  We probe the theoretical connection among three different approaches to
analyze the entanglement of identical particles, i.e., the first quantization
language (1QL), elementary-symmetric/exterior products (which has the
mathematical equivalence to no-labeling approaches), and the algebraic approach
based on the GNS construction. Among several methods to quantify the
entanglement of identical particles, we focus on the computation of reduced
density matrices, which can be achieved by the concept of \emph{symmetrized
partial trace} defined in 1QL. We show that the symmetrized partial trace
corresponds to the interior product in symmetric and exterior algebra (SEA),
which also corresponds to the subalgebra restriction in the algebraic approach
based on GNS representation. Our research bridges different viewpoints for
understanding the quantum correlation of identical particles in a consistent
manner.
