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The system-bath entanglement theorem (SBET) was established in terms oflinear response functions [J. Chem. Phys. 152, 034102 (2020)] and generalizedto correlation functions [arXiv: 2312.13618 (2023)] in our previous works. Thistheorem connects the entangled system-bath properties to the local system andbare bath ones. In this work, firstly we extend the SBET to field-dressedconditions with multiple bosonic Gaussian environments at differenttemperatures. Not only the system but also environments are considered to be ofoptical polarizability, as in reality. With the aid of the extended SBETdeveloped here, for the evaluation of the nonlinear spectroscopy such as thepump-probe, the entangled system-bath contributions can be obtained uponreduced system evolutions via certain quantum dissipative methods. The extendedSBET in the field-free condition and its counterpart in the classical limit isalso presented. The SBET for fermionic environments is elaborated within thetransport scenarios for completeness.

Extended system-bath entanglement theorem for multiple bosonic or fermionic environments