  Using higher-order derivative with respect to the parameter, we will give
lower bounds for variance of unbiased estimators in quantum estimation
problems. This is a quantum version of the Bhattacharyya inequality in the
classical statistical estimation. Because of non-commutativity of operator
multiplication, we obtain three different types of lower bounds; Type S, Type R
and Type L. If the parameter is a real number, the Type S bound is useful. If
the parameter is complex, the Type R and L bounds are useful. As an
application, we will consider estimation of polynomials of the complex
amplitude of the quantum Gaussian state. For the case where the amplitude lies
in the real axis, a uniformly optimum estimator for the square of the amplitude
will be derived using the Type S bound. It will be shown that there is no
unbiased estimator uniformly optimum as a polynomial of annihilation and/or
creation operators for the cube of the amplitude. For the case where the
amplitude does not necessarily lie in the real axis, uniformly optimum
estimators for holomorphic, antiholomorphic and real-valued polynomials of the
amplitude will be derived. Those estimators for the holomorphic and real-valued
cases attains the Type R bound, and those for the antiholomorphic and
real-valued cases attains the Type L bound. This article clarifies what is the
best method to measure energy of laser.
