Visibility-Based Hypothesis Testing Using Higher-Order Optical Interference

Many quantum information protocols rely on optical interference to compare data sets with efficiency or security unattainable by classical means. Standard implementations exploit first-order coherence between signals whose preparation requires a shared phase reference. Here, we analyze and experimentally demonstrate the binary discrimination of visibility hypotheses based on higher-order interference for optical signals with a random relative phase. This provides a robust protocol implementation primitive when a phase lock is unavailable or impractical. With the primitive cost quantified by the total detected optical energy, optimal operation is typically reached in the few-photon regime.

Figure 1

An interferometric primitive for visibility-based hypothesis testing with phase-keyed signals. Input data in possession of two parties $A$ and $B$ are mapped onto phase patterns ${\varphi }_1^A,{\varphi }_2^A,\dots ,{\varphi }_m^A$ and ${\varphi }_1^B,{\varphi }_2^B,\dots ,{\varphi }_m^B$ used to modulate sequences of $m$ pulses described by a family of normalized temporal waveforms $u_1(t),u_2(t),\dots ,u_m(t)$. Generated optical signals can be viewed as prepared in collective modes described, respectively, by $u(t)=\sum _{j=1}^m{u}_j(t)\exp (i{\varphi }_j^A)/\sqrt{m}$ and $v(t)=\sum _{j=1}^m{u}_j(t)\exp (i{\varphi }_j^B)/\sqrt{m}$. The signals are brought to interference at a $50:50$ beam splitter whose output ports are monitored by photodetectors. The outcome of a single repetition of an interferometric measurement is a pair of integers $k$, $k^{\prime }$ specifying the number of counts registered by each of the detectors over the duration of the signals.

Figure 2

Discrimination between a pair of hypotheses encoded in interference visibilities ${\mathcal{V}}_1$, ${\mathcal{V}}_2$ for the coherent and the random-phase scenario. (a) Information gained from a single photodetection event for a fixed phase between interfering signals. (b) Maximum information per one detected photon $C^{\mathrm{rnd}}/(\eta \overline{n})$ for signals with a random global phase. (c) Optimal $\eta {\overline{n}}^{*}$ maximizing the ratio $C^{\mathrm{rnd}}/(\eta \overline{n})$. White squares on the diagonal in (b) and (c) represent the case $|{\mathcal{V}}_1|=|{\mathcal{V}}_2|$ when the two hypotheses are indistinguishable.

Figure 3

Chernoff information per one detected photon $C^{\mathrm{rnd}}/(\eta \overline{n})$ as a function of the average photon number $\eta \overline{n}$ in a single realization of the visibility measurement with a random global phase, depicted for the pair of visibilities ${\mathcal{V}}_1=0.98$ and ${\mathcal{V}}_2=0.56$. The scenario based on full photocount statistics $K\to \infty$ (blue line) is compared with limited photon number resolution $K=2$ (green line), and inference based only on the photocount difference $\mathrm{\Delta }k=k^{\prime }-k$ (black line).

Figure 4

(a) The simplified scheme of the experimental setup. NDF, neutral density filters. POL, polarizer. HWP, half-wave plate. QWP, quarter-wave plate. APD, avalanche photodiode. FPGA, time tagger based on field-programmable gate array. (b) Experimental joint photocount distributions $P_{k{k}^{\prime }}^{\mathrm{rnd}}$ obtained from approximately $1.5\times {10}^6$ outcomes $k{k}^{\prime }$ for visibilities ${\mathcal{V}}_1=0.98$ (left), ${\mathcal{V}}_2=0.56$ (right). (c) The error probability $ϵ$ in hypothesis testing as a function of the data set size $N$, determined for each $N$ from $1.5\times {10}^4$ repetitions of the Neyman-Pearson test on independent sets of experimental data. The results are compared with the standard Chernoff bound (green, solid line) and its refined version (gray, dashed line). The error bars account for 1 standard deviation. The gray shaded region corresponds to the error probability of testing hypotheses ${\mathcal{V}}_1=0.98$ vs ${\mathcal{V}}_2$ from the range $0\le {\mathcal{V}}_2\le 0.56$ using the Neyman-Pearson test designed for ${\mathcal{V}}_2=0.56$.