  Quantum embedding is an appealing route to fragment a large interacting
quantum system into several smaller auxiliary `cluster' problems to exploit the
locality of the correlated physics. In this work we critically review
approaches to recombine these fragmented solutions in order to compute
non-local expectation values, including the total energy. Starting from the
democratic partitioning of expectation values used in density matrix embedding
theory, we motivate and develop a number of alternative approaches, numerically
demonstrating their efficiency and improved accuracy as a function of
increasing cluster size for both energetics and non-local two-body observables
in molecular and solid state systems. These approaches consider the
$N$-representability of the resulting expectation values via an implicit global
wave~function across the clusters, as well as the importance of including
contributions to expectation values spanning multiple fragments simultaneously,
thereby alleviating the fundamental locality approximation of the embedding. We
clearly demonstrate the value of these introduced functionals for reliable
extraction of observables and robust and systematic convergence as the cluster
size increases, allowing for significantly smaller clusters to be used for a
desired accuracy compared to traditional approaches in ab initio wave~function
quantum embedding.
