Limitation of independent and identical distribution to capture the hierarchy of nonlocality

Importance of considering independent and identically distributed (iid) scenario is well-established in quantum theory. From entanglement detection to entangled state transformation, this scenario is useful in many tasks. Nevertheless, here we focus on the limitation of iid scenario. For this purpose, we consider the notion of nonlocality associated with local state discrimination problems (LSDPs). Here, the term `nonlocality' denotes the difference between the local and the global abilities of discriminating quantum states. It is natural to consider LSDPs to study the aforesaid limitation because iid copies of quantum states are considered as resource in these problems. However, we first explore entanglement as resource in LSDPs where the states are pure. Then, we generalize our findings considering mixed states. In present LSDPs, our findings make the teleportation-based protocols sub-optimal. In fact, our results shed light on Schmidt rank as resource in LSDPs. Interestingly, we find instances where the states cannot be locally discriminated even if finite number of iid copies of the states are available. But local discrimination of these states can be done consuming quite less resource when we consider entanglement-assisted LSDPs. This allows us to compare the degrees of nonlocality associated with similar LSDPs within the same Hilbert space. Thus, we report a hierarchy of nonlocality which cannot be captured through iid scenario but it can be observed when entanglement is used as resource in certain LSDPs. Moreover, the difference in nonlocality may increase with increasing dimension. Finally, we show an application of our theory in secure locking of information and its resource-efficient extraction.