Zero-error quantum hypothesis testing in finite time with quantum error correction

Under the unitary evolutions it is always possible to distinguish different Hamiltonians with zero-error probability within a finite time. At the presence of noises, however, the error probability does not always go to zero within a finite time. In this article, we give a necessary and sufficient condition for Markovian dynamics to achieve zero-error quantum hypothesis testing in a finite time. We show that when the condition fails, the maximal fidelity of the output states under two different Hamiltonians, aided with arbitrary control operations, is always lower bounded by an exponential function, which remains positive at any finite time; zero-error quantum hypothesis testing thus cannot be achieved within any finite time. However, when the condition holds, quantum error corrections can be used to correct the noises and partially maintain the coherent evolutions, zero-error quantum hypothesis testing can then be achieved within a finite time.