ThreadState correspondence from bit threads to qubit threads Yi-Yu Lin12and Jie-Chen Jin3y 1Beijing Institute of Mathematical Sciences and Applications BIMSA Beijing 101408 China

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Thread/State correspondence: from bit threads to qubit threads

Yi-Yu Lin1,2∗and Jie-Chen Jin3†

1Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, 101408, China

2Yau Mathematical Sciences Center (YMSC),

Tsinghua University, Beijing, 100084, China and

3School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275, China

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.

∗Electronic address: yiyu@bimsa.cn

†Electronic address: jinjch5@mail2.sysu.edu.cn

arXiv:2210.08783v1 [hep-th] 17 Oct 2022

Contents

I. Introduction 2

II. Background review 5

A. Bit threads and locking bit thread conﬁgurations 5

B. Surface/state correspondence 9

III. Motivation and thread/state prescription 10

A. Motivation: constructing the SS states of bulk extremal surfaces by “SS bit model” 10

B. Prescription: thread/state correspondence 15

IV. holographic qubit threads model 21

A. Reading SS states from holographic qubit threads model 21

B. Constructing holographic qubit threads model from locking bit thread conﬁgurations 25

V. Relations with other holographic models 31

A. Relation with holographic tensor network models: qubit threads make disentangling

easy 31

B. Relation with kinematic space: qubit threads as CMI threads 35

VI. Bit threads and the connectivity of spacetime 39

VII. Conclusions and discussions 43

Acknowledgement 44

A. proof based on strong duality of convex program 44

References 47

I. INTRODUCTION

Advances in recent decades suggest that we may have found an important and marvelous clue

or “belief” to the mystery between gravity and quantum mechanics, namely “It from Qubit” [1–

4]. In this view, spacetime is not a fundamental object, but rather emerges from a structure of

quantum entanglement. The clue is successively related to the concepts of the black hole area

entropy [5–8], the holographic principle (especially AdS/CFT duality) [9–11], and the RT formula

of the holographic entanglement entropy [12–14], which have built a bridge between quantum

mechanics and general relativity. In particular, the RT formula shows that the entanglement

entropy that characterizes quantum entanglement between diﬀerent parts of a particular class of

(i.e., “holographic”) quantum systems can be equivalently (i.e., “dually”) described by the area of

an extremal surface in a corresponding curved spacetime [12–14].

More recently, many enlightening ideas from other ﬁelds, such as condensed matter physics,

quantum information theory, network ﬂow optimization theory, etc., have entered and beneﬁted

the study of the holographic gravity. One of the most striking examples is that inspired by the ten-

sor network method originally used in condensed matter physics as a numerical simulation tool to

investigate the wave functions of quantum many-body systems, various holographic tensor network

(TN) models have been constructed as toy models of the holographic duality, such as MERA (mul-

tiscale entanglement renormalization ansatz) tensor network [15–18], perfect tensor network [19],

random tensor network [20], p-adic tensor network [21–23], OSED (one-shot entanglement distil-

lation) tensor network [24–26] and so on. For more research on tensor networks in the holographic

context, see e.g. [27–43]. Furthermore, inspired by the continuous version of the holographic tensor

network models, especially the construction of cMERA (continuous MERA) tensor network [44],

[45, 46] proposed the so-called SS correspondence (surface/state correspondence) as a more speciﬁc

mechanism of the holographic principle. The SS duality refers to the duality between a codimension

two convex surface Σ in the holographic bulk spacetime and a quantum state described by a density

matrix ρ(Σ), which is deﬁned on the Hilbert space of the quantum theory dual to the Einstein’s

gravity. This can be understood intuitively in the context of tensor networks. For a convex closed

surface Σ, we can always contract the indices of tensors contained in the region enveloped by Σ to

obtain a state ρ(Σ).

Another idea to further explore the profound connection between spacetime geometry and quan-

tum entanglement is inspired by the optimization problem in network ﬂow theory. [47–49] developed

the optimization theory of ﬂows on manifolds, endowed with the name of “bit threads”, and pro-

posed the concept of bit threads can equivalently formulate the RT formula. Although bit threads

are visually intuitive and usually interpreted implicitly as the bell pairs distilled from the boundary

quantum systems, they are mostly used merely as a mathematical tool to study diﬀerent aspects

of holographic principles (for the recent developments of bit threads see e.g. [50–75]). Among these

developments, it is worth noting that in [51] bit threads are related with the holographic entangle-

ment distillation tensor networks, and in [52], the component ﬂow ﬂuxes in the locking bit thread

conﬁgurations are shown to explain the partial entanglement entropies of the boundary quantum

systems. In this paper 1, by studying the connection between bit threads and SS duality, we pro-

pose a natural and novel physical property of bit threads, dubbed “thread/state correspondence”,

that is, in a so-called locking bit thread conﬁguration [51–54], each bit thread is in a quantum su-

perposition state of two orthogonal states. Our thread/state rules explicitly accentuate for the ﬁrst

time the implied meaning of the name “bit threads”, namely, that these threads, which are used

mathematically to recover the RT formula, can be physically assigned a meaning closely related to

the concept of “bits”. Moreover, since the ﬂuxes of bit threads are directly related to the geometric

quantities of a holographic spacetime, while this newly proposed thread-state property can cleverly

characterize quantum entanglement as one will see, it can be expected to be a very useful advance

in the study of the relationship between quantum entanglement and spacetime geometry.

More explicitly, this paper will show that, using thread/state rules, we can do (but expect

to do more than) the following things: ﬁrst we can construct the explicit expressions for the SS

states corresponding to a set of bulk extremal surfaces in the SS duality, and nicely characterize

their entanglement structure; use the locking bit thread conﬁgurations to construct a new toy

model of the holographic principle and actually, we explicitly give the relationship between this

model and the holographic tensor network models; naturally understand the so-called kinematic

space [76, 77], endow it with the interpretation of microscopic states such that to explain that the

entropy is proportional to volume therein; in some sense quantitatively characterize the famous

“It from qubit” thought experiment [1], that is, by removing the entanglement in the boundary

quantum system, the bulk spacetime will accordingly deform, or even break up.

The idea of thread-state will provide a complementary perspective distinct from the local ten-

sors used in holographic TN models. Moreover, our thread/state prescription for locking thread

conﬁgurations is an enlightening step towards the issue of spacetime emergence. It is intriguing to

ﬁnd the similar rules for the more general non-locking bit thread conﬁgurations, then it is possible

to further read the SS states of the general bulk surfaces. It is even more tantalizing to completely

reconstruct the bulk geometry merely from the properties of the quantum state assigned to the

bit threads. In addition, it is also interesting to consider how the thread/state rules adapt to

the covariant bit threads [55] of the covariant RT formula [14], the quantum bit threads [56, 57]

that can account for the bulk quantum corrections to the RT formula [78, 79], the Lorentzian bit

threads [58, 59] that can characterize the holographic complexity [80, 81], the hyperthreads [60, 61]

1For a brief pre-release version containing the core of this work, see [50].

that can study the multipartite entanglement, and so on, and may provide useful insights on all of

these topics.

The structure of this paper is as follows: Section II is the background review of the basic

knowledge of bit threads and the surface/state correspondence. Section III presents the motivation

for this work. In Section III A we study the problem of constructing SS states of bulk extremal

surfaces, then in Section III B we propose the thread/state correspondence as a natural and eﬃcient

prescription for this problem. Based on the interesting connection between bit threads and SS

correspondence, in Section IV we construct a holographic qubit threads model using locking bit

thread conﬁgurations, wherein the bit threads can be understood as qubit threads or CMI threads.

In Section V A and V B, we discuss its close connection with the holographic tensor network model

and the holographic kinematic space respectively. In Section VI we use our qubit threads model to

discuss the connectivity of spacetime, and show that qubit threads can play the roles of “sewing”

a spacetime in a sense.

II. BACKGROUND REVIEW

This section is a brief review of bit thread, locking bit thread conﬁguration and surface/state

duality. Readers who are familiar with these aspects may skip to section III.

A. Bit threads and locking bit thread conﬁgurations

In the framework of holographic principle, bit threads are unoriented bulk curves which end on

the boundary and subject to the rule that the thread density is less than 1/4GNeverywhere [47–

49]. 2Mathematically, this thread density bound implies that the number of threads passing

through the minimal surface γ(A) that separates a boundary subregion Aand its complement Ac

cannot exceed its area Area (γ(A)), hence the ﬂux of bit threads F lux (A) connecting Aand its

complement Acdoes not exceed Area (γ(A)):

F lux (A)≤1

4GN·Area (γ(A)) .(1)

Borrowing terminology from the theory of ﬂows on networks, a thread conﬁguration is said to lock

the region Awhen the bound (1) is saturated. Actually, this bound is tight: for any A, there does

2When one takes the Hodge dual of bit threads one gets calibrated geometries, which mathematicians (geometers)

use to identify minimal surfaces. This is a viewpoint adopted in [82].

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Thread/Statecorrespondence:frombitthreadstoqubitthreadsYi-YuLin1;2andJie-ChenJin3y1BeijingInstituteofMathematicalSciencesandApplications(BIMSA),Beijing,101408,China2YauMathematicalSciencesCenter(YMSC),TsinghuaUniversity,Beijing,100084,Chinaand3SchoolofPhysicsandAstronomy,SunYat-SenUniversity,Guangz...

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ThreadState correspondence from bit threads to qubit threads Yi-Yu Lin12and Jie-Chen Jin3y 1Beijing Institute of Mathematical Sciences and Applications BIMSA Beijing 101408 China

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