Entanglement-enhanced classical communication without a shared frame of reference

Two parties, Alice and Bob, share a communication channel but lack a shared reference frame. Alice's task is to communicate a message to Bob, and she does so by preparing an object in a state that represents the message, for example as a rotation, and transmitting this object to Bob who measures the state of the object to reveal the message. Due to the lack of a shared reference frame, Bob may not be able to perform the appropriate measurement to learn the message. For example Bob may be lacking the reference angle against which to measure the rotation. Here we tackle the problem of how two parties, lacking a shared reference frame, could prepare and measure a message in order to communicate successfully. We deem a prepare-and-measure procedure to be successful if it minimizes the average error over all received messages. In our communication protocol the parties circumvent the lack of a shared reference frame by preparing and sending two objects such that the message is the relative transformation parameter from the state of the rst object into the state of the second object. Bob performs joint measurements on the pair of received objects to infer the message from the measurement outcomes. Our aim is to devise a prepare-and-measure scheme that ensures the highest average success rate for sending messages as relative transformation parameters between two objects. We use Schur's lemmas, group representation theory, and quantum estimation theory to derive optimal measurements given constraints imposed on Alice's preparations. We can nd closed-form solutions for prepare-and-measure schemes for some constraints and employ numerical methods to obtain optimal protocols in the more general cases. In particular we discover that, whereas preparing objects in an entangled state is sucient for success, entanglement is not always necessary. Our theory lays the groundwork for circumventing a lack of reference frames between parties by sending messages through the parameter of a relative transformation between two objects.