  We derived the equation of the density operator for generalized entropy and
generalized expectation value with quantum analysis when conserved quantities
exist. The derived equation is simplified when the conventional expectation
value is employed. The derived equation is also simplified when the commutation
relations, $[\hat{\rho}, \hat{H}]$ and $[\hat{\rho}, \hat{Q}^{[a]}]$, are the
functions of the density operator $\hat{\rho}$, where $\hat{H}$ is the
Hamiltonian, and $\hat{Q}^{[a]}$ is the conserved quantity. We derived the
density operators for the von Neumann entropy, the Tsallis entropy, and the
R\'enyi entropy in the case of the conventional expectation value. We also
derived the density operators for the Tsallis entropy and the R\'enyi entropy
in the case of the escort average (the normalized $q$-expectation value), when
the density operator commutes with the Hamiltonian and the conserved
quantities. We found that the argument of the density operator for the
canonical ensemble is simply extended to the argument for the generalized Gibbs
ensemble in the case of the conventional expectation value, even when conserved
quantities do not commute. The simple extension of the argument is also shown
in the case of the escort average, when the density operator $\hat{\rho}$
commutes with the Hamiltonian $\hat{H}$ and the conserved quantity
$\hat{Q}^{[a]}$: $[\hat{\rho}, \hat{H}] = [\hat{\rho}, \hat{Q}^{[a]}]=0$. These
findings imply that the argument of the density operator for the canonical
ensemble is simply extended to the argument for the generalized Gibbs ensemble
in some systems.
