  In quantum state discrimination, the observers are given a quantum system and
aim to verify its state from the two or more possible target states. In the
local quantum state marking as an extension of quantum state discrimination,
there are N composite quantum systems and N possible orthogonal target quantum
states. Distant Alice and Bob are asked to correctly mark the states of the
given quantum systems via local operations and classical communication. Here we
investigate the local state marking with N 4 ${\otimes}$ 4 systems, N=4, 5, 6,
and 7. Therein, Alice and Bob allow for three local operations: measuring the
local observable either ${\sigma}_{z}$ or ${\sigma}_{x}$ simultaneously, and
entanglement swapping. It shows that, given arbitrary four 4 ${\otimes}$ 4
systems, Alice and Bob can perform the perfect local quantum state marking. In
the N=5, 6 cases, they can perform perfect local state marking with specific
target states. We conjecture the impossibility of the local quantum state
marking given any seven target states since Alice and Bob cannot fulfill the
task in the simplest case.
