  By means of the technique of integration within an ordered product of
operators and Dirac notation, we introduce a new kind of asymmetric integration
projection operators in entangled state representations. These asymmetric
projection operators are proved to be the Hermitian operator. Then, we
rigorously demonstrate that they correspond 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 representation. As applications, we recover the previous
results of the parity measurement in quantum metrology by our formalism.
