Abstract
This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.
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