  Correlations of fluctuations are the driving forces behind the dynamics and
thermodynamics in quantum many-body systems. For qubits embedded in a quantum
bath, the correlations in the bath are the key to understanding and combating
decoherence -a critical issue in quantum information technology. However, there
is no systematic method for characterizing the many-body correlations in
quantum baths beyond the second order or the Gaussian approximation. Here we
present a scheme to characterize the correlations in a quantum bath to
arbitrary order. The scheme employs weak measurement of the bath via projective
measurement of a central system. The bath correlations, including both the
"classical" and the "quantum" parts, can be reconstructed from the correlations
of the measurement outputs. The possibility of full characterization of
many-body correlations in a quantum bath forms the basis for optimizing quantum
control against decoherence in realistic environments, for studying the quantum
characteristics of baths, and for quantum sensing of correlated clusters in
quantum baths.
