I am studying this book: Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems (https://arxiv.org/abs/1508.02595).
In chapter 7, it introduces the ideas of generalized local unitary (gLU) and generalized stochastic local (gSL) transformations to define different phases. One important difference is that a gLU transformation can't connect the GHZ state to the product state. Therefore, the gapped liquid phases defined by gLU transformation contains symmetry-breaking phases. On the other hand, a gSL transformation can change the GHZ state to the product state with a finite probability. Using gSL transformation to study topological order is more appropriate.
Box 7.20: Short/long-range entanglement: A state is short-range entangled (SRE) if it is convertible to a product state by a gSL transformation. Otherwise, the state is long-range entangled (LRE).
I am happy with their definition so far. However, I have questions about the following phase diagram in the next subsection.
In this figure, they use gLU transformation to define SRE phases. This makes me very confused. In the case without any symmetry involved, are gLU and gSL transformations equivalent? It is not obvious to me that these two definitions are the same.