Asymptotic Performance of Port-Based Teleportation
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Communications in
Mathematical Physics
Asymptotic Performance of Port-Based Teleportation Matthias Christandl1 , Felix Leditzky2,3 , Christian Majenz4,5,7 , Graeme Smith2,3,6 , Florian Speelman4,7,10 , Michael Walter4,5,8,9 1 2 3 4 5 6 7 8 9 10
QMATH, Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark JILA, University of Colorado/NIST, Boulder, USA. E-mail: [email protected] Center for Theory of Quantum Matter, University of Colorado Boulder, Boulder, CO, USA QuSoft, Amsterdam, The Netherlands. E-mail: [email protected]; [email protected] Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands Department of Physics, University of Colorado Boulder, Boulder, CO, USA. CWI, Amsterdam, The Netherlands Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands Informatics Institute, University of Amsterdam, Amsterdam, The Netherlands
Received: 29 March 2019 / Accepted: 23 June 2020 © The Author(s) 2020
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. Contents 1.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Quantum teleportation protocols . . . . . . . . . . . . . 1.2 Summary of main results . . . . . . . . . . . . . . . . . 1.3 Structure of this paper . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation and definitions . . . . . . . . . . . . . . . . . . 2.2 Representation theory of the symmetric and unitary group Port-Based Teleportation . . . . . . . . . . . . . . . . . . . . 3.1 Deterministic PBT . . . . . . . . . . . . . . . . . . . . 3.2 Probabilistic PBT . . . . . . . . . . . . . . . . . . . . .
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