  Distributed quantum metrology can enhance the sensitivity for sensing
spatially distributed parameters beyond the classical limits. Here we
demonstrate distributed quantum phase estimation with discrete variables to
achieve Heisenberg limit phase measurements. Based on parallel entanglement in
modes and particles, we demonstrate distributed quantum sensing for both
individual phase shifts and an averaged phase shift, with an error reduction up
to 1.4 dB and 2.7 dB below the shot-noise limit. Furthermore, we demonstrate a
combined strategy with parallel mode entanglement and multiple passes of the
phase shifter in each mode. In particular, our experiment uses six entangled
photons with each photon passing the phase shifter up to six times, and
achieves a total number of photon passes N=21 at an error reduction up to 4.7
dB below the shot-noise limit. Our research provides a faithful verification of
the benefit of entanglement and coherence for distributed quantum sensing in
general quantum networks.
