Abstract Topological quantum field theories (TQFTs) provide a general, minimal‐assumption language for describing quantum‐state preparation and measurement. They therefore provide a general language in which to express multi‐agent communication protocols, e.g., local operations, classical communication (LOCC) protocols. Here, LOCC protocols are constructed using TQFT and it is shown that LOCC protocols generically induce quantum error‐correcting codes (QECCs). Using multi‐observer scenarios described by quantum Darwinism and Bell/EPR experiments as examples, it is shown how these LOCC‐induced QECCs effectively convert entanglement into classical redundancy. In the accompanying Part II, it is shown that such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on the boundaries between interacting systems. The connection between inter‐agent communication and spacetime using BF and Chern‐Simons theories, and then using topological M‐theory is investigated.
Abstract Topological quantum field theories (TQFTs) provide a general, minimal‐assumption language for describing quantum‐state preparation and measurement. They therefore provide a general language in which to express multi‐agent communication protocols, e.g., local operations, classical communication (LOCC) protocols. Here, LOCC protocols are constructed using TQFT and it is shown that LOCC protocols generically induce quantum error‐correcting codes (QECCs). Using multi‐observer scenarios described by quantum Darwinism and Bell/EPR experiments as examples, it is shown how these LOCC‐induced QECCs effectively convert entanglement into classical redundancy. In the accompanying Part II, it is shown that such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on the boundaries between interacting systems. The connection between inter‐agent communication and spacetime using BF and Chern‐Simons theories, and then using topological M‐theory is investigated.
Abstract Topological quantum field theories (TQFTs) provide a general, minimal‐assumption language for describing quantum‐state preparation and measurement. They therefore provide a general language in which to express multi‐agent communication protocols, e.g., local operations, classical communication (LOCC) protocols. In the accompanying Part I, we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error‐correcting codes (QECCs) on the agent‐environment boundary. Such QECCs can be regarded as implementing or inducing the emergence of spacetimes on such boundaries. Here connection between inter‐agent communication and spacetime is investigated, by exploiting different realizations of TQFT. The authors delved into TQFTs that support on their boundaries spin‐networks as computational systems: these are known as topological quantum neural networks (TQNNs). TQNNs, which have a natural representation as tensor networks, implement QECC. The HaPPY code is recognized to be a paradigmatic example. How generic QECCs, as bulk‐boundary codes, induce effective spacetimes is then shown. The effective spatial and temporal separations that take place in QECC enables LOCC protocols between spatially separated observers. The implementation of QECCs in BF and Chern‐Simons theories are then considered, and QECC‐induced spacetimes are shown to provide the classical redundancy required for LOCC. Finally, the topological M‐theory is considered as an implementation of QECC in higher spacetime dimensions.
Topological quantum field theories (TQFT) encode quantum correlations in topological features of spaces. In this work, we leverage this feature to explore how information encoded in TQFTs can be stored and retrieved in the presence of local decoherence affecting its physical carriers. TQFT states' inherent nonlocality, redundancy, and entanglement position them as natural error-correcting codes. We demonstrate that information recovery protocols can be derived from the principle that protected information must be uniformly distributed across the system and from interpreting correlations in terms of space connectivity. Specifically, we employ a topological framework to devise erasure error correction protocols, showing that information can be successfully recovered even when parts of the system are corrupted.
Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the W entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.
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.
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.
We explore the intersection of studies in quantum error correction and quantum local operations and classical communication (LOCC). We consider one-way LOCC measurement protocols as quantum channels and investigate their error correction properties, emphasizing an operator theory approach to the subject, and we obtain new applications to one-way LOCC state distinguishability as well as new derivations of some established results. We also derive conditions on when states that arise through the stabilizer formalism for quantum error correction are distinguishable under one-way LOCC.