  In this work we study how entanglement of purification (EoP) and the new
quantity of "complexity of purification" are related to each other using the
$E_P=E_W$ conjecture. First, we consider two strips in the same side of a
boundary and study the relationships between the entanglement of purification
of this mixed state and the parameters of the system such as dimension,
temperature, length of the strips and the distance between them. Next, using
the same setup, we introduce two definitions for the complexity of mixed
states, complexity of purification (CoP) and the interval volume (VI). We study
their connections to other parameters similar to the EoP case. Then, we extend
our study to more general examples of BTZ black holes solution in massive
gravity, charged black holes and multipartite systems. Finally, we give various
interpretations of our results using resource theories such as LOCC and also
bit thread picture.
