Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing

Abstract

We derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing. Using an information-spectrum method, we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0. As an asymptotically optimal measurement, we propose a projection measurement characterized by the irreducible representation theory of the special linear group SL(H). Specially, in spin 1/2 system, it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction. As a byproduct, we obtain another proof of quantum Stein's lemma.