  Identifying entanglement-based order parameters characterizing topological
systems, in particular topological superconductors and topological insulators,
has remained a major challenge for the physics of quantum matter in the last
two decades. Here we show that the end-to-end, long-distance, bipartite
squashed entanglement between the edges of a many-body system, defined in terms
of the edge-to-edge quantum conditional mutual information, is the natural
nonlocal order parameter for topological superconductors in one dimension as
well as in quasi one-dimensional geometries. For the Kitaev chain in the entire
topological phase, the edge squashed entanglement is quantized to log(2)/2,
half the maximal Bell-state entanglement, and vanishes in the trivial phase.
Such topological squashed entanglement exhibits the correct scaling at the
quantum phase transition, is stable in the presence of interactions, and is
robust against disorder and local perturbations. Edge quantum conditional
mutual information and edge squashed entanglement defined with respect to
different multipartitions discriminate topological superconductors from
symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and
the spin-1/2 Ising model in transverse field. For systems featuring multiple
topological phases with different numbers of edge modes, like the quasi 1D
Kitaev ladder, topological squashed entanglement counts the number of Majorana
excitations and distinguishes the different topological phases of the system.
In fact, we show that the edge quantum conditional mutual information and the
edge squashed entanglement remain valid detectors of topological
superconductivity even for systems, like the Kitaev tie with long-range
hopping, featuring geometrical frustration and a suppressed bulk-edge
correspondence.
