Experimentally detecting a quantum change point via the Bayesian inference

Detecting a change point is a crucial task in statistics that has been recently extended to the quantum realm. A source state generator that emits a series of single photons in a default state suffers an alteration at some point and starts to emit photons in a mutated state. The problem consists in identifying the point where the change took place. In this Rapid Communication, we build a pseudo-on-demand single-photon source to prepare the photon sequences, and consider a learning agent that applies Bayesian inference on experimental data to solve this problem. This learning machine adjusts the measurement over each photon according to the past experimental results and finds the change position in an online fashion. Our results show that the local-detection success probability can be largely improved by using such a machine-learning technique. This protocol provides a tool for improvement in many applications where a sequence of identical quantum states is required.

Figure 1

Schematic diagram of the three different detection protocols. The optimal measurement is constituted by four steps: the source state preparation, a quantum memory for storing each photon, global detection, and decision making. For the simple local strategy, such as basic local (BL) detection, the quantum memory model is removed and global detection is replaced by a local detection. The measurement base is fixed in the BL method, but is changed in the Bayesian inference (BI) strategy. The Helstrom measurement in this method is decided by the prior probability obtained from the last step (see Appendix pp1). Meanwhile, the priors are also updated according to the measurement results using the Bayesian update rule. The change point is determined by choosing the position with the largest prior probability after the last step.

Figure 2

(a) Experimental setup. This setup consists of two main parts: the source state preparation part at the Alice side and the detecting agent part controlled by Bob. Preparation part: The pseudo-on-demand single photon is generated by the SPDC process in BBO crystal associated with the postselection method and a chopper, which divides the photon sequence into $n$ equal discrete time bins [30]. The photons then pass through a PBS to produce the default states $|H\rangle$ and an EOM after that creating the change point at some certain place in the sequence. Detection part: The measurement device consists of an electronic-controlled HWP, a PBS, and a classical learning agent (personal computer). The measurement basis realized by the PBS and the electronic-controlled HWP can respond rapidly when the learning agent calculates the new basis from the latest priors (see Appendix pp1). (b) The time sequence diagram. The first line is the trigger signal (generated by AWG), which has a 100-ms interval. The second line denotes the time bins created by the chopper. Each time bin has a 2.5-ms width and the red balls represent signal photons postselected by the logic unit [30]. The third and fourth lines are the EOM signal and the folding signal by EOM and chopper, respectively. In the picture, we show a case where the change point is set at the fifth photon, and the arrows on the red balls denote the photon polarization.

Figure 3

(a) The results of prior probabilities at each step when $c^2=0.604$ and $k=5$. From the first step to the last ($21\mathrm{st}$) step, the priors fluctuate depend on the measurement results and begin to converge at the sixth step. At the last step, the highest column is located at $k=5$, which corresponds to a prior probability of 0.569. This result corresponds to the correct change point in this experiment. (b) The relationship between success probability and change point position. Here, Alice sets $c^2=0.604$, but varies each change position. The purple triangles correspond to the success probabilities for the optimal (global) strategy [8], the red squares are the numerical simulation results for the BI strategy, and the circles are the actual experimental results. The simulation average value of the success probabilities are denoted by the dashed red line. The black solid line and the gray region are the average values and associated errors obtained from experiments, respectively. It is clear that the probabilities derived from the BI protocol are all beyond the BL one, denoted as the solid orange line.

Figure 4

Comparison of the BI, BL, and optimal measurement for every overlap. The red dots are the experimental results obtained using BI detection, and the solid light red line is the corresponding simulation value. The orange triangles are the results obtained from BL detection, and the solid purple line is its theoretical value. The success probability for the optimal measurement is shown as the dashed black line. Although the BI approach cannot surpass the optimal (global) detection, it nearly covers the gap between the BL and the optimal measurement.