Asymptotic Performance of Port-Based Teleportation

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.

Optimal port-based teleportation

“…The right hand side is obtained by using assumed form of X A in (B4) to second constraint in (15). Observing that F µ (α) ⊂ P µ , we deduce p 2 µ (α)λ µ (α) ≤ c µ .…”

“…The bar denotes the complex wise conjugation and U is an element of unitary group U (d). This specific kind of symmetry implies that all the basic elements describing considered here teleportation protocols, like signals, kPBT operator, and optimal measurements [15,21] are elements of the algebra of partially transposed permutation operators A (k) n (d) with respect to k last subsystems [20,22,23]. To describe the above connection more quantitatively we need a few additional concepts.…”

“…Solution of the primal problem Since our goal is to find optimal set of effects {Π i } i∈I , we expect from them to have the same type of symmetries implied by the resource state ρ from (19) and signals {σ i } i∈I given in (4) (see also papers [15,19,21]). Having this in mind we assume that optimal measurements are the elements of the algebra A k d (n) studied in [17] and briefly highlighted in Section V. We know from [16], that in non-optimal case of the multi-port based teleportation protocol, Alice's measurements are of the form of square-root measurements:…”

“…For example in deterministic scheme the entanglement fidelity F scales as 1 − 1/N 2 rather than 1 − 1/N, while in probabilistic version the probability of success p scales as 1 − 1/N instead of 1 − 1/ √ Nin every variant we have square improvement. For the full analysis of asymptotic performance of PBT scheme in all variants we refer to [15].…”

Tracking cryptographic keys and encrypted data using position verification

“…This port‐based teleportation is also asymptotic [18], meaning there is a chance that the teleportation fails. There is a lower bound on this failure rate and an optimised version of a multi‐port scheme has been discovered and published [19].…”