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arXiv:2012.05737 · Physical review. D/Physical review. D.

Bit thread, entanglement distillation, and entanglement of purification

Yi-Yu Lin, Jia-Rui Sun, Yuan Sun

Published Jun 1, 2021• 23 citations
arXivHTMLPDF

Abstract

We investigate the relations between bit thread, entanglement distillation, and entanglement of purification in the holographic framework. Specifically, we give a bit thread interpretation for the one-shot entanglement distillation (OSED) tensor network, which can be understood as reconstructing the geometric structure of the bulk spacetime from the entanglement information of the boundary quantum system through the ``surface growth scheme.'' Moreover, by showing that the holographic entanglement of purification (EoP) process can be regarded as a special case of our ``surface growth scheme,'' we naturally obtain the bit thread interpretation of the holographic EoP in our framework, which turns out to be different from the existing interpretations. Since our interpretation for holographic EoP is obtained in a more general and physical framework, the advantage of our version is that it is more natural, and possibly more reasonable, to reflect the real physical entanglement structures.

DOI
10.1103/physrevd.103.126002

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We investigate the relations between bit thread, entanglement distillation, and entanglement of purification in the holographic framework. Specifically, we give a bit thread interpretation for the one-shot entanglement distillation (OSED) tensor network, which can be understood as reconstructing the geometric structure of the bulk spacetime from the entanglement information of the boundary quantum system through the surface growth scheme.'' Moreover, by showing that the holographic entanglement of purification (EoP) process can be regarded as a special case of our surface growth scheme,'' we naturally obtain the bit thread interpretation of the holographic EoP in our framework, which turns out to be different from the existing interpretations. Since our interpretation for holographic EoP is obtained in a more general and physical framework, the advantage of our version is that it is more natural, and possibly more reasonable, to reflect the real physical entanglement structures.
Abstract The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a “bit thread” formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.
Bit threads, arising as the convex dual of the minimal-surface formula for holographic entanglement entropy, are line-like structures with nontrivial bulk trajectories. Their physical picture has long been associated with Bell-pair interpretations, yet the meaning of their detailed trajectories, particularly their nonuniqueness, remains unclear. We propose to apply Coecke's notion of quantum information flow (QIF) in protocol-network geometry, developed within categorical quantum mechanics, to a holographic setup, and show that its trajectories within the discrete holographic bulk---the tensor network---obey the divergence-free and density-bound conditions of bit threads. From this process-centered perspective, bit-thread nonuniqueness becomes natural: for a given resource state, different protocols realizing the same distillation task can give rise to different QIF trajectories. We further analyze stabilizer-type holographic tensor networks using ZX string diagrams and show that QIF trajectories can probe finer entanglement structure beyond the level captured by entanglement entropy.
A bstract We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.
Abstract In this work we study how entanglement of purification (EoP) and the new quantity of “complexity of purification” are related to each other using theEP=EWconjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.
Abstract Starting from an interesting coincidence between the bit threads and SS (surface/state) correspondence, both of which are closely related to the holographic RT formula, we introduce a property of bit threads that has not been explicitly proposed before, which can be referred to as thread/state correspondence (see [50] for a brief pre-release version). Using this thread/state correspondence, we can construct the explicit expressions for the SS states corresponding to a set of bulk extremal surfaces in the SS correspondence, and nicely characterize their entanglement structure. Based on this understanding, we use the locking bit thread configurations to construct a holographic qubit threads model as a new toy model of the holographic principle, and show that it is closely related to the holographic tensor networks, the kinematic space, and the connectivity of spacetime.
Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.