Optimal Multi-port-based Teleportation Schemes
Marek Mozrzymas, Michał Studziński, Piotr Kopszak
Abstract
In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>, the number of teleported systems<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>, and local dimension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>of ports increases as<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.