Asymptotic performance of port-based teleportation

TL;DR

This paper establishes the fundamental asymptotic limit on port-based quantum teleportation performance, showing error scales inversely with the square of the number of ports, and connects representation theory with random matrix theory.

Contribution

It derives the optimal asymptotic error scaling for port-based teleportation and improves bounds by linking optimization to spectral properties of the Laplacian on the simplex.

Findings

Optimal error scales as 1/N^2 for large N

Derived bounds using Dirichlet eigenvalues and representation theory

Connected quantum teleportation analysis with random matrix theory

Abstract

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.