Joint multi-party remote preparation of arbitrary qudit states via a <italic>d</italic>-level six-qudit Cluster state

<p indent="0mm">We propose two protocols for joint, multi-party remote preparation of qudit states by using a <italic>d</italic>-level six-qudit Cluster state as the quantum channel. The first protocol is used for the joint remote preparation of an arbitrary two-qudit state via the <italic>d</italic>-level six-qudit Cluster state. The second is used for bi-directional joint remote preparation of an arbitrary single-qudit state via a <italic>d</italic>-level six-qudit Cluster state. All senders share information about the prepared state. The first sender applies Positive Operator-Valued Measurement (POVM) on her entangled particles in accordance with her information about the prepared state, while the second sender performs projective measurements in accordance with the information about the prepared state and the first sender’s measurement results. The receiver can prepare the original state by performing a corresponding unitary operation according to the measurement results from all the senders. These two protocols have the advantage of being able to provide a high channel capacity by using a <italic>d</italic>-level Cluster state as the quantum channel.