Protecting an optical qubit against photon loss

We consider quantum error-correction codes for multimode bosonic systems, such as optical fields, that are affected by amplitude damping. We demonstrate that the most accessible method of transforming optical systems with the help of passive linear networks has limited usefulness in preparing and manipulating such codes. These limitations stem directly from the recoverability condition for one-photon loss. We introduce a three-photon code protecting against the first order of amplitude damping, i.e., a single photon loss, and discuss its preparation using linear optics with single-photon sources and conditional detection. Quantum state and process tomography in the code subspace can be implemented using passive linear optics and photon counting. An experimental proof-of-principle demonstration of elements of the proposed quantum error correction scheme for a one-photon erasure lies well within present technological capabilities.

Figure 1

The Bloch sphere of the encoded qubit. The logic state $\mid H⟩$ can be transformed deterministically using linear optics into any of three states indicated with dots. All linear optics transformations that do not involve auxiliary photons are given by the rotation group of the tetrahedron, generated from a $2\pi ∕3$ rotation ${\Gamma }_3$ around the vertical axis and a $\pi$ rotation ${\Gamma }_2$ around the axis passing through the midpoints of the opposite edges of the tetrahedron.

Figure 2

(Color online) A universal circuit for encoding a single qubit in the dual-rail representation using four auxiliary photons. Three of the auxiliary photons are sent through a tritter ${\Gamma }_2$ to generate the state $\mid T_3⟩$. One of the output modes from the tritter is combined with the remaining photon and the input qubit. The output is accepted upon detecting a combination 110. Filled circles denote phase shifts, and horizontal bars are beam splitters with real amplitude transmission and reflection coefficients. The convention is that the beam reflected off the thickened side of a beam splitter acquires a minus sign. The sign change is introduced also by four mirrors. The amplitude reflection coefficients for the beam splitters are $r_1=0.586$, $r_2=0.728$, $r_3=0.448$, $r_4=0.625$, $r_5=0.837$, and $r_6=0.984$.

Figure 3

(Color online) A network for conditional manipulations of the three-photon code. The three upper modes carry the encoded qubit. One of the modes undergoes a linear transformation with two auxiliary modes carrying respectively one and zero photons, shown within a dashed box. The transformation network involves beam splitters with amplitude reflectivities $r_1$ and $r_2$ and a phase delay $\varphi$. The output is accepted when the auxiliary modes generate one- and zero photon events on the detectors. The three signal modes are then attenuated by beam splitters with complex amplitude transmissivities ${\eta }_1$, ${\eta }_2$, and ${\eta }_3$, and the gate is successful when none of the detectors monitoring the reflected beams registers any photons.

Figure 4

A contour plot of the maximal attainable success rate $S={\parallel \widehat{A}\parallel }^2$, expressed in percent, of the conditional transformation of the logic qubit given by an operator $\widehat{A}$ parametrized according to Eq. (24).