A special feature of the ground state in a topologically ordered phase is the existence of large-scale correlations depending only on the topology of the regions. These correlations can be detected by the topological entanglement entropy or by a measure called irreducible correlation. We show that these two measures coincide for states obeying an area law and having zero correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols. This establishes an information-theoretical approach to multipartite correlations in topologically ordered systems.
Strongly interacting systems can be described in terms of correlation functions at various orders. A quantum analog of high-order correlations is the topological entanglement in topologically ordered states of matter at zero temperature, usually quantified by topological entanglement entropy (TEE). In this work, we propose a statistical interpretation that unifies the two under the same information-theoretic framework. We demonstrate that the existence of a non-zero TEE can be understood in the statistical view as the emergent
nth order mutual information
In (for arbitrary integer
n≥3) reflected in projectively measured samples, which also makes explicit the equivalence between the two existing methods for its extraction -- the Kitaev-Preskill and the Levin-Wen construction. To exploit the statistical nature of
In, we construct a restricted Boltzmann machine (RBM) which captures the high-order correlations and correspondingly the topological entanglement that are encoded in the distribution of projected samples by representing the entanglement Hamiltonian of a local region under the proper basis. Furthermore, we derive a closed form which presents a method to interrogate the trained RBM, making explicit the analytical form of arbitrary order of correlations relevant for
In. We remark that the interrogation method for extracting high-order correlation can also be applied to the construction of auxiliary fields that disentangle many-body interactions relevant for diverse interacting models.
An anyonic system not only has potential applications in the construction of topological quantum computers but also presents a unique property known as topological entanglement entropy (TEE) in quantum many-body systems. Recently, Bonderson et al. [Ann. Phys. (NY) 385, 399 (2017)] have formally defined the entropy of anyonic charge entanglement (ACE) in anyonic states that has been shown to be able to derive TEE. We give it operational meaning from the perspective of quantum information theory. Specifically, for an anyonic bipartite system, we define an operational measure of topological correlation based on the principle of maximum entropy, where the topological correlation is the information that cannot be accessed by local operations constrained by anyonic superselection rules (SSRs) and classical communication. For a given anyonic bipartite state with maximal rank, we prove that its topological correlation is equal to its entropy of ACE. This measure can be extended to measure nonlocal resources of other compound quantum systems in the presence of SSRs and can provide a more refined classification of correlations in a multipartite system with SSRs.
Topologically ordered systems exhibit large-scale correlation in their ground states, which may be characterized by quantities such as topological entanglement entropy. We propose that the concept of irreducible many-body correlation (IMC), the correlation that cannot be implied by all local correlations, may also be used as a signature of topological order. In a topologically ordered system, we demonstrate that for a part of the system with holes, the reduced density matrix exhibits IMCs which become reducible when the holes are removed. The appearance of these IMCs then represents a key feature of topological phase. We analyze the many-body correlation structures in the ground state of the toric code model in external magnetic fields, and show that the topological phase transition is signaled by the IMCs.
Topological order represents a profound state of matter characterized by exotic properties robust against local perturbations, such as fractionalized excitations and ground state degeneracy dependent on system topology. While extensively studied in closed quantum systems, the detection and characterization of topological order in open quantum systems, which inevitably interact with their environment, remain a significant challenge. Environmental coupling typically introduces decoherence and dissipation, potentially obscuring or even destroying fragile quantum correlations. This paper investigates the concept of "entropic fingerprints" as a robust diagnostic tool for identifying topological order in systems subjected to environmental influences. We explore how various entanglement measures, particularly the topological entanglement entropy and related information-theoretic quantities, can be adapted and reinterpreted in the context of non-equilibrium steady states or dynamically evolving open systems. Through a theoretical framework employing Lindblad master equations and tensor network methods, we demonstrate that universal signatures of topological order, though potentially modified, can persist in the steady-state entanglement properties of open systems. Our findings suggest that specific patterns in the bipartite entanglement entropy, conditional entropies, and mutual information, robust against a finite range of environmental parameters, can serve as reliable indicators of underlying topological phases. This research not only offers a pathway for experimentally probing topological materials in realistic, noisy environments but also deepens our fundamental understanding of how quantum information is preserved or degraded under open system dynamics.
It is well known that the topological entanglement entropy (
Stopo) of a topologically ordered ground state in two spatial dimensions can be captured efficiently by measuring the tripartite quantum information (
I3) of a specific annular arrangement of three subsystems. However, the nature of the general
N-partite information (
IN) and correlation of a topologically ordered ground state remains unknown. In this work, we study such
IN measure and its nontrivial dependence on the arrangement of
N subsystems. For the collection of subsystems (CSS) forming a closed annular structure, the
IN measure ($N\ensuremath{\ge}3$) is a topological invariant equal to the product of
Stopo and the Euler characteristic of the CSS embedded on a planar manifold, $|{I}^{N}|=\ensuremath{\chi}{S}_{\mathrm{topo}}$. Importantly, we establish that
IN is robust against several deformations of the annular CSS, such as the addition of holes within individual subsystems and handles between nearest-neighbor subsystems. While the addition of a handle between further neighbor subsystems causes
IN to vanish, the multipartite information measures of the two smaller annular CSS emergent from this deformation again yield the same topological invariant. For a general CSS with multiple holes (
nh>1), we find that the sum of the distinct, multipartite information measured on the annular CSS around those holes is given by the product of ${S}_{\mathrm{topo}}, \ensuremath{\chi}$ and ${n}_{h}, {\ensuremath{\sum}}_{{\ensuremath{\mu}}_{i}=1}^{{n}_{h}}|{I}_{{\ensuremath{\mu}}_{i}}^{{N}_{{\ensuremath{\mu}}_{i}}}|={n}_{h}\ensuremath{\chi}{S}_{\mathrm{topo}}$. This constrains the concomitant measurement of several multipartite information on any complicated CSS. The
Nth order irreducible correlations for an annular CSS of
N subsystems is also found to be bounded from above by
∣IN∣, which shows the presence of correlations among subsystems arranged in the form of closed loops of all sizes. Thus, our results offer important insight into the nature of the many-particle entanglement and correlations within a topologically ordered state of matter.