  In this paper, we extend the standard formalism of quantum mechanics to a
quantum theory for a total system including one internal measuring apparatus.
The internality of the measuring apparatus implies that different decomposition
of a given density operator for the internal measuring apparatus into mixture
of pure states may have different physical implications. We use `specified
mixed-state description' to call a density operator with a specified
decomposition into mixture of pure states. The proposed theory has three basic
assumptions, which roughly speaking have the following contents: (i) Physical
states of the total system can be associated with vectors in the total Hilbert
space; (ii) the dynamical evolution of a state vector obeys Schr\"{o}dinger
equation; and (iii) under a principle of compatible description and certain
non-transition condition, a pure-vector description of the total system may
imply the existence of certain specified mixed-state description. The principle
of compatible description states that different mathematical descriptions for
the same physical state of the total system must give consistent predictions
for results of measurements performed by the internal measuring apparatus. This
principle imposes a restriction to vectors in the Hilbert space and this may
effectively break the time-reversal symmetry of Schr\"{o}dinger equation.
