Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors

Abstract

Identifying entanglement-based order parameters characterizing topological systems has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that squashed entanglement between the system edges, defined in terms of the edge-edge quantum conditional mutual information, is the natural order parameter for topological superconductors and for systems with edge modes. In the topological phase, the long-distance edge-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, discriminates between topological order and ordered phases associated to broken symmetries, counts the number of Majorana excitations for systems of increasing geometrical complexity, and is robust against the effects of disorder and local perturbations. In fact, it turns out to be a valid signature of topological order even for systems with weakened bulk-edge correspondence. Squashed entanglement is defined for all quantum states, reducing to the von Neumann entanglement entropy on pure states. As such, it provides a unified framework for the characterization of topological order in any dimension and, moreover, can be generalized to any setting, including finite temperature, multipartite correlations, and nonequilibrium. Squashed entanglement and quantum conditional mutual information are defined in terms of linear combinations of reduced entropies and can thus be probed using currently available experimental setups and techniques.

Edge states, Majorana fermions and topological order in superconducting wires with generalized boundary conditions

“…Indeed, such topological nonlocal edge-toedge correlations are faithfully quantified by a specific measure of bipartite entanglement [84,85], the squashed entanglement (SE) E 0 SQ between the edges, obtained by a suitable quadripartition of the system, and by its natural upper bound, the edge-to-edge quantum conditional mutual information (QCMI) I (4) , as recently shown in Refs. [67,68]. The edge-edge QCMI provides an upper bound on the long-distance, edge-edge squashed entanglement obtained by four-partitioning the system into the two edges and a bipartition of the bulk.…”

“…The QCMI upper bound on the SE is then obtained by a suitable combination of reduced von Neumann entropies stemming from the quadripartition; this combination squashes out the classical contributions leaving only the genuinely quantum ones. Being the topological order encoded in the edges, the edge-edge SE E 0 SQ identifies unequivocally the topologically ordered phases, also fulfilling all the criteria of a genuine nonlocal order parameter [67,68]. In particular, the edge-edge SE assumes the quantized value E 0 SQ = log 2/2, i.e.…”

“…We derive a general four-parameters classification of the boundary effects and demonstrate that particle-hole and reflection symmetries can either be broken or be preserved depending on the values taken by the boundary parameters. We investigate systematically the robustness of the edge modes under the influence of the different boundary conditions by using several estimators, including the long-distance edge-to-edge entanglement measuring the nonlocal correlation of the Majorana excitations [67,68].…”

Topological Phases of an Interacting Majorana Benalcazar–Bernevig–Hughes Model

“…We will thus need to consider entanglement measures able to quantify the nonlocal correlations between the edges and, in particular, the long-distance topological entanglement between corner Majorana modes. In fact, such a measure, the topological squashed entanglement, exists and was recently introduced and successfully applied [44] to the unambiguous characterization of topological order in some basic models of topological superconductivity, including the Kitaev chain, the two-leg Kitaev ladder [45], and the Kitaev tie [46]. We plan to report in the near future the results of a similar investigation along the same lines for the interacting Majorana BBH model.…”

Topological phases of an interacting Majorana Benalcazar-Bernevig-Hughes model

“…We will thus need to consider entanglement measures able to quantify the nonlocal correlations between the edges and, in particular, the long-distance topological entanglement between corner Majorana modes. In fact, such a measure, the topological squashed entanglement, exists and has been recently introduced and successfully applied [43] to the unambiguous characterization of topological order in some basic models of topological superconductivity, including the Kitaev chain, the two-leg Kitaev ladder [44], and the Kitaev tie [45]. We plan to report in the near future the results of a similar investigation along the same lines for the interacting Majorana BBH model.…”