Quantum advantage in a unified scenario and secure detection of resources

Abstract: Quantum resources may provide advantage over their classical counterparts. We say this as quantum advantage. Here we consider a single communication task to study different approaches of observing quantum advantage. We say this setting as a unified scenario. In our task, there are three parties - the Manager, Alice, and Bob. The Manager sends a value of a random variable to Alice and at the same time Bob receives some partial information regarding that value. Initially, neither Alice nor Bob knows the input of the other, received from the Manager. The goal of the task is achieved if and only if the value of the random variable, sent to Alice by the Manager, is identified by Bob with success probability greater than half all the time. Here non-zero error probability is allowed. However, to help Bob, Alice sends a limited amount of classical or quantum information to him (cbit or qubit). We show that the goal of the task can be achieved when Alice sends a qubit. On the other hand, a cbit communication is not sufficient for achieving the goal. Thus, it establishes quantum advantage. We further show that the optimal success probability in the overall process for a qubit communication can be higher than the optimal success probability for a cbit communication. In fact, this success probability is higher for qubit communication even if along with cbit communication Alice and Bob share randomness. Clearly, it demonstrates a more prominent non-classical feature. Actually, we obtain higher success probability compared to other tasks. This also suggests the experiment friendly nature of our task. We then connect our task with semi-device-independence and show how our task can be useful to detect quantumness of a communication in a secure way. With increasing dimension of the random variable, to achieve the goal, the separation between the quantum and the classical communication also increases.