Although quantum metrology allows us to make precision measurements beyond the standard quantum limit, it mostly works on the measurement of only one observable due to the Heisenberg uncertainty relation on the measurement precision of noncommuting observables for one system. In this paper, we study the schemes of joint measurement of multiple observables which do not commute with each other using the quantum entanglement between two systems. We focus on analyzing the performance of a SU(1,1) nonlinear interferometer on fulfilling the task of joint measurement. The results show that the information encoded in multiple noncommuting observables on an optical field can be simultaneously measured with a signal-to-noise ratio higher than the standard quantum limit, and the ultimate limit of each observable is still the Heisenberg limit. Moreover, we find a resource conservation rule for the joint measurement.

Photon pairs produced by the pulse-pumped nonlinear parametric processes have been a workhorse of quantum information science (QIS). Engineering the spectral property of quantum states is crucial for practical QIS applications. However, photon pairs with desirable spectral properties are currently achieved with specially engineered optical hardware but with severely limited flexibility in tuning the spectral properties of the sources. Here, we demonstrate a spectrally programmable photon pair source by exploiting a two-stage nonlinear interferometer scheme with a computer-controlled phase device. The phase-control device can introduce phase shifts for spectral engineering by a programmable phase function that can be arbitrarily defined. When the phase function is properly designed, the output spectrum of the source can be freely customized and changed without replacing any hardware component in the system. Using this approach, we are able to program photon pairs with factorable positively correlated and negatively correlated spectra. In addition, we also realize a source of multi-dimensional three-channel spectrally factorable photon pairs. Our investigation provides a flexible and powerful new approach for engineering the mode profile of photon pairs and should find wide applications in QIS.