# Definition for long range entanglement (LRE) by generalized local unitary (gLU) and generalized stochastic local (gSL) transformations

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.

I think it is a typo.

It is helpful to sketch a short history:

1. In 2010, Chen, Gu and Wen wrote a paper (arXiv:1004.3835) where they introduced the notion of local unitary evolution and used it to talk about the classification. In this paper, you can find the following figure on p6:

This paper does not discuss stochastic evolution/transformation. So indeed their caption is not strictly true for every (cat state) ground state, but perhaps they were thinking of taking the symmetry-broken ground state---in which case it becomes true.

2. In 2012, Xiao-Gang Wen wrote a stackexchange post asking how to characterize symmetry-breaking in quantum systems. You can see that the posts/discussions in that thread led Wen to characterize such phases in terms of their entanglement. Presumably, the notion of 'stochastic evolution' was subsequently developed to deal with the fact that while symmetry-breaking had long-range entanglement in its cat state, it is less entangled than topologically-ordered phases.

3. In 2015 (with updates over the next years), Zeng, Chen, Zhou and Wen wrote the book that you are referencing. It seems that the figure you are asking about was borrowed from their earlier work (with modifications), and the caption was not updated to include the more subtle notions of unitary-vs-stochastic.

Perhaps you can contact one of the authors so that they can correct the typo in a next version.