Gottesman-Kitaev-Preskill Encoding in Continuous Modal Variables of Single Photons

GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous variable logical qubits that can be corrected for errors caused by phase space displacements. Their experimental realization is challenging, in particular, using propagating fields, where quantum information is encoded in the quadratures of the electromagnetic field. However, traveling photons are essential in many applications of GKP codes involving the long-distance transmission of quantum information. We introduce a new method for encoding GKP states in propagating fields using single photons, each occupying a distinct auxiliary mode given by the propagation direction. The GKP states are defined as highly correlated states described by collective continuous modes, as time and frequency. We analyze how the error detection and correction protocol scales with the total photon number and the spectral width. We show that the obtained code can be corrected for displacements in time-frequency phase space, which correspond to dephasing, or rotations, in the quadrature phase space and to photon losses. Most importantly, we show that generating two-photon GKP states is relatively simple, and that such states are currently produced and manipulated in several photonic platforms where frequency and time-bin biphoton entangled states can be engineered.

Figure 1

Left: physical TF GKP state with peak spacing $2{\mathrm{\Omega }}_o$ in the collective variable ${\mathrm{\Omega }}_1$. ${\mathrm{\Omega }}_{\perp }$ is a collective variable orthogonal to ${\mathrm{\Omega }}_1$. Right: a displacement of $2\sqrt{n}{\mathrm{\Omega }}_o$ in the local variable ${\omega }_j$ also displaces variable ${\mathrm{\Omega }}_1$. The displacement in the orthogonal direction ${\mathrm{\Omega }}_{\perp }$ is not relevant since we ignore the information it contains.