Quantum Theory of Percolation.

Richard P. Wool

Gesualdo Delfino A

Liang Cao, M. Jeng, Jennifer M. Schwarz

Haitao Wang, Omar Malik, Jinchuan Hou +3 more

Understanding entanglement percolation across quantum networks (QNs) has been a persistent challenge. Concurrence percolation theory (ConPT) provides a statistical-physics framework for this, drawing parallels between entanglement measure (concurrence) and classical percolation measure (probability). However, ConPT’s reach has been confined to idealized, pure-state entanglement resources, overlooking realistic mixed states “contaminated” by stochastic noise. This work extends into this realistic domain, introducing a mixed-state ConPT that reveals a percolation threshold lower than the pure-state counterpart. This paradoxical finding seemingly suggests that stochastic contamination is less detrimental than purity-preserving unitary noise, a notion that challenges conventional belief. We resolve this finding by highlighting an operational detail: implementing mixed-state ConPT effectively requires quantum memory to buffer the probabilistic nature of mixed-state protocols. We find evidence that without quantum memory, mixed-state ConPT consistently underperforms pure-state ConPT. This reinforces our understanding that stochastic noise poses a more significant impediment, a critical consideration for QN design. Understanding how entanglement spreads across realistic quantum networks, which are inevitably affected by environmental noise, is a key challenge. The authors show that a noisy network only appears to outperform an ideal one if it is supported by quantum memory, confirming that noise remains a critical barrier.

Konrad Kieling, Jens Eisert

Xiangyi Meng, Jianxi Gao, Shlomo Havlin

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 and 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 percolationlike 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.