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An unextendible product basis (UPB) is a set of orthogonal product stateswhich span a subspace of a given Hilbert space while the complementary subspacecontains no product state. These product bases are useful to produce boundentangled (BE) states. In this work we consider reducible and irreducible UPBsof maximum size, which can produce BE states of minimum rank. From a reducibleUPB, it is possible to eliminate one or more states locally, keeping thepost-measurement states orthogonal. On the other hand, for an irreducible UPB,the above is not possible. Particularly, the UPBs of the present size areimportant as they might be useful to produce BE states, having ranks of thewidest variety, which satisfy the range criterion. Here we talk about such BEstates. We also provide other types of BE states and analyze certain propertiesof the states. Some of the present BE states are associated with the tilestructures. Furthermore, we provide different UPBs corresponding to the presentBE states of minimum rank and discuss important properties of the UPBs.

Unextendible product bases, bound entangled states, and the range criterion