No. 990

Title:

Bhattacharyya inequality for quantum state estimation

Author(s):

TSUDA, Yoshiyuki (Institute of Statistical Mathematics)

Key words:

quantum Cramer-Rao inequality; quantum Gaussian state; uniformly minimum variance unbiased estimator;

Abstract:

    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. The Type S bound is used if the parameter is a real number, and so do the others if it is a complex number. 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 Minimum Variance Unbiased Estimator (UMVUE) for the square of the amplitude will be derived using the Type S bound. It will be shown that there is no UMVUE as a polynomial of creation/annihilation operators for the cube of the amplitude. For the case where the amplitude does not necessarily lie in the real axis, UMVUEs for holomorphic, antiholomorphic and real-valued polynomials of the amplitude will be derived. The UMVUEs 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.