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

Abstract

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.

Figures (12)

• Figure 1: (a) Schematic illustration of a single-photon Dirac comb state in the frequency basis (Eq. \ref{['eq:freq_sp_comb']}) and the same state in the temporal basis (Eq. \ref{['eq:time_sp_comb']}). Each spike represents a Dirac delta function. (b) GKP encoding of the three basis states of a logical qutrit in the frequency basis of a single-photon Dirac comb, by discrete frequency shifts of $\frac{\Delta\Omega}{3}$ that translate the entire comb. The red comb represents the logical $\ket{\overline{0}}$ state, the green comb represents $\ket{\overline{1}}$ and the yellow comb represents $\ket{\overline{2}}$.
• Figure 2: Schematic representation of the superdense coding protocol based on FBS decoder. The final measurements are of the arrival-time of one photon and the frequency of the other.
• Figure 3: Schematic illustration of the second step of the encoding phase of our superdense coding protocol, described in Section \ref{['sec:sdc_encoding']}, showing how the correlations in the biphoton frequency comb are affected by the frequency shift operator. In each panel, the top line represents photon $H$ and the bottom line represents photon $V$. Pairs of spikes linked by vertical dotted lines represent the pairs of frequency eigenstates of the form $\ket{\omega_1}^{\mathrm{freq}}\ket{\omega_2}^{ \mathrm{freq}}$ that appear as summands in the EPR-like biphoton frequency comb. The top panel represents the state $\ket{\varphi_{0j}}$, before the frequency shift is applied ($\ket{\varphi_{0j}}$ includes a time-shift, but in the frequency basis this just applies a phase to each spike, so is not explicitly represented in the figure). The bottom panel represents the state $\ket{\varphi_{kj}}$ after a frequency shift, which acts on each summand as $\ket{\omega_1}^{\mathrm{freq}}\ket{\omega_2}^{ \mathrm{freq}} \mapsto \ket{\omega_1 + \frac{k\Delta\Omega}{n}}^{\mathrm{freq}}\ket{\omega_2}^{ \mathrm{freq}}$, showing how arbitrary frequency shifts are distinguishable in the idealized case. In practice, the range of distinguishable shifts is limited by the available bandwidth.
• Figure 4: Modeling the probability distributions of a biphoton frequency comb (EPR-like entangled TFGKP state) in both the time and frequency domains. (a) Schematic of a biphoton frequency comb generation. (b) The signal-idler temporal correlation in the comb state with a 20 GHz FSR cavity. Frequency spectrum of the comb state, with phase-matching FWHM bandwidth of (c) 250 GHz and (d) 7.4 THz.
• Figure 5: Frequency beamsplitter (FBS) operation via a phase modulator. $\omega_1$ and $\omega_2$ are the two input and output spectral modes that belong to the computational-space of FBS operation. The dark green frequency bins at the output of FBS indicate the undesired sideband population, causing this scheme to be probabilistic Imany2018. Here we consider the sandwiched scheme Lu2018Seshadri2022 for near-deterministic FBS operation. The optical loss of our proposed scheme can be 5 dB Nussbaum2022.