Environment-assisted quantum-information correction for continuous variables

Quantum-information protocols are inevitably affected by decoherence which is associated with the leakage of quantum information into an environment. In this article we address the possibility of recovering the quantum information from an environmental measurement. We investigate continuous-variable quantum information, and we propose a simple environmental measurement that under certain circumstances fully restores the quantum information of the signal state although the state is not reconstructed with unit fidelity. We implement the protocol for which information is encoded into conjugate quadratures of coherent states of light and the noise added under the decoherence process is of Gaussian nature. The correction protocol is tested using both a deterministic as well as a probabilistic strategy. The potential use of the protocol in a continuous-variable quantum-key distribution scheme as a means to combat excess noise is also investigated.

Figure 1

Schematic of the investigated protocol. Sender: Information is encoded in conjugate quadratures (or in a single quadrature) of a coherent state. Environment: The environment is simulated by a variable beam splitter with injected Gaussian noise, and the environmental measurement is carried out with a heterodyne detector (or homodyne detector). Receiver: The information is measured with a heterodyne (or homodyne) detector.

Figure 2

Schematic of the experimental setup. Information is encoded by using an amplitude modulator (AM) and a phase modulator (PM) operating at 14.3 MHz. The beam splitter coupling is controlled by combining two polarizing beam splitters (PBS) with a half-wave plate, and the efficiency of the environmental measurement is likewise controlled by a PBS and a half-wave plate. The environmental noise modes are generated by two modulators fed by independent Gaussian noise. Measurements are carried out by simultaneously detecting conjugate quadratures as described in the main text.

Figure 3

The results for the deterministic approach. The added noise, $V_{S,\mathrm{add}}^{\mathrm{wff}}$, given in shot noise units, is plotted against the amount of excess noise from the environment $V_{E,\mathrm{in}}$. (a) The weak coupling regime with $\eta =0.9.$ (b) The strong coupling regime with $\eta =0.1$. The solid line is associated with the ideal noise cancellation, and the dashed line takes into account the measurement imperfections in the environment. The nonunity measurement efficiency of the environment plays a bigger role for the strong coupling regime, due to the fact that more information will in this case leak into the environment. Without employing the corrective action, the added noise ($V_{S,\mathrm{add}}^{\mathrm{woff}}$) would have been between 2.2 and 6.1 for $10<V_{E,\mathrm{in}}<45$ in the weak coupling regime and between 19.9 and 91 for $1.1<V_{E,\mathrm{in}}<9$ in the strong coupling regime.

Figure 4

The added noise, $V_{S,\mathrm{add}}^{\mathrm{opt}}$, in shot noise units, is plotted against the efficiency of the environmental measurement, $\gamma$. The blue squares and red dots correspond to the amplitude and phase quadrature measurements, respectively. The solid and the dashed lines correspond to the theoretical prediction for the optimized approach with respect to noise minimization [Eq. (20) with $V_{E,\mathrm{in}}=25$ and $\eta =0.9$] and the approach for which the excess noise is removed [Eq. (15)]. The added noise when measured directly without any correction is $V_{\mathrm{add}}=2.77$. The inset shows the channel gain as a function of $\gamma$, and the solid curve is associated with theory [Eq. (22)].

Figure 5

The added noise of the receiving heterodyne detector in shot noise units as a function of the logarithm of the success probability for the amplitude quadrature (left figure) and the phase quadrature (right figure). The solid curve corresponds to numerical calculations using Wigner function analysis, and the upper straight lines are the added noises without feedforward corrections. These variances are $V_{S,\mathrm{add}}^{\mathrm{wo}\mathrm{\hspace{0.5em}}\mathrm{ff}}=4.55$ and $V_{S,\mathrm{add}}^{\mathrm{wo}\mathrm{\hspace{0.5em}}\mathrm{ff}}=3$ for the left and right figures, respectively. The lower lines correspond to perfect noise cancellation.