Channel purification via continuous-variable quantum teleportation with Gaussian postselection

We present a protocol based on continuous-variable quantum teleportation and Gaussian postselection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian operations.

Figure 1

(a) Initial quantum channel, with a transmission $T$ and equivalent input added noise $\epsilon$, such that an input state of variance $V_{\mathrm{in}}$ is transformed to an output state of variance $V_{\mathrm{out}}=T(V_{\mathrm{in}}+\frac{1-T}{T}+\epsilon )$ [38]. The channel can be modeled using a beam splitter of transmission $T$ and a thermal state $\widehat{\rho }\left({\lambda }_{\epsilon }\right)$ of variance $1+\frac{T}{1-T}\epsilon$ in the second input mode. (b) Teleportation protocol with Gaussian postselection: Bob sends one half of an EPR state through the imperfect channel and keeps the other half. Alice performs a dual-homodyne measurement (dual HD) and postselection with the state to be transmitted, and communicates the result to Bob, who applies a corrective displacement (AM PM). (c) Equivalent effective system: the input state is noiselessly amplified (or attenuated) with a gain $g_{\mathrm{eff}}$, and sent through a quantum channel of transmission $\eta$ and added noise $\mathrm{\Delta }$.

Figure 2

Bell test with a maximally entangled state $|\psi \rangle$ (12). (a) Initial lossy channel. (b) Channel purified using teleportation and postselection. (c) Effective picture of (b), in the unit transmission regime ($\eta =1$). Two channels are pictured as $|\psi \rangle$ has two independent modes $H$ and $V$.

Figure 3

(a) CHSH quantities $S^{\mathrm{tele}}$ (solid line) and $S^{\mathrm{loss}}$ (dashed line). (b) Probability of success of the two postselection for the input state $|\psi \rangle$, obtained using (7). For both figures, the optimal postselection gain (8) is used for each values of $\chi$, and $T=0.5$.

Figure 4

(a) Illustration of the relation (A2), without the phase factor. (b) Equivalence between the postselection using the filter function $Q\left(2\gamma \right)$ with $2\gamma =\frac{x+ip}{\sqrt{2}}$, and an NLA. (c) Equivalences (a) and (b) used to compute the output state (B17).