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.
Highlights
It was only known that Fd∗(N ) = 1 − (N −2) as a consequence of an explicit converse bound [19]. We prove that this asymptotic scaling is achievable, and we provide a converse with improved dependency on the local dimension, see Corollary 1.6
We completed the picture of the asymptotic performance of port-based teleportation (PBT) in the important regime when the input dimension is fixed while the number of ports tends to infinity
We determined the asymptotic performance of deterministic PBT in the fully optimized setting, showing that the optimal infidelity decays as (1/N 2) with the number of ports N
Summary
For deterministic port-based teleportation using a maximally entangled resource and the pretty good measurement, a closed expression for the entanglement fidelity was derived in [21], but its asymptotics for fixed d > 2 and large N remained undetermined. We derive the asymptotics of deterministic port-based teleportation using a maximally entangled resource and the pretty good measurement, which we call the standard protocol. We again use an expression for the entanglement fidelity of the optimal deterministic PBT protocol derived in [22] The asymptotics of this formula for fixed d and large N have remained undetermined so far. Our second main result, derived, concerns the deterministic setting in arbitrary dimension using EPR pairs, for which we compute the asymptotics of the lower optimal entanglement fidelity Fdstd bound on the entanglement fidelity (Theorem 1.2).
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