Abstract
In this chapter, we review continuous-variable quantum states and quantum state manipulations. First, we show several formalisms of quantum states including state vectors, stabilizers, nullifiers and covariance matrices. Second, we show their transformations by unitary operations. Since we mainly handle Gaussian operations in this thesis, which are members of a subgroup of unitary operations, we then discuss Gaussian operations by focusing on their matrix-based formalisms and examples of their unitary operators. As the last issue, we discuss entangled states and entanglement criteria. We show an extension of the van Loock-Furusawa entanglement criteria, which enables us to detect entanglement in cluster states of arbitrary graphs.
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Notes
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In the following subsections, we show it as transformation rules for nullifiers.
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Beam splitter matrices and operators are usually defined just as the author pleases in each paper. In some cases, the definitions of beam splitter matrices are not mentioned explicitly. Such ambiguity of their definition might confuse the readers. In this thesis, we define the four-types of beam splitters for convenience. By explicitly declaring the type of its matrix when we use a beam splitter in theory, we can avoid confusion derived from the ambiguity of the beam splitter matrix.
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Ukai, R. (2015). Quantum States and Quantum State Manipulations. In: Multi-Step Multi-Input One-Way Quantum Information Processing with Spatial and Temporal Modes of Light. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55019-8_3
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