New kind of asymmetric projection integral operators and parity measurement

By means of the technique of integration within an ordered product of operators and Dirac notation, we introduce a kind of Hermitian operator constructed by entangled state representations. For such a Hermitian operator, we rigorously prove that it corresponds to a parity measurement combined with a beam splitter when any two-mode quantum state passes through such device. Therefore we obtain a new relation between a Hermitian operator and the entangled state representations.