  A single qubit driven by an appropriate sequence of control pulses can serve
as a spectrometer of local noise affecting its energy splitting. We show that
by driving and observing two spatially separated qubits, it is possible to
reconstruct the spectrum of cross-correlations of noises acting at various
locations. When the qubits are driven by the same sequence of pulses, real part
of cross-correlation spectrum can be reconstructed, while applying two distinct
sequence to the two qubits allows for reconstruction of imaginary part of this
spectrum. The latter quantity contains information on either causal
correlations between environmental dynamics at distinct locations, or on the
occurrence of propagation of noisy signals through the environment. We
illustrate the former case by modeling the noise spectroscopy protocol for
qubits coupled to correlated two-level systems. While entanglement between the
qubits is not necessary, its presence enhances the signal from which the
spectroscopic information is reconstructed.
