Spectroscopy with entanglement has shown great potential to break limitations of traditional spectroscopic measurements, yet the role of entanglement in spectroscopic multiparameter joint measurement, particularly in the infrared optical range, remains elusive. Here, we find an uncertain relation that constrains the precision of infrared spectroscopic multiparameter measurements using entangled photon pairs. Under such a relation, we demonstrate a trade-off between the measurement precisions of the refractive index and the absorption coefficient of the medium in the infrared range, and also illustrate how to balance their respective estimation errors. Our work shall provide guidance toward future experimental designs and applications in entanglement-assisted spectroscopy.

• Laser spectroscopy
• Light-matter interaction
• Photon pairs & parametric down-conversion
• Quantum description of light-matter interaction
• Quantum states of light
• Infrared techniques