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.
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.
As promising candidates for realizing fault-tolerant quantum computing, systems composed of anyons feature nontensor product state space due to their distinctive fusion rules, resulting in entanglement properties that differ fundamentally from those of conventional quantum systems. However, a quantitative characterization of entanglement in such systems remains elusive. In this Letter, we address this issue in bipartite settings using the framework of quantum resource theory. We propose three measures that quantify total entanglement, conventional entanglement, and anyonic charge entanglement (ACE) of a bipartite anyonic state, respectively. Our central result is summarized as a theorem: Total entanglement is composed of ACE and conventional entanglement. Geometrically, ACE measures the distance between the state and its charge-decorrelated version, while conventional entanglement is the minimal distance between the latter and the set of separable states, characterizing Bell nonlocality in the conventional sense. Our result confirms the long-standing intuition that ACE stems from the reduced dimensionality of the separable state space due to superselection and fusion constraints. Furthermore, we prove that ACE is equivalent to a previously proposed probe termed the entropy of ACE, thereby providing rigorous theoretical support for the latter’s monotonicity as an entanglement measure and extending the known correspondence between the geometric and operational characterizations of bipartite correlations to anyonic systems.
Anyonic systems are modeled by topologically protected Hilbert spaces which obey complex superselection rules restricting possible operations. These Hilbert spaces cannot be decomposed into tensor products of spatially localized subsystems, whereas the tensor product structure is a foundation of the standard entanglement theory. We formulate bipartite entanglement theory for pure anyonic states and analyze its properties as a nonlocal resource for quantum information processing. We introduce a new entanglement measure, asymptotic entanglement entropy (AEE), and show that it characterizes distillable entanglement and entanglement cost similarly to entanglement entropy in conventional systems. AEE depends not only on the Schmidt coefficients but also on the quantum dimensions of the anyons shared by the local subsystems. Moreover, it turns out that AEE coincides with the entanglement gain by anyonic excitations in certain topologically ordered phases.
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.
Non-Abelian anyons, a promising platform for fault-tolerant topological quantum computation, adhere to the charge super-selection rule (cSSR), which imposes restrictions on physically allowed states and operations. However, the ramifications of cSSR and fusion rules in anyonic quantum information theory remain largely unexplored. In this study, we unveil that the information-theoretic characteristics of anyons diverge fundamentally from those of non-anyonic systems such as qudits, bosons, and fermions and display intricate structures. In bipartite anyonic systems, pure states may have different marginal spectra, and mixed states may contain pure marginal states. More striking is that in a pure entangled state, parties may lack equal access to entanglement. This entanglement asymmetry is manifested in quantum teleportation employing an entangled anyonic state shared between Alice and Bob, where Alice can perfectly teleport unknown quantum information to Bob, but Bob lacks this capability. These traits challenge conventional understanding, necessitating new approaches to characterize quantum information and correlations in anyons. We expect that these distinctive features will also be present in non-Abelian lattice gauge field theories. Our findings significantly advance the understanding of the information-theoretic aspects of anyons and may lead to realizations of quantum communication and cryptographic protocols where one party holds sway over the other.
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.
Abstract Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These “planar connectome” states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.