Christandl Matthias, Leditzky Felix, Majenz Christian, Smith Graeme, Speelman Florian, Walter Michael
QMATH, Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark.
JILA, University of Colorado/NIST, Boulder, USA.
Commun Math Phys. 2021;381(1):379-451. doi: 10.1007/s00220-020-03884-0. Epub 2020 Nov 20.
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 of ports, the error of the optimal protocol is proportional to the inverse square of . 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.
量子隐形传态是量子香农理论的基本组成部分之一。普通隐形传态简单高效，而基于端口的隐形传态（PBT）则可实现诸如通用可编程量子处理器、即时非局域量子计算以及对基于位置的量子密码学的攻击等应用。在这项工作中，我们确定了PBT性能的基本极限：对于任意固定的输入维度和大量端口，最优协议的误差与端口数量的平方成反比。我们通过推导一个可达界来证明这一点，该可达界是通过将相应的优化问题与有序单纯形上拉普拉斯算子的最低狄利克雷特征值相关联而得到的。我们还给出了在端口数量上具有匹配阶数的改进逆界。此外，我们确定了根据最大纠缠资源态定义的PBT变体的主导阶渐近性。这些结果的证明依赖于将最近推导的表示理论公式与随机矩阵理论联系起来。在此过程中，我们改进了约翰松关于舒尔 - 外尔分布涨落的收敛结果，这可能具有独立的研究价值。
