in characterizing quantum systems and foreshadows novel device-independent protocols of quantum information applications.

Figure 1

Schematic diagram of quantum tomography and self-characterization. (a) Tomography of quantum measurements demands a set of known probe states, whereas tomography of quantum states demands well-calibrated quantum detectors. This forms a loop paradox in calibrating quantum systems. (b) In contrast, quantum detector self-characterization only uses the detector outcome probabilities from the measurement part itself to reconstruct the response range of the measurement, with unknown states, which thus can break the aforementioned loop paradox.

Figure 2

Experimental setup. (a) Heralded single photons are generated via spontaneous parametric down-conversion, followed which a set of probe states are prepared by three electronically controlled wave plates and directed towards the measurement device (b) or (c). (b) The MUB device is composed of two wave plates followed by a beam displacer (BD) to perform projection on a certain basis. (c) The SIC device is a four-outcome general measurement realized by wave plates, BDs and single photon counting modules (SPCMs). BBO, $\beta$-barium borate crystal; KDP, potassium di-hydrogen phosphate; HWP, half wave plate; QWP, quarter wave plate.

Figure 3

Results of quantum detector self-characterization (QDSC). (a), (c) The reconstructed $Q$ and $t$ (chromatic bars) for the MUB and SIC devices, respectively. The corresponding results of quantum detector tomography (transparent bars with solid line edges) are also plotted for comparison. (b), (d) Left: the estimated response range (blue region) and the measured data (points), illustrated in the probability space despite the linear dependencies of the measurement operators, for the MUB and SIC devices, respectively. Right: the detailed results represented in terms of the values of $\mathcal{L}$ in Eq. (3). Error bars are standard uncertainties derived from 40 runs of the experiment.

Figure 4

Comparison of quantum detector tomography (QDT) and self-characterization (QDSC). The results are represented via tests of (a) the MUB device (red dot) and (b) the SIC device (purple triangle). Each marker represents the measured values of $\mathcal{L}$ in Eq. (3) averaged over 40 runs for a same probe state. The marginal distributions in the horizontal (QDT) and vertical (QDSC) axes, represented by histograms from the $50\times 40$ data before average and the corresponding Kernel fittings (red dashed lines), reflect the deviations of the measured data from the bound (black dashed lines).