  Establishing long-distance quantum entanglement, i.e., entanglement
transmission, in quantum networks (QN) is a key and timely challenge for
developing efficient quantum communication. Traditional comprehension based on
classical percolation assumes a necessary condition for successful entanglement
transmission between any two infinitely distant nodes: they must be connected
by at least a path of perfectly entangled states (singlets). Here, we relax
this condition by explicitly showing that one can focus not on optimally
converting singlets but on establishing concurrence -- a key measure of
bipartite entanglement. We thereby introduce a new statistical theory,
concurrence percolation theory (ConPT), remotely analogous to classical
percolation but fundamentally different, built by generalizing bond percolation
in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance
network analysis, we determine the path connectivity by series/parallel rules
and approximate higher-order rules via star-mesh transforms. Interestingly, we
find that the entanglement transmission threshold predicted by ConPT is lower
than the known classical-percolation-based results and is readily achievable on
any series-parallel networks such as the Bethe lattice. ConPT promotes our
understanding of how well quantum communication can be further systematically
improved versus classical statistical predictions under the limitation of QN
locality -- a "quantum advantage" that is more general and efficient than
expected. ConPT also shows a percolation-like universal critical behavior
derived by finite-size analysis on the Bethe lattice and regular
two-dimensional lattices, offering new perspectives for a theory of criticality
in entanglement statistics.
