††institutetext: Institut für Theoretische Physik und Astrophysik
and Würzburg-Dresden Cluster of Excellence ct.qmat,
Julius-Maximilians-Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

# Entwinement as a possible alternative to complexity

Johanna Erdmenger  Marius Gerbershagen

###### Abstract

Unlike the standard entanglement entropy considered in the holographic
context, _entwinement_ measures entanglement between degrees of freedom that
are not associated to a spatial subregion. Entwinement is defined for two-
dimensional CFTs with a discrete ${\mathbb{Z}}_{N}$ gauge symmetry. Since the
Hilbert space of these CFTs does not factorize into tensor products, even the
entanglement entropy associated to a spatial subregion cannot be defined as
the von Neumann entropy of a reduced density matrix. While earlier works
considered embedding the density matrix into a larger, factorizing Hilbert
space, we apply a gauge invariant approach by using a density matrix uniquely
defined through its relation to the local algebra of observables. We
furthermore obtain a fully gauge invariant definition of entwinement valid for
general CFTs with ${\mathbb{Z}}_{N}$ gauge symmetry in terms of all
observables acting on the degrees of freedom considered. Holographically,
entwinement is dual to the length of non-minimal geodesics present for conical
defects or black holes. In this context, we propose a definition of
entwinement for thermal states dual to the BTZ black hole. Our results show
that “entwinement is enough” to describe the full bulk geometry for the
conical defect and provide strong hints that the same holds true for the BTZ
black hole. Thus, it provides an alternative to holographic complexity for the
theories considered.

###### Keywords:

AdS-CFT Correspondence, Gauge-gravity Correspondence, Entanglement Entropy,
Entwinement

##  1 Introduction

The AdS/CFT correspondence Maldacena:1997re  is one of the most influential
developments in the recent history of theoretical physics. Due to the
holographic nature of the correspondence, the encoding of the bulk geometry in
the boundary field theory is of particular interest, a question that has
received a large amount of attention in the last years. A strong indication
that entanglement must play an important role in this encoding was provided by
the seminal work of Ryu and Takayanagi Ryu:2006bv . It connects the
entanglement entropy $S(A)$ of a subregion $A$ – a quantum information
quantity in the boundary field theory – to a geometric quantity in the bulk,
namely the area of the minimal codimension two bulk surface $\gamma_{A}$
anchored at $\partial A$ on the boundary,

$S(A)=\frac{\text{Area}(\gamma_{A})}{4G_{N}}.$ (1)

Entanglement entropy is defined for a bipartition of the Hilbert space into
tensor factors

${\mathcal{H}}={\mathcal{H}}_{A}\otimes{\mathcal{H}}_{A^{c}}$ (2)

as the von Neumann entropy of the reduced density matrix associated to
${\mathcal{H}}_{A}$,

$S(A)=-{\mathrm{Tr}}_{{\mathcal{H}}_{A}}(\rho_{A}\log\rho_{A})\hskip
28.45274pt\text{where}\leavevmode\nobreak\
\rho_{A}={\mathrm{Tr}}_{{\mathcal{H}}_{A^{c}}}\rho,$ (3)

while $\gamma_{A}$ is termed the Ryu-Takayanagi (RT) surface. The simple and
direct connection between the boundary field theory and the bulk geometry
provided by the Ryu-Takayanagi proposal has motivated the conjecture that the
bulk spacetime is emergent from entanglement, summarized in the slogan
“entanglement builds geometry” Swingle:2009bg ; VanRaamsdonk:2010pw ;
Bianchi:2012ev .

However, there are features of the bulk spacetime that entanglement entropy of
spatial subregions $A$ cannot describe. Among these are regions known as
entanglement shadows at a finite distance from a naked singularity, for
instance for conical defects. These regions are not probed by the RT surfaces
and thus evade a holographic reconstruction of bulk spacetime from the
boundary CFT data. Further examples include event horizons of black holes that
are not penetrated by any RT surface Balasubramanian:2014sra ;
Freivogel:2014lja . The AdS Schwarzschild and BTZ black holes have an
entanglement shadow of thickness of order the AdS scale surrounding the
horizon Hubeny:2013gta . A further example is the growth of wormholes with
time in two-sided black hole spacetimes, which entanglement entropy cannot
capture Susskind:2014moa ; Susskind:2014rva . While the length of the wormhole
between the two asymptotic boundaries continues to grow for a long time, the
entanglement entropy of a subregion consisting of two parts on both asymptotic
boundaries quickly saturates to a constant value Hartman:2013qma .

AdS3/CFT2 is a particular simple context where these issues appear, since 2+1
dimensional gravity admits only solutions with constant curvature. Therefore
all spacetimes with negative curvature are quotients of pure AdS3. These
quotients are obtained as follows. In pure AdS3, called the covering space,
multiple subregions are identified with each other. These subregions are
referred to as fundamental domains. For a 2+1 dimensional gravity theory, RT
surfaces are geodesics on a constant time slice that end on two points
delimiting an interval $A$ on the boundary. Geodesics on the covering space
descend to geodesics on the quotient space. Since a single point on the
quotient space maps to multiple points on the covering space, a single
boundary interval $A$ has multiple geodesics attached to it. The RT surface is
the geodesic with minimum length.

In this work, we mainly focus on conical defects, for which the quotient group
is ${\mathbb{Z}}_{N}$. In this case, there are $N$ geodesics attached to every
boundary interval $A$. As discussed in Balasubramanian:2014sra , the minimal
geodesic or RT surface penetrates only to a finite distance into the bulk,
leaving an entanglement shadow around the naked singularity at the origin,
where there is a region of spacetime not probed by the minimal surfaces that
compute spatial entanglement in the dual field theory. The minimal (RT)
geodesics thus do not probe the full spacetime. On the other hand, as
displayed in figure fig. 1, the non-minimal geodesics do probe the full
spacetime. This can be most easily seen from the quotienting picture and the
fact that geodesics in the covering space correspond to geodesics in the
conical defect. Since the full covering space is probed by geodesics, there
are no regions in the conical defect that are not probed by (possibly non-
minimal) geodesics. To completely specify the encoding of the bulk geometry in
the boundary field theory from geodesics, it is therefore necessary to obtain
a field theory dual to the length of both minimal and non-minimal geodesics.
While it is well established that this dual is the entanglement entropy for
minimal geodesics, much less is known about the field theory dual to the
length of non-minimal geodesics for which the term _entwinement_ has been
coined in Balasubramanian:2014sra .

$\tilde{B}^{0}_{0}$$\tilde{B}^{1}_{0}$$\tilde{B}^{2}_{0}$$\tilde{B}^{0}_{1}$$\tilde{B}^{0}_{2}$
Figure 1: LHS: Constant time slice of the conical defect geometry in AdS3 for
$N=3$ (i.e. for the quotient group $\mathbbm{Z}_{3}$), together with minimal
(in red) and non-minimal geodesics (in blue and green). The entanglement
shadow indicated in grey is not penetrated by any minimal geodesic. RHS: Some
of the corresponding geodesics together with their associated boundary
intervals $\tilde{B}^{i}_{k}$ on a constant time slice of the covering space.

Entwinement is a quantum information theoretic quantity in the boundary field
theory. It is defined in Balasubramanian:2014sra  as the entanglement entropy
$E^{i}_{k}$ of a boundary interval $\tilde{B}^{i}_{k}$ on the covering space,
symmetrized over translations of $\tilde{B}^{i}_{k}$ by elements of the
quotient group ${\mathbb{Z}}_{N}$,

$E^{\prime}_{k}=\sum_{i\in{\mathbb{Z}}_{N}}E^{i}_{k}.$ (4)

Here, $k$ is a free parameter encoding the size of $\tilde{B}^{i}_{k}$. To be
precise, $\tilde{B}^{i}_{k}$ is given as the union of $k$ fundamental domains
with an interval $B$ contained in a single fundamental domain (see fig. 1).
While the degrees of freedom in consideration are localized in
$\tilde{B}^{i}_{k}$ on the covering space, they are not localized in a single
subregion in the quotient space. Thus, entwinement is proposed as a measure
for entanglement between non-spatially organized degrees of freedom
Balasubramanian:2014sra , i.e. degrees of freedom that are not associated to a
spatial subregion on the boundary.

There is however an important subtlety that a definition of any field theory
quantity needs to address, namely all quantities must be gauge invariant. In
the conical defect case, the boundary field theory naturally aquires a
${\mathbb{Z}}_{N}$ gauge symmetry Balasubramanian:2014sra , for which the
factorization (2) of the Hilbert space into tensor factors does not hold and
reduced density matrices for spatial subregions are not well defined a priori.
Hence, even finding a field theory dual to the length of a minimal geodesic
first requires determining a suitable generalization of the reduced density
matrix and of entanglement entropy. Furthermore, turning back to entwinement
we note that while the definition (4) correctly reproduces the length of a
non-minimal geodesic, it requires introducing unphysical
non-${\mathbb{Z}}_{N}$ invariant states that live in the field theory on the
boundary of the covering space. We also note that while we have defined
entwinement for holographic theories, it is in principle a well-defined
quantity for any field theory with ${\mathbb{Z}}_{N}$ gauge symmetry.

In this paper, we develop fully gauge invariant techniques to characterize
entanglement for CFTs with discrete ${\mathbb{Z}}_{N}$ gauge symmetries. These
techniques are based on the sets of local operators acting on the degrees of
freedom considered. They apply to both entanglement entropy for spatial
subregions and to entwinement and do not rely on introducing unphysical gauge
invariant states. For the entanglement entropy associated to a spatial
subregion, the set of local operators forms an algebra. This enables us to
apply well-known techniques (see e.g. Harlow:2016vwg ; OhyaPetz1993  for a
review) to define a unique reduced density matrix, whose von Neumann entropy
reproduces the RT formula. For entwinement, the set of operators does not form
an algebra. Instead, it forms only a linear subspace that is closed under
addition but not multiplication, as was already observed in
Balasubramanian:2018ajb . We develop a new technique to define an entropy
associated to this linear subspace. We define entwinement as the minimum of
the entropy of a probability distribution obtained from projective
measurements using operators from the linear subspace. We show that this new
definition gives the same results as those of Balasubramanian:2014sra
described above. Therefore, we obtain a gauge invariant field theory dual to
the length of both minimal and non-minimal geodesics in the conical defect.

Moreover, we propose a definition of entwinement for thermal states of the CFT
dual to the BTZ black hole. The BTZ black hole is dual to a thermal state,
i.e. to an Euclidean CFT on the torus of spatial size $2\pi$. We propose that
entwinement is given as the entanglement entropy of an interval on a “large”
torus of spatial size $2\pi M$, where $M\in{\mathbb{N}}$. We show that its
bulk dual probes not only the entanglement shadows around the horizon, but
also the growth of the wormhole for the two-sided black hole. Under some
assumptions about the microscopic nature of the field theory description of
black holes, the proposed definition of entwinement is a well-defined quantity
in the field theory on the small torus with spatial size $2\pi$ and in accord
with the definition using a linear subspace.

This result is particularly interesting in light of the holographic complexity
conjectures of Susskind:2014moa ; Susskind:2014rva . Complexity is a quantity
from quantum information which measures how difficult it is to prepare a
target state from a given reference state by applying certain unitary gates.
Several possible holographic duals to complexity have been proposed in
Susskind:2014rva ; Stanford:2014jda ; Brown:2015bva ; Brown:2015lvg . It is,
however, still unclear which complexity measures in the field theory reproduce
these proposals, despite much work in this direction (see e.g. Chapman:2017rqy
; Jefferson:2017sdb ; Khan:2018rzm ; Hackl:2018ptj ; Chapman:2018hou ;
Goto:2018iay  and references therein). It has been conjectured that complexity
can explain the features of the bulk geometry not captured by the entanglement
entropy of spatial subregions, in particular the growth of the wormhole in
two-sided black hole geometries Susskind:2014moa ; Susskind:2014rva .
Entwinement as defined in this paper probes the same features of the bulk
geometry that complexity is conjectured to explain. In particular, these
features include the wormhole growth with time in the two-sided black hole
case. Therefore, at least for the geometries considered – conical defects in
AdS3 and the BTZ black hole – it provides an alternative to complexity.

Other work on entwinement includes Balasubramanian:2016xho ;
Balasubramanian:2018ajb . In Balasubramanian:2016xho , entwinement was
calculated for symmetric product orbifolds using a replica trick and agreement
with the length of non-minimal bulk geodesics was found. The authors of
Balasubramanian:2018ajb  pointed out similarities between entwinement and
entanglement for indistinguishable particles. They also suggested that
entwinement is naturally associated to a linear subspace of observables, a
proposal that we verify in our calculations.

Our paper is organized as follows: We begin in sec. 2 with a review of CFTs
with discrete gauge symmetry. In sec. 3 we apply the algebraic approach to
defining entanglement entropy to these CFTs. We show equivalence of this
approach both with a naive calculation using unphysical non-gauge invariant
states and for holographic CFTs with the Ryu-Takayanagi proposal. In sec. 4,
we turn to the study of entwinement. We introduce a gauge invariant definition
of entwinement based on a linear subspace of observables. We show that this
method is in agreement both with earlier calculations of
Balasubramanian:2014sra ; Balasubramanian:2016xho  using non-gauge invariant
states and with the length of non-minimal geodesics in the bulk. Sec. 5
contains a proposal for entwinement in the thermal state of the CFT dual to
the BTZ black hole. We specify the assumptions underlying this proposal and
show that it implies that entwinement probes not only the entanglement shadows
around the horizon, but also the growth of the wormhole for the two-sided
black hole. In sec. 6, we briefly comment on $S_{n}$ orbifolds arising from
the D1-D5 system. Finally in sec. 7, we present our conclusions.

##  2 Conformal field theories with discrete gauge symmetry

This section serves as an introduction to the conformal field theories whose
entanglement structure will be determined below. In particular, we will
introduce tensor product theories and permutation orbifolds and present their
Hilbert space structure.

We begin by considering the conical defect as a quotient of pure AdS3. This
pure AdS3, the covering space, is partitioned into $N$ fundamental domains
which are identified under the action of the quotient group. This implies that
the boundary splits into $N$ fundamental domains as well. On each fundamental
domain lives a conformal field theory with central charge $\tilde{c}$, which
we will call the _seed CFT_. Taking $N$ non-interacting copies of this seed
CFT without changing the boundary conditions obeyed by the fields of the CFTs
gives a _tensor product CFT_ denoted as CFTN.

Coming back from the holographic setting to general conformal field theories,
we consider as an example the seed CFT given by $\tilde{c}$ free bosons
$X^{m}(t,\phi)$ on a circle with action

$S_{\text{CFT}}[X]=\sum_{m=0}^{\tilde{c}-1}\int
dt\int_{0}^{2\pi}d\phi\,\partial X^{m}(t,\phi)\bar{\partial}X^{m}(t,\phi),$
(5)

and periodic boundary conditions $X^{m}(t,\phi+2\pi)=X^{m}(t,\phi)$. The
tensor product theory is a CFT of $c=N\tilde{c}$ free bosons $X^{m,n}(t,\phi)$
with action

$S_{\text{CFT}^{N}}[X]=\sum_{m=0}^{\tilde{c}-1}\sum_{n=0}^{N-1}\int
dt\int_{0}^{2\pi}d\phi\,\partial
X^{m,n}(t,\phi)\bar{\partial}X^{m,n}(t,\phi),$ (6)

and boundary conditions $X^{m,n}(t,\phi+2\pi)=X^{m,n}(t,\phi)$.

In general, the Hilbert space of the tensor product theory is a $N$-fold
tensor product of the Hilbert space of the seed CFT,

${\mathcal{H}}_{\text{CFT}^{N}}=({\mathcal{H}}_{\text{CFT}})^{\otimes N}.$ (7)

To implement the quotient, we have to demand that our theory be invariant
under ${\mathbb{Z}}_{N}$ permutations of the copies. It is well known how to
construct such ${\mathbb{Z}}_{N}$ invariant CFTs, which go under the name of
_permutation orbifolds_ 111Such orbifolds can be constructed for general
permutation groups. We comment on entanglement in the $S_{n}$ case in sec. 6.
(for an introduction to CFT orbifolds in general see for example
Ginsparg:1988ui ; Dixon:1986qv ). We will be mainly interested in the Hilbert
space ${\mathcal{H}}_{\text{orb}}$ of the orbifold theory, which splits up
into twisted sectors ${\mathcal{H}}_{\text{orb},k}$ labeled by
$k\in{\mathbb{Z}}_{N}$,

${\mathcal{H}}_{\text{orb}}=\bigoplus_{k=0}^{N-1}{\mathcal{H}}_{\text{orb,k}}\,,$
(8)

in which the fields obey the boundary conditions

$X^{m,n}(t,\phi+2\pi)=X^{m,n+k}(t,\phi),$ (9)

where $n+k$ is understood to be modulo $N$. As is important for the
consideration of entanglement in these CFTs, each of the
${\mathcal{H}}_{\text{orb},k}$ sectors contains only ${\mathbb{Z}}_{N}$
symmetric states. This means that in each ${\mathcal{H}}_{\text{orb},k}$
sector we apply the projection operator

$P=\frac{1}{N}\sum_{k\in{\mathbb{Z}}_{N}}g^{k},$ (10)

where $g^{k}$ are the ${\mathbb{Z}}_{N}$ generators acting on the fields as

$g^{k}X^{m,n}g^{-k}=X^{m,n+k}.$ (11)

Only ${\mathbb{Z}}_{N}$ symmetric states with $g$ eigenvalue 1 survive the
projection. The conical defect geometry is identified with the ground state of
the $k=1$ twisted sector. The central charge $c$ of the orbifold theory is
unchanged as compared to the covering theory.

A further theory will prove useful later on, although not directly describing
the actual physical systems we are interested in. This theory is obtained from
the tensor product theory by changing the boundary conditions of the fields to
be periodic up to permutations

$X^{m,n}(t,\phi+2\pi)=X^{m,n+1}(t,\phi).$ (12)

The theory in which the fields obey (12) is called the _covering CFT_. It has
the same action as the tensor product theory, which in our example using
boundary conditions (12) can be equivalently written as the theory of
$c/N=\tilde{c}$ free bosons $\tilde{X}^{m}(t,\phi)$ on a $N$ times larger
spatial circle,

$S_{\text{covering CFT}}[X]=\sum_{m=0}^{\tilde{c}-1}\int dt\int_{0}^{2\pi
N}d\phi\,\partial\tilde{X}^{m}(t,\phi)\bar{\partial}\tilde{X}^{m}(t,\phi).$
(13)

While this theory can describe aspects of states contained in the $k=1$
twisted sector of the orbifold theory, it is not equivalent to the orbifold
theory restricted to the $k=1$ twisted sector. In particular, its Hilbert
space contains non-${\mathbb{Z}}_{N}$ invariant states. The $k=1$ twisted
sector of the orbifold is only equivalent to the covering theory when the
${\mathbb{Z}}_{N}$ symmetry of the action (13) is promoted to a gauge
symmetry.

##  3 Entanglement entropy

Entanglement entropy is commonly defined as (3) using the bipartition (2) of
the Hilbert space into tensor factors. A direct application of this definition
is not possible in our case, since the Hilbert space of the orbifold theory
does not decompose into tensor factors. After showing this for the case of a
${\mathbb{Z}}_{2}$ symmetric Hilbert space, we explain two methods for dealing
with this problem and prove that they give equivalent results. Finally, we
generalize to the ${\mathbb{Z}}_{N}$ case. It is interesting to note that
similar problems occur in lattice gauge theories and analogous methods to the
ones presented here have been employed to define entanglement entropy in these
systems222We would like to thank Mari-Carmen Bañuls for pointing this out to
us. (see for example Buividovich:2008gq ; Donnelly:2011hn ; Casini:2013rba ;
Radicevic:2014kqa ; Aoki:2015bsa ; Ghosh:2015iwa ; Soni:2015yga ).

The Hilbert space of the CFT2 tensor product theory is given by

${\mathcal{H}}=\left\\{\,{|X,Y\rangle}\,\right\\},$ (14)

where we have denoted field eigenstates as ${|X,Y\rangle}$ such that

$X^{m,1}(\phi){|X,Y\rangle}=X^{m}(\phi){|X,Y\rangle},\leavevmode\nobreak\
\leavevmode\nobreak\ X^{m,2}(\phi){|X,Y\rangle}=Y^{m}(\phi){|X,Y\rangle}.$
(15)

The ${\mathbb{Z}}_{2}$ symmetry generators are $g^{0}=\mathbb{1}$ and $g$ with
action

$g{|X,Y\rangle}={|Y,X\rangle}.$ (16)

${\mathcal{H}}$ decomposes in ${\mathbb{Z}}_{2}$ symmetric and antisymmetric
states obtained by applying the orthogonal projection operators
$P^{\pm}=\frac{1}{2}(\mathbb{1}\pm g)$,

$P^{\pm}{|XY\rangle}=\frac{1}{2}({|X,Y\rangle}\pm{|Y,X\rangle}),$ (17)

with $P^{\pm}P^{\mp}=0$ and $(P^{\pm})^{2}=P^{\pm}$. Hence

${\mathcal{H}}={\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-}$ (18)

with

${\mathcal{H}}^{\pm}=P^{\pm}{\mathcal{H}}=\left\\{\leavevmode\nobreak\
({|X,Y\rangle}\pm{|Y,X\rangle})/2\leavevmode\nobreak\ \right\\}.$ (19)

The orbifold twisted sectors are given by ${\mathcal{H}}^{+}$ with appropriate
boundary conditions (9). We will restrict to states in a single sector and
drop the $k$ index labeling the different sectors. This does not introduce any
problems, since the states we are interested in lie in a single twisted
sector. Moreover, states in different twisted sectors are orthogonal to each
other, thus in traces of the density matrix over the Hilbert space
${\mathcal{H}}_{\text{orb}}$, only the twisted sector containing the state we
are interested in contributes.

The Hilbert spaces of the tensor product and covering theories decompose into
tensor factors associated to a subregion $A$,

${\mathcal{H}}={\mathcal{H}}_{A}\otimes{\mathcal{H}}_{A^{c}},$ (20)

with $A^{c}$ the complement of $A$. We denote the field eigenstates spanning
${\mathcal{H}}_{A}$ by ${|X_{A},Y_{A}\rangle}$ and analogously for
${\mathcal{H}}_{A^{c}}$, such that a state in ${\mathcal{H}}$ can
alternatively be written as
${|X,Y\rangle}={|X_{A}X_{A^{c}},Y_{A}Y_{A^{c}}\rangle}$.333This notation just
means that
$X^{m,1}(\phi){|X_{A}X_{A^{c}},Y_{A}Y_{A^{c}}\rangle}=\left\\{\begin{aligned}
X_{A}(\phi)\leavevmode\nobreak\ &,\leavevmode\nobreak\ \leavevmode\nobreak\
\phi\in A\\\ X_{A^{c}}(\phi)\leavevmode\nobreak\ &,\leavevmode\nobreak\
\leavevmode\nobreak\ \phi\in A^{c}\end{aligned}\right.$ (21) and analogously
for $X^{m,2}$. The ${\mathcal{H}}^{+}$ Hilbert space of the orbifold, on the
other hand, does not decompose into tensor factors. This can be seen by
considering the Hilbert space of states that are ${\mathbb{Z}}_{2}$
(anti-)symmetrized only in the subregion $A$ or $A^{c}$,

${\mathcal{H}}_{A}^{\pm}=\left\\{\leavevmode\nobreak\
({|X_{A},Y_{A}\rangle}\pm{|Y_{A},X_{A}\rangle})/2\leavevmode\nobreak\
\right\\},\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{H}}_{A^{c}}^{\pm}=\left\\{\leavevmode\nobreak\
({|X_{A^{c}},Y_{A^{c}}\rangle}\pm{|Y_{A^{c}},X_{A^{c}}\rangle})/2\leavevmode\nobreak\
\right\\}.$ (22)

Tensoring a state of ${\mathcal{H}}_{A}^{+}$ with a state of
${\mathcal{H}}_{A^{c}}^{+}$ gives

$\displaystyle\frac{1}{4}({|X_{A},Y_{A}\rangle}+{|Y_{A},X_{A}\rangle})\otimes({|X^{\prime}_{A^{c}},Y^{\prime}_{A^{c}}\rangle}+{|Y^{\prime}_{A^{c}},X^{\prime}_{A^{c}}\rangle})$
(23)
$\displaystyle=\frac{P^{+}}{2}({|X_{A}X^{\prime}_{A^{c}},Y_{A}Y^{\prime}_{A^{c}}\rangle}+{|X_{A}Y^{\prime}_{A^{c}},Y_{A}X^{\prime}_{A^{c}}\rangle})$

which is manifestly in ${\mathcal{H}}^{+}$. But tensoring a state of
${\mathcal{H}}_{A}^{-}$ with one of ${\mathcal{H}}_{A^{c}}^{-}$ also gives a
state of ${\mathcal{H}}^{+}$,

$\displaystyle\frac{1}{4}({|X_{A},Y_{A}\rangle}-{|Y_{A},X_{A}\rangle})\otimes({|X^{\prime}_{A^{c}},Y^{\prime}_{A^{c}}\rangle}-{|Y^{\prime}_{A^{c}},X^{\prime}_{A^{c}}\rangle})$
(24)
$\displaystyle=\frac{P^{+}}{2}({|X_{A}X^{\prime}_{A^{c}},Y_{A}Y^{\prime}_{A^{c}}\rangle}-{|X_{A}Y^{\prime}_{A^{c}},Y_{A}X^{\prime}_{A^{c}}\rangle}).$

From the explicit form of the r.h.s. of (23), (24), it is obvious that pairs
of these states span all of ${\mathcal{H}}^{+}$, hence ${\mathcal{H}}^{+}$ is
given as a direct sum

${\mathcal{H}}^{+}=({\mathcal{H}}_{A}^{+}\otimes{\mathcal{H}}_{A^{c}}^{+})\oplus({\mathcal{H}}_{A}^{-}\otimes{\mathcal{H}}_{A^{c}}^{-}).$
(25)

Similarly,

${\mathcal{H}}^{-}=({\mathcal{H}}_{A}^{+}\otimes{\mathcal{H}}_{A^{c}}^{-})\oplus({\mathcal{H}}_{A}^{-}\otimes{\mathcal{H}}_{A^{c}}^{+}).$
(26)

Therefore, the physical Hilbert space ${\mathcal{H}}^{+}$ of
${\mathbb{Z}}_{2}$ symmetric states decomposes only into a direct sum of
tensor factors.

###  3.1 Embedding the state into a larger Hilbert space

A simple way of dealing with the problem of a non-factorizing Hilbert space is
to embed states into the enlarged Hilbert space ${\mathcal{H}}$, which
factorizes along spatial degrees of freedom. Explicitely, we use the density
matrix $\rho\oplus 0$ in
${\mathcal{H}}={\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-}$ to define a reduced
density matrix and an entanglement entropy $S(A)$ by

$S(A)={\mathrm{Tr}}_{{\mathcal{H}}_{A}}(\rho_{A}\log\rho_{A})\,,\hskip
28.45274pt\text{where}\leavevmode\nobreak\
\rho_{A}={\mathrm{Tr}}_{{\mathcal{H}}_{A^{c}}}(\rho\oplus 0).$ (27)

In practice, this means that we compute $S(A)$ as the entanglement entropy of
the union of all copies $\tilde{A}^{i}$ of $A$ in the covering theory,

$S(A)=\tilde{S}(\bigcup_{i=0}^{N-1}\tilde{A}^{i}),$ (28)

where the $\tilde{S}$ notation is a reminder that this entanglement entropy is
computed in the covering theory.

$=$$\tilde{A}^{0}$$\tilde{A}^{1}$$\tilde{A}^{2}$$=$ Figure 2: The entanglement
entropy of a subregion $A$ in the conical defect is given by the entanglement
entropy of the union of all copies of $A$ in the covering space or
equivalently its complement. For a small interval, the RT surface is the union
of the RT surfaces of the copies of $A$, while for a large interval it is the
union of the RT surfaces of the complement. This exchange of dominance is the
reason for the emergence of the entanglement shadow, depicted in grey on the
left.

This procedure has been used to define entanglement entropy for the conical
defect in Balasubramanian:2014sra  and for holographic CFTs reproduces the RT
proposal (see fig. 2). However, it requires introducing unphysical states that
are traced over to obtain the reduced density matrix. As we now show, there
exists a different approach which bypasses these problems and in fact leads to
the same results.

###  3.2 Algebraic entanglement entropy

We begin with a review of the algebraic approach to defining entanglement
entropy Harlow:2016vwg ; OhyaPetz1993  for general systems, before applying it
to the ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{N}$ orbifold theories introduced
in sec. 2. The basic idea of this approach is to consider the unique density
matrix $\rho_{M}\in M$ associated to a von Neumann algebra $M$ by the
requirement that it gives the same expectation values as the global density
matrix for all operators from $M$,

${\mathrm{Tr}}_{{\mathcal{H}}^{0}}(\rho{\mathcal{O}})={\mathrm{Tr}}_{{\mathcal{H}}^{0}}(\rho_{M}{\mathcal{O}})\leavevmode\nobreak\
\leavevmode\nobreak\ \forall{\mathcal{O}}\in M.$ (29)

Here we have denoted the gauge invariant Hilbert space as ${\mathcal{H}}^{0}$
(the special case of a ${\mathbb{Z}}_{2}$ gauge symmetry for which
${\mathcal{H}}^{0}\equiv{\mathcal{H}}^{+}$ is described in the subsection
3.2.1). Then, the entanglement entropy associated to $M$ is given by the von
Neumann entropy for $\rho_{M}$,

$S(M)=\hat{\mathrm{Tr}}_{{\mathcal{H}}^{0}}(\rho_{M}\log\rho_{M}),$ (30)

where $\hat{\mathrm{Tr}}$ indicates that the trace is normalized differently
than the ${\mathrm{Tr}}$ functional, essentially due to the fact that we trace
over the whole Hilbert space instead of only a tensor factor as in (3).

For an introduction to algebraic entanglement entropy for finite-dimensional
Hilbert spaces see Harlow:2016vwg , whose conventions we will use below. The
statements that we will make in the following are valid only for finite-
dimensional Hilbert spaces (type I von Neumann algebras). This does not
introduce any additional problems, since the entanglement entropy is strictly
infinite for type II and type III von Neumann algebras on infinite dimensional
Hilbert spaces Witten:2018lha ; OhyaPetz1993 , thus a regularization is needed
in any case. A general, rigorous treatment of algebraic entanglement entropy
can be found in OhyaPetz1993 .

The algebras of local operators that we consider are von Neumann algebras,
i.e. sets of operators that are closed under Hermitean conjugation, addition
and multiplication and contain the identity. The set of operators
$Z_{M}\subset M$ that commute with all elements of a von Neumann algebra $M$
is called the center of $M$. A von Neumann algebra with trivial center is
called a factor. The importance of factor algebras is that they define an
associated Hilbert space decomposition into tensor factors,

${\mathcal{H}}={\mathcal{H}}_{A}\otimes{\mathcal{H}}_{A^{c}}.$ (31)

The elements of a factor algebra $M_{A}$ of operators localized in $A$ act
trivially on ${\mathcal{H}}_{A^{c}}$, i.e. $M_{A}$ is given as

$M_{A}={\mathcal{L}}({\mathcal{H}}_{A})\otimes\mathbb{1},$ (32)

where ${\mathcal{L}}({\mathcal{H}}_{A})$ is the set of linear operators acting
on the Hilbert space ${\mathcal{H}}_{A}$. Hence, if $M_{A}$ has trivial center
the algebraic definition reduces to the usual one and the density matrix
associated to $M_{A}$ is given the reduced density matrix on
${\mathcal{H}}_{A}$ times the identity on ${\mathcal{H}}_{A^{c}}$,

$\rho_{M_{A}}=\rho_{A}\otimes\frac{\mathbb{1}_{{\mathcal{H}}_{A^{c}}}}{|{\mathcal{H}}_{A^{c}}|},$
(33)

with $\rho_{A}={\mathrm{Tr}}_{{\mathcal{H}}_{A^{c}}}\rho$. The
$1/|{\mathcal{H}}_{A^{c}}|$ factor is necessary for the correct normalization
${\mathrm{Tr}}\rho_{M_{A}}=1$.

For von Neumann algebras $M_{A}$ with non-trivial center acting on the Hilbert
space of gauge invariant states ${\mathcal{H}}^{0}$, $Z_{M}$ is in general
spanned by a set of projection operators $Q^{\alpha}$. Since these are
mutually commuting, they can be simultaneously diagonalized. Therefore, the
projection operators can be taken to be mutually orthogonal,
$Q^{\alpha}Q^{\beta}=Q^{\alpha}$ for $\alpha=\beta$ and
$Q^{\alpha}Q^{\beta}=0$ otherwise. These operators project onto factors, i.e.
$Q^{\alpha}M_{A}Q^{\alpha}$ is a factor on $Q^{\alpha}{\mathcal{H}}^{0}$ while
$Q^{\alpha}M_{A}Q^{\beta}=0$ for $\alpha\neq\beta$. This implies a
factorization of the Hilbert space

${\mathcal{H}}^{0}=\bigoplus_{\alpha}({\mathcal{H}}^{\alpha}_{A}\otimes{\mathcal{H}}^{\alpha}_{A^{c}}),$
(34)

while $M_{A}$ decomposes as

$M_{A}=\bigoplus_{\alpha}({\mathcal{L}}({\mathcal{H}}^{\alpha}_{A})\otimes\mathbb{1}_{{\mathcal{H}}^{\alpha}_{A^{c}}})\,.$
(35)

This decomposition is just the explicit form of the statement that $M_{A}$
acts locally in $A$ and maps ${\mathcal{H}}^{0}$ into itself. The reduced
density matrix $\rho_{M_{A}}$ associated to $M_{A}$ is given by

$\rho_{M_{A}}=\bigoplus_{\alpha}\left(\rho^{\alpha}_{A}\otimes\frac{\mathbb{1}_{{\mathcal{H}}^{\alpha}_{A^{c}}}}{|{\mathcal{H}}^{\alpha}_{A^{c}}|}\right)\,,$
(36)

where

$\rho^{\alpha}_{A}={\mathrm{Tr}}_{{\mathcal{H}}^{\alpha}_{A^{c}}}(Q^{\alpha}\rho\,Q^{\alpha}).$
(37)

Remember that $Q^{\alpha}$ is the projection onto
${\mathcal{H}}^{\alpha}_{A}\otimes{\mathcal{H}}^{\alpha}_{A^{c}}$, i.e.
$Q^{\alpha}{\mathcal{H}}^{0}={\mathcal{H}}^{\alpha}_{A}\otimes{\mathcal{H}}^{\alpha}_{A^{c}}$.
Therefore, the entanglement entropy associated to $M_{A}$ is given by

$S(M_{A})=-\hat{\mathrm{Tr}}_{{\mathcal{H}}^{0}}(\rho_{M_{A}}\log\rho_{M_{A}})=-\sum_{\alpha}{\mathrm{Tr}}_{{\mathcal{H}}^{\alpha}_{A}}(\rho^{\alpha}_{A}\log\rho^{\alpha}_{A}).$
(38)

The effect of the modified trace is to cancel out the
$1/|{\mathcal{H}}^{\alpha}_{A^{c}}|$ factors, which were needed to ensure the
correct normalization of $\rho_{M_{A}}$. We can already see that this
modification is necessary by considering the case that $M_{A}$ is a factor. If
we had not cancelled the $1/|{\mathcal{H}}_{A^{c}}|$ factor, the definition
(30) would not agree with (3).

####  3.2.1 ${\mathbb{Z}}_{2}$ case

In the case we are considering, $\alpha\in\\{+,-\\}$ and the projections are
given by

$Q^{\pm}=\frac{1}{2}(\mathbb{1}\pm g_{A})=\frac{1}{2}(\mathbb{1}\pm
g_{A^{c}}),$ (39)

where $g_{A},g_{A^{c}}$ are ${\mathbb{Z}}_{2}$ generators acting only in the
subregion $A,A^{c}$ ,

$g_{A}{|X_{A}X_{A^{c}},Y_{A}Y_{A^{c}}\rangle}={|Y_{A}X_{A^{c}},X_{A}Y_{A^{c}}\rangle},\leavevmode\nobreak\
\leavevmode\nobreak\
g_{A^{c}}{|X_{A}X_{A^{c}},Y_{A}Y_{A^{c}}\rangle}={|X_{A}Y_{A^{c}},Y_{A}X_{A^{c}}\rangle}.$
(40)

We expand $\rho$ in field eigenstates 444 $\rho(X,Y,X^{\prime},Y^{\prime})$ is
a functional of the functions $X,Y,X^{\prime},Y^{\prime}$. The integration
measures $dXdY$ and $dX^{\prime}dY^{\prime}$ are assumed to be appropriately
normalized.,

$\rho=\int dXdY\int
dX^{\prime}dY^{\prime}\rho(X,Y,X^{\prime},Y^{\prime}){|X,Y\rangle}{\langle
X^{\prime},Y^{\prime}|},$ (41)

where $\rho(X,Y,X^{\prime},Y^{\prime})$ obeys

$\rho(X,Y,X^{\prime},Y^{\prime})=\rho(Y,X,X^{\prime},Y^{\prime})=\rho(X,Y,Y^{\prime},X^{\prime})=\rho(Y,X,Y^{\prime},X^{\prime})$
(42)

due to the fact that $\rho$ is a density matrix for states in
${\mathcal{H}}^{+}$ and thus $\rho=g\rho=\rho g$. We obtain for
$\rho_{A}^{\pm}$ that

$\displaystyle\rho_{A}^{\pm}={\mathrm{Tr}}_{{\mathcal{H}}^{\pm}_{A^{c}}}(Q^{\pm}\rho
Q^{\pm})={\mathrm{Tr}}_{{\mathcal{H}}^{\pm}_{A^{c}}}(Q^{\pm}\rho)$ (43)
$\displaystyle=\frac{1}{8}\int dXdY\int dX^{\prime}_{A}dY^{\prime}_{A}$
$\displaystyle(\rho(X,Y,X^{\prime}_{A}X_{A^{c}},Y^{\prime}_{A}Y_{A^{c}})\pm\rho(X,Y,X^{\prime}_{A}Y_{A^{c}},Y^{\prime}_{A}X_{A^{c}}))$
$\displaystyle\times({|X_{A},Y_{A}\rangle}\pm{|Y_{A},X_{A}\rangle})({\langle
X^{\prime}_{A},Y^{\prime}_{A}|}\pm{\langle Y^{\prime}_{A},X^{\prime}_{A}|})$

and therefore

$\displaystyle\rho_{A}$ $\displaystyle=\rho_{A}^{+}\oplus\rho_{A}^{-}$ (44)
$\displaystyle=\frac{1}{4}\int dXdY\int
dX^{\prime}_{A}dY^{\prime}_{A}\,{|X_{A},Y_{A}\rangle}{\langle
X^{\prime}_{A},Y^{\prime}_{A}|}\,\begin{aligned}
(&\rho(X_{A}X_{A^{c}},Y_{A}Y_{A^{c}},X^{\prime}_{A}X_{A^{c}},Y^{\prime}_{A}Y_{A^{c}})\\\
+&\rho(Y_{A}X_{A^{c}},X_{A}Y_{A^{c}},Y^{\prime}_{A}X_{A^{c}},X^{\prime}_{A}Y_{A^{c}})\\\
+&\rho(X_{A}X_{A^{c}},Y_{A}Y_{A^{c}},Y^{\prime}_{A}Y_{A^{c}},X^{\prime}_{A}X_{A^{c}})\\\
+&\rho(Y_{A}X_{A^{c}},X_{A}Y_{A^{c}},X^{\prime}_{A}Y_{A^{c}},Y^{\prime}_{A}X_{A^{c}}))\end{aligned}$
$\displaystyle=\int dXdY\int
dX^{\prime}_{A}dY^{\prime}_{A}\,\rho(X_{A}X_{A^{c}},Y_{A}Y_{A^{c}},X^{\prime}_{A}X_{A^{c}},Y^{\prime}_{A}Y_{A^{c}}){|X_{A},Y_{A}\rangle}{\langle
X^{\prime}_{A},Y^{\prime}_{A}|}\,,$

where in the last line we have used (42). Eq. (38) together with
${\mathcal{H}}_{A}={\mathcal{H}}_{A}^{+}\oplus{\mathcal{H}}_{A}^{-}$ then
implies that the von Neumann entropy of $\rho_{A}$ is equal to the
entanglement entropy associated to $M_{A}$ in the algebraic approach.
Furthermore, from the last line of (44), it is clear that $\rho_{A}$ is
nothing but the reduced density matrix obtained in section 3.1 by first
embedding $\rho$ in ${\mathcal{H}}={\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-}$
as $\rho\oplus 0$ and then taking the partial trace over
${\mathcal{H}}_{A^{c}}$ 555Compare for example to the definition of a partial
trace of a density matrix $\rho=\rho_{ij,kl}{|ij\rangle}{\langle kl|}$ of a
two qubit system, $\rho_{A}=\rho_{ik,jk}{|i\rangle}{\langle j|}$. The last
line of (44) has exactly the same structure, where the sum over $k$ is the
equivalent of the integral over $X_{A^{c}},Y_{A^{c}}$.. Therefore, we have
shown that for the ${\mathbb{Z}}_{2}$ case, the algebraic approach and the
embedding procedure of the last subsection give the same results.

####  3.2.2 ${\mathbb{Z}}_{N}$ case

The generalization of the above procedure to the ${\mathbb{Z}}_{N}$ case is
straightforward. As above, the Hilbert space ${\mathcal{H}}$ of the covering
theory decomposes into a direct sum
${\mathcal{H}}=\bigoplus_{n=0}^{N-1}{\mathcal{H}}^{n}$ of Hilbert spaces
${\mathcal{H}}^{n}$ with fixed $g$ eigenvalues of $e^{-2\pi in/N}$. The
${\mathcal{H}}^{n}$ are obtained from ${\mathcal{H}}$ by applying the
projection operators

$P^{n}=\frac{1}{N}\sum_{j=0}^{N-1}e^{2\pi inj/N}g^{j}\,.$ (45)

As for the $P^{\pm}$ above, the $P^{n}$ are pairwise orthogonal and square to
themselves, $P^{n}P^{m}=P^{n}\delta^{nm}$. The Hilbert space of the
${\mathbb{Z}}_{N}$ orbifold is given as a direct sum of ${\mathcal{H}}^{0}$
Hilbert spaces with appropriate twisted boundary conditions. Each
${\mathcal{H}}^{n}$ factor decomposes into a direct sum of tensor products

${\mathcal{H}}^{n}=\bigoplus_{m=0}^{N-1}{\mathcal{H}}_{A}^{m}\otimes{\mathcal{H}}_{A^{c}}^{n-m}\,,$
(46)

where the ${\mathcal{H}}_{A}^{n}$ and ${\mathcal{H}}_{A^{c}}^{n}$ factors are
obtained from ${\mathcal{H}}_{A}$ and ${\mathcal{H}}_{A^{c}}$ by applying the
projection operators

$Q_{A,A^{c}}^{n}=\frac{1}{N}\sum_{j=0}^{N-1}e^{2\pi inj/N}g_{A,A^{c}}^{j}.$
(47)

Applying (37) together with (46) for $n=0$, we obtain

$\rho_{A}^{n}={\mathrm{Tr}}_{{\mathcal{H}}_{A^{c}}^{N-n}}(Q_{A}^{n}\rho
Q_{A}^{n})\,.$ (48)

For $\rho\in{\mathcal{H}}^{0}$ we again obtain

$\bigoplus_{n=0}^{N-1}\rho_{A}^{n}=\rho_{A}\,,$ (49)

where $\rho_{A}$ is given by embedding $\rho$ into ${\mathcal{H}}$ and then
tracing out ${\mathcal{H}}_{A^{c}}$. Hence also for the ${\mathbb{Z}}_{N}$
case, we obtain equality between the entanglement entropy in the orbifold and
covering theory.

This justifies the calculation procedure for the entanglement entropy in the
conical defect as the entanglement entropy of the union of all copies of $A$
in the covering space from (28). It also implies that in the specific example
that we studied – even though a decomposition of the Hilbert space into tensor
factors ${\mathcal{H}}_{A}\otimes{\mathcal{H}}_{A^{c}}$ does not exist – the
RT proposal remains applicable if we identify the entanglement entropy $S(A)$
with the algebraic entanglement entropy (38).

##  4 Entwinement

Sec. 3 aimed at answering the question how entangled the degrees of freedom in
a subregion $A$ are with those in its complement $A^{c}$. However, this is not
the only question that one can ask about the entanglement structure of a
density matrix. Other interesting questions are about the entanglement
structure of a non-spatially organized subset of the degrees of freedom, e.g.
the fields of one of the $N$ copies of the seed CFT, with the remainder of the
system666This subset and its complement is an example of a bipartition of the
target space, i.e. the space of field values of the theory. Entanglement with
respect to target space bipartitions has recently been studied in
Mazenc:2019ety .. In particular, for entwinement we are concerned with the
entanglement between degrees of freedom localized in different subregions for
each copy of the seed CFT.

From the definition (4) of Balasubramanian:2014sra , entwinement is given by a
sum of entanglement entropies $E^{i}_{k}$ in the covering space. All
$E^{i}_{k}$ in the sum give the same contribution, therefore it is sufficient
to obtain just one of them for calculating entwinement. In fact as we will see
later on, a gauge invariant definition of entwinement can not distinguish the
$E^{i}_{k}$ for different $i$. All $E^{i}_{k}$ are obtained from the same
linear subspace of gauge invariant operators, thus the physical quantity that
the $E^{i}_{k}$ represent is the same for all $i$. Therefore, we will drop the
$i$ index in the following and define entwinement to be the quantity

$E_{k}=E^{i}_{k}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{with}\leavevmode\nobreak\
i=0,...,N-1\leavevmode\nobreak\ \text{arbitrary}.$ (50)

As a consequence, our definition (50) differs by a factor of $1/N$ from the
definition (4) of Balasubramanian:2014sra .

As before, the Hilbert space does not decompose into tensor factors for the
degrees of freedom whose entanglement entropy $E^{i}_{k}$ represents. To solve
this problem, we adopt the same methods as for entanglement entropy in sec. 3.
It turns out that the first method described above of embedding the state into
a larger, factorizing Hilbert space works fine, while the second method of
using the algebraic entanglement entropy does not work in the entwinement
case. As already observed in Balasubramanian:2018ajb , the set of observables
associated to entwinement only forms a linear subspace instead of an algebra.
Going beyond the work of Balasubramanian:2018ajb , we develop a gauge
invariant definition of entwinement from measurements with operators of the
aforementioned linear subspace.

###  4.1 Embedding the state into a larger Hilbert space

As for ordinary entanglement entropy, we embed the density matrix into the
Hilbert space ${\mathcal{H}}=\bigoplus_{n=0}^{N-1}{\mathcal{H}}^{n}$ of all
states regardless of their gauge invariance properties. Field eigenstates of
${\mathcal{H}}$ are denoted as ${|X^{0}...X^{N-1}\rangle}$. Since we do not
perform any symmetrization on these states, the corresponding Hilbert space
factorizes,

${\mathcal{H}}={\mathcal{H}}_{\tilde{B}_{k}}\otimes{\mathcal{H}}_{\tilde{B}^{c}_{k}},$
(51)

where $\tilde{B}_{k}=\tilde{B}^{i}_{k}$ for any $i=0,...,N-1$. A state that is
unentangled with respect to this factorization is given by

${|X^{0}...X^{N-1}\rangle}={|X^{0}...X^{k-1}X^{k}_{B}\rangle}\otimes{|X^{k}_{B^{c}}X^{k+1}...X^{N-1}\rangle},$
(52)

where as before ${|X^{n}_{B}\rangle}$ denotes a field eigenstate for the
$X^{m,n}$ fields defined in the subregion $B$. Due to the factorization
property the reduced density matrix

$\rho_{\tilde{B}_{k}}={\mathrm{Tr}}_{{\mathcal{H}}_{\tilde{B}^{c}_{k}}}(\rho\oplus
0\oplus...\oplus 0)$ (53)

is well-defined and thus $E_{k}$ is obtained as

$E_{k}={\mathrm{Tr}}_{{\mathcal{H}}_{\tilde{B}_{k}}}(\rho_{\tilde{B}_{k}}\log\rho_{\tilde{B}_{k}})\,.$
(54)

The important difference between $\rho_{\tilde{B}_{k}}$ and a reduced density
matrix $\rho_{A}$ for ordinary entanglement entropy is that in (53) we have
traced out different fields over different subregions, while in (27) all
fields were traced out over the same subregion. Of course, this calculation is
unsatisfactory for the same reason as that for ordinary entanglement entropy:
In taking the partial trace, we trace out unphysical non-gauge invariant
states.

###  4.2 Entwinement from linear subspace of observables

In contrast to the entanglement entropy studied in the last subsection, for
entwinement there is no associated algebra of observables in the orbifold
theory. Starting from the simple ${\mathbb{Z}}_{2}$ case, we first determine
that the set of operators acting on the degrees of freedom associated to
entwinement forms only a linear subspace that is closed under addition, but
not multiplication. Then we show that nevertheless there is an entropy
associated to this linear subspace which is equal to the entwinement $E_{k}$.

####  4.2.1 ${\mathbb{Z}}_{2}$ case

We focus again on the simple example with $N=2$. In the covering theory, we
can associate an algebra $M_{\tilde{B}_{k}}$ of observables to entwinement,
which contains operators acting in the full spatial circle on $k$ fields and
in the subregion $B$ on one field. Explicitly,

$M_{\tilde{B}_{k}}={\mathcal{L}}({\mathcal{H}}_{\tilde{B}_{k}})\otimes\mathbb{1},$
(55)

where ${\mathcal{L}}({\mathcal{H}}_{\tilde{B}_{k}})$ is the set of linear
operators acting on ${\mathcal{H}}_{\tilde{B}_{k}}$. For the example of $k=1$,
${\mathcal{L}}({\mathcal{H}}_{\tilde{B}_{1}})$ is given by operators of the
form

${\mathcal{O}}_{\tilde{B}_{1}}=\int dXdY_{B}\int
dX^{\prime}dY^{\prime}_{B}\,{\mathcal{O}}(X,Y_{B},X^{\prime},Y^{\prime}_{B}){|X,Y_{B}\rangle}{\langle
X^{\prime},Y^{\prime}_{B}|}.$ (56)

Since the elements of the $M_{\tilde{B}_{k}}$ algebra are operators in the
covering theory, the algebra doesn’t map ${\mathcal{H}}^{+}$ into itself. To
obtain operators that map ${\mathcal{H}}^{+}$ into itself, we have to project
onto operators that are block diagonal with respect to the factorization
${\mathcal{H}}={\mathcal{H}}^{+}\oplus{\mathcal{H}}^{-}$:777The algebra
$M_{A}$ of operators in $A$ acting on ${\mathcal{H}}^{+}$ introduced in the
last subsection is obtained in the same way from the algebra of operators in
$A$ acting on ${\mathcal{H}}$.

$M_{B}=P^{+}M_{\tilde{B}_{k}}P^{+}+P^{-}M_{\tilde{B}_{k}}P^{-}=\frac{1}{2}\left({\mathcal{L}}({\mathcal{H}}_{\tilde{B}_{k}})\otimes\mathbb{1}+g({\mathcal{L}}({\mathcal{H}}_{\tilde{B}_{k}})\otimes\mathbb{1})g\right).$
(57)

The $+$ in (57) means that every element of $M_{B}$ is a sum of an element of
$M_{\tilde{B}_{k}}$ with the corresponding ${\mathbb{Z}}_{2}$ translated
element of $gM_{\tilde{B}_{k}}g$, i.e.

$M_{B}=\left\\{\leavevmode\nobreak\
\frac{1}{2}\left({\mathcal{O}}_{\tilde{B}_{k}}\otimes\mathbb{1}+g({\mathcal{O}}_{\tilde{B}_{k}}\otimes\mathbb{1})g\right)\leavevmode\nobreak\
\right\\}.$ (58)

The basic problem now is that $M_{B}$ does not form an algebra, since it is
not closed under multiplication. As already observed in
Balasubramanian:2018ajb , $M_{B}$ forms a linear subspace that is closed only
under addition and Hermitean conjugation. Therefore, the method which we used
in the previous subsection to define the entanglement entropy no longer
applies. A direct generalization of eq. (29) to linear subspaces is also not
possible due to the structure of the Hilbert space. Unlike the case of
ordinary entanglement entropy, the ${\mathbb{Z}}_{2}$ symmetric Hilbert space
${\mathcal{H}}^{+}$ does not possess a decomposition as a tensor product or a
direct sum of tensor products in terms of which $\rho_{\tilde{B}_{k}}$ is a
density matrix for some subset of states in ${\mathcal{H}}^{+}$.

However, $\rho_{\tilde{B}_{k}}$ still is associated to the linear subspace
$M_{B}$ in the sense that $\rho_{\tilde{B}_{k}}$ is the part of the full
density matrix $\rho$ that can be measured using operators from $M_{B}$.
Before showing this, we would like to briefly remind the reader of some basic
concepts from quantum information theory. Every density matrix $\rho$ can
written in terms of a spectral decomposition

$\rho=\sum_{i}p_{i}{|\psi_{i}\rangle}{\langle\psi_{i}|},$ (59)

where $p_{i}\in[0,1]$ is the probability that the state of the quantum system
is ${|\psi_{i}\rangle}$. The ${|\psi_{i}\rangle}$ form a complete orthonormal
basis for the Hilbert space of the system. Performing a projective measurement
with a Hermitean operator
${\mathcal{O}}=\sum_{j}m_{j}{|\chi_{j}\rangle}{\langle\chi_{j}|}$ yields the
state

$\rho_{\mathcal{O}}=\sum_{i,j}p_{i}|\langle\chi_{j}|\psi_{i}\rangle|^{2}{|\chi_{j}\rangle}{\langle\chi_{j}|}=\sum_{j}p_{{\mathcal{O}},j}{|\chi_{j}\rangle}{\langle\chi_{j}|},$
(60)

where $p_{{\mathcal{O}},j}$ is the probability of measuring the eigenvalue
$m_{j}$ of ${\mathcal{O}}$. The von Neumann entropy of $\rho_{\mathcal{O}}$ is
in general larger than the von Neumann entropy of $\rho$ NielsenChuang ,

$S(\rho_{\mathcal{O}})=S(\mathbf{p}_{\mathcal{O}})=-\sum_{j}p_{{\mathcal{O}},j}\log
p_{{\mathcal{O}},j}\geq S(\rho)=S(\mathbf{p})=-\sum_{i}p_{i}\log p_{i}.$ (61)

Equality is obtained when the eigenbases of ${\mathcal{O}}$ and $\rho$ agree,
i.e. when ${|\psi_{i}\rangle}={|\chi_{i}\rangle}$. In this case, we have full
knowledge of the density matrix $\rho$. Hence, the von Neumann entropy of
$\rho$ can be obtained by looking for the infimum of $S(\rho_{\mathcal{O}})$
over the set of all Hermitean operators or equivalently over the set of bases
${|\chi_{j}\rangle}$,

$S(\rho)=\inf\\{S(\mathbf{p}_{\mathcal{O}})\leavevmode\nobreak\
|\leavevmode\nobreak\ \text{$\\{{|\chi_{j}\rangle}\\}$ is a basis for the
Hilbert space of $\rho$}\\}.$ (62)

The same procedure can be applied if one has access only to the linear
subspace $M_{B}$ of Hermitean operators. $M_{B}$ is spanned by a set of basis
operators of the form

${\mathcal{O}}_{j}=\frac{1}{2}\left({|\chi_{j}\rangle}{\langle\chi_{j}|}\otimes\mathbb{1}+g({|\chi_{j}\rangle}{\langle\chi_{j}|}\otimes\mathbb{1})g\right),$
(63)

where the ${|\chi_{j}\rangle}$ form a complete orthonormal basis for the
covering Hilbert space factor ${\mathcal{H}}_{\tilde{B}_{k}}$. Again,
measurements using the operators ${\mathcal{O}}_{j}$ yield a probability
distribution given by

$p_{{\mathcal{O}},j}={\mathrm{Tr}}_{{\mathcal{H}}^{+}}(\rho{\mathcal{O}}_{j})$
(64)

and a corresponding entropy
$S(\mathbf{p}_{\mathcal{O}})=-\sum_{j}p_{{\mathcal{O}},j}\log
p_{{\mathcal{O}},j}$. We define entwinement to be the infimum of
$S(\mathbf{p}_{\mathcal{O}})$ over all bases $\\{{\mathcal{O}}_{j}\\}$,

$E_{k}=S(M_{B})=\inf\\{S(\mathbf{p}_{\mathcal{O}})\leavevmode\nobreak\
|\leavevmode\nobreak\ \text{$\\{{\mathcal{O}}_{j}\\}$ is a basis for
$M_{B}$}\\}.$ (65)

Due to (62), to show that the definition (65) agrees with (54), we only have
to show that the probability distribution (64) agrees with the probability
distribution $\tilde{p}_{{|\chi_{j}\rangle}{\langle\chi_{j}|}}$ for the basis
${|\chi_{j}\rangle}$ with respect to $\rho_{\tilde{B}_{k}}$. This follows
directly from the fact that $g\rho g=\rho$ and $\rho{\mathcal{H}}^{-}=0$,

$p_{{\mathcal{O}},j}={\mathrm{Tr}}_{{\mathcal{H}}^{+}}\left(\rho\,{\mathcal{O}}_{j}\right)={\mathrm{Tr}}_{\mathcal{H}}(\rho\,{|\chi_{j}\rangle}{\langle\chi_{j}|}\otimes\mathbb{1})={\mathrm{Tr}}_{{\mathcal{H}}_{\tilde{B}_{k}}}(\rho_{\tilde{B}_{k}}{|\chi_{j}\rangle}{\langle\chi_{j}|})=\tilde{p}_{{|\chi_{j}\rangle}{\langle\chi_{j}|}}.$
(66)

Thus we see that entwinement is obtained as the minimal entropy of the
probability distribution of a measurement with an operator from $M_{B}$.
Therefore, entwinement is a measure for the amount of information that can be
obtained about the density matrix $\rho$ from measurements with operators in
the linear subspace $M_{B}$.

We further note that this definition of entwinement reduces to the
entanglement entropy from the algebraic approach if the linear subspace closes
into an algebra, at least for algebras on finite-dimensional Hilbert spaces
considered in the last section. This follows directly from the fact that the
algebras $M_{A}$ we consider are linear subspaces as well, therefore we can
apply the same techniques as above. For each ${\mathcal{H}}^{\alpha}_{A}$
factor of the decomposition (34), we choose a basis of states, which yields a
basis for the Hermitean operators of $M_{A}$. Due to (29), the probability
distributions for a measurement of these operators with respect to
$\rho_{M_{A}}$ and $\rho$ agree. Hence, the minimal entropy of this
probability distribution is equal to the von Neumann entropy of $\rho_{M_{A}}$
given by eq. (30). Therefore, the definition (65) of an entropy associated to
a linear subspace of operators is a generalization of the algebraic approach,
which itself is a generalization of the usual definition of entanglement
entropy using a partial trace.

####  4.2.2 ${\mathbb{Z}}_{N}$ case

The generalization of the above method to the ${\mathbb{Z}}_{N}$ case follows
immediately. The projection of the elements of the algebra $M_{\tilde{B}_{k}}$
acting on ${\mathcal{H}}$ to operators that are block-diagonal for the
${\mathcal{H}}=\bigoplus_{n}{\mathcal{H}}^{n}$ decomposition gives the linear
subspace

$M_{B}=\sum_{n=0}^{N-1}P^{n}M_{\tilde{B}_{k}}P^{n}=\frac{1}{N}\sum_{j=0}^{N-1}g^{j}M_{\tilde{B}_{k}}g^{N-j}.$
(67)

As before, the Hilbert space ${\mathcal{H}}^{0}$ does not decompose into
tensor products for which a reduced density matrix associated to $M_{B}$ could
be defined. Due to $g\rho=\rho g=\rho$, expectation values w.r.t. $\rho$ for
elements

${\mathcal{O}}_{j}=\frac{1}{N}\sum_{j=0}^{N-1}g^{j}({|\chi_{j}\rangle}{\langle\chi_{j}|}\otimes\mathbb{1})g^{N-j}$
(68)

of $M_{B}$ are equal to expectation values for
${|\chi_{j}\rangle}{\langle\chi_{j}|}$ w.r.t. $\rho_{\tilde{B}_{k}}$,

$p_{{\mathcal{O}}_{j}}={\mathrm{Tr}}_{{\mathcal{H}}^{0}}(\rho{\mathcal{O}}_{j})={\mathrm{Tr}}_{\mathcal{H}}(\rho{|\chi_{j}\rangle}{\langle\chi_{j}|}\otimes\mathbb{1})={\mathrm{Tr}}_{{\mathcal{H}}_{\tilde{B}_{k}}}(\rho_{\tilde{B}_{k}}{|\chi_{j}\rangle}{\langle\chi_{j}|})=\tilde{p}_{{|\chi_{j}\rangle}{\langle\chi_{j}|}}.$
(69)

Here ${|\chi_{j}\rangle}$ is a basis for ${\mathcal{H}}_{\tilde{B}_{k}}$. Thus
we see that also in the ${\mathbb{Z}}_{N}$ case, $E_{k}$ is given as the
minimal entropy of the probability distribution of measurements with operators
from $M_{B}$.

Since $M_{B}$ is independent of the choice of subregion $\tilde{B}^{i}_{k}$ in
the covering space on which ${|\chi_{j}\rangle}{\langle\chi_{j}|}$ acts,
$E_{k}$ is independent of this choice as well. Alternatively, we see that from
the perspective of measurements with gauge invariant operators, all
$E^{i}_{k}$ factors from (4) are associated to the same linear subspace and
thus represent the same physical quantity. Hence, the length of a non-minimal
geodesic is in fact not directly equal to the entropy associated to $M_{B}$
divided by $4G_{N}$, but only up to a prefactor of $1/N$.

So far we have calculated entwinement associated to the degrees of freedom (up
to the ${\mathbb{Z}}_{N}$ symmetry) $X^{0},...,X^{k-1},X_{B}^{k}$, i.e. $k$
fields on the full space and one field on the subregion $B$. However, the
definition of entwinement given above is sufficiently general to allow for
different subsets of the total degrees of freedom $X^{0},...,X^{N-1}$. For
example, consider the subset consisting of the union of $N/M$ subsets
$X^{Mi},...,X^{Mi+k-1},X_{B}^{Mi+k}$ labeled by $i\in{0,...,N/M-1}$, where
$M\in{\mathbb{N}}$ divides $N$ and $k<M$. Holographically, the corresponding
quantity is the union of $N/M$ geodesics on the covering space. From the
results of the last section, it is clear that this is equivalent to a single
minimal geodesic in a conical defect geometry that arises from quotienting the
covering space by a ${\mathbb{Z}}_{N/M}$ group. The corresponding central
charge is $\tilde{c}N/M=c/M$. Of course, the geodesic will be non-minimal in
the original ${\mathbb{Z}}_{N}$ quotient conical defect if $k\geq 1$888Note
however that now the possibility of a phase transition arises, just as for the
ordinary entanglement entropy.. Thus we see that using another covering space
which is obtained by unrolling the spatial coordinate $N/M$ times instead of
$N$ times will still give us the length of a non-minimal geodesic, altough
with a different prefactor $c/M$ instead of $c/N$ for $E_{k}$. For $k=0$ and
$M=1$, the geodesic becomes minimal and we recover the ordinary entanglement
entropy.

####  4.2.3 Relation to existing results

Generalizations of entanglement similar to the one derived above have been
considered before in 2005IJTP…44.2127B ; 2003PhRvA..68c2308B . There,
entanglement has been defined with respect to a convex cone $C$ of states,
i.e. a set of operators representing density matrices that is closed under
taking convex linear combinations $\lambda x_{1}+(1-\lambda)x_{2}$, where
$\lambda\in[0,1]$. Pure states are unit trace elements of $C$ that cannot be
written as convex combinations of other elements of $C$. Furthermore, a second
cone $D$ is introduced together with a map $\pi$ from $C$ to $D$. Then, a pure
state $x\in C$ is termed generalized unentangled relative to $D$ if $\pi(x)$
is pure as well. The authors of 2005IJTP…44.2127B ; 2003PhRvA..68c2308B  also
introduce an entanglement measure on states $y\in D$ as

$S(y)=\inf\\{S(\mathbf{p})\leavevmode\nobreak\ |\leavevmode\nobreak\
y=\sum_{i}p_{i}y_{i}\leavevmode\nobreak\ \text{with $y_{i}$ pure}\\},$ (70)

where $S(\mathbf{p})$ is a Schur concave entropy measure on the probability
distribution $\\{p_{i}\\}$, e.g. the Shannon entropy
$S(\mathbf{p})=-\sum_{i}p_{i}\log p_{i}$. This definition is equivalent to the
ordinary entanglement if $D$ is the set of operators acting on a tensor factor
of the Hilbert space that $C$ acts on and if $\pi$ is the partial trace999The
proof of this statements works by showing that the probability distribution of
any decomposition of $y$ is a transformation of the probability distribution
of the spectral decomposition of $y$ by a doubly stochastic matrix, which
increases the value of all Schur concave functions.. It is also equivalent to
entwinement if we identify $C$ with the set of operators acting on
${\mathcal{H}}^{0}$ and $D$ with $M_{B}$. In this case, the map $\pi$ takes an
element of $C$ to an element of $D$ by embedding it in the enlarged Hilbert
space ${\mathcal{H}}$, taking the partial trace over
${\mathcal{H}}_{\tilde{B}_{k}^{c}}$ and projecting onto a ${\mathbb{Z}}_{N}$
invariant operator as in (67). Using the convex cone formalism, one can in
fact show in general that a restriction to measurements using a subspace of
observables implies the existence of a pair of cones $C$ and $D$ for which the
generalized entanglement described above can be defined 2005IJTP…44.2127B .

The entanglement measure (70) is extended to mixed states in 2005IJTP…44.2127B
; 2003PhRvA..68c2308B  using the convex hull construction familiar from
entanglement of formation, i.e. the entanglement measure for mixed states is
given by a second infimum over convex decompositions of the full state in $C$.
We do not extend the entwinement definition in the same way, since we do not
aim at separating quantum correlations due to entanglement from classical
correlations due to a mixed global density matrix. Rather, we are interested
in quantifying the total amount of correlations measured using operators from
a linear subspace. The reason for that stems from the intuition from ordinary
entanglement entropy: The Ryu-Takayanagi formula applies for both mixed and
pure states, although ordinary entanglement entropy quantifies the total
amount of correlations in a given subsystem and not those from entanglement
alone.

##  5 Application to the BTZ black hole

As for the conical defect, the BTZ black hole Banados:1992wn  is a simple
quotient of pure AdS3. Therefore we expect to be able to apply the same
techniques as in the previous section. In the following we propose a
definition of entwinement for thermal states and show that for simple examples
of boundary subregions it reproduces the length of non-minimal geodesics in
the dual BTZ black hole spacetime. These geodesics probe important features of
the bulk geometry, in particular not only the entanglement shadow but also the
growth of the wormhole for the two-sided black hole. Therefore entwinement
offers a possible alternative to complexity to describe these bulk geometry
features from boundary data. The definition we give rests on some assumptions
about the microscopic nature of the dual CFT state, which we will detail but
not attempt to verify. We only focus on the non-rotating black hole.

#### The covering theory

Since black holes are dual to thermal states, the corresponding CFT is
Euclidean with compactified time, i.e. the CFT lives on the torus. The BTZ
black hole arises from a quotient of pure AdS3 with ${\mathbb{Z}}$ quotient
group. The starting point is pure AdS3 in coordinates for which the metric is

$ds^{2}=-\left(\frac{r^{2}}{L^{2}}-m\right)dt^{2}+\left(\frac{r^{2}}{L^{2}}-m\right)^{-1}dr^{2}+r^{2}d\phi^{2},$
(71)

where $m$ is the mass of the black hole and $\phi\in{\mathbb{R}}$. The BTZ
black hole is obtained by identifying $\phi\sim\phi+2\pi$. Prior to this
identification, the boundary theory is in the Rindler vacuum with respect to
the $t,\phi$ coordinates Maldacena:1998bw . Thus we expect in the BTZ case the
covering theory to be a CFT on a line (spatial coordinate
$\phi\in{\mathbb{R}}$) in a thermal state. Therefore, we propose the covering
theory to be the Euclidean CFT on a cylinder with periodic time direction.
From the ${\mathbb{Z}}$ quotient group, the covering theory acquires a
${\mathbb{Z}}$ gauge symmetry. The cylinder with periodic time can be thought
of as being formed by an infinite number of tori cut open at a fixed value of
the spatial coordinate and glued together. Then, the ${\mathbb{Z}}$ gauge
symmetry acts by permuting these tori with each other.

For the massless black hole, which can be obtained either as the $m\to 0$
limit of the BTZ geometry or the $N\to\infty$ limit of the conical defect, the
above definition for the covering field theory is in accord with the one for
the conical defect. The $m\to 0$ limit implies that $\beta$, the radius of the
circle in the time direction, goes to infinity. Thus we are left with a CFT on
a line with non-compact time and ${\mathbb{Z}}$ symmetry permuting strips with
infinite temporal and finite spatial extent with each other, which is
precisely the $N\to\infty$ limit of the covering theory for the conical
defect.

There is however a problem in defining the covering theory since this requires
unwrapping the spatial direction an infinite number of times. Therefore, for
fixed $c$ the central charge $\tilde{c}=c/N$ of the covering theory is zero,
leading to an ill-defined theory. But as we have seen in the last section, it
is also possible to use a covering space which covers the original geometry
$M<N$ times, where $N$ is the number of copies in the full covering space
which is infinite in the BTZ case. On the CFT side, this implies that instead
of unwrapping the torus an infinite number of times to a cylinder, we only
take a finite number $M\ll c$ of torus copies. To summarize, for the covering
theory to be well-defined, we take it to be a CFT on a torus which is $M$
times larger in the space direction than the one of the quotient theory. Since
$\tilde{c}=c/M\gg 1$, this _finite covering CFT_ is still in the large $c$
limit.

For this definition to be in accord with the one from sec. 4, however, we have
to make some assumptions about the microscopic nature of the field theory
description of the BTZ black hole. The BTZ spacetime is not dual to a single
field theory state, but to an ensemble of microstates. The above definition is
only well-defined if (on average) each microstate is in a twisted sector of
the theory, such that the set of fields of the theory split into $M$ subsets
which are continuously connected by the boundary conditions. Furthermore,
computing entwinement in each microstate and averaging over the result must
reproduce the definition given above. For the massless BTZ black hole, these
assumptions have been verified to hold for $M\lesssim\sqrt{c}$ in the CFT of
the D1/D5-brane system at the orbifold point Balasubramanian:2016xho . In the
massive black hole case, we leave the question for which values of $M$ this
assumption is justified, or whether it is justified at all, for future work.
Instead we will explore the implications of our entwinement proposal for the
dual bulk description in terms of geodesics.

For simplicity, we restrict our calculation of entwinement to entangling
intervals in the finite covering theory that are small compared to the size of
the spatial direction of the torus. This means that we work in the $k\ll M$
limit, where $k$ is the parameter of (50). In this limit, the results for the
entanglement entropy in the finite covering CFT will match those of the
entanglement entropy of the same interval in a CFT with the same finite
$\tilde{c}$ on the cylinder with periodic time. Therefore, we can equivalently
take our finite covering theory to be the theory on the cylinder with periodic
time and finite $\tilde{c}$. Then we simply calculate the entanglement entropy
of an interval in this finite covering theory.

#### One-sided black hole

Concentrate first on the one-sided BTZ black hole. As described above, for
$k\ll M$ to obtain entwinement we only need to calculate the entanglement
entropy in the finite covering theory, i.e. a CFT on the cylinder with
periodic time and finite $\tilde{c}$. Thus, we have to compute the
entanglement entropy of a single interval of length $L=|B|+2\pi k$ on the
cylinder with periodic time, where $|B|<2\pi$ is the length of the interval
$B$ and $k\in{\mathbb{N}}$. The result for this entanglement entropy was
derived in Calabrese:2004eu  and is given by, up to a regularization constant

$E_{k}=\frac{\tilde{c}}{3}\log\left(\frac{\beta}{2\pi}\sinh\left(\frac{\pi
L}{\beta}\right)\right),$ (72)

where $\beta$ is the inverse temperature. The interpretation of this result is
as follows. Eq. (72) is equal to the entanglement entropy on the cylinder with
periodic time for any $k\in{\mathbb{N}}$. Eq. (72) can also be identified with
entwinement, but as explained above, this identification holds only for $k\ll
M\ll c$. This can be checked by referring to the entanglement entropy on the
finite covering space (the large torus with period $2\pi M$). The entanglement
entropy in this case has been computed in Barrella:2013wja  and as expected
agrees with (72) as long as $k\ll M$. Finally, eq. (72) is equal to
$1/4G_{N}M$ times the length of a geodesic associated to a boundary interval
of length $|B|$ and winding $k$ times around the black hole horizon. The
equality between geodesic lengths and eq. (72) holds for all
$k\in{\mathbb{N}}$. However, for entwinement $k\ll M\ll c$, hence there is an
upper bound on the winding number of those geodesics that are dual to
entwinement. For high winding numbers, the geodesics come very close to the
black hole horizon. Therefore, the breakdown between geodesic lengths and
entwinement in this limit might be a signal that bulk quantum corrections must
be taken into account to describe the region near the black hole horizon.

#### Two-sided black hole

We now turn to the case of the two-sided BTZ black hole, which is dual to the
thermofield double state Maldacena:2001kr ,

${|\Omega\rangle}=\frac{1}{\sqrt{Z}}\sum_{n,m}e^{-\beta
E_{n}/2}{|E_{n}\rangle}_{1}{|E_{n}\rangle}_{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ {|E_{n}\rangle}\leavevmode\nobreak\ \text{an
energy eigenstate}.$ (73)

The same arguments as for the one-sided case lead to the finite covering
theory being the CFT on a line with central charge $\tilde{c}$ in the
thermofield double state. The reason is that correlation functions in the
thermofield double state (including correlation functions of twist operators
that yield entanglement entropy via the replica trick) are calculated by
correlation functions of the Euclidean theory in a background with periodic
time $\tau$. The difference to the thermal case is that operators on the first
boundary are inserted at $\tau=0$, while operators on the second boundary are
inserted at $\tau=i\beta/2$. Therefore, we can apply the same mapping to the
covering theory as for the one-sided case described in detail above. The only
difference to be taken into account is that in correlation functions,
operators on the second asymptotic boundary are inserted at $\tau=i\beta/2$.
For simplicity, we will also restrict to the same small interval limit $k\ll
M$ as above.

Since the entanglement entropy in the thermofield double state for an interval
living on only one boundary reduces to the value in the thermal state,
interesting entangling intervals are those that consist of two components on
different asymptotic boundaries. The entanglement entropy for a CFT on a line
in the thermofield double state was calculated in Hartman:2013qma . For an
interval consisting of two subintervals of length $L=|B|+2\pi k$ at the same
position on both asymptotic boundaries, the authors of Hartman:2013qma  obtain
a growth of the entanglement entropy for a time of order $t\sim L/2$ followed
by a saturation to a constant value,

$E_{k}=\left\\{\begin{aligned}
\frac{2\tilde{c}}{3}\log\left(\frac{\beta}{2\pi}2\sinh\left(\frac{\pi
L}{\beta}\right)\right)\leavevmode\nobreak\ &,\leavevmode\nobreak\
\leavevmode\nobreak\ t\gg L/2\,,\\\
\frac{2\tilde{c}}{3}\log\left(\frac{\beta}{2\pi}2\cosh\left(\frac{2\pi
t}{\beta}\right)\right)\leavevmode\nobreak\ &,\leavevmode\nobreak\
\leavevmode\nobreak\ t\ll L/2\,.\\\ \end{aligned}\right.$ (74)

Here, time evolution is defined by evolving forwards in time in both copies,

${|\Omega(t)\rangle}=e^{i(H_{1}+H_{2})t}{|\Omega\rangle},$ (75)

where $H_{1},H_{2}$ are the Hamiltonians of the two copies101010Evolving with
$e^{i(H_{1}-H_{2})t}$ leaves ${|\Omega\rangle}$ unchanged..

For $t\ll L/2$, (74) is equal to the length of two bulk geodesics stretching
between the two boundaries of the two-sided black hole and ending at the
endpoints of the subintervals at both boundaries. For $t\gg L/2$, on the other
hand, we have two bulk geodesics of the same kind as for the one sided case,
i.e. geodesics stretching between two endpoints of the same subinterval on one
asymptotic boundary. As for the one sided-case, these geodesics wind $k$ times
around the horizon and do not enter the black hole interior111111Since there
is a one-to-one map between geodesics in the BTZ geometry and geodesics on its
covering space (pure AdS3), we may equally well think of all of these
geodesics as geodesics in pure AdS3. Since the action of the bulk quotient
does not change the local geometry, but only the global topology, the geodesic
lengths in both cases agree..

Thus, eq. (74) shows that entwinement captures the growth of the wormhole for
a time that depends linearly on $k\propto L$. While the equality between
geodesic lengths and (74) is exact and holds for any $k\in{\mathbb{N}}$, for
entwinement $k$ is bounded from above by the condition $k\ll M\ll c$ that was
necessary for the calculation to be well-defined. Hence, the growth of
entwinement with time is limited by $M$, whose value is itself limited by the
central charge $c$. However, for large enough $k$, it still continues for a
much longer time than the growth of the entanglement entropy. The upper limit
on the time that entwinement can describe the wormhole growth imposed by $k\ll
M\ll c$ is in agreement with the expectation that quantum corrections lead to
a breakdown of the wormhole growth at late times Susskind:2014moa . Of course,
stating this argument properly requires detailed knowledge of the microstates
of the ensemble and cannot be done in the averaged description that we base
our definition of entwinement on.

We also note that in the late time limit, the entanglement entropy in the
covering theory saturates at the maximal value allowed by the positivity of
mutual information121212Since entwinement reduces to ordinary entanglement
entropy in an enlarged Hilbert space which respects positivity of mutual
information, we know that positivity of mutual information also holds for
entwinement. between the two components of the entangling interval (the
entanglement entropy of one component is given by eq. (72)). Since the CFT in
question is holographic and in the large central charge limit, the Ryu-
Takayanagi formula applies and the phase transition at time $t=L/2$ is a first
order transition.

In appendix A we show that with a slight generalization of the calculation of
Hartman:2013qma , entwinement captures the length of goedesics that not only
stretch between different asymptotic boundaries of the two-sided black hole
but also wind a non-vanishing amount of times around the spatial
circle131313Note that these geodesics are different from the winding geodesics
dual to entwinement introduced above. The winding geodesics described in sec.
5 are contained fully in one asymptotic region. In contrast, the geodesics
derived in app. A pass through the wormhole from one asymptotic region to the
other, while also winding $k$ times around the spatial circle..

Therefore, in summary we see that the definition of entwinement given in this
section implies that the dual geodesics probe a region of spacetime which is
much larger than the one probed by entanglement entropy. The size of the bulk
region probed by entwinement is limited by the value of the central charge $c$
in the boundary theory. When the dual geodesics come close to the black hole
horizon or singularity, the relation between geodesic lengths and entwinement
breaks down. This may be related to the fact that bulk quantum corrections
could be necessary to describe the geometry in these regions.

##  6 $S_{n}$ Orbifolds

A further application of the techniques for defining entanglement entropy and
entwinement of sec. 3 and 4 is possible in the D1/D5-brane system. At a
certain point in the moduli space the field theory description of this system
is a $(T^{4})^{n}/S_{n}$ symmetric product orbifold, where $n=n_{1}n_{5}$,
$n_{1}$ is the number of D1-branes, $n_{5}$ the number of D5-branes and
$c=6n_{1}n_{5}$ the central charge Strominger:1996sh ; deBoer:1998kjm ;
Seiberg:1999xz ; Larsen:1999uk . States of the $S_{n}$ orbifold are denoted by

${|X^{1},...,X^{n}\rangle}_{h},$ (76)

where $X^{i}$ is the eigenvalue of the field operator $\hat{X}^{i}$ 141414When
the difference between field operators and the corresponding eigenvalues in
the states is not obvious, we use a hat on the field operators to distinguish
them. and the $h$ subscript indicates the boundary conditions
$\hat{X}^{i}(\phi+2\pi)=\hat{X}^{h(i)}(\phi)$151515We suppress he second index
labeling the different fields in each of the $n$ copies (four free bosons and
fermions in the $(T^{4})^{n}/S_{n}$ case).. A $S_{n}$ generator $g$ acts by
permuting $\hat{X}^{i}$ with $\hat{X}^{g(i)}$. The boundary conditions change
as

$\hat{X}^{g(i)}(\phi+2\pi)=\hat{X}^{gh(i)}(\phi)\leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\
\hat{X}^{i}(\phi+2\pi)=\hat{X}^{ghg^{-1}(i)}(\phi).$ (77)

Therefore the action of $g$ is obtained as

$g{|X^{1},...,X^{n}\rangle}_{h}={|X^{g^{-1}(1)},...,X^{g^{-1}(n)}\rangle}_{ghg^{-1}},$
(78)

where we have written $g$ in a slight abuse of notation for both the
permutation operators acting on states in the Hilbert space as well as on
numbers from 1 to $n$. $S_{n}$ invariant states formed by applying the
projection operator $P=\frac{1}{n!}\sum_{g\in S_{n}}g$ necessarily contain all
states with boundary conditions in the same conjugacy class

$C(h)=\\{\,ghg^{-1}\leavevmode\nobreak\ |\leavevmode\nobreak\ g\in
S_{n}\,\\},$ (79)

so twisted sectors are labeled not by a $S_{n}$ element $h$, but by a
conjugacy class $C(h)$. The group of $S_{n}$ elements commuting with a given
$g$ is called the stabilizer subgroup $N(g)$. The sum over $S_{n}$ splits up
into a sum over $g\in C(h)$ that changes the boundary conditions and a sum
over $N(g)$ that permutes some of the fields with fixed boundary conditions
with each other.

Note that although the sum over elements of the conjugacy class $C(h)$
permutes some of the fields $\hat{X}^{i}$ with each other, it does not change
the boundary conditions for the corresponding eigenvalues $X^{i}$. The
eigenvalues in ${|X^{g^{-1}(1)},...,X^{g^{-1}(n)}\rangle}_{ghg^{-1}}$ on the
right hand side of (78) obey boundary conditions

$X^{g^{-1}(i)}(\phi+2\pi)=X^{g^{-1}ghg^{-1}(i)}(\phi)\leavevmode\nobreak\
\Leftrightarrow\leavevmode\nobreak\ X^{i}(\phi+2\pi)=X^{h(i)}(\phi)$ (80)

which are equivalent to those obeyed by the eigenvalues in
${|X^{1},...,X^{n}\rangle}_{h}$ on the left hand side of (78). The situation
in Balasubramanian:2018ajb  in which eigenvalues that were continuously
connected by the boundary conditions became disconnected due to permutations
does not occur in our case.

The orbifold Hilbert space ${\mathcal{H}}_{\text{orb}}$ is a direct sum of
Hilbert spaces consisting of elements with boundary conditions in the same
conjugacy class. Conjugacy classes for the $S_{n}$ group are specified by a
partition of $n$ into integers,

$n=\sum_{N=1}^{n}Nn_{N}.$ (81)

A generic state contains $n_{N}$ cycles of length $N$. The corresponding
stabilizer subgroups contain amongst other $S_{n}$ elements ${\mathbb{Z}}_{N}$
permutations of the fields inside all cycles of length $N$. Thus, for a state
with fixed boundary conditions $h\in S_{n}$ there exists an enlarged Hilbert
space $\tilde{\mathcal{H}}_{g}$ in which the fields inside the cycles of
length $N$ are not ${\mathbb{Z}}_{N}$ symmetrized. Embedding the density
matrix into this enlarged Hilbert space and taking partial traces for tensor
factors of $\tilde{\mathcal{H}}_{g}$ gives a definition of entwinement. From
the results of sec. 4, it is clear that this partial trace definition of
entwinement can be reproduced from a definition employing a linear subspace of
operators.

A particular example for a state of the $S_{n}$ orbifold is the one dual to
the conical defect, which is the ground state of the twisted sector in which
the fields are sewn together in $n/N$ cycles of length $N$
Balasubramanian:2005qu . In this case, $N(g)$ is given by the $S_{n}$ elements
consisting of ${\mathbb{Z}}_{N}$ permutations inside the cycles of length $N$
and $S_{n/N}$ permutations of the cycles with each other
Balasubramanian:2014sra . The enlarged non-${\mathbb{Z}}_{N}$ invariant
Hilbert space $\tilde{\mathcal{H}}_{g}$, in which the $N(g)$ invariant states
are embedded, is comprised of states invariant under the $S_{n/N}$. This
$S_{n/N}$ symmetry is the symmetry of states in the untwisted (vacuum) sector
of a $S_{n/N}$ orbifold with central charge $\tilde{c}=6n/N=c/N$. Therefore,
also for the $S_{n}$ orbifold entwinement in the state dual to the conical
defect is given by the ordinary entanglement entropy in the vacuum state of
the covering CFT with central charge $\tilde{c}=c/N$.

It would be very interesting to perform an analysis of entanglement entropy
(see Asplund:2011cq ; Giusto:2014aba ; Giusto:2015dfa ; Bombini:2019vuk  for
some work in this direction) and particularly of entwinement for microstates
of the BTZ black hole in the D1/D5-brane system. These microstates were
studied for instance in the context of the fuzzball program Skenderis:2008qn ;
Mathur:2008nj . This would provide a strong test of the proposal of sec. 5.

##  7 Conclusions

In this paper we have explored the entanglement structure of CFTs with
discrete gauge symmetries, in particular for holographic CFTs dual to conical
defects and the black holes in AdS3. We have shown that suitable
generalizations of the concept of entanglement exist for observers that can
measure only a subset of observables such as an algebra or a linear subspace
of local operators. These generalizations agree with naive calculations that
work by embedding the density matrix into a larger factorizing Hilbert space
containing unphysical states. Futhermore, the corresponding entropies are
equal to lengths of dual bulk geodesics. In this way, we obtain an explicit
construction of a in principle measureable quantity in the boundary field
theory dual to the length of a non-minimal geodesic in bulk. This enables us
to obtain the full bulk geometry, including the entanglement shadow, from the
field theory side. Thus, entwinement offers a description of the features of
the bulk geometry conjectured to be explained by complexity Susskind:2014moa ;
Susskind:2014rva .

Entwinement, the quantity that measures entanglement between non-spatially
organized degrees of freedom, has an analogue for systems of indistinguishable
particles, as already observed in Balasubramanian:2018ajb . To see this,
consider a system of $N$ indistinguishable qubits. It is not possible to
assign labels to the different qubits and to ask how a specific set of qubits,
e.g. qubit 1, 2, … $k$, is entangled with the remainder. However it is a valid
question to ask how are $k$ particles entangled with the remaining $N-k$. This
is analogous to the question entwinement aims to answer.

Applying the same techniques as in sec. 4, a generalized entanglement mesaure
$E_{k}$ between $k$ qubits and the remaining $N-k$ in analogy to entwinement
is defined as follows. Consider performing measurements with all possible
Hermitean operators that act on $k$ qubits simultaneously. Since the qubits
are indistinguishable, it is not possible to determine on which qubits these
operators act on, but it is of course well-defined to state that they act on
$k$ of them. These measurements lead to probability distributions of states.
We identify the minimum of the entropy of the probability distributions with
$E_{k}$. We say that $k$ qubits are unentangled with the remaining $N-k$ if
this entropy vanishes, i.e. if we measure a unique pure state or equivalently
if $E_{k}=0$.

This definition, which for pure states is equivalent to the more general
proposal of 2003PhRvA..68c2308B ; 2004PhRvL..92j7902B ; 2005IJTP…44.2127B ,
does not measure entanglement alone but also contains classical correlations
from global mixed states and ${\mathbb{Z}}_{N}$ symmetry of the states. In
contrast, other proposals for quantifying entanglement in systems of
indistinguishable particles (see amongst others 2001PhRvA..64e4302L ;
2002AnPhy.299…88E ; 2001PhRvA..64d2310P ; 2014PhRvL.112o0501K ;
2003PhRvA..67b4301S ; 2003PhRvL..91i7902W ; 2004PhRvA..70a2109G ;
2009PhRvL.102q0503T ) often aim at differentiating correlations due to
statistics and global mixed states from genuine quantum correlations (see e.g.
2002AnPhy.299…88E ). Thus, the analogue of entwinement described above is not
equivalent to such proposals. Instead of measuring only the entropy of quantum
correlations, it measures the entropy of all correlations observable from
measurements acting on $k$ qubits simultaneously. The motivation for this
definition comes from ordinary entanglement entropy which – although it
measures a mixture of classical and quantum correlations – has a simply
geometric bulk dual for holographic CFTs.

We also proposed a definition of entwinement for thermal states on a circle
dual to the BTZ black hole. Like entwinement for the state dual to the conical
defect, the definition reduces to the entanglement entropy in a covering CFT
on an $M$ times larger spatial circle with central charge $c/M$. We take the
covering theory to be also in a thermal state. Then, entwinement calculates
the length of a non-minimal geodesic in the bulk that probes the entanglement
shadow around the event horizon of the black hole. Furthermore, generalizing
to the two-sided black hole case, the results of Hartman:2013qma  imply that
entwinement also captures the growth of the wormhole with time in the dual
bulk geometry.

While this definition yields a simple calculation procedure for entwinement,
it rests on some assumptions about the microscopic nature of the degrees of
freedom of the CFT state dual to the BTZ black hole. Namely we assume that
there exist some $M\in{\mathbb{N}}$ such that the fields of the theory can be
separated into $M$ subsets that are continuously connected by the boundary
conditions, i.e. we assume that the state is in an approriate twisted sector
of an orbifold. The black hole is dual to an ensemble of states, so this
condition needs to hold only after averaging over the elements of the
ensemble. The larger the maximum possible value of $M$ is, the more of the
bulk geometry is probed by the geodesics dual to entwinement. It is an
important next step to check this assumption in concrete examples. An obvious
candidate for this is the $S_{n}$ orbifold dual to the D1-D5 system, for which
a large number of microstates have been identified as part of the fuzzball
program (see Skenderis:2008qn ; Mathur:2008nj  for a review).

However if these assumptions do hold, the results summarized in sec. 5 imply
that entwinement probes a subset of the bulk spacetime that is much larger
than that probed by the entanglement entropy of spatial subregions. In
particular, a striking implication is that entwinement is sensitive to the
growth of the wormhole in the two-sided BTZ spacetime – and thus sensitive to
physics inside the horizon – for a much longer time than entanglement entropy
of spatial subregions. Hence, the assertion that “entanglement is not enough”
Susskind:2014moa  to describe the wormhole growth in two-sided black hole
geometries is called into question, at least if one includes entanglement of
non-spatially organized degrees of freedom. In this way, entwinement provides
an alternative to complexity proposed in Susskind:2014moa ; Susskind:2014rva
that for AdS3/CFT2 could describe the same physics. The advantage of
entwinement is that unlike for complexity not only its definition in the
boundary CFT is clear, but also its bulk dual. In particular there are no free
parameters in our proposal, whereas for complexity a clear picture for which
reference state and gate set to choose to match holographic complexity
definitions is still missing.

Our work opens the door to a large number of future directions. First of all,
it would be interesting to apply the notion of generalized entanglement to
other examples such as higher dimensional asymptotically AdS geometries or
more exotic bulk geometries such as the Bañados geometry Banados:1998gg . A
further direction worth exploring is a possible generalization of entwinement
to find a dual for geodesics not affixed to a constant time slice, in the
spirit of the HRT proposal Hubeny:2007xt  (see also Hubeny:2014qwa  for the
related concept of residual entropy).

Moreover, for the state dual to the conical defect, the derived field theory
dual to the lengths of non-minimal geodesics completes the field-theory
description of the kinematic space of Czech:2015qta  comprised of boundary-
anchored geodesics. This kinematic space provides the basis for the
reconstruction of geometric objects in the bulk from field-theory data using
methods from integral geometry. Our results thus pave the way to a new
description of geometric objects such as curves, points or areas in the
conical defect in terms of a field-theoretic quantity. In particular, they
allow for a derivation of a field-theory dual to holographic complexity for
the conical defect, in generalization of the results of Abt:2017pmf ;
Abt:2018ywl  that for the vacuum state provide an expression of the volume in
the complexity=volume proposal in terms of the entanglement entropy, using the
kinematic space approach.

A further direction to explore is a possible application of entwinement to the
explanation of quantum chaos from black holes in holography (see e.g.
Shenker:2013pqa ; Shenker:2013yza ; Leichenauer:2014nxa ; Roberts:2014isa ).
The generalized entanglement of 2005IJTP…44.2127B ; 2003PhRvA..68c2308B
equivalent to our entwinement definition for pure states was already proposed
to be a good indicator of quantum chaos 2006EL…..76..746W .

###### Acknowledgements.

We would like to thank Mari-Carmen Bañuls, Pascal Fries, René Meyer, Christian
Northe and Ignacio A. Reyes for discussions. MG acknowledges financial support
from the DFG through Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 39085490).

##  Appendix A Winding geodesics in the two-sided black hole geometry

We now derive that entwinement as defined in sec. 5 can as well compute the
length of geodesics between different asymptotic boundaries of the wormhole in
a two-sided BTZ black hole geometry with non-vanishing winding number around
the spatial circle. We first consider the field theory calculation in
generalization of Hartman:2013qma  and then the dual geodesics.

###  A.1 Field-theory calculation

Correlation functions in the TFD state ${|\Omega(t)\rangle}$ are calculated by
an analytical continuation from Euclidean correlation functions on a
background with periodic time direction Hartman:2013qma . Operators $\phi_{1}$
on the first boundary are inserted at $\tau=0$, while operators $\phi_{2}$ on
the second boundary are inserted at $\tau=i\beta/2$ and analytically continued
to $\tau=i\beta/2-2t$.

Entwinement is given by the entanglement entropy in the covering space, which
we will calculate from the Rényi entropy. This entropy is given by a
correlation function of twist operators,

${\mathrm{Tr}}\rho_{A}^{n}=\langle\sigma_{n}(z_{1},\bar{z}_{1})\tilde{\sigma}_{n}(z_{2},\bar{z}_{2})\tilde{\sigma}_{n}(z_{3},\bar{z}_{3})\sigma_{n}(z_{4},\bar{z}_{4})\rangle.$
(82)

We want to calculate the most general entanglement entropy of an entangling
interval consisting of two parts on both asymptotic boundaries. Therefore, the
entangling interval consist of a part with size $L_{1}$ on the first boundary
and another part on the second boundary with size $L_{2}$ and relative
position $\Delta L$ compared to the first part (see fig. 3). Thus,

$\displaystyle z_{1}=\bar{z}_{1}=0\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle z_{3}=L_{2}+\Delta
L+2t+i\beta/2\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{z}_{3}=L_{2}+\Delta L-2t-i\beta/2$ (83) $\displaystyle
z_{2}=\bar{z}_{2}=L_{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle z_{4}=-\bar{z}_{4}=\Delta L+2t+i\beta/2.$

$\phi$$\tau$$i\beta$$L$ | $\phi$$\tau$$i\beta$$\Delta L$$L_{2}$$L_{1}$
---|---
Figure 3: Twist operator insertions on the Euclidean cylinder for the
computation of the entanglement entropy of an interval which consists of two
disconnected components on the two asymptotic boundaries. LHS: computation of
Hartman:2013qma , both components of the entangling interval are of the same
size and at the same position; RHS: position and size of both parts is
arbitrary.

Mapping to the plane with $w=e^{2\pi z/\beta}$ and performing a conformal
transformation maps the insertions to $0,x,1,\infty$, where $x$ is the
conformally invariant cross-ratio. For $L_{1}+L_{2}\gg|L_{1}-L_{2}|$ and
$L_{1}+L_{2}\gg|L_{1}-L_{2}-2\Delta L|$, $x$ asymptotes to 0 for late times
($t\gg|L_{1}-L_{2}-2\Delta L|/4$) and to 1 for early times
($t\ll|L_{1}-L_{2}-2\Delta L|/4$). Expanding the four-point function in
conformal blocks and the blocks in a power series in $x$ resp. $1-x$, we see
that the leading term is

$\langle\sigma_{n}(0,0)\tilde{\sigma}_{n}(x,\bar{x})\tilde{\sigma}_{n}(1,1)\sigma_{n}(\infty,\infty)\rangle=\left\\{\begin{aligned}
x^{-2h}\bar{x}^{-2h}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ t\gg|L_{1}-L_{2}-2\Delta L|/4\\\
(1-x)^{-2h}(1-\bar{x})^{-2h}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ t\ll|L_{1}-L_{2}-2\Delta L|/4\end{aligned}\right.$ (84)

where $h=\bar{h}=\frac{\tilde{c}}{24}(n-1/n)$ is the conformal weight of the
twist operators. Putting everything together, we obtain

$E^{i}_{k}=\left\\{\begin{aligned}
S_{1}(L_{1})+S_{1}(L_{2})\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ t\gg|L_{1}-L_{2}-2\Delta L|/4\\\ S_{2}(\Delta
L)+S_{2}(\Delta L+L_{2}-L_{1})\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ t\ll|L_{1}-L_{2}-2\Delta L|/4\\\ \end{aligned}\right.,$
(85)

where

$S_{1}(L)=\frac{\tilde{c}}{3}\log\left(\frac{\beta}{2\pi}2\sinh\left(\frac{\pi
L}{\beta}\right)\right)$ (86)

is equal to the entanglement entropy for an interval of length $L$ in the
thermal state and proportional to the length of a geodesic whose endpoints lie
on the same asymptotic boundary whereas

$S_{2}(L)=\frac{\tilde{c}}{6}\log\left[\left(\frac{\beta}{2\pi}\right)^{2}\left(2\cosh\left(\frac{2\pi
L}{\beta}\right)+2\cosh\left(\frac{4\pi t}{\beta}\right)\right)\right]$ (87)

is proportional to the length of a geodesic stretching between the two
asymptotic boundaries and winding $\lfloor L/2\pi\rfloor$ times around the
black hole, as we will see in the next subsection. The behaviour changes if
the conditions $L_{1}+L_{2}\gg|L_{1}-L_{2}|,|L_{1}-L_{2}-2\Delta L|$ are not
fulfilled, e.g. for $L_{1}+L_{2}\gg|L_{1}-L_{2}|$ but
$L_{1}+L_{2}\ll|L_{1}-L_{2}-2\Delta L|$ (large distance in the relative
position) entwinement is given by $S_{1}(L_{1})+S_{2}(L_{2})$ for all times.

The observed growth of entanglement entropy in a thermofield double state on a
line is in fact universal for every CFT. The expansion in eq. (84) for early
and late time is independent of the field content of the CFT.

###  A.2 Dual geodesics

To derive the length of geodesics dual to entwinement, we only need to
calculate them in the covering geometry of the black hole geometry (i.e. pure
AdS3), since the BTZ identification maps geodesics to geodesics. Pure AdS3 is
given by the surface

$-Y_{-1}^{2}-Y_{0}^{2}+Y_{1}^{2}+Y_{2}^{2}=-1$ (88)

in the embedding space ${\mathbb{R}}^{2,2}$ with coordinates $Y_{i}$. The time
parameter which we have used in the previous calculations is defined w.r.t.
global AdS3 coordinates given by

$\displaystyle Y_{-1}=\cosh x\cosh\rho\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{0}=\sinh t\sinh\rho$ (89) $\displaystyle Y_{2}=-\sinh
x\cosh\rho\leavevmode\nobreak\ \leavevmode\nobreak$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ Y_{1}=\cosh
t\sinh\rho.$

For simplicity, we will calculate the length of geodesics in Poincaré patch
coordinates

$\displaystyle
Y_{-1}=\frac{1}{2z}(1+(z^{2}-x_{0}^{2}+x_{1}^{2}))\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{0}=\frac{x_{0}}{z}$ (90) $\displaystyle
Y_{2}=\frac{1}{2z}(1-(z^{2}-x_{0}^{2}+x_{1}^{2}))\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{1}=\frac{x_{1}}{z}.$

Finally, the winding number is defined w.r.t the usual BTZ coordinates

$\displaystyle Y_{-1}=\frac{r}{r_{+}}\cosh(r_{+}\phi)\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{0}=\sqrt{\frac{r^{2}}{r_{+}^{2}}-1}\sinh(r_{+}\tau)$ (91) $\displaystyle
Y_{2}=\sqrt{\frac{r^{2}}{r_{+}^{2}}-1}\cosh(r_{+}\tau)\leavevmode\nobreak\
\leavevmode\nobreak$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
Y_{1}=\frac{r}{r_{+}}\sinh(r_{+}\phi).$

The event horizon is located at $r_{+}=\frac{2\pi}{\beta}$. To simplify the
notation, we set $r_{+}=1\Leftrightarrow\beta=2\pi$ in the following. The
point $(t,x,\rho)=(t_{b},L,\infty)$ maps to $(x_{0},x_{1},z)=(e^{-L}\sinh
t_{b},\pm e^{-L}\cosh t_{b},0)$ Hartman:2013qma , where the sign in the
$x_{1}$ coordinate depends on which boundary of the wormhole the point is
located at. Therefore, we have to calculate the length of geodesics between
the points

$\displaystyle P_{1}=e^{-\Delta L}(\sinh t_{b},-\cosh
t_{b},0)\leavevmode\nobreak\ \leavevmode\nobreak$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ P_{3}=e^{-\Delta
L-L_{2}}P_{1}$ (92) $\displaystyle P_{2}=(\sinh t_{b},\cosh
t_{b},0)\leavevmode\nobreak\ \leavevmode\nobreak$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ P_{4}=e^{-L_{1}}P_{2}$

in Poincaré patch coordinates. Geodesics between two points
$P^{(1)}=(x_{0}^{(1)},x_{1}^{(1)},0)$ and
$P^{(2)}=(x_{0}^{(2)},x_{1}^{(2)},0)$ in this coordinate system are semi-
circles that extend in the bulk up to a maximal radial coordinate
$z^{*}=\frac{1}{2}\sqrt{(x_{1}^{(1)}-x_{1}^{(2)})^{2}-(x_{0}^{(1)}-x_{0}^{(2)})^{2}}$.
The length of a geodesic is only dependent on $z^{*}$ and the UV cutoff
$\epsilon$,

$l_{\gamma}=2\int_{\epsilon}^{z^{*}}\frac{dz}{z}\frac{1}{\sqrt{1-(z/z^{*})^{2}}}\approx
2\log\frac{2z^{*}}{\epsilon}.$ (93)

The dual bulk geodesics can either connect $P_{1}$ with $P_{3}$ and $P_{2}$
with $P_{4}$ or $P_{1}$ with $P_{2}$ and $P_{3}$ with $P_{4}$. Given an
appropriate cutoff and restoring the $\beta$ dependence, we obtain

$\displaystyle l_{P_{1}\leftrightarrow P_{3}}$
$\displaystyle\sim\log\left(\sinh\left(\frac{\pi L_{1}}{\beta}\right)\right)$
(94) $\displaystyle l_{P_{2}\leftrightarrow P_{4}}$
$\displaystyle\sim\log\left(\sinh\left(\frac{\pi L_{2}}{\beta}\right)\right)$
$\displaystyle l_{P_{1}\leftrightarrow P_{2}}$
$\displaystyle\sim\log\left(\cosh\left(\frac{2\pi\Delta
L}{\beta}\right)+\cosh\left(\frac{4\pi t_{b}}{\beta}\right)\right)$
$\displaystyle l_{P_{3}\leftrightarrow P_{4}}$
$\displaystyle\sim\log\left(\cosh\left(\frac{2\pi(\Delta
L+L_{2}-L_{1})}{\beta}\right)+\cosh\left(\frac{4\pi
t_{b}}{\beta}\right)\right).$

The sum of these lengths $l_{P_{1}\leftrightarrow
P_{3}}+l_{P_{2}\leftrightarrow P_{4}}$ and $l_{P_{1}\leftrightarrow
P_{2}}+l_{P_{3}\leftrightarrow P_{4}}$ is equal - up to non-universal cutoff
dependent factors - to the entwinement result (85). At early times
$|t|<t_{b}^{*}$, the $l_{P_{1}\leftrightarrow P_{2}}$,
$l_{P_{3}\leftrightarrow P_{4}}$ geodesics form the Ryu-Takayanagi surface,
while at late times $|t|>t_{b}^{*}$ the $l_{P_{1}\leftrightarrow P_{3}}$,
$l_{P_{2}\leftrightarrow P_{4}}$ geodesics dominate. The case $t_{b}^{*}=0$ in
which the RT surface is constant for all times is also possible, for example
for large $\Delta L$ and $L_{1}\approx L_{2}$. To derive the winding number of
the geodesics, we use the coordinate transformation to BTZ coordinates

$\cosh(r_{+}\phi)=\frac{1}{2}\left(\frac{1}{\sqrt{z^{2}-x_{0}^{2}+x_{1}^{2}}}+\sqrt{z^{2}-x_{0}^{2}+x_{1}^{2}}\right).$
(95)

Thus $\phi_{P_{1}}=-\Delta L$, $\phi_{P_{2}}=0$, $\phi_{P_{3}}=-\Delta
L-L_{2}$, $\phi_{P_{4}}=-L_{1}$. We use the difference $|\Delta\phi|$ in the
$\phi$ coordinate at the endpoints of a geodesic to define its winding number
$w=\lfloor|\Delta\phi|/2\pi\rfloor$. At times $t_{b}<t_{b}^{*}$, the winding
numbers are given by $w_{P_{1}\leftrightarrow P_{2}}=\lfloor\Delta
L/2\pi\rfloor$ and $w_{P_{3}\leftrightarrow P_{4}}=\lfloor(\Delta
L+L_{2}-L_{1})/2\pi\rfloor$, while after the phase transition at
$t_{b}>t_{b}^{*}$ the geodesics wind $\lfloor L_{1,2}/2\pi\rfloor$ times
around the spatial circle.

## References

  * (1) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113–1133, [hep-th/9711200].
  * (2) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602, [hep-th/0603001].
  * (3) B. Swingle, _Entanglement Renormalization and Holography_ , _Phys. Rev._ D86 (2012) 065007, [0905.1317].
  * (4) M. Van Raamsdonk, _Building up spacetime with quantum entanglement_ , _Gen. Rel. Grav._ 42 (2010) 2323–2329, [1005.3035].
  * (5) E. Bianchi and R. C. Myers, _On the Architecture of Spacetime Geometry_ , _Class. Quant. Grav._ 31 (2014) 214002, [1212.5183].
  * (6) V. Balasubramanian, B. D. Chowdhury, B. Czech and J. de Boer, _Entwinement and the emergence of spacetime_ , _JHEP_ 01 (2015) 048, [1406.5859].
  * (7) B. Freivogel, R. Jefferson, L. Kabir, B. Mosk and I.-S. Yang, _Casting Shadows on Holographic Reconstruction_ , _Phys. Rev._ D91 (2015) 086013, [1412.5175].
  * (8) V. E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, _Holographic entanglement plateaux_ , _JHEP_ 08 (2013) 092, [1306.4004].
  * (9) L. Susskind, _Entanglement is not enough_ , _Fortsch. Phys._ 64 (2016) 49–71, [1411.0690].
  * (10) L. Susskind, _Computational Complexity and Black Hole Horizons_ , _Fortsch. Phys._ 64 (2014) 44–48, [1403.5695].
  * (11) T. Hartman and J. Maldacena, _Time Evolution of Entanglement Entropy from Black Hole Interiors_ , _JHEP_ 05 (2013) 014, [1303.1080].
  * (12) D. Harlow, _The Ryu–Takayanagi Formula from Quantum Error Correction_ , _Commun. Math. Phys._ 354 (2017) 865–912, [1607.03901].
  * (13) M. Ohya and D. Petz, _Quantum entropy and its use_.  Texts and monographs in physics. Springer, Berlin u.a., 1993.
  * (14) V. Balasubramanian, B. Craps, T. De Jonckheere and G. Sárosi, _Entanglement versus entwinement in symmetric product orbifolds_ , _JHEP_ 01 (2019) 190, [1806.02871].
  * (15) D. Stanford and L. Susskind, _Complexity and Shock Wave Geometries_ , _Phys. Rev._ D90 (2014) 126007, [1406.2678].
  * (16) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Holographic Complexity Equals Bulk Action?_ , _Phys. Rev. Lett._ 116 (2016) 191301, [1509.07876].
  * (17) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Complexity, action, and black holes_ , _Phys. Rev._ D93 (2016) 086006, [1512.04993].
  * (18) S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski, _Toward a Definition of Complexity for Quantum Field Theory States_ , _Phys. Rev. Lett._ 120 (2018) 121602, [1707.08582].
  * (19) R. Jefferson and R. C. Myers, _Circuit complexity in quantum field theory_ , _JHEP_ 10 (2017) 107, [1707.08570].
  * (20) R. Khan, C. Krishnan and S. Sharma, _Circuit Complexity in Fermionic Field Theory_ , _Phys. Rev._ D98 (2018) 126001, [1801.07620].
  * (21) L. Hackl and R. C. Myers, _Circuit complexity for free fermions_ , _JHEP_ 07 (2018) 139, [1803.10638].
  * (22) S. Chapman, J. Eisert, L. Hackl, M. P. Heller, R. Jefferson, H. Marrochio et al., _Complexity and entanglement for thermofield double states_ , 1810.05151.
  * (23) K. Goto, H. Marrochio, R. C. Myers, L. Queimada and B. Yoshida, _Holographic Complexity Equals Which Action?_ , _JHEP_ 02 (2019) 160, [1901.00014].
  * (24) V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere and F. Galli, _Entwinement in discretely gauged theories_ , _JHEP_ 12 (2016) 094, [1609.03991].
  * (25) P. H. Ginsparg, _APPLIED CONFORMAL FIELD THEORY_ , in _Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena Les Houches, France, June 28-August 5, 1988_ , pp. 1–168, 1988, hep-th/9108028.
  * (26) L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, _The Conformal Field Theory of Orbifolds_ , _Nucl. Phys._ B282 (1987) 13–73.
  * (27) P. V. Buividovich and M. I. Polikarpov, _Entanglement entropy in gauge theories and the holographic principle for electric strings_ , _Phys. Lett._ B670 (2008) 141–145, [0806.3376].
  * (28) W. Donnelly, _Decomposition of entanglement entropy in lattice gauge theory_ , _Phys. Rev._ D85 (2012) 085004, [1109.0036].
  * (29) H. Casini, M. Huerta and J. A. Rosabal, _Remarks on entanglement entropy for gauge fields_ , _Phys. Rev._ D89 (2014) 085012, [1312.1183].
  * (30) D. Radicevic, _Notes on Entanglement in Abelian Gauge Theories_ , 1404.1391.
  * (31) S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, _On the definition of entanglement entropy in lattice gauge theories_ , _JHEP_ 06 (2015) 187, [1502.04267].
  * (32) S. Ghosh, R. M. Soni and S. P. Trivedi, _On The Entanglement Entropy For Gauge Theories_ , _JHEP_ 09 (2015) 069, [1501.02593].
  * (33) R. M. Soni and S. P. Trivedi, _Aspects of Entanglement Entropy for Gauge Theories_ , _JHEP_ 01 (2016) 136, [1510.07455].
  * (34) E. Witten, _APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory_ , _Rev. Mod. Phys._ 90 (2018) 045003, [1803.04993].
  * (35) E. A. Mazenc and D. Ranard, _Target Space Entanglement Entropy_ , 1910.07449.
  * (36) M. A. Nielsen and I. L. Chuang, _Quantum computation and quantum information_.  Cambridge Univ. Press, Cambridge, 2007.
  * (37) H. Barnum, G. Ortiz, R. Somma and L. Viola, _A Generalization of Entanglement to Convex Operational Theories: Entanglement Relative to a Subspace of Observables_ , _International Journal of Theoretical Physics_ 44 (Dec, 2005) 2127–2145, [quant-ph/0506099].
  * (38) H. Barnum, E. Knill, G. Ortiz and L. Viola, _Generalizations of entanglement based on coherent states and convex sets_ , _Phys. Rev._ A68 (Sep, 2003) 032308, [quant-ph/0207149].
  * (39) M. Banados, C. Teitelboim and J. Zanelli, _The Black hole in three-dimensional space-time_ , _Phys. Rev. Lett._ 69 (1992) 1849–1851, [hep-th/9204099].
  * (40) J. M. Maldacena and A. Strominger, _AdS(3) black holes and a stringy exclusion principle_ , _JHEP_ 12 (1998) 005, [hep-th/9804085].
  * (41) P. Calabrese and J. L. Cardy, _Entanglement entropy and quantum field theory_ , _J. Stat. Mech._ 0406 (2004) P06002, [hep-th/0405152].
  * (42) T. Barrella, X. Dong, S. A. Hartnoll and V. L. Martin, _Holographic entanglement beyond classical gravity_ , _JHEP_ 09 (2013) 109, [1306.4682].
  * (43) J. M. Maldacena, _Eternal black holes in anti-de Sitter_ , _JHEP_ 04 (2003) 021, [hep-th/0106112].
  * (44) A. Strominger and C. Vafa, _Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. Lett._ B379 (1996) 99–104, [hep-th/9601029].
  * (45) J. de Boer, _Six-dimensional supergravity on S**3 x AdS(3) and 2-D conformal field theory_ , _Nucl. Phys._ B548 (1999) 139–166, [hep-th/9806104].
  * (46) N. Seiberg and E. Witten, _The D1 / D5 system and singular CFT_ , _JHEP_ 04 (1999) 017, [hep-th/9903224].
  * (47) F. Larsen and E. J. Martinec, _U(1) charges and moduli in the D1 - D5 system_ , _JHEP_ 06 (1999) 019, [hep-th/9905064].
  * (48) V. Balasubramanian, P. Kraus and M. Shigemori, _Massless black holes and black rings as effective geometries of the D1-D5 system_ , _Class. Quant. Grav._ 22 (2005) 4803–4838, [hep-th/0508110].
  * (49) C. T. Asplund and S. G. Avery, _Evolution of Entanglement Entropy in the D1-D5 Brane System_ , _Phys. Rev._ D84 (2011) 124053, [1108.2510].
  * (50) S. Giusto and R. Russo, _Entanglement Entropy and D1-D5 geometries_ , _Phys. Rev._ D90 (2014) 066004, [1405.6185].
  * (51) S. Giusto, E. Moscato and R. Russo, _AdS 3 holography for 1/4 and 1/8 BPS geometries_, _JHEP_ 11 (2015) 004, [1507.00945].
  * (52) A. Bombini and G. Fardelli, _Holographic entanglement entropy and complexity of microstate geometries_ , 1910.01831.
  * (53) K. Skenderis and M. Taylor, _The fuzzball proposal for black holes_ , _Phys. Rept._ 467 (2008) 117–171, [0804.0552].
  * (54) S. D. Mathur, _Fuzzballs and the information paradox: A Summary and conjectures_ , 0810.4525.
  * (55) H. Barnum, E. Knill, G. Ortiz, R. Somma and L. Viola, _A Subsystem-Independent Generalization of Entanglement_ , _Phys. Rev. Lett._ 92 (Mar, 2004) 107902, [quant-ph/0305023].
  * (56) Y. S. Li, B. Zeng, X. S. Liu and G. L. Long, _Entanglement in a two-identical-particle system_ , _Phys. Rev._ A64 (Nov, 2001) 054302, [quant-ph/0104101].
  * (57) K. Eckert, J. Schliemann, D. Bruß and M. Lewenstein, _Quantum Correlations in Systems of Indistinguishable Particles_ , _Annals of Physics_ 299 (Jul, 2002) 88–127, [quant-ph/0203060].
  * (58) R. Paškauskas and L. You, _Quantum correlations in two-boson wave functions_ , _Phys. Rev._ A64 (Oct, 2001) 042310, [quant-ph/0106117].
  * (59) N. Killoran, M. Cramer and M. B. Plenio, _Extracting Entanglement from Identical Particles_ , _Phys. Rev. Lett._ 112 (Apr, 2014) 150501, [1312.4311].
  * (60) Y. Shi, _Quantum entanglement of identical particles_ , _Phys. Rev._ A67 (Feb, 2003) 024301, [quant-ph/0205069].
  * (61) H. M. Wiseman and J. A. Vaccaro, _Entanglement of Indistinguishable Particles Shared between Two Parties_ , _Phys. Rev. Lett._ 91 (Aug, 2003) 097902, [quant-ph/0210002].
  * (62) G. Ghirardi and L. Marinatto, _General criterion for the entanglement of two indistinguishable particles_ , _Phys. Rev._ A70 (Jul, 2004) 012109, [quant-ph/0401065].
  * (63) G. Tóth and O. Gühne, _Entanglement and Permutational Symmetry_ , _Phys. Rev. Lett._ 102 (May, 2009) 170503, [0812.4453].
  * (64) M. Banados, _Three-dimensional quantum geometry and black holes_ , _AIP Conf. Proc._ 484 (1999) 147–169, [hep-th/9901148].
  * (65) V. E. Hubeny, M. Rangamani and T. Takayanagi, _A Covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062, [0705.0016].
  * (66) V. E. Hubeny, _Covariant Residual Entropy_ , _JHEP_ 09 (2014) 156, [1406.4611].
  * (67) B. Czech, L. Lamprou, S. McCandlish and J. Sully, _Integral Geometry and Holography_ , _JHEP_ 10 (2015) 175, [1505.05515].
  * (68) R. Abt, J. Erdmenger, H. Hinrichsen, C. M. Melby-Thompson, R. Meyer, C. Northe et al., _Topological Complexity in AdS 3/CFT2_, _Fortsch. Phys._ 66 (2018) 1800034, [1710.01327].
  * (69) R. Abt, J. Erdmenger, M. Gerbershagen, C. M. Melby-Thompson and C. Northe, _Holographic Subregion Complexity from Kinematic Space_ , _JHEP_ 01 (2019) 012, [1805.10298].
  * (70) S. H. Shenker and D. Stanford, _Black holes and the butterfly effect_ , _JHEP_ 03 (2014) 067, [1306.0622].
  * (71) S. H. Shenker and D. Stanford, _Multiple Shocks_ , _JHEP_ 12 (2014) 046, [1312.3296].
  * (72) S. Leichenauer, _Disrupting Entanglement of Black Holes_ , _Phys. Rev._ D90 (2014) 046009, [1405.7365].
  * (73) D. A. Roberts, D. Stanford and L. Susskind, _Localized shocks_ , _JHEP_ 03 (2015) 051, [1409.8180].
  * (74) Y. S. Weinstein and L. Viola, _Generalized entanglement as a natural framework for exploring quantum chaos_ , _EPL (Europhysics Letters)_ 76 (Dec, 2006) 746–752, [quant-ph/0603071].