Higher-dimensional performance of port-based teleportation

TL;DR

This paper develops a graph-theoretic algebraic method to explicitly analyze the performance of port-based quantum teleportation in higher dimensions, overcoming previous computational limitations.

Contribution

It introduces a graphical algebra framework that enables explicit calculation of PBT performance metrics for arbitrary dimensions and small port numbers.

Findings

Fidelity approaches N/d^2 as dimension increases

Provides explicit success probability calculations for low port numbers

Confirms existing lower bounds on teleportation fidelity

Abstract

Port-based teleportation (PBT) is a variation of regular quantum teleportation that operates without a final unitary correction. However, its behavior for higher-dimensional systems has been hard to calculate explicitly beyond dimension $d=2$. Indeed, relying on conventional Hilbert-space representations entails an exponential overhead with increasing dimension. Some general upper and lower bounds for various success measures, such as (entanglement) fidelity, are known, but some become trivial in higher dimensions. Here we construct a graph-theoretic algebra (a subset of Temperley-Lieb algebra) which allows us to explicitly compute the higher-dimensional performance of PBT for so-called "pretty-good measurements" with negligible representational overhead. This graphical algebra allows us to explicitly compute the success probability to distinguish the different outcomes and fidelity for arbitrary dimension $d$ and low number of ports $N$, obtaining in addition a simple upper bound. The results for low $N$ and arbitrary $d$ show that the fidelity asymptotically approaches ${N}/{d^2}$ for large $d$, confirming the performance of one lower bound from the literature.