Wick's theorem for matrix product states

TL;DR

This paper extends Wick's theorem to matrix product states, showing that N-point functions can be reconstructed from two- and three-point functions, which simplifies the analysis of correlations in low-entanglement quantum systems.

Contribution

It demonstrates that N-point functions in matrix product states are fully determined by lower-order correlations, providing a new framework for state reconstruction and analysis.

Findings

N-point functions are characterized by two- and three-point functions.

Reconstruction of unknown states from correlation measurements is possible.

Potential applications in perturbative approaches to interacting quantum theories.

Abstract

Matrix-product states and their continuous analogues are variational classes of states that capture quantum many-body systems or quantum fields with low entanglement; they are at the basis of the density-matrix renormalization group method and continuous variants thereof. In this work we show that, generically, N-point functions of arbitrary operators in discrete and continuous translationally invariant matrix product states are completely characterized by the corresponding two- and three-point functions. Aside from having important consequences for the structure of correlations in quantum states with low entanglement, this result provides a new way of reconstructing unknown states from correlation measurements, e.g., for one-dimensional continuous systems of cold atoms. We argue that such a relation of correlation functions may help in devising perturbative approaches to interacting theories.