  Quantum coherent superpositions of states with different energies, i.e.,
states with coherence with respect to energy basis, are important resource for
modern quantum technologies. States with small coherence can be obtained either
autonomously, due to the effect of a weak coherent drive or, potentially, due
to the presence of an environment. In this paper, we propose a
measurement-based protocol for quantum coherence synthesis from individual
systems (with low initial coherence) into a global (and higher) coherence of
the joint system. As an input, it uses $N$ non-interacting copies of two-level
systems (TLS), with low initial energy and coherence. These can be supplied by,
e.g., a weak external drive or can result from an interaction with a bath. This
protocol conditionally synthesizes an output state with higher energy and
coherence than the initial state had, representing an universal process whose
rules have not been well studied, yet. In addition to energy and coherence, we
study the quantity called mutual coherence, showing increase after the protocol
application, as well. This approach is based on application of sequential
pairwise projective measurements on TLS pairs (conditionally removing their
ground states), that are diagonal in the TLS energy basis. The functionality of
the coherence synthesis is robust with respect to dephasing effects of the TLS
environment on the system. Our approach may show its benefits in quantum
sensing, quantum batteries charging, or other applications where synthesis of a
larger coherent system from smaller (weaker) resources is useful.
