Loss-tolerant quantum secure positioning with weak laser sources

Quantum position verification (QPV) is the art of verifying the geographical location of an untrusted party. Recently, it has been shown that the widely studied Bennett & Brassard 1984 (BB84) QPV protocol is insecure after the 3 dB loss point assuming local operations and classical communication (LOCC) adversaries. Here, we propose a time-reversed entanglement swapping QPV protocol (based on measurement-device-independent quantum cryptography) that is highly robust against quantum channel loss. First, assuming ideal qubit sources, we show that the protocol is secure against LOCC adversaries for any quantum channel loss, thereby overcoming the 3 dB loss limit. Then, we analyze the security of the protocol in a more practical setting involving weak laser sources and linear optics. In this setting, we find that the security only degrades by an additive constant and the protocol is able to verify positions up to 47 dB channel loss.

Figure 1

Relativistic constraints. We assume that all quantum and classical signals travel at the speed of light and that the speed of light is normalized to unity. In this case, the time required to send a message from one position to another position is equal to the Euclidean distance between them. More specifically, the Euclidean distance between ${\mathsf{pos}}_1$ and ${\mathsf{pos}}^{*}$ is defined as $d({\mathsf{pos}}_{\mathsf{1}},{\mathsf{pos}}^{*})$ where $d(·,·)$ is the distance measure in $\mathbb{R}$. The protocol is based on a $N$-fold sequential repetition setting, where the verifiers only send out their qubit states at intervals of ${\mathsf{t}}_{i+1}-{\mathsf{t}}_i=2d({\mathsf{pos}}_{\mathsf{1}},{\mathsf{pos}}^{*})=2d({\mathsf{pos}}^{*},{\mathsf{pos}}_2)$. Note that, for simplicity, we assume the prover is located at the center.

Figure 2

BSM based on linear optics. A successful Bell-state measurement corresponds to the following detection patterns: a coincident detection in $D_{1H}$ and $D_{2V}$, or in $D_{1V}$ and $D_{2H}$, indicates a projection into the Bell state $|{\mathrm{\Psi }}^{-}\rangle$, while a click in $D_{1H}$ and $D_{1V}$, or in $D_{2H}$ and $D_{2V}$, reveals a projection into the Bell state $|{\mathrm{\Psi }}^{+}\rangle$

Figure 3

The upper bound of the estimated single-photon error rate versus overall loss between Alice and Bob. The simulation assumes a baseline QBER of $0.1\%$. The the detectors are assumed to have an efficiency of 64% and a dark count rate of $2.5\times {10}^{-6}$. The starting cutoff point is about 6.8 dB, which is the total loss in the BSM. The numerical results are obtained using $N={10}^x$ with $x=10,11,12,13$ (from left to right).