Minimal pole representation and analytic continuation of matrix-valued correlation functions

We present a minimal pole method for analytically continuing matrix-valued imaginary frequency correlation functions to the real axis, enabling precise access to off-diagonal elements and thus improving the interpretation of self-energies and susceptibilities in quantum simulations. Traditional methods for matrix-valued analytic continuation tend to be either noise-sensitive or make ad-hoc positivity assumptions. Our approach avoides these issues via the construction of a compact pole representation with shared poles through exponential fits, expanding upon prior work focused on scalar functions. We test our method across various scenarios, including fermionic and bosonic response functions, with and without noise, and for both continuous and discrete spectra of real materials and model systems. Our findings demonstrate that this technique addresses the shortcomings of existing methodologies, such as artificial broadening and positivity violations. The paper is supplemented with a sample implementation in Python.