Quantum field tomography and continuous matrix-product states

A. Steffens, M. Friesdorf, C. Riofrío, R. Hübener, and J. Eisert

We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of 
unknown quantum fields based on data of correlation functions. At the basis of the analysis is the 
concept of continuous matrix product states, a complete set of variational states grasping states 
in quantum field theory. We innovate a practical method, making use of and developing tools in 
estimation theory applied in the context of compressed sensing such as Prony methods and matrix 
pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation 
functions. In the absence of a phase reference, we highlight how specific higher order correlation 
functions can still be predicted. We exemplify the functioning of the approach by reconstructing 
randomised continuous matrix product states from their correlation data and study the robustness of 
the reconstruction for different noise models. Furthermore, we apply the method to data generated 
by simulations based on continuous matrix product states and using the time-dependent variational 
principle. The presented approach is expected to open up a new window into experimentally studying 
continuous quantum systems, such as encountered in experiments with ultra-cold atoms on top of atom 
chips. By virtue of the analogy with the input-output formalism in quantum optics, it also allows 
for studying open quantum systems.