Witnessing the Effective Entanglement in the COW Protocol

TL;DR

The paper addresses the problem of certifying effective entanglement in the Coherent One-Way (COW) QKD protocol from prepare-and-measure data. It introduces a two-parameter entanglement witness $W$ built from correlations in the $Z$ and $X$ bases and leverages the EB-P&M equivalence to relate observed statistics to the witness expectation $\mathrm{tr}(\rho_{eff} W)$. The authors derive explicit admissible regions for the parameters $(a,b)$ that ensure $W$ is a valid witness and show how to compute $\mathrm{tr}(\rho_{eff} W)$ from COW data, including a renormalization step to focus on relevant detection events. Application to experimental data demonstrates clear signatures of effective entanglement under realistic channel losses, connecting entanglement witnessing with practical QKD security considerations and outlining a path to bound entanglement from observable statistics.

Abstract

We present a rigorous mathematical framework for verifying effective entanglement in a Coherent One-Way (COW) quantum key distribution setup. In particular, we introduce a two-parameter family of entanglement witnesses, identify the parameter ranges where they constitute valid witnesses, and demonstrate their ability to reveal effective entanglement in the COW protocol. Additionally, we analyze previously obtained experimental data from a COW implementation and report clear signatures of effective entanglement.

Figures (3)

• Figure 1: A schematic representation of the COW protocol. Alice prepares three types of quantum states: two signal states and one decoy state. On the receiver’s side, Bob retrieves the key from the signal states through the data line and verifies the coherence of the decoy states using the monitoring line.
• Figure 2: Schematic representation of the admissible region in the $(a,b)$ parameter space. The red area corresponds to the values of $a$ and $b$ for which $W$ qualifies as a valid entanglement witness, while the blue subset highlights the region where $W$ detects effective entanglement in the experimental data obtained under a channel loss of $14\,\mathrm{dB}$.
• Figure 3: Expectation value of the witness operator proposed in Eq. (\ref{['effective witness']}), with $a = -\tfrac{\sqrt{3}}{2}$ and $b = -\tfrac{\sqrt{3}}{2}$, for different channel losses