Abstract
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 (cMPS), a complete set of variational states grasping states in one-dimensional quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used 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 randomized cMPS 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 cMPS 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 those 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.
Highlights
In this work, we are concerned with one-dimensional quantum fields with fast decaying spatial correlations
We have introduced the concept of quantum field tomography
In spite of the inherent difficulties of attempting to reconstruct a continuous system, i.e., a system with infinite degrees of freedom, we have shown that this task can be done when only a relevant class of naturally occurring states is considered
Summary
Quantum theory predicts probability distributions of outcomes in anticipated quantum measurements. A moment of thought reveals that to think about quantum field tomography in the sense of trying to “fill an infinite table with numbers” is rather ill-guided This is not the actual problem one aims at solving in any practical context—one again needs to identify the appropriate model and the right “sparsity structure”. The basis of the analysis are low-order multi-point correlation functions directly accessible in many common current experiments This approach opens up a new window into grasping the physics of continuous quantum systems in equilibrium and non-equilibrium. III, we will describe in great technical detail how to reconstruct a field state from its low order correlation functions and give a complete matrix product state description of it
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