Discrimination of binary coherent states using a homodyne detector and a photon number resolving detector

We investigate quantum measurement strategies capable of discriminating two coherent states probabilistically with significantly smaller error probabilities than can be obtained using nonprobabilistic state discrimination. We apply a postselection strategy to the measurement data of a homodyne detector as well as a photon number resolving detector in order to lower the error probability. We compare the two different receivers with an optimal intermediate measurement scheme where the error rate is minimized for a fixed rate of inconclusive results. The photon number resolving (PNR) receiver is experimentally demonstrated and compared to an experimental realization of a homodyne receiver with postselection. In the comparison, it becomes clear that the performance of the PNR receiver surpasses the performance of the homodyne receiver, which we prove to be optimal within any Gaussian operations and conditional dynamics.

Figure 1

(a) Schematics of the homodyne receiver, showing that the signal (S) is interfered with a local oscillator (LO) on a 50:50 beam splitter. The photocurrents of two photodiodes are subtracted, resulting in a quadrature measurement along the encoding quadrature. (b) Marginal distribution of the two signal states. In the example, we assume a signal with $\left|\alpha \right|{}^2=0.24$. According to the measurement result the correct answer ($-,?,+$) is guessed. (c) Schematics of the photon number resolving (PNR) receiver. The signal (S) is interfered with an auxiliary oscillator (AO) on a highly transmissive beam splitter. Finally, the signal is measured by a photon number resolving detector (PNRD). (d) Photon number distribution of two signal states. According to the measurement outcome of the PNRD the correct answer is guessed. In the example, we assume a signal with $\left|\alpha \right|{}^2=0.24$ and a displacement of $\beta =1$. Results for $n=1$ are considered inconclusive. (e) Comparison of PNR receivers with $m=1$–$3$ (solid lines) with homodyne receiver (dashed lines) at equal success rates. The PNR receiver outperforms the homodyne receiver for all signal amplitudes. The dot-dashed lines show the error rate of the optimal measurement.

Figure 2

Simplified scheme of the experiment, where the abbreviated components are a fiber mode cleaner (FMC), beam splitters (BSs, 50:50), a polarizing beam splitter (PBS), a half-wave plate (HWP), and a photon number resolving detector (PNRD). The graphs show modulation and the corresponding recorded quadrature measurements and detection events of the PNRD.

Figure 3

(a) Dependence of the acceptance probability for ideal homodyne detection on the signal’s mean photon number $\left|\alpha \right|{}^2$ and the postselection threshold $B$. The dashed line shows the postselection threshold $B$ for which the homodyne detection and USD have equal acceptance probability. (b) The error probabilities of ideal homodyne detection shown in a logarithmic contour plot. The dashed line shows the error rate of homodyne detection for success rates equal to USD. (c) Experimentally measured acceptance probability. (d) Experimentally measured error probabilities. In (c) and (d), we corrected for the quantum efficiency of the receiver.

Figure 4

Comparison between the effective and ideal response functions of the homodyne receiver.

Figure 5

(a) Experimental data showing the effect of the displacement $\beta$ on the error rate $p_{E,\mathrm{PNR}}$ for a given signal amplitude $\left|\alpha \right|{}^2=0.24$ (corrected for quantum efficiency) and different numbers of dropped results $m$. The theoretically predicted error probabilities for the receiver without imperfections are shown with dashed lines. The error rates at the optimal displacement ${\beta }_{\mathrm{opt}}$ for different receivers are marked. (b) Error rates for varying $m$ and optimized ${\beta }_{\mathrm{opt}}$. The error bars reflect the standard deviations of repeated measurements, which are larger than the statistical errors. The experimental data are compared to ideal receivers (dashed lines). (c) Experimental data for acceptance rates (points) and theoretical predictions (dashed lines). Below the curve for an optimal USD device (solid gray), states can be discriminated without error in principle. We corrected for the quantum efficiencies of the receiver.

Figure 6

(a) Comparison of experimental error rates and acceptance rates for the two receiver schemes. For this comparison the success rate of both schemes is fixed to the one that is theoretically reached by the PNR receiver. Experimental data are shown for PNR receivers with $m=1$ and $2$ (filled circles and triangles) and the homodyne receiver (open circles and triangles). Additionally, theoretical predictions for the homodyne receiver (dashed line), the PNR receiver (solid line), and the optimal intermediate measurement (dot-dashed lines) are shown. The mean photon number is varied for the different receivers. The PNR receiver again outperforms the homodyne receiver, and we find a relatively good agreement of experimental data and theoretical predictions. (b) Error rate for various signal amplitudes. The PNR receiver surpasses the homodyne receiver. Statistical error bars show standard deviation of the random process. We corrected for the quantum efficiencies of the receivers.

Figure 7

(a) Key rate $G$ on logarithmic scale (log${}_{10}$G) as a function of the channel transmittance $\eta$. The solid curve corresponds to the optimized PNR receiver. The dashed curve shows the secret key rate for the decoding strategy based on optimized homodyne detection [47, 48]. The other figures show the optimized (b) threshold number $m$, (c) signal amplitude $\alpha$, and (d) displacement $\beta$ as functions of the channel transmittance.