Quantum filtering of optical coherent states

We propose and experimentally demonstrate nondestructive and noiseless removal (filtering) of vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors, i.e., vacuum states in the quantum information are diagnosed through a weak measurement, and on that basis, probabilistically filtered out. We consider three different filters based on on-off detection, phase stabilized, and phase randomized homodyne detection. We find that on-off detection, optimal in the ideal theoretical setting, is superior to the homodyne strategy also in a practical setting.

Figure 1

Application of the quantum filter device. The filter F is placed between two quantum channels connecting sender S and receiver R. We assume, that the channels have non-Gaussian on-off (first part) and Gaussian properties (last part). The on-off behavior of the first channel will be masked by excess noise in the second channel (e.g., $\mid \alpha ⟩⟨\alpha \mid \to \mid 0⟩⟨0\mid \to {\rho }_{\mathit{th}}$). However, a quantum filter in the intermediate station can sense the channel break and reject the noisy state by sending information over a classical channel to R.

Figure 2

Schematic illustration of a coherent state quantum filter for the non-Gaussian channel: (a) Filter device with verification measurement; (b) (left-hand side) filter using APD as a detector, (b) (right-hand side) using homodyne detection with a local oscillator (LO).

Figure 3

Marginal distribution for the perturbed state $(p=0.02)$ (circles), the vacuum state (crosses) and the filtered state using an APD filter (triangles). The solid line and the dotted-dashed line correspond to the theoretical performance of a filter with APD $({\eta }_{\mathrm{APD}}=1)$ and with homodyne detector $({\eta }_{\mathrm{HDS}}=1)$, respectively. The mean photon number in the filter is $R{\mid \alpha \mid }^2=1.65$ and the error probabilities are identical $E_{\mathrm{APD}}=E_{\mathrm{HDS}}=5.3\times {10}^{-3}$.

Figure 4

Acceptance probability for mean photon number $R{\mid \alpha \mid }^2$ impinging on the filter detector. The triangles, circles, and squares show experimental data for APD and homodyne detection with and without stabilized LO, respectively. The solid lines are theoretical predictions for detectors with unit quantum efficiency. $E_{\mathrm{APD}}=E_{\mathrm{HDS}}=E_{\mathrm{HDR}}=5.3\times {10}^{-3}$.

Figure 5

In all figures the triangles are experimental data points for a filter using APD, the circles and squares show data for filters using homodyne detection with and without stabilized LO, respectively; the solid lines are theoretical predictions for filters using unit quantum efficiency detectors, $\eta =1$ [APD: $S∕R={(1-p_d)}^2$ for $\eta =1$]. (a) Sensitivity as a function of error probability. The dashed line should guide the eye to the error rate where the detectors are compared $E_{\mathrm{APD}}=E_{\mathrm{HDS}}=E_{\mathrm{HDR}}=5.3\times {10}^{-3}$. (b) Gain $G$ as a function of mean photon number $R{\mid \alpha \mid }^2$ impinging on the filter detector. Signal probability fixed to $p=2\%$. (c) Parametric plot of Gain $G$ and success probability $P_S$ for $R{\mid \alpha \mid }^2∊[0,1.65]$. The performance decreases from APD to HDS and HDR. The error bars show the statistical errors ($3\sigma$ error bars) and are much smaller than the experimental errors.