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I know long-range entanglement is the essence of nontrivial topological ordered states. (Trivial refers to short range entangled and nontrivial refers to long range.) So, entanglement measure at large scale compared with correlation length is a tool to distinguish topological trivial and nontrivial states. One of the ways to measure entanglement is through topological entanglement entropy, $S_{top}=log D$, where $D$ is the quantum dimension. The state is topological trivial if and only if $D=1$. But it seems that this method can not be applied to distinguish nontrivial states. For example, for FQH state with filling factor $v=1/q$, $D=\sqrt{q}$, indicating that it is nontrivial. But we know that FQH ground state has $q$ fold degeneracy, all of them are nontrivial. My question is how to distinguish between long range entangled states in a quantitative way.

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"How to distinguish between long range entangled states in a quantitative way?"

I know four quantitative ways to describe long-range entangled states.

1) Modular transformation (ground state topological degeneracy and non-Abelian geometric phases) for 2+1D topological orders. (This may be most general.)

2) K-matrix for all Abelian topological orders.

3) String-net (spherical fusion category) for 2+1D non-chiral topological orders

4) Patterns of zeros an $Z_n$ chiral algebra for chiral topological orders.

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A stupid question. Can point (1) be used to classify long-range entangled states? If so, how? Should one find all possible inequivalent representations of the modular group? –  Heidar Nov 25 '12 at 10:11
    
Another stupid question. All these points depend on special features of 2+1 dimensions it seems. Besides point (3), it is not clear to me if any of them can be generalized to 3+1D. Are there any attempts for higher-dimensions? –  Heidar Nov 25 '12 at 10:14
    
Very good points! Yes all the points are for 2+1D. Point (1) may be the most general but less developed. –  Xiao-Gang Wen Nov 25 '12 at 18:54
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The quantum dimension is one such measure. The different ground states of a particular $1/q$ FQHE are not topologically distinct. In fact they aren't distinct from one another at all -- they simply span a space such that any linear combination of them is also a ground state. The existence of a ground state degeneracy that depends on the genus of the manifold that the state is constructed on is instead an indication of topological order.

The states which are topologically distinct (but don't differ in long-range correlations of a local order parameter, as in the Landau paradigm) are the states at different values of $q$, which do in fact have different values of $D$.

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Thanks, but what about the hierarchy FQH states, where filling factor $v=\frac{p}{q}=\frac{r^2\tilde{p}}{s^2\tilde{q}}$. The quantum dimension is not related one by one with degeneracy $\tilde{q}$. –  a0087946gy Sep 26 '12 at 14:05
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