Improving bidirectional controlled remote preparation of arbitrary number of qudits in noisy environment

By elaborately selecting entangled channel, we put forward a universal scheme to achieve bidirectional remote preparation of qudit states with arbitrary numbers of particles under the control of the supervisor. Each sender needs to perform a positive-operator-value measurement and a projective measurement, while the supervisor requires to execute a projective measurement. Based on their measurement outcomes, the receivers can recover the desired states concurrently by carrying out single-qudit generalized Pauli operations which are given by general formulas. A point to highlight is that classical communication cost is greatly reduced at the supervisor’s broadcast channel by taking advantage of network coding. In addition, we analyze the influence of amplitude-damping and phase-damping noises, and utilize weak measurement and environment-assisted measurement to suppress noise.