Topological entanglement entropy, which is defined as the conditional mutual information of a state for an annular region in a topologically ordered system, is widely used to detect characteristic topological correlations in topologically ordered phases. We investigate an information-theoretical meaning of the topological entanglement entropy by analyzing a multi-partite correlation measure called irreducible correlation. We prove that the topological entanglement entropy is equivalent to the irreducible correlation if the state has zero-correlation length. We further show that for zero-correlation length states the irreducible correlation is equal to the maximum communication rate of a particular class of secret sharing protocols. Therefore, we provide an operational interpretation of the topological entanglement entropy.
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.
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.
A characterization of topological order in terms of bi-partite entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid. 96, 110405 (2006)]. It was argued that in a topological phase there is a universal additive constant in the entanglement entropy, called the topological entanglement entropy, which reflects the underlying gauge theory for the topological order. In the present paper, we evaluate numerically the topological entanglement entropy in the ground states of a quantum dimer model on the triangular lattice, which is known to have a dimer liquid phase with
Z2 topological order. We examine the two original constructions to measure the topological entropy by combining entropies on plural areas, and we observe that in the large-area limit they both approach the value expected for
Z2 topological order. We also consider the entanglement entropy on a topologically nontrivial ``zigzag'' area and propose to use it as another way to measure the topological entropy.
We consider the problem of identifying a topological order based on bulk entanglement of the ground-state wavefunction. Previous work showed that some universal information can be extracted from multi-entropy measures, a class of multipartite entanglement measures obtained by applying permutation operators exchanging the degrees of freedom between different replicas of the wavefunction. It remains an open question to what extent such entanglement measures can be used to extract any universal information from the ground state. Here we show that the topological partition function
Z(M) of a manifold
M can be extracted provided that
M satisfies a topological condition which we term ``local achirality". We show that locally-achiral manifolds can be used to extract universal properties of 2+1d topological phases that go beyond the
S and
T matrices. As a first step towards classifying locally-achiral manifolds, we show that, in four dimensions, such manifolds have vanishing Pontryagin number. We relate this property to the existence of beyond-cohomology time-reversal symmetry protected topological order (T-SPT) in four dimensions. Finally, we present an entanglement measure that detects this nontrivial T-SPT.
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.