  We show through Thermofield Dynamics approach that the action of the
thermalized quantum logic gate on the thermalized state is equivalent to
thermalization of the state that arise from the application of the
non-thermalized quantum logic gate. In particular, we study the effect of
temperature on a mixed state associated to a system capable of implementing a
CNOT quantum logic gate. According to a proposal in the literature, a way of
implementing such a logic gate is by using a representation of the qubit states
as elements of the Fock space of a bosonic system. We consider such a proposal
and use the Thermofield Dynamics to determine the thermalized qubit states. The
temperature acts as a quantum noise on pure states, making them a statistical
mixture. In this context we analyze the fidelity as a function of the
temperature and using the Mandel parameter, we determine temperature ranges for
which the statistics of the system becomes subpoissonian, poissonian and
superpoissonian. Finally, we calculate the Wigner function, allowing an
analysis of the thermal state in phase space, and we obtain that the increase
of temperature decreases non-classical properties of the system. The
temperature range where one has a subpoissonian statistics and high fidelity is
determined.
