**Abstract**
We study the error exponents in a quantum hypothesis testing problem when both the null-hypothesis and the counter-hypothesis can be correlated states of a spin chain. We prove the theorems for the quantum Chernoff and Hoeffding bounds for a class of states that include certain finitely correlated states as well as the global Gibbs states of translation-invariant finite-range interactions [4,5]. Our method is based on reducing the correlated case to an i.i.d.~problem, for which the problems of Chernoff and Hoeffding bounds have recently been solved [1,2,3,6,7].

**References:**

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[2] K.M.R. Audenaert, M. Nussbaum, A. Szkola, F. Verstraete: Asymptotic Error Rates in Quantum Hypothesis Testing; arXiv:0708.4282.

[3] M. Hayashi: Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding; quant-ph/0611013.

[4] F. Hiai, M. Mosonyi, T. Ogawa: Large deviations and Chernoff bound for certain correlated states on the spin chain; arXiv:0706.2141.

[5] F. Hiai, M. Mosonyi, T. Ogawa: Quantum Hoeffding bound for states on a spin chain; arXiv:0707.2020.

[6] H. Nagaoka: The Converse Part of the Theorem for Quantum Hoeffding Bound; quant-ph/0611289.

[7] M. Nussbaum, A. Szko\l a: A lower bound of Chernoff type for symmetric quantum hypothesis testing; quant-ph/0607216.