Degradation of the resource state in port based teleportation scheme
Port based teleportation (PBT) is a protocol of quantum teleportation in which a receiver does not have to apply correction to the transmitted state. In this protocol two spatially separated parties can teleport an unknown quantum state only by exploiting joint measurements on shared d dimensional maximally entangled states (resource state) together with a state to be teleported and one way classical communication. In this paper we analyse degradation of the resource state after one round of PBT and implications for the recycling protocol for deterministic PBT introduced earlier. In the recycling protocol the main idea is to re use the remaining resource state after one or many rounds of PBT for further processes of teleportation. It was stated by other authors that the recycling protocol is effective by arguing that the resource state does not degrade too much after each round of teleportation process. In particular, there is a claim that the fidelity between ideal resource state and its real version, each of them after one round of PBT, reaches asymptotically 1 when the number of shared entangled pairs tends to infinity. Here, considering original setup for the recycling protocol, we disprove these claims. We show the resource state is heavily distorted after even one round of PBT with fidelity not exceeding the value 1/d. This bound was obtained by referring only to Schwarz inequality and general properties of measurements exploited in the protocol. As additional results we present explicit formula for the mentioned fidelity involving group theoretic parameters describing irreducible representations in the Schur Weyl duality. For the first time, we also analyse the degradation of the resource state for the optimal PBT scheme and show its substantial distortion for all d 2. In the both versions, the qubit case is discussed separately resulting in compact expression for fidelity, depending only on the number of shared entangled pairs. Additionally, we present arguments that fidelity between the ideal and the real state after one round of PBT is not the quantity which judges about the usefulness of the post measurement state for the next rounds of PBT.