Controlled Bidirectional Remote Implementation of Arbitrary Unknown Single-Qubit Operations via a High-Dimensional Quantum Channel

Abstract

Consider two remote laboratories: Alice holds an unknown input |ξ⟩ and a private single-qubit unitary UA, while Bob holds |η⟩ and UB. We present an exact local-operations-and-classical-communication protocol that produces UB|ξ⟩ at Alice’s site and UA|η⟩ at Bob’s site only after the controller labels are released. A five-qubit channel first relocates the input states; operation delivery then uses a five-particle channel of local dimension d, a 2d-outcome nonsymmetric qubit–qudit measurement, and label-dependent inverse maps. We give the qutrit case explicitly and one correction formula for arbitrary d. A dual-layer construction lets Charlie authorize both directions, while David and Fiona authorize one direction each. In the ideal lossless model, all admissible branches have recovery maps and their probabilities sum to unity. The basic qutrit protocol uses ten entangled carriers and 13 fixed-length classical bits; the dual-layer protocol uses fourteen carriers and 19 bits. Thus, dimensional flexibility and direction-selective control are obtained at increased resource-protection and signaling costs. This is an exact theoretical remote-operation construction, not an experimentally mature or composably secure communication protocol.