Hybrid Analog Teleportation-Direct Transmission in Noisy Bosonic Channels

Abstract

Quantum teleportation uses a shared entangled resource, local operations, and a digitally error-corrected classical channel to transfer quantum states between distant parties. We introduce a hybrid teleportation-direct transmission protocol for state transfer that still exploits entanglement, but replaces classical communication and digital error correction with an analog feedforward through a noisy quantum channel. We show that quantum teleportation outperforms this protocol if the communication channel reduces the entanglement of all bipartite states having the same amount of entanglement as the resource; otherwise, the hybrid protocol is optimal. We apply our result to the state transfer of a uniformly distributed coherent-states codebook, highlighting experimentally relevant scenarios where our protocol is most effective. Our findings are directly relevant to both optical and superconducting microwave channels, where analog feedforward techniques have been recently implemented.

Figures (5)

• Figure 1: Sketch of quantum teleportation and the analog protocol. Both protocols leverage an entangled resource state, distributed by Charlie, to transfer a state from Alice to Bob obliviously. In the analog protocol, we exploit the quantum description of the communication channel $\mathcal{G}$ to implement an optimal encoding/decoding scheme, which replaces the measure/feedforward operations of quantum teleportation.
• Figure 2: Analog scheme for quantum state transfer. Alice aims to transfer an unknown state $\rho_{\rm T}$ to Bob using the pre-shared entangled state $\rho_{\rm AB}$ as a resource. She applies a two-mode squeezing operation, parametrized by the linear gain ${d\geq1}$ (see Eq. \ref{['encoding']}), to her share of $\rho_{\rm AB}$ and the state $\rho_{\rm T}$ (with first moment $v_T$ and covariance matrix $\Gamma_{\rm T}$). She transmits the mode with first moment $\sqrt{d}\,v_{\rm T}$ to Bob through the communication channel $\mathcal{G}$, while discarding the other mode, which corresponds to its complex conjugate. Finally, Bob performs a decoding operation using his part of the entangled state, obtaining $\rho_{\rm out}$, which is a "noisy" version of the state $\rho_{\rm T}$. For infinite $d$, this is an Analog Quantum Teleportation protocol. For ${d=1}$, it is direct state transfer. For intermediate values of $d$, it is an HTDT protocol.
• Figure 3: Average fidelities, optimal encoding, and no-cloning threshold.a) Average fidelity $\mkern 1.5mu\overline{\mkern-1.1mu\mathcal{F}\mkern-0.3mu}\mkern 1.5mu$ of HTDT optimized over $d$ (red), quantum teleportation (blue), and entanglement-free case (green), for the state transfer of a codebook of uniformly distributed coherent states. We consider a quantum-limited attenuator with transmissivity $x$ as a communication channel, and the optimal triplet for quantum teleportation ${(a,b,c)=(\cosh(2r),\cosh(2r),\sinh(2r))}$ as resource. In b), we plot the infidelities ratio ${\delta=(1-\mkern 1.5mu\overline{\mkern-1.1mu\mathcal{F}\mkern-0.3mu}\mkern 1.5mu_{\rm ef})/(1-\mkern 1.5mu\overline{\mkern-1.1mu\mathcal{F}\mkern-0.3mu}\mkern 1.5mu_{\rm an})}$, showing the advantage of the quantum teleportation (blue) and HTDT (red) over the entanglement-free case. c) Plot of the optimal $d$ for HTDT, i.e., ${d_{\rm opt}=[x-\tanh^2(r) ]/[x^2-\tanh^2(r)]}$. In a) and c) we set ${2r=\ln(2)\simeq 0.693}$, such that ${\mkern 1.5mu\overline{\mkern-1.1mu\mathcal{F}\mkern-0.3mu}\mkern 1.5mu_{\rm qt}=2/3}$ matches the no-cloning threshold Cerf2000. Changing $r$ results in qualitatively similar plots. d) Plot of the logarithmic negativity $2r$ needed to reach the no-cloning threshold for the optimal analog protocol (continuous) and quantum teleportation (black dashed) protocols.
• Figure 4: Alice–Bob–Charlie configuration. We consider a configuration in which Charlie is equidistant from Alice and Bob, i.e., ${D_{\rm CA} = D_{\rm CB} \equiv D_{\rm C}}$. All channels have the same loss rate $\gamma$, so that ${x_{\rm CA}=x_{\rm CB}\equiv x_{\rm C}}$. The quantity $h_{\rm C}$ denotes Charlie’s distance from the Alice–Bob axis.
• Figure 5: Average fidelity versus Charlie’s position. We plot the average fidelities for the analog protocol optimized over the encoding squeezing (red), quantum teleportation (blue), and the entanglement-free case (green) for various values of Charlie's logarithmic negativity ($2r_{\rm C}$). The results are shown for the configuration in Fig. \ref{['Fig4']} ($D_{\rm CA} = D_{\rm CB} \equiv D_{\rm C}$) as a function of Charlie's vertical position ($h_{\rm C}$), for parameter values ${\gamma=10^{-3}}$ dB/m, ${x_{\rm AB}=0.7}$ (corresponding to ${D_{\rm AB}\simeq 1.5}$ km), and a symmetric pure state at Charlie. The red curves are optimized for a finite encoding squeezing, so HTDT always outperforms quantum teleportation in the shown regimes.

Theorems & Definitions (2)

• Theorem
• proof