**Anilesh Mohari**

Institute of Mathematical Sciences, Chennai

September 3, 2015

**Isomorphism theorem for Kolmogorov states in quantum spin chain**:
We prove translation dynamics on a two sided quantum spin chain
with two invariant Kolmogorov states are weakly isomorphic. For a class of Kolmogorov states of two sided quantum spin chain, we prove further that they give isomorphic translation dynamics once their restrictions to the two sided infinite tensor product of diagonal matrices with respect to an orthonormal basis are isomorphic. We also give an overview how these problems are related to various properties of translation invariant ground states of Heisenberg type of Hamiltonian in one dimensional quantum spin chain.