Experimental Shot-by-Shot Estimation of Quantum Measurement Confidence

We demonstrate the single-shot confidence estimation for individual quantum measurement outcomes using the continuous measurement theory of the quantum counting process applied to the quantum state identification problem. We experimentally obtain single-shot and average confidences for quantum measurements and show that they favorably compare to that of the idealized classical measurement. Finally, we demonstrate that single-shot confidence estimations correctly represent observed experimental outcomes for a large ensemble of measurements.

Figure 1

(a) A constellation diagram of $M=4$ coherent states that differ by phase. Fuzzy circles represent uncertainty due to shot noise. Black dot—a possible outcome of a homodyne measurement. Arrows—difference between the value obtained with a homodyne and the expected states (used in calculating ${\overrightarrow{P}}_C$, see text); (b) an example of the experimentally measured $\overrightarrow{P}(t)$ in a single-shot measurement. Here, single photon detections occur three times at $t_1$, $t_2$, and $t_3$; (c) Experimental setup. For each signal pulse, the quantum measurement determines the most likely input state and the confidence of discrimination (Bayesian probability of each of the possible states based on the measurement record, see text). The optical input for system efficiency characterization is defined at the input port of the fiber beam splitter (BS) used for displacement and includes detection efficiency of the single-photon detector (SPD).

Figure 2

A visual comparison of single-shot confidence estimates in quantum and classical measurements. Left: the original image of $128\times 128\mathrm{pixels}$ where four primary (black, yellow, cyan, and magenta) colors correspond to the different symbols $|{\varphi }_s⟩$. Middle: the same image experimentally obtained from continuous quantum measurement record. Right: the same image reconstructed from the simulated ideal homodyne measurement. The color for each pixel is calculated as a sum of primary colors weighted with the probabilities $\overrightarrow{P}$ (middle) and ${\overrightarrow{P}}_C$ (right) of corresponding symbols.

Figure 3

Probability of the measurement outcome with a certain fidelity for faint input ($⟨n⟩=2.68$ photons per signal pulse) prepared in one of $M=4$ states. Orange bars—simulation of the ideal homodyne receiver adjusted for system efficiency of 74.5%. Red bars—simulation of the ideal homodyne receiver. Purple bars—experimental data. Blue bars—simulation of the idealized experimental receiver. Histogram bin width is 0.1, bin center positions are shown on the horizontal axis.

Figure 4

Experimentally measured single-shot confidence vs the experimental ensemble average probability of a successful state discrimination obtained for faint input ($⟨n⟩=2.68$ photons per signal pulse) prepared in one of $M=4$ states. Vertical error bars correspond to 1 standard deviation, horizontal error bars show histogram bin size.