Entanglement-assisted quantum error correction with linear optics

We construct a theory of continuous-variable entanglement-assisted quantum error correction. We present an example of a continuous-variable entanglement-assisted code that corrects for an arbitrary single-mode error. We also show how to implement encoding circuits using passive optical devices, homodyne measurements, feedforward classical communication, conditional displacements, and off-line squeezers.

Figure 1

The above figure demonstrates the operation of a continuous-variable entanglement-assisted code. Lines with bars through them denote multiple modes. Thin lines denote quantum information and thick lines denote classical information. Alice possesses states $|\psi ⟩$, $|0⟩$, and half of the entangled modes $|\Phi ⟩$. Bob possesses the other half of entangled modes $|\Phi ⟩$. The unitary $U$ encodes the multimode state $|\psi ⟩$ with the help of several position-quadrature squeezed ancillas $|0⟩$ and entangled modes $|\Phi ⟩$. Alice sends her modes over a noisy quantum channel. The entanglement-assisted communication paradigm assumes that the noisy channel affects Alice’s modes only. Bob measures all the modes to diagnose the errors and corrects them with a recovery operator $R$. Bob can perform these measurements with homodyne detection.