  We introduce an entanglement branching operator to split a composite
entanglement flow in a tensor network which is a promising theoretical tool for
many-body systems. We can optimize an entanglement branching operator by
solving a minimization problem based on squeezing operators. The entanglement
branching is a new useful operation to manipulate a tensor network. For
example, finding a particular entanglement structure by an entanglement
branching operator, we can improve a higher-order tensor renormalization group
method to catch a proper renormalization flow in a tensor network space. This
new method yields a new type of tensor network states. The second example is a
many-body decomposition of a tensor by using an entanglement branching
operator. We can use it for a perfect disentangling among tensors. Applying a
many-body decomposition recursively, we conceptually derive projected entangled
pair states from quantum states that satisfy the area law of entanglement
entropy.
