GKP-inspired high-dimensional superdense coding with energy-time entanglement

Superdense coding, the application of entanglement to boost classical communication capacity, is a cornerstone of quantum communication. In this paper, we propose a high-dimensional superdense coding protocol using energy-time entangled states. These states are biphoton frequency combs, an example of entangled time-frequency Gottesman-Kitaev-Preskill (TFGKP) states or time-frequency grid states. Inspired by GKP codes, our protocol involves discretizing the continuous time and frequency degrees of freedom and encoding information by time-frequency displacements. This approach leverages the inherently large Hilbert space found in quantum frequency combs, with resilience against both temporal and spectral errors. In addition to describing the theoretical structure of the protocol, we propose an experimental implementation using standard telecommunication components, time-resolving single-photon detectors and a frequency beamsplitter. We also analyze the effect of experimental noise and errors on the channel capacity of the protocol. We demonstrate that for realistic experimental parameters, contemporary technologies satisfy the prerequisites for superdense coding with biphoton frequency combs, achieving a transmission rate of approximately 8.91 bits per transmitted photon (equivalent to 481 distinguishable messages with asymptotically vanishing errors). This more than doubles the previously highest transmission rate of 4 bits achieved by the Kwiat-Weinfurter scheme, while also having competitive optical loss. Furthermore, our results beat the rate achievable using a single-photon frequency comb with identical parameters by 4.6 times. Our protocol thus represents an experimentally feasible application of time-frequency grid states to entanglement-assisted communication, contributing to the active fields of continuous-variable and high-dimensional quantum information.