Demonstration of Coherent-State Discrimination Using a Displacement-Controlled Photon-Number-Resolving Detector

We experimentally demonstrate a new measurement scheme for the discrimination of two coherent states. The measurement scheme is based on a displacement operation followed by a photon-number-resolving detector, and we show that it outperforms the standard homodyne detector which we, in addition, prove to be optimal within all Gaussian operations including conditional dynamics. We also show that the non-Gaussian detector is superior to the homodyne detector in a continuous variable quantum key distribution scheme.

Figure 1

(a) Schematics of the homodyne receiver. The signal (S) is interfered with a local oscillator (LO). The photocurrents are subtracted resulting in a quadrature measurement. (b) Marginal distribution of the two signal states with intervals where the answers $\{-,?,+\}$ are guessed. (c) Schematics of the photon-number-resolving (PNR) receiver. The signal (S) is interfered with an auxiliary oscillator (AO). Finally, the signal is measured by a photon-number-resolving detector (PNRD). (d) Photon number distribution of two signal states. In the examples, we assume a signal with $|\alpha {|}^2=0.24$ and a displacement of $\beta =1$.

Figure 2

(a) Scheme of the experimental implementation of the receivers in Figs. 1a, 1c, where the abbreviated components are a fiber mode cleaner (FMC), beam splitters (BS, 50-50), a polarizing beam splitter (PBS), a piezomounted mirror (PZT), a half wave plate (HWP), and a photon-number-resolving detector (PNRD). (b) (top) Modulation pattern of the electro-optical modulators in the state preparation (green dots) and the displacement of the PNR receiver (red horizontal line); (middle) simultaneously recorded quadrature measurements; (bottom) detection events of the APD. Shaded areas show inconclusive results for increasing postselection parameters $B$ and $m$.

Figure 3

(a) Error rates for the Kennedy receiver ($\beta =\alpha$) and the PNR receiver with varying $m$ and optimized ${\beta }_{\mathrm{opt}}$. Error bars reflect the standard deviations of repeated measurements, which are larger than the statistical errors. Experimental data are compared to ideal receivers (dashed lines). The experimental data have been corrected for the detector inefficiency in all figures. (b) Experimental error rates versus acceptance rates with increasing signal amplitudes for the PNR receiver and the homodyne (HD) receiver. For this comparison the success rate of both schemes is fixed to the one, that is theoretically reached by the PNR receiver. The theoretical predictions for the homodyne receiver (gray dashed line), the PNR receiver (solid line), and the optimal intermediate measurement (dot-dashed lines) are shown with statistical error bars. The new receiver outperforms the homodyne receiver for several data points. For details, see 28. (c) (left-hand scale) Key rate $G$ in logarithmic scale as a function of the channel transmittance $\eta$ using the PNR receiver (solid curve) and the homodyne receiver [31, 32] (dashed curve); (right-hand scale) optimized threshold $m$ (dotted line). Photon number resolution for high photon number, e.g., $m=10$, was demonstrated in 33.