Single-Qubit Loss-Tolerant Quantum Position Verification Protocol Secure against Entangled Attackers.

We give a tight characterization of the relation between loss tolerance and error rate of the most popular protocol for quantum position verification, which is based on BB84 states. Combining it with classical information, we show, using semidefinite programming, for the first time a fault-tolerant protocol that is secure against attackers who preshare a linear amount of entanglement (in the classical information), arbitrarily slow quantum information and that tolerates a certain amount of photon loss. We also extend this analysis to the case of more than two bases, showing even stronger loss tolerance for that case. Finally, we show that our techniques can be applied to improve the analysis of one-sided device-independent quantum key distribution protocols.

Asymptotic Performance of Port-Based Teleportation.

Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur-Weyl distribution by Johansson, which might be of independent interest.

Quantum communication complexity advantage implies violation of a Bell inequality.

We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.