Adaptive optimal measurement for the squeezed vacuum and coherent state

作者：Chen Kun; Chen Shu-Xin^*; Wu De-Wei; Yang Chun-Yan; Wang Xi; Li Xiang; Wu Hao; Liu Zhuo-Wei

来源：Acta Physica Sinica, 2016, 65(19): 194203.

DOI：10.7498/aps.65.194203

• quantum balanced homodyne technique
• positive operator-valued measure
• quantum Cramer-Rao lower bound
• maximum likelihood estimator

The output phase of the Sagnac interferometer has been measured with quantum balanced homodyne technique when coherent light and squeezed vacuum light are fed into the Sagnac interferometer simultaneously [Chen Kun et al., Acta Phys. Sin. 65 054203(2016)]. Nevertheless, there exist two deficiencies: 1) the phase sensitivity is related to the phase itself; 2) there are strict requirements for the phases of local oscillator light, coherent light and squeezed vacuum light.
For overcoming these deficiencies, an adaptive optimal measurement scheme is suggested for the phase estimation. Firstly, we calculate that the quantum Fisher information (QFI) of the squeezed vacuum and coherent state is sinh(2) r + vertical bar alpha vertical bar(2)e(2r) by treating them as a quantum pure state, for they satisfy a condition of the quantum pure state, namely rho(theta) = rho(theta)(2) The QFI is related to quantum Cramer-Rao lower bound which can be used to evaluate the performance of the estimator. Secondly, we make an analysis of positive operator-valued measure (POVM) and design a set of the optimal measurement operators for reaching the quantum Cramer-Rao lower bound, whereas the optimal measurement operators depend on the true value of the phase which is what we want to estimate. In order to solve the problem and estimate the parameter effectively, an adaptive method is suggested. We set an initial value of the phase parameter to obtain a set of measurement operators which are not optimal at the first step. And then the initial measurement operators are used for POVM and to obtain a conditional probability function, from which we can obtain a new value of the phase with maximum likelihood estimator. Therefore, the measurement operators and conditional probability function will be updated with the new value. As the measurement operators and conditional probability function are updated step by step, we can estimate the value adaptively. In fact, the results of the maximum likelihood estimator will converge at the true value of the phase parameter gradually, which is then proved with the theoretical analysis. All in all, an adaptive measurement method of estimating the phase parameter of the squeezed vacuum and coherent state in Sagnac interferometer is suggested, and is proved theoretically to be that the scheme will converge at the true value of the phase with a probability of 1 and can reach the quantum Cramer-Rao lower bound.

• 出版日期2016-10-5
• 单位中国人民解放军空军工程大学