# Target Space Entanglement Entropy

Edward Mazenc & Daniel Ranard

###### Abstract

We define a notion of target space entanglement entropy. Rather than
partitioning the base space on which the theory is defined, we consider
partitions of the target space. This is the physical case of interest for
first-quantized theories, such as worldsheet string theory. We associate to
each subregion of the target space a suitably chosen sub-algebra of
observables $\mathcal{A}$. The entanglement entropy is calculated as the
entropy of the density matrix restricted to $\mathcal{A}$. As an example, we
illustrate our framework by computing spatial entanglement in first-quantized
many-body quantum mechanics. The algebra $\mathcal{A}$ is chosen to reproduce
the entanglement entropy obtained by embedding the state in the fixed particle
sub-sector of the second-quantized Hilbert space. We then generalize our
construction to the quantum field-theoretical setting.

Department of Physics, Stanford University,
Stanford, CA 94305-4060, USA

###### Contents

  1. 1 Introduction
  2. 2 Preview: Particle on a line & the need for subalgebras
  3. 3 Review of the algebraic definition of entanglement entropy
  4. 4 Algebraic entanglement entropy for first-quantized systems
    1. 4.1 Single-particle warm-up
    2. 4.2 General target spaces
    3. 4.3 Multiple indistinguishable particles
      1. 4.3.1 Bosons
      2. 4.3.2 Fermions
  5. 5 Comparison with embedding into second-quantized theory
  6. 6 Competing notions of locality in relativistic field theories
  7. 7 Computation of entanglement entropy for finite-particle states: “Fock”- vs. “field”-based factorization
    1. 7.1 Proof
  8. 8 Target space entanglement entropy for field theories
  9. 9 Finiteness of the entanglement entropy
  10. 10 Discussion
    1. 10.1 Worldlines, worldsheets & reparametrization invariance
    2. 10.2 Different factorizations and the c=1 matrix quantum mechanics
  11. 11 Acknowledgements
  12. 12 Appendix: The algebra for bosons

##  1 Introduction

Previous studies of entanglement in field theory mostly address entanglement
with respect to partitions of the base space [1, 2]. Recall that in quantum
field theory we speak of both the base space and the target space. For
instance, in standard $d+1$ scalar field theory, the field $\phi(\vec{x})$
take values in the target space $\mathbb{R}$. The base space $\mathbb{R}^{d}$
parametrizes instead which degree of freedom we are speaking of: it labels the
$\vec{x}\in\mathbb{R}^{d}$ of $\phi(\vec{x})$. Colloquially, we refer to the
base space as the space the field “lives on.” To calculate the entanglement
entropy of a spatial partition, we partition the base space $\mathbb{R}^{d}$;
see Figure 1.

Meanwhile, in promising theories of quantum gravity, a “spatial” partition may
not be associated with a partition of the base space, but rather a partition
of the target space. For example, in first-quantized string theory, a
spacetime subregion corresponds to a restriction of the embedding coordinates
of the string. It is thus a partition of the target space, not a partition of
the base worldsheet.

Figure 1: Distinction between partitions of the base space (left) versus
target space (right). We illustrate the case of 2+1d scalar field theory for
concreteness. At every point of the base space ($\cong\mathbb{R}^{2}$) labeled
by $\vec{x}$, the local degree of freedom takes values in the target space
($\cong\mathbb{R}$) with coordinate $\phi$. Previous studies of entanglement
entropy have focused on subregions of the base space where the values of
$\vec{x}$ are restricted. Instead, we consider restrictions on the values of
the field $\phi$.

Likewise, Matrix theory is a $0+1$ dimensional theory that describes, in a
particular frame, quantum gravity in an 11-dimensional spacetime [3]. In this
case, the “base space” is nothing more than a point. Clearly, it is senseless
to partition it. Instead, subregions of the physical spacetime correspond to a
subspace of the moduli space of $D0$ branes [4, 5].

Motivated by these examples, the main goal of this paper is to define reduced
density matrices and entanglement entropies of states with respect to
subregions in target space. The first challenge is that target space
partitions do not correspond to tensor factorizations of the Hilbert space,
whereas the usual framework for entanglement entropy hinges upon such a
factorization. We therefore leverage the powerful algebraic framework, which
defines a reduced density matrix relative to a subalgebra of observables, and
treats tensor factorizations as a special case. This algebraic framework is by
no means new, though it has only recently gained widespread traction in the
high energy community via the work of [6, 7, 8], which offer excellent
introductions to the subject. Our task therefore reduces to finding which
subalgebra most accurately reflects an agent having access only to observables
confined to the spatial subregion of interest. We then define target space
entanglement entropy as the entropy of the state restricted to this
subalgebra.

Figure 2: (Top) For many of our most promising theories of quantum gravity,
such as worldsheeet string theory or the BFSS matrix quantum mechanics, the
emergent physical spacetime of interest is encoded in the target space of the
theory. (Bottom) To each subregion of the target space, we associate a
particular subalgebra of observables $\mathcal{A}$. The reduced density matrix
$\rho_{\mathcal{A}}$ is defined via restriction of the state to the subalgebra
$\mathcal{A}$. We define the target space entanglement entropy as the entropy
of $\rho_{\mathcal{A}}$.

A single particle on a line furnishes the simplest toy model. We may think of
the position of the particle $x(t)$ as 0+1-dimensional QFT. The base space is
a single point, and the target space $\mathbb{R}$ is the physical space the
particle is moving in. To define a notion of spatial entanglement, we must
partition the “target space.” First-quantized many body quantum mechanics
provides the ideal testing ground for our proposed definition because we have
a firm grasp on its second quantized formulation - plain old QFT - where we
understand entanglement entropy well. Making a similar comparison in the
string theory context would require the intricacies of string field theory
(see [9] for recent work directly in that context).

In the non-relativistic case, our definition of entanglement entropy for the
above quantum-mechanical system agrees with the the standard field theory
definition. For relativistic quantum field theory, we find two ostensibly
natural notions of locality and discuss their relative merits. An explicit
computation for one-particle excited states shows to what extent the
entanglement entropy associated to these different “spatial” partitionings can
be compared.

Further, we stress our framework is by no means limited to quantum mechanics.
We generalize our construction to partitions of the target spaces of arbitrary
sigma models and interacting field theories. We compute the entanglement
entropy for a half (target) space partition in the the simplest example, a
massive scalar field on two spatial lattice sites.

We conclude by discussing the important role played by reparametrization
invariance in theories such as worldsheet string theory and point out the
limitations of our framework. We then apply the lessons learned in Section 6
to the case of 2d string theory and the “baby cousin” of the AdS/CFT
correspondence, the holographic $c=1$ matrix model. There, we face multiple
notions of emergent locality [5], and we sketch the possible role of different
factorizations of the Hilbert space.

##  2 Preview: Particle on a line & the need for subalgebras

To handle partitions of target space, we will need the algebraic framework for
entanglement entropy. However, let us motivate it further by exploring in more
detail the example of particle on a line.

The Hilbert space
$L^{2}(\mathbb{R})=\textrm{span}\\{|x\rangle:x\in\mathbb{R}\\}$ may also be
considered as the Hilbert space of a 0+1-dimensional QFT, where the base space
is a single point, and
$L^{2}(\mathbb{R})=\textrm{span}\\{|\phi\rangle:\phi\in\mathbb{R}\\}$ is the
space of field values $\phi$ at that point. From the latter perspective, we
will call $\mathbb{R}$ the “target space.” Alternatively, from the perspective
of quantum mechanics on the line, $\mathbb{R}$ is simply the space on which
the particle lives.

We can partition the target space $\mathbb{R}$ into a region $A$ and its
complement $\bar{A}$; for instance, we might choose half-spaces
$A\equiv\\{x:x\leq x_{0}\\}$ and $\bar{A}\equiv\\{x:x>x_{0}\\}$. This bi-
partition induces a decomposition of the Hilbert space into a direct sum,

$\mathcal{H}=L^{2}(A\cup\bar{A})=\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}}$
(1)

where

$\displaystyle\mathcal{V}_{A}$
$\displaystyle\equiv\textrm{span}\\{\ket{x}:x\in A\\},$ (2)
$\displaystyle\mathcal{V}_{\bar{A}}$
$\displaystyle\equiv\textrm{span}\\{\ket{x}:x\in\bar{A}\\}.$

We emphasize that this decomposition is not a tensor factorization. Therefore
one might wonder how to define a subsystem, a partial trace and reduced
density matrix, or an entanglement entropy. To proceed, we thus review a more
general notion of subsystems, based on sub-algebras rather than tensor
factors.

##  3 Review of the algebraic definition of entanglement entropy

Traditionally, one defines the entanglement of a state
$|\psi\rangle\in\mathcal{H}$ relative to some bi-partition of the Hilbert
space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$, where we have
divided the degrees of freedom into subsystem $A$ and its complement
$\bar{A}$. We will consider pure states on $\mathcal{H}$, in which case the
entanglement can be quantified by the the von Neumann entanglement entropy of
the reduced state
$\rho_{A}=\operatorname{Tr}_{\bar{A}}(|\psi\rangle\langle\psi|)$.

There are many ways to factorize a Hilbert space $\mathcal{H}$ as
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$, and different
factorizations may be appropriate for different purposes. Given a
factorization, it is natural to consider the algebra $\mathcal{A}$ of
operators local to $A$, i.e. operators of the the form
$O_{A}\otimes\mathds{1}_{\bar{A}}$. These operators represent the observables
and operations available to an observer confined to subsystem $A$.

Even without a factorization of the Hilbert space, we can still choose a sub-
algebra of “accessible” observables $\mathcal{A}$ and use this to define the
subsystem $A$. Recall that an algebra $\mathcal{A}$ of operators on a Hilbert
space $\mathcal{H}$ is a subset $\mathcal{A}\subset L(\mathcal{H})$ where
$L(\mathcal{H})$ denotes the space of all linear operators; here we consider
finite-dimensional “von Neumann algebras,” required to be closed under
addition, multiplication, scaling, and Hermitian conjugation.111Though we
often refer to it as an “algebra of observables,” not all of the elements need
be Hermitian. Here we will require an algebra to include the identity element.
There are additional subtleties in infinite dimensions that we will not
immediately discuss, though the results presented above all hold for finite
direct sums of Type $I$ factors. See Section 9 for more discussion of infinite
dimensions. By identifying any subalgebra $\mathcal{A}\subset L(\mathcal{H})$
with an abstract “subsystem,” we generalize the notion of a subsystem beyond
tensor factors.

We review a few key facts about algebras of operators. The most important
theorem is that given any algebra $\mathcal{A}\subset L(\mathcal{H})$, there
exists a decomposition of the Hilbert space as a direct sum of tensor
products,

$\mathcal{H}=\bigoplus_{i}\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},i}$ (3)

such that the operators $\mathcal{O}_{A}\in\mathcal{A}$ are precisely those
which take the form

$\mathcal{O}_{A}=\sum_{i}\mathcal{O}_{A,i}\otimes\mathds{1}_{\mathcal{H}_{\bar{A},i}}$
(4)

for some $\mathcal{O}_{A,i}\in L(\mathcal{H}_{A,i})$. This follows from a
pedestrian version of the Artin-Wedderburn theorem. Schematically, we can
write

$\mathcal{A}=\bigoplus_{i}L(\mathcal{H}_{A,i})\otimes\mathds{1}_{\mathcal{H}_{\bar{A},i}}.$
(5)

In the case that the above sum has only one term, $\mathcal{A}$ is called a
“factor,” and indeed the Hilbert space tensor factorizes as
$\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$. In infinite dimensions, the
existence of this tensor factorization hinges upon the “Type” of algebra; see
Section 9 for more.

Given an algebra $\mathcal{A}\subset L(\mathcal{H})$, an important related
algebra is the commutant $\mathcal{A}^{\prime}\subset L(\mathcal{H})$, the set
of operators that commute with all the operators on $\mathcal{A}$. Given the
above decomposition (3), Schur’s lemma allows us to easily write down the
commutant, which takes the form

$\displaystyle\mathcal{A}^{\prime}=\bigoplus_{i}\mathds{1}_{\mathcal{H}_{A,i}}\otimes
L(\mathcal{H}_{\bar{A},i}).$ (6)

We can also define the center
$Z(\mathcal{A})\equiv\mathcal{A}\cap\mathcal{A}^{\prime}$, the set of
operators on $\mathcal{A}$ that commute with all operators on $\mathcal{A}$.
The center may be expressed as

$\displaystyle Z(\mathcal{A})=\textrm{span}\\{\Pi_{i}\\}_{i}$ (7)

where $\Pi_{i}$ are the projectors onto the direct sum sectors
$\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},i}$. In practice, we often start
out with $\mathcal{A}$, then determine the minimal projectors spanning its
center. This in turn allows us to actually find the block decomposition of the
Hilbert space laid out in Eqn. 3. Note that when there is only one sector,
i.e. $\mathcal{A}$ is a factor, the center contains only multiples of the
identity.

With these ingredients in hand, we can easily define the reduced density
matrix with respect to a sub-algebra. Say we have a state $\rho$ and want to
define a reduced density state $\rho_{A}$ with respect to $\mathcal{A}$. First
recall that for an ordinary tensor factorization
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$, the reduced state
$\rho_{A}$ can be defined as the unique state on $\mathcal{H}_{A}$ such that
$\operatorname{Tr}(\rho_{A}\mathcal{O}_{A})=\operatorname{Tr}(\rho\mathcal{O}_{A})$
for all $\mathcal{O}_{A}\in L(\mathcal{H})$. With that definition, one can
show $\rho_{A}$ is given by the familiar partial trace.

Analogously, for the case of an algebra, we will define $\rho_{\mathcal{A}}$
to be the unique element of $\mathcal{A}$ such that

$\operatorname{Tr}(\rho_{\mathcal{A}}\mathcal{O}_{A})=\operatorname{Tr}(\rho\mathcal{O}_{A})$
(8)

for all $\mathcal{O}_{A}\in\mathcal{A}$. Given the decomposition of Eqn. 3, it
turns out one can easily express $\rho_{\mathcal{A}}$ by using partial traces
on each sector. Let

$\displaystyle p_{i}$
$\displaystyle\equiv\operatorname{Tr}(\Pi_{i}\rho\Pi_{i}),$ (9)
$\displaystyle\rho_{i}$
$\displaystyle\equiv\frac{1}{p_{i}}\Pi_{i}\rho\Pi_{i}.$ (10)

Then one can show $\rho_{A}$ must be given by

$\rho_{\mathcal{A}}=\sum_{i}p_{i}\operatorname{Tr}_{\bar{A},i}(\rho_{i})\otimes\frac{\mathds{1}_{\mathcal{H}_{\bar{A},i}}}{\textrm{dim}(\mathcal{H}_{\bar{A},i})},$
(11)

where $\rho_{i}$ is a state living on the $i$’th sector
$\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},i}$. The partial traces on each
sector are well-defined because each sector factorizes individually.

To define the entanglement entropy of $\rho$, we further consider the state

$\tilde{\rho}_{\mathcal{A}}=\sum_{i}p_{i}\operatorname{Tr}_{\bar{A},i}(\rho_{i})$
(12)

on the Hilbert space $\mathcal{H}_{A}\equiv\bigoplus_{i}\mathcal{H}_{A,i}$,
where we have simply stripped off the identity factors. Then we define the
entanglement entropy of $\rho$ with respect to $\mathcal{A}$ as the ordinary
von Neumann entropy of the state $\tilde{\rho}_{A}$ on the Hilbert space
$\mathcal{H}_{A}$.222This definition differs from the naive definition
$\operatorname{Tr}_{\mathcal{H}}\left(\rho_{\mathcal{A}}\log\rho_{\mathcal{A}}\right)$
by the term $\Delta
S=\sum_{i}p_{i}log\left({\textrm{dim}(\mathcal{H}_{\bar{A},i})}\right)$. To
reproduce the standard entropy for the case of a factor, we must use the
definition outlined in the main text. See Appendix A.7.2 of [7] or more
broadly [10]. To avoid any confusion: throughout the text, when we refer to
the von Neumann entropy of $\rho_{\mathcal{A}}$, we really mean
$S(\rho,\mathcal{A})$, or equivalently $S(\tilde{\rho}_{\mathcal{A}})$. That
is, we define

$\displaystyle S(\rho,\mathcal{A})$ $\displaystyle\equiv
S(\tilde{\rho}_{\mathcal{A}})$ (13)
$\displaystyle=S\left(\sum_{i}p_{i}\operatorname{Tr}_{\bar{A},i}(\rho_{i})\right)$
(14) $\displaystyle=-\sum_{i}p_{i}\log(p_{i})+\sum_{i}p_{i}S(\rho_{i})$ (15)
$\displaystyle\equiv
S(\rho,\mathcal{A})_{classical}+S(\rho,\mathcal{A})_{quantum}.$ (16)

We find that the entanglement entropy breaks into two pieces, a “classical”
piece and a “quantum” piece. $S(\rho,\mathcal{A})_{classical}$ is the Shannon
entropy of the classical probability distribution $\\{p_{i}\\}$ over the
different blocks (often referred to as “superselection sector”).
$S(\rho,\mathcal{A})_{quantum}$ on the other hand is the weighted sum of the
quantum von Neumann entropy of the reduced density matrices $\rho_{i}$ within
each block [6, 11].

Another simple way to define $S(\rho,\mathcal{A})$ is to embed $\mathcal{H}$
into an extended Hilbert space $\mathcal{H}_{ext}$ which does have a tensor
factorization. In particular, we define

$\displaystyle\mathcal{H}_{A}$
$\displaystyle\equiv\bigoplus_{i}\mathcal{H}_{A,i},$ (17)
$\displaystyle\mathcal{H}_{\bar{A}}$
$\displaystyle\equiv\bigoplus_{i}\mathcal{H}_{\bar{A},i},$

so that we can define the extended Hilbert space

$\displaystyle\mathcal{H}_{ext}$
$\displaystyle\equiv\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$
$\displaystyle=\bigoplus_{i,j}\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},j}$
$\displaystyle\supset\bigoplus_{i}\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},i}=\mathcal{H}.$
(18)

Therefore we can also view the state $\rho$ on $\mathcal{H}$ as a state
$\rho_{ext}$ on the extended Hilbert space $\mathcal{H}_{ext}$. Then
$S(\rho,\mathcal{A})$ is then precisely the “ordinary” entanglement entropy
obtained by taking the partial trace of $\rho$ over $\mathcal{H}_{\bar{A}}$
and then computing the von Neumann entropy of this reduced density
matrix333Readers familiar with the literature on entanglement entropy in gauge
theory might object that the “extended Hilbert space” and algebraic
definitions famously disagree. However, the extended Hilbert space
construction in gauge theory differs from the one in Eq. 3. , i.e.

$\displaystyle
S(\rho,\mathcal{A})=\operatorname{Tr}_{\mathcal{H}_{A}}\left(\rho_{A}\log\rho_{A}\right)$
(19) $\displaystyle\text{with}\
\rho_{A}\equiv\operatorname{Tr}_{\mathcal{H}_{\bar{A}}}\left(\rho_{ext}\right)$

In the case that $\mathcal{A}$ corresponds to the set of operators on a tensor
factor, the quantity $S(\rho,\mathcal{A})$ agrees with the standard von
Neumann entanglement entropy. However, beyond agreement with the “ordinary”
case, what motivates this definition of $S(\rho,\mathcal{A})$?

We might first ask the motivation for ordinary von Neumann entanglement
entropy. Besides proving a useful tool for analyzing field theories and many-
body physics, the entanglement entropy affords several operational or
information-theoretic interpretations. For instance, the von Neumann
entanglement entropy between subsystems $A$ and $\bar{A}$ also equals the
“distillable entanglement,” the number of Bell pairs that can be distilled by
observers on $A$ and $\bar{A}$ using only local operations on $A,\bar{A}$ and
classical communication. The entanglement entropy with respect to an algebra
affords an analogous interpretation as distillable entanglement, where
observers on $A$ and $\bar{A}$ are restricted to using operations associated
to $\mathcal{A}$ and $\mathcal{A}^{\prime}$, respectively. However, it turns
out the distillable entanglement is equal to the quantum piece
$S(\rho,\mathcal{A})_{quantum}$ alone [12, 13, 14]. See Equation 46 for an
elaboration of the operational interpretation.

##  4 Algebraic entanglement entropy for first-quantized systems

We first apply the algebraic definition of entanglement entropy to our example
of a particle on a line, spelling out the details on this first pass. We then
proceed more generally to the first-quantized quantum mechanics of $N$
particles. Later in Section 5, we will confirm that our framework gives the
same spatial entanglement entropy had we instead embedded the first-quantized,
$N$-particle Hilbert space into the $N$-particle sector of a second-quantized
Fock space, and defined the entanglement entropy using the tensor
factorization associated to the Fock space. Nonetheless, we develop the
algebraic approach as a general tool, applicable even when no obvious second-
quantized theory exists.

###  4.1 Single-particle warm-up

We return to the particle on a line, introduced in Section 2. The Hilbert
space is simply $\mathcal{H}=L^{2}(\mathbb{R})$, and we can think of
$\mathbb{R}$ alternately as the space on which the particle lives, or the
target space of a 0+1-dimensional QFT. For this section, we primarily adopt
the language of the former.

Partitioning the line $\mathbb{R}$ into a region $A\subset\mathbb{R}$ and its
complement $\bar{A}$, we obtain the decomposition of Eqn. 1,

$\mathcal{H}=\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}},$ (20)

where $\mathcal{V}_{A}=\textrm{span}\\{\ket{x}:x\in A\\}$ and likewise
$\mathcal{V}_{\bar{A}}=\textrm{span}\\{\ket{x}:x\in\bar{A}\\}$.

Now we choose an algebra $\mathcal{A}\subset L(\mathcal{H})$ to associate to
the region $A$. We propose the following algebra,

$\mathcal{A}=\bigg{\langle}\\{\ket{x}\bra{x^{\prime}}:x,x^{\prime}\in
A\\}\cup\mathds{1}_{\mathcal{H}}\bigg{\rangle}.$ (21)

The angular brackets denote “the algebra generated by,” i.e. the algebra of
all operators generated by addition, multiplication, and scaling of the
operators within the brackets. To physically motivate this choice, note the
Hermitian operators in $\mathcal{A}$ correspond to what observers situated in
the region $A$ of the line could measure. Including the identity is crucial.
Physically, it corresponds to the fact that an observer should be able to act
trivially on the system.

It will also be useful to define the projector

$\Pi_{A}=\int_{x\in A}dx\ket{x}\bra{x},$ (22)

which acts on the subspace $\mathcal{V}_{A}$ as the identity $\mathds{1}_{A}$.
We denote the orthogonal complement as $\Pi_{\bar{A}}=\mathds{1}_{\bar{A}}$.

Written in the position basis, with the basis partitioned into elements in $A$
and $\bar{A}$, all operators $\mathcal{O}\in\mathcal{A}$ take the following
form

$\mathcal{O}=\left(\begin{array}[]{cc}\mathcal{O}_{A}&0\\\
0&c_{0}\mathds{1}_{\bar{A}}\end{array}\right)$ (23)

where $c_{0}$ is an arbitrary constant and

$\displaystyle\mathcal{O}_{A}$ $\displaystyle=$
$\displaystyle\Pi_{A}\mathcal{O}\Pi_{A}$ $\displaystyle=$
$\displaystyle\int_{x,x^{\prime}\in
A}dx\,dx^{\prime}\,\mathcal{O}(x,x^{\prime})\ket{x}\bra{x^{\prime}}.$

To analyze the structure of this algebra, we compute the commutant
$\mathcal{A}^{\prime}$, again the set of operators that commute with all those
in $\mathcal{A}$. By Schur’s Lemma, $\mathcal{A}^{\prime}$ is given by all
operators of the form

$\mathcal{O}^{\prime}=\left(\begin{array}[]{cc}c_{1}\mathds{1}_{A}&0\\\
0&\mathcal{O}_{\bar{A}}\end{array}\right)$ (24)

with $c_{1}$ some other arbitrary constant. Thus we could also denote
$\mathcal{A}^{\prime}$ as the algebra $\bar{\mathcal{A}}$ corresponding to the
complementary region $\bar{A}$, with analogous definition

$\mathcal{A}^{\prime}=\bar{\mathcal{A}}\equiv\bigg{\langle}\\{\ket{x}\bra{x^{\prime}}:x,x^{\prime}\in\bar{A}\\}\cup\mathds{1}_{\mathcal{H}}\bigg{\rangle}.$
(25)

Hence the center $\mathcal{Z}=\mathcal{A}\cap\mathcal{A}^{\prime}$ is given by

$\mathcal{Z}=\bigg{\langle}\int_{x\in
A}dx\ket{x}\bra{x}\cup\mathds{1}_{\mathcal{H}}\bigg{\rangle}=\textrm{span}\\{\Pi_{A},\Pi_{\bar{A}}\\}.$
(26)

The center being non-trivial simply reflects the fact that $\mathcal{A}$ does
not induce a simple tensor factorization.

As guaranteed by the theorem of Eqn. 3, the algebra $\mathcal{A}$ induces a
decomposition of the Hilbert space. The decomposition is apparent from the
form of $\mathcal{A}$ and $\mathcal{A}^{\prime}$ in Eqns. 23, 24 above. We
have

$\displaystyle\mathcal{H}$
$\displaystyle=\bigoplus_{i=0,1}\mathcal{H}_{A,i}\otimes\mathcal{H}_{\bar{A},i}.$
(27)

where

$\displaystyle\mathcal{H}_{A,0}$ $\displaystyle=\mathbb{C}$ (28)
$\displaystyle\mathcal{H}_{\bar{A},0}$
$\displaystyle=\mathcal{V}_{\bar{A}}=\textrm{span}\\{|x\rangle:x\in\bar{A}\\}$
(29) $\displaystyle\mathcal{H}_{A,1}$
$\displaystyle=\mathcal{V}_{A}=\textrm{span}\\{|x\rangle:x\in A\\}$ (30)
$\displaystyle\mathcal{H}_{\bar{A},1}$ $\displaystyle=\mathbb{C}$ (31)

so that

$\displaystyle\mathcal{H}$
$\displaystyle=(\mathbb{C}\otimes\mathcal{V}_{\bar{A}})\oplus(\mathcal{V}_{A}\otimes\mathbb{C})$
(32) $\displaystyle=\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}},$ (33)

recovering Eqn. 1.

The decomposition here is slightly trivial, because the Hilbert spaces
$\mathcal{H}_{A,0}$ and $\mathcal{H}_{\bar{A},1}$ happen to be the trivial
space $\mathbb{C}$. Each sector corresponds to the number of particles in $A$.
For example, $\mathcal{H}_{A,1}\otimes\mathcal{H}_{\bar{A},1}$ is the sector
where the particle is within $A$. It is the tensor product of
$\mathcal{H}_{A,1}$, the space of wavefunctions on $A$, with the trivial space
$\mathcal{H}_{\bar{A},1}$, whose single ray represents the state of $\bar{A}$
with zero particles. Likewise, we can think of the sector
$\mathcal{H}_{A,0}\otimes\mathcal{H}_{\bar{A},0}$ as the sector where the
particle is within $\bar{A}$.

Now we compute the reduced density matrix of a state with respect to
$\mathcal{A}$. Let $\rho=|\psi\rangle\langle\psi|$ be a general pure state for

$\displaystyle|\psi\rangle$ $\displaystyle\equiv\int
dx\,\psi(x)|x\rangle=\int_{x\in
A}dx\,\psi(x)|x\rangle+\int_{x\in\bar{A}}dx\,\psi(x)|x\rangle$ (34)
$\displaystyle\equiv|\psi_{A}\rangle+|\psi_{\bar{A}}\rangle.$ (35)

We compute the reduced density matrix $\rho_{A}$ using Eqn. 11, taking note of
the decomposition in Eqn. 27. First we project the density matrix $\rho$ into
each of the two sectors, yielding $|\psi_{A}\rangle\langle\psi_{A}|$ and
$|\psi_{\bar{A}}\rangle\langle\psi_{\bar{A}}|$. Following Eqn. 9, define

$\displaystyle p_{0}$
$\displaystyle\equiv\operatorname{Tr}(|\psi_{A}\rangle\langle\psi_{A}|)=\langle\psi_{A}|\psi_{A}\rangle,$
(36) $\displaystyle p_{1}$
$\displaystyle\equiv\operatorname{Tr}(|\psi_{\bar{A}}\rangle\langle\psi_{\bar{A}}|)=\langle\psi_{\bar{A}}|\psi_{\bar{A}}\rangle,$
(37)

and

$\displaystyle\rho_{0}$
$\displaystyle=\frac{1}{p_{0}}|\psi_{\bar{A}}\rangle\langle\psi_{\bar{A}}|,$
(38) $\displaystyle\rho_{1}$
$\displaystyle=\frac{1}{p_{1}}|\psi_{A}\rangle\langle\psi_{A}|.$ (39)

Finally, plugging these into Equations 11 and 13, we obtain

$S(\rho,\mathcal{A})=-p_{0}\log(p_{0})-p_{1}\log(p_{i}),$ (40)

and we find the entanglement entropy has a contribution only from the
classical term. This classical piece is the Shannon entropy associated to the
probabilities of the single particle appearing in $A$ or $\bar{A}$. In the
multi-particle case, we will see that there is generically a quantum piece as
well.

###  4.2 General target spaces

Nothing in our construction relied on properties of the simple target line
$\mathbb{R}$. Indeed, we may consider a particle moving on some general
$d$-dimensional target space $T$, with coordinates $\vec{x}$. The Hilbert
space is given by $L^{2}(T)$ and admits the same decomposition
$\mathcal{H}=\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}}$ where
$A\cup\bar{A}=T$. We can take $A$ to as complicated a region as we would like.
We define the relevant subalgebra as

$\mathcal{A}=\bigg{\langle}\\{\ket{\vec{x}}\bra{\vec{x}^{\prime}}:\vec{x},\vec{x}^{\prime}\in
A\\}\cup\mathds{1}_{\mathcal{H}}\bigg{\rangle},$ (41)

which will again have non-trivial center.

All subsequent steps follow through straightforwardly.

###  4.3 Multiple indistinguishable particles

We consider now the general set-up of $N$ particles propagating on a general
target space $T$, for instance $T=\mathbb{R}^{d}$. A large literature exists
on the entanglement of identical particles [15], including e.g. an algebraic
approach in [16]. However, here we will study the entanglement with respect to
partitions of $T$, not the set of particles.

####  4.3.1 Bosons

We first study bosons. Denoting the single particle Hilbert space by
$\mathcal{H}=L^{2}(T)$, the physical Hilbert space is the symmetric quotient

$\mathcal{H}_{N}\equiv\operatorname{Sym}(\mathcal{H}^{\otimes
N})\equiv\frac{\mathcal{H}^{\otimes N}}{S_{N}}$ (42)

where the $S_{N}$ quotient arises from the indistinguishability of the $N$
particles. That is, $\mathcal{H}_{N}$ consists of permutation-symmetric
wavefunctions $\psi(\vec{x}_{1},...,\vec{x}_{N})$.

Given a partition of the target space $T$ into complementary regions
$A,\bar{A}$, we want to associate a sub-algebra of observables
$\mathcal{A}\subset L(\mathcal{H}_{N})$ to $A$.

We propose the following algebra,

$\mathcal{A}\equiv\bigg{\langle}\\{P_{S_{N}}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}_{1}\otimes\mathds{1}_{2}\otimes...\otimes\mathds{1}_{N}\right)P_{S_{N}}:\vec{x},\vec{x}^{\prime}\in
A\\}\cup\mathds{1}_{\mathcal{H}_{N}}\bigg{\rangle},$ (43)

where $P_{S_{N}}$ is the projection onto the symmetric subspace of
$\mathcal{H}^{\otimes N}$,

$P_{S_{N}}\equiv\frac{1}{N_{!}}\sum_{\sigma\in S_{N}}P_{\sigma},$ (44)

and where $P_{\sigma}$ permutes the subsystems according to the permutation
$\sigma\in S_{N}$. The appearance of $P_{S_{N}}$ in Eqn. 43 is crucial for
generating all multi-particle operators. The subscripts on the kets are
particle labels, denoting which copy of $\mathcal{H}$ within
$\mathcal{H}^{\otimes N}$ the operator acts on. For instance, unpacking the
notation for the case of $N=2$, we have

$P_{S_{2}}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}_{1}\otimes\mathds{1}_{2}\right)P_{S_{2}}=\frac{1}{2!}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}_{1}\otimes\mathds{1}_{2}+\mathds{1}_{1}\otimes\ket{\vec{x}}_{2}\bra{\vec{x}^{\prime}}_{2}\right).$
(45)

To motivate this algebra operationally, note that in ordinary quantum
mechanics, if an external apparatus $\mathcal{X}$ situated in region $A$ were
coupled to the system of identical particles $\mathcal{H}_{N}$ in a way that
respected permutation symmetry and particle-number conservation, the apparatus
$\mathcal{X}$ could only be coupled with a Hamiltonian of the form

$\displaystyle H_{int}=\sum_{i}O_{i}^{X}\otimes O_{i}^{A}$ (46)

for operators $O_{i}^{X}\in L(\mathcal{X})$ and $O_{i}^{A}\in\mathcal{A}$.
(The fact that interactions must take this form may be more obvious from the
form of the algebra in Eqn. 4.3.1.) If observers on $A$ and $\bar{A}$ are
allowed to perform operations only using such apparatuses, the amount of
entanglement distillable through local operations and classical communication
(LOCC) will be equal to the (quantum piece of the) entanglement entropy with
respect to $\mathcal{A}$. This operational interpretation follows as a
corollary to the discussions in [12, 13, 14].

To better understand the above algebra, note that we can decompose

$\displaystyle\mathcal{H}_{N}$ $\displaystyle\equiv\frac{\mathcal{H}^{\otimes
N}}{S_{N}}=\frac{(\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}})^{\otimes
N}}{S_{N}}$ $\displaystyle=\bigoplus^{N}_{k=0}\frac{\mathcal{V}_{A}^{\otimes
k}}{S_{k}}\otimes\frac{\mathcal{V}_{\bar{A}}^{\otimes N-kk}}{S_{N-k}}.$ (47)

where we define $\mathcal{V}_{A}^{0}=\mathcal{V}_{\bar{A}}^{0}=\mathbb{C}$.
The sectors indexed by $k$ in the sum correspond to states with $k$ particles
in $A$ and $N-k$ particles in $\bar{A}$. It turns out that when our algebra
$\mathcal{A}$ of Eqn. 43 above is decomposed in the general way of Eqn. 3, we
obtain precisely the above decomposition. That is, schematically, we have

$\displaystyle\mathcal{A}=\bigoplus^{N}_{k=0}L\left(\frac{\mathcal{V}_{A}^{\otimes
k}}{S_{k}}\right)\otimes\mathds{1}_{\operatorname{Sym}(\mathcal{V}_{\bar{A}}^{\otimes
N-k})}.$ (48)

To justify the above using the definition in Eqn. 43, see Appendix A. The
above demonstrates the algebra decomposes according to the particle number
“superselection” sectors reviewed in [15].

We may now write down the reduced density matrix. Let $\Pi_{k}$ be the
projector onto the $k$’th sector in the decomposition of Eqn. 4.3.1; for an
explicit expression, see Appendix A. Following the definition for the reduced
density matrix in Eqn. 11, we have

$\rho_{A}=\sum_{k=0}^{N}p_{k}\operatorname{Tr}_{\textrm{Sym}(\mathcal{V}_{\bar{A}}^{\otimes
N-k})}\rho_{k}\otimes\frac{\mathds{1}_{\textrm{Sym}(\mathcal{V}_{\bar{A}}^{\otimes
N-k})}}{\textrm{dim}(\textrm{Sym}(\mathcal{V}_{\bar{A}}^{\otimes N-k}))}$ (49)

where

$p_{k}\equiv\operatorname{Tr}_{\mathcal{H}_{N}}\left(\Pi_{k}\rho\Pi_{k}\right)$
(50)

and

$\rho_{k}\equiv\frac{1}{p_{k}}\left(\Pi_{k}\rho\Pi_{k}\right)$ (51)

In particular, for a pure state, we have

$p_{k}=\binom{N}{k}\int_{A}dx_{1}...dx_{k}\int_{\bar{A}}dx_{k+1}...dx_{N}|\psi(x_{1},...,x_{N})|^{2},$
(52)

which is the probability of finding $k$ particles in $A$.

We can immediately compute the classical part of the entanglement entropy,
$\sum_{k}-p_{k}\log(p_{k})$, corresponding to the Shannon entropy for finding
varying numbers of particles in $A$ and $\bar{A}$. Meanwhile, unlike for the
case of a single particle, here the blocks $\rho_{k}$ of the density matrix
are generically entangled between $A$ and $\bar{A}$, so the quantum term of
the entanglement entropy is nonzero.

####  4.3.2 Fermions

All the machinery we have built generalizes quite simply to fermions. In that
case, we need to consider the algebra

$\mathcal{A}_{F}=\bigg{\langle}\\{P_{\text{Asym}_{N}}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}_{1}\otimes\mathds{1}_{2}\otimes...\otimes\mathds{1}_{N}\right)P_{\text{Asym}_{N}}:\vec{x},\vec{x}^{\prime}\in
A\\}\cup\mathds{1}_{\mathcal{H}_{N}}\bigg{\rangle}$ (53)

where $P_{\text{Asym}_{N}}$ is the projector onto the anti-symmetric subspace
of $\mathcal{H}^{\otimes N}$, defined via

$P_{\text{asym}_{N}}=\frac{1}{N!}\sum_{\sigma\in
S_{N}}(-1)^{\sigma}P_{\sigma}.$ (54)

##  5 Comparison with embedding into second-quantized theory

To define entanglement in a first-quantized theory of many particles, rather
than use an algebraic definition, we could also translate to the second-
quantized picture where a natural tensor factorization does exist. Here, we
consider the latter approach and confirm that it agrees with the calculations
of the previous section.

In the second-quantized approach, we consider the $N$-particle Hilbert space

$\mathcal{H}_{N}\equiv\operatorname{Sym}(\mathcal{H}^{\otimes N})$ (55)

as a subspace of the Fock space

$\mathcal{H}_{F}\equiv\bigoplus_{N=0}^{\infty}\operatorname{Sym}(\mathcal{H}^{\otimes
N}).$ (56)

Let us re-phrase the familiar process of second quantization as the process
whereby, given a basis of the single-particle Hilbert space $\mathcal{H}$, we
induce a tensor factorization of the Fock space $\mathcal{H}_{F}$. For
instance, choosing the position basis of $\mathcal{H}$, we write the Fock
space as

$\displaystyle\mathcal{H}_{F}=\bigotimes_{\vec{x}}\mathcal{H}_{\vec{x}}$ (57)

where
$\mathcal{H}_{\vec{x}}=\textrm{span}\\{|0\rangle_{\vec{x}},|1\rangle_{\vec{x}},...\\}$
is the countably infinite-dimensional Hilbert space whose basis states
$|n\rangle_{\vec{x}}$ indicate $n$ particles occupying position
$\vec{x}$.444The above tensor product is purely formal; it’s a continously
indexed tensor product. However, if we chose a countable basis for the single-
particle Hilbert space $\mathcal{H}$, rather than the naive basis of position
kets, the above tensor product would be countably indexed, so that it could be
made rigorous.

For concreteness, using the factorization of Eqn. 57, the zero-particle state
in the Fock space looks like
$\bigotimes_{\vec{x}}|0\rangle_{\vec{x}}\in\mathcal{H}_{F}$, and the single-
particle state $|\vec{y}\rangle\in\mathcal{H}_{1}$ embeds into the Fock space
as
$\left(\bigotimes_{\vec{x}\neq\vec{y}}|0\rangle_{\vec{x}}\right)\otimes|1\rangle_{\vec{y}}\in\mathcal{H}_{F}$.

More generally, by defining raising and lower operators
$a^{\dagger}_{\vec{x}},a_{\vec{x}}$ for each factor $\mathcal{H}_{\vec{x}}$
such that
$\ket{n}_{\vec{x}}=\frac{a^{\dagger}_{\vec{x}}}{\sqrt{n!}}\ket{0}_{\vec{x}}$,
we can neatly rewrite the embedding of the state

$\ket{\psi}=\int
d\vec{x}_{1}...d\vec{x}_{N}\psi(\vec{x}_{1},....,\vec{x}_{N})\ket{x_{1}}\otimes....\otimes\ket{x_{N}}\in\mathcal{H}_{N}$
(58)

as a state in the Fock space

$\ket{\psi}_{F}=\int
d\vec{x}_{1}...d\vec{x}_{N}\psi(\vec{x}_{1},....,\vec{x}_{N})a^{\dagger}_{\vec{x}_{1}}...a^{\dagger}_{\vec{x}_{N}}\left(\otimes_{\vec{x}}\ket{0}_{\vec{x}}\right)\in\mathcal{H}_{F}$
(59)

The “target space” of the first-quantized theory thus becomes the base space
of the second-quantized theory. The partition of the target space in the
first-quantized theory becomes an ordinary partition of the base space for the
second-quantized theory. Given a region $A$, we want to check that the
algebraic entanglement entropy $S(\rho,\mathcal{A})$ of a pure state $\rho$
living on $\mathcal{H}_{N}$ matches the ordinary entanglement entropy of
$\rho$ when viewed as a state on $\mathcal{H}_{F}$.

For a region $A$, we decompose the single-particle Hilbert space as

$\displaystyle\mathcal{H}=\mathcal{V}_{A}\oplus\mathcal{V}_{\bar{A}}.$ (60)

Then we can define a Fock space for $\mathcal{V}_{A}$,

$\displaystyle(\mathcal{V}_{A})_{F}$
$\displaystyle\equiv\bigoplus_{N=0}^{\infty}\operatorname{Sym}(\mathcal{V}_{A}^{\otimes
N})$ (61) $\displaystyle=\bigotimes_{\vec{x}\in A}\mathcal{H}_{\vec{x}},$

and likewise for $\bar{A}$.

The entire Fock space therefore factorizes as

$\displaystyle\mathcal{H}_{F}$
$\displaystyle=(\mathcal{V}_{A})_{F}\otimes(\mathcal{V}_{\bar{A}})_{F}$ (62)
$\displaystyle=\left(\bigoplus_{N=0}^{\infty}\operatorname{Sym}(\mathcal{V}_{A}^{\otimes
N})\right)\otimes\left(\bigoplus_{N=0}^{\infty}\operatorname{Sym}(\mathcal{V}_{\bar{A}}^{\otimes
N})\right)$ (63) $\displaystyle=\
\bigoplus_{N,M=0}^{\infty}\operatorname{Sym}(\mathcal{V}_{A}^{\otimes
N})\otimes\operatorname{Sym}(\mathcal{V}_{\bar{A}}^{\otimes M})$ (64)
$\displaystyle\supset\bigoplus^{N}_{k=0}\operatorname{Sym}(\mathcal{V}_{A}^{\otimes
k})\otimes\operatorname{Sym}(\mathcal{V}_{\bar{A}}^{\otimes
N-k})=\mathcal{H}_{N}.$ (65)

where the last line uses the decomposition of Eqn. 4.3.1. Thus we embed
$\mathcal{H}_{N}\subset\mathcal{H}_{F}$ in a way neatly compatible with the
factorization into $A$, $\bar{A}$.

Combining the above embedding with Eqn. 48, which illustrates the structure of
the algebra $\mathcal{A}$ in the first-quantized picture, and recalling the
definition of $S(\rho,\mathcal{A})$ in either Eqn. 13 or 19, we conclude that

$\displaystyle S(\rho,\mathcal{A})=S(\rho_{F})$ (66)

where $\rho_{F}$ indicates the state $\rho$ embedded in the Fock space
$\mathcal{H}_{F}$. $S(\rho_{F})$ is the ordinary von Neumann entanglement
entropy with respect to the factorization of Eqn. 62, i.e.

$S(\rho_{F})=\operatorname{Tr}_{A}\left(\rho_{A}\log\rho_{A}\right).$ (67)

##  6 Competing notions of locality in relativistic field theories

In Section 5 we saw that the algebraic entanglement entropy in the first-
quantized setting agrees with the ordinary entanglement entropy in the second-
quantized setting. However, we must take care with relativistic field
theories, where we find two competing notions of locality. While one appears
quite natural from the first quantized perspective, the other serves as the
standard in most QFT calculations of entanglement entropy.

Consider the free scalar field in $d+1$ dimensions. We will discuss two
alternative factorizations of the Hilbert space, given a spatial partition.
Similar discussion appears already in [17]. Afterward, we return to the
subject of algebraic entanglement entropy.

Let us start by reviewing the “ordinary” spatial factorization of a quantum
field theory, ignoring subtleties associated to the continuum [8]. While the
content may be familiar, we must be explicit to avoid confusion between the
alternative factorizations.

The Hilbert space formally factorizes as

$\displaystyle\mathcal{H}_{QFT}=\bigotimes_{\vec{x}\in\mathbb{R}^{d}}\mathcal{P}_{\vec{x}}$
(68)

where

$\displaystyle\mathcal{P}_{\vec{x}}\equiv\textrm{span}\\{|\phi\rangle_{\vec{x}}:\phi\in\mathbb{R}\\}\cong
L^{2}(\mathbb{R})$ (69)

is the Hilbert space associated to the field degree of freedom living at base
point $\vec{x}$. This is the ordinary tensor factorization of a field theory.
When free field theory is viewed as a collection of coupled harmonic
oscillators, $\mathcal{P}_{\vec{x}}$ is the Hilbert space of the harmonic
oscillator ”living" at $\vec{x}$.

The field operator $\hat{\phi}(\vec{x})$ living at a point $\vec{x}$ is local
to the tensor factor $\mathcal{P}_{\vec{x}}$, and it acts on states
$\phi|\phi\rangle_{\vec{x}}\in\mathcal{P}_{\vec{x}}$ as

$\displaystyle\hat{\phi}(\vec{x})|\phi\rangle_{\vec{x}}=\phi|\phi\rangle_{\vec{x}}.$
(70)

Given a field configuration $\phi:\mathbb{R}^{d}\to\mathbb{R}$ denoted
$\phi(x)$, one can then define a field ket $|\phi\rangle\in\mathcal{H}_{QFT}$
as the simultaneous eigenstate of the field operators $\hat{\phi}(\vec{x})$
with respective eigenvalues $\phi(x)$. That is,

$\displaystyle|\phi\rangle\equiv\bigotimes_{\vec{x}\in\mathbb{R}^{d}}|\phi(x)\rangle_{\vec{x}}.$
(71)

Finally, the wavefunctional $\Psi[\phi]$ expands an arbitrary state in
$\mathcal{H}_{QFT}$ in terms of field kets $|\phi\rangle$. Given a region
$A\subset\mathbb{R}^{d}$ and complementary region $\bar{A}$, we obtain a
bipartite factorization

$\displaystyle\mathcal{H}_{QFT}=\mathcal{H}_{QFT,A}\otimes\mathcal{H}_{QFT,\bar{A}}$
(72)

where

$\displaystyle\mathcal{H}_{QFT,A}=\bigotimes_{\vec{x}\in
A}\mathcal{P}_{\vec{x}}$ (73)

and likewise for $\mathcal{H}_{QFT,\bar{A}}$.

We will call this factorization of the Hilbert space the “ordinary” or “field-
based” factorization. It is the usual factorization used to define
entanglement in relativistic field theories, wherein the vacuum exhibits an
area law divergence in entanglement entropy. (Again, for a continuum field
theory, this ordinary factorization does not actually exist as a tensor
product [8].)

Meanwhile, we also have a “Fock-based” factorization of the Hilbert space,
akin to the factorization expressed in Eqn. 57. We utilize the Fock struture
of the free theory,

$\displaystyle\mathcal{H}_{QFT}\cong\mathcal{H}_{F}\equiv\bigoplus_{N=0}^{\infty}\operatorname{Sym}(\mathcal{H}^{\otimes
N}),$ (74)

where $\mathcal{H}$ is the single-particle Hilbert space. How do we identify
the two Hilbert spaces above? We can use the momentum basis
$\mathcal{H}=\textrm{span}\\{|\vec{p}\rangle\\}$ for the single-particle
space. Let $a_{\vec{p}},a_{\vec{p}}^{\dagger}$ be the ladder operators that
raise/lower the occupancy of the $|\vec{p}\rangle$ state in the Fock space. If
we identify $|\vec{p}\rangle\in\mathcal{H}$ with the single-particle momentum
eigenstate in the free field theory, $|\vec{p}\rangle\in\mathcal{H}_{QFT}$,
then the ladder operators are related to the field operators in the usual way
(taking $d+1=3+1$ for simplicity)

$\displaystyle\hat{\phi}(\vec{x})$
$\displaystyle=\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\vec{p}}}}\left(a_{\vec{p}}e^{i\vec{p}\cdot\vec{x}}+a_{\vec{p}}^{\dagger}e^{-i\vec{p}\cdot\vec{x}}\right),$
(75) $\displaystyle E_{\vec{p}}$
$\displaystyle\equiv\sqrt{\vec{p}^{2}+m^{2}},$ (76)

using the normalization conventions of [18].

If we instead choose the position basis for the single-particle Hilbert space
$\mathcal{H}=\textrm{span}\\{|\vec{x}\rangle\\}$, with the momentum and
position basis related by the ordinary Fourier transform, we can define the
coresponding ladder operators that raise/lower the occupancy of the
$|\vec{x}\rangle$ state in the Fock space. These are given by

$\displaystyle a_{\vec{x}}$
$\displaystyle\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}e^{i\vec{p}\cdot\vec{x}}a_{\vec{p}},$
(77) $\displaystyle a_{\vec{x}}^{\dagger}$
$\displaystyle\equiv\int\frac{d^{3}}{(2\pi)^{3}}\vec{p}e^{-i\vec{p}\cdot\vec{x}}a_{\vec{p}}^{\dagger},$
(78)

and these ladder operators are local to the factors of the tensor
factorization in Eqn. 57,

$\displaystyle\mathcal{H}_{F}=\bigotimes_{\vec{x}}\mathcal{H}_{\vec{x}},$ (79)

where the local Hilbert spaces
$\mathcal{H}_{\vec{x}}=\textrm{span}\\{|0\rangle_{\vec{x}},|1\rangle_{\vec{x}},...\\}$
have basis states$|n\rangle_{\vec{x}}$ that count the number of particles
occupying single-particle state $\vec{x}$.

This defines the “Fock-based” factorization referred to above. The ladder
operators $a_{\vec{x}}\dagger,a_{\vec{x}}$ raise and lower the particle number
of the free theory. In this factorization, the vacuum is just the zero-
particle state $\bigotimes_{\vec{x}}|0\rangle_{\vec{x}}$. Note this is a
product state! The vacuum is unentangled with respect to the Fock
factorization. Clearly, the Fock-based factorization differs from the ordinary
field-based factorization.

To sharpen the distinction between the factorizations, let us define ladder
operators $\alpha_{\vec{x}},\alpha^{\dagger}_{\vec{x}}$ associated to the
“harmonic oscillator” Hilbert space $\mathcal{P}_{\vec{x}}$, the local degrees
of freedom in the ordinary tensor factorization. That is, take

$\displaystyle\alpha_{\vec{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\hat{\phi}(x)+i\hat{\pi}(\vec{x})),$ (80)
$\displaystyle\alpha^{\dagger}_{\vec{x}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\hat{\phi}(x)-i\hat{\pi}(\vec{x}))$

were $\hat{\pi}(\vec{x})$ is the canonical conjugate of the field operator
$\hat{\phi}(\vec{x})$, acting as $-i\frac{\delta}{\delta\phi(\vec{x})}$ on the
wavefunctional. Note these are not the same as the ladder operators
$a_{\vec{x}}\dagger,a_{\vec{x}}$ associated to the Fock-based factorization.

Any operator local to $\vec{x}$ in the ordinary factorization should commute
with $\phi(\vec{y})$ for all $\vec{y}\neq\vec{x}$, whereas

$\displaystyle[a_{\vec{x}},\hat{\phi}(\vec{y})]\propto K(\vec{x},\vec{y}),$
(81)

where $K(\vec{x},\vec{y})$ is the convolution kernel

$\displaystyle
K(\vec{x},\vec{y})\equiv\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}e^{i\vec{p}\cdot(\vec{x}-\vec{y})}\frac{1}{\sqrt{2E_{\vec{p}}}},$
(82)

emphasizing that the operators $a_{\vec{x}}\dagger,a_{\vec{x}}$ local in the
Fock-based factorization are slightly non-local in the ordinary field-based
factorization.

In one sense, the two factorizations are “close,” because the kernel $K(x,y)$
is peaked near $\vec{x}\sim\vec{y}$. Thus an operator local to a region $A$ in
the Fock-based factorization will be well-approximated by an operator local to
a sufficiently larger $B\supset A$ in the ordinary factorization.

In another sense, the alternatives yield drastically different entanglement
entropies: the vacuum is unentangled in the Fock-based factorization, while it
exhibits diverging entanglement in the ordinary factorization. It turns out
that multi-particle excited states yield a middle ground: if the wavefunctions
of the particles are sufficiently spread, the two factorizations will yield
approximately equal entanglement entropies, up to a correction which is
precisely the vacuum entanglement.

Which factorization is “correct”? Of course they merely constitute different
choices. If we want to leverage the operational interpretation of entanglement
entropy, we must ask which algebra of observables is available to an observer
who “has access to region $A$” ? We will not further pursue this question, but
a point in favor of the ordinary factorization is that the Hamiltonian is
truly local with respect to this factorization. Moreover, only in the ordinary
factorization is there a strict lightcone, i.e. exact commutation of
spacelike-separated Heisenberg operators.

##  7 Computation of entanglement entropy for finite-particle states: “Fock”-
vs. “field”-based factorization

Our algebraic setup calculates the entanglement entropy relative to the Fock-
based tensor product factorization of the QFT. In this section, we show there
is a sense in which the entanglement entropy of a multi-particle state
decomposes into the universal, divergent area-law piece and an additive
contribution we can associate to the particles’ wavefunction. The extra
entanglement due to the excitations has been called the “excess of
entanglement” above the vacuum [19, 20]. We will address the simple case of
single-particle excitations, but the account of finitely multi-particle
excitations is similar.

Our calculation of the entanglement entropy of single-particle excitations
(with respect to the ordinary tensor factorization) has precedent in the
related calculations of [19, 20]. However, those arguments only apply to
momentum eigenstates, rather than to excited states with more general
wavefunctions. The argument sketched here has a different scope.

In the Fock basis, we can describe a single particle state as

$\ket{\psi}=\int
d^{3}x\psi(x)a^{\dagger}_{\vec{x}}\ket{0}=\int\frac{d^{3}p}{(2\pi)^{3}}\tilde{\psi}(p)a^{\dagger}_{\vec{p}}\ket{0}$
(83)

The entanglement entropy for a spatial subgion $A$, can be immediately
computed as

$H(\\{p_{A}^{(a_{x})},1-p_{A}^{(a_{x})}\\})=-p_{A}^{(a_{x})}\log(p_{A}^{(a_{x})})-(1-p_{A}^{(a_{x})})\log(1-p_{A}^{(a_{x})})$
(84)

with

$p_{A}^{(a_{x})}=\int_{A}d^{3}x|\psi(x)|^{2}$ (85)

We wish now wish to compare this to the entanglement entropy computed relative
to the field-based factorization. First, let us rewrite the state $\ket{\psi}$
as

$\ket{\psi}=\int
d^{3}xf(x)\hat{\phi}(\vec{x})\ket{0}=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\tilde{f}(p)}{\sqrt{2E_{\vec{p}}}}a^{\dagger}_{\vec{p}}\ket{0}$
(86)

Eqn. 83 therefore identifies
$\tilde{\psi}(p)=\frac{\tilde{f}(p)}{\sqrt{2E_{\vec{p}}}}$, or alternatively,
in position space $\psi(\vec{x})=\int{d\vec{y}}K(\vec{x},\vec{y})f(y)$.

Below, we give a proof (on the lattice) that we may well approximate the
entanglement entropy as

$\displaystyle S(\rho_{A})\approx S_{0}+H(\\{p_{A},1-p_{A}\\})$ (87)

where $S_{0}$ is the entanglement of the vacuum and
$H(\\{p_{A},1-p_{A}\\})=-p_{A}\log(p_{A})-(1-p_{A})\log(1-p_{A})$ is the
Shannon entropy of the classical probability distribution, but now with
$p_{A}=\int_{A}d^{3}x|f(\vec{x})|^{2}$.

Before delving into the mechanics of the proof, we stress we may meaningfully
compare the Shannon entropies $H(\\{p_{A},1-p_{A}\\})$ and
$H(\\{p_{A}^{(a_{x})},1-p_{A}^{(a_{x})}\\})$. When the kernel
$K(\vec{x},\vec{y})$ is narrowly peaked near $\vec{x}\sim\vec{y}$, and the
regions $A$ are taken sufficiently large, these quantities are in fact close
(at least on the lattice). In fact,

$\displaystyle H(\\{p_{A}^{(a_{x})},1-p_{A}^{(a_{x})}\\})\to
H(\\{p_{A},1-p_{A}\\})$ (88)

precisely in the limit described in Section 7.1 below, as the wavefunctions
are spread over large regions.

###  7.1 Proof

Consider a free, massive scalar field on a finite square lattice in $d$
spatial dimensions. The discretized field theory Hamiltonian is that of
coupled harmonic oscillators,

$\displaystyle H=\sum_{x_{i}}\phi(x_{i})^{2}+\pi(x_{i})^{2}+\sum_{\langle
x_{i},x_{j}\rangle}m^{2}(\phi(x_{i})-\phi(x_{j}))^{2},$ (89)

for fields $\phi(x_{i})$ at site $i$ and conjugate momenta $\pi(x_{i})$. We
consider the single-particle excitation

$\displaystyle|\psi\rangle=\sum_{x_{i}}f(x_{i})\phi(x_{i})|\Omega\rangle,$
(90)

not necessarily an energy or momentum eigenstate, where $|\Omega\rangle$ is
the vacuum, and $f(x_{i})$ is some “wavefunction” of the discrete positions
$x_{i}$, normalized so that the overall state is normalized.555However, note
the norm of $|\psi\rangle$ is not given by $\sum_{i}|f(x_{i})|^{2}$, because
the states $\phi(x_{i})|\Omega\rangle$ are not orthogonal for distinct $i$.
Partition the lattice into complementary, contiguous regions $A,\bar{A}$, and
consider the reduced state $\rho_{A}$. We want to show that

$\displaystyle S(\rho_{A})\approx S_{0}+H(\\{p_{A},1-p_{A}\\})$ (91)

where $S_{0}$ is the entanglement of the vacuum,
$H(\\{p_{A},1-p_{A}\\})=-p_{A}\log(p_{A})-(1-p_{A})\log(1-p_{A})$ is the
Shannon entropy of the classical probability distribution, and where

$\displaystyle p_{A}\equiv\sum_{x_{i}\in A}|f(x_{i})|^{2}$ (92)

is essentially the probability of finding the particle in $A$ (at least for
large $A$, due to subtleties about measuring particle position in this
context). Eqn. 91 will hold with small error when the system has large volume
and the wavefunction $f_{i}$ is not too concentrated around the boundary of
$A,\bar{A}$. To be more precise, let $X_{R}\subset A$ be the sub-region of $A$
consisting of sites at a distance larger than $R$ lattice units from the
boundary $\partial A$, and let $B_{R}=A\backslash X_{R}$ be the buffer region
between $X$ and $A$. We can quantify the amount of the wavefunction $f(x_{i})$
concentrated in the buffer region as

$\displaystyle p_{B_{R}}\equiv\sum_{x_{i}\in B_{R}=A\backslash
X_{R}}|f(x_{i})|^{2}.$ (93)

We will prove that for $p_{B_{R}}$ sufficiently small for a choice of buffer
size $R$ sufficiently large, and for total lattice volume sufficiently large,
Eqn. 91 holds to arbitrarily good approximation. That is, we show Eqn. 91
holds exactly in the limit of a sequence of systems and wavefunctions where
$|A|,|\bar{A}|\to\infty$, and $p_{B_{R}}\to 0$ for a choice of buffer sizes
$R\to\infty$. One could also prove the result with more fine-grained error
analysis, but proving the simple limit will serve our illustration.

Now we sketch the proof.

Proof sketch. Divide the state into two terms

$\displaystyle|\psi\rangle$ $\displaystyle=\sum_{x_{i}\in
A}f(x_{i})\phi(x_{i})|\Omega\rangle+\sum_{x_{i}\in\bar{A}}f(x_{i})\phi(x_{i})|\Omega\rangle$
(94) $\displaystyle\equiv|\psi_{A}\rangle+|\psi_{\bar{A}}\rangle.$ (95)

We can approximate the state instead as

$\displaystyle|\psi\rangle$
$\displaystyle\approx|\widetilde{\psi}\rangle\equiv\sum_{x_{i}\in
X_{R}}f(x_{i})\phi(x_{i})|\Omega\rangle+\sum_{x_{i}\in\bar{A}}f(x_{i})\phi(x_{i})|\Omega\rangle$
(96) $\displaystyle\equiv|\psi_{X_{R}}\rangle+|\psi_{\bar{A}}\rangle.$ (97)

Then

$\displaystyle\langle\widetilde{\psi}|\psi\rangle=1-\langle\psi_{B_{R}}|\psi\rangle\to
1$ (98)

in the given limit where $p_{B_{R}}\to 0$, so $|\widetilde{\psi}\rangle$
approaches $|\psi\rangle$ in trace-distance. Then the entanglement entropy of
$|\widetilde{\psi}\rangle$ approaches the entanglement entropy of
$|\psi\rangle$, using the continuity of the entanglement entropy with respect
to trace distance [21]. The continuity result of [21] requires the same
assumptions as those discussed in Section 9, which the single-particle states
here satisfy.666In a more detailed argument, some care must be taken with how
the continuity bound depends on lattice size. Thus we can examine the
entanglement entropy of $|\widetilde{\psi}\rangle$ rather than $|\psi\rangle$.
The reduced density matrix has four terms

$\displaystyle\operatorname{Tr}_{\bar{A}}(|\widetilde{\psi}\rangle\langle\widetilde{\psi}|)=\operatorname{Tr}_{\bar{A}}\left(|\psi_{B_{R}}\rangle\langle\psi_{B_{R}}|+|\psi_{B_{R}}\rangle\langle\psi_{\bar{A}}|+|\psi_{\bar{A}}\rangle\langle\psi_{B_{R}}|+|\psi_{\bar{A}}\rangle\langle\psi_{\bar{A}}|\right).$
(99)

Let’s start with the fourth term, call it
$\sigma_{A}\equiv\operatorname{Tr}_{\bar{A}}\left(|\psi_{\bar{A}}\rangle\langle\psi_{\bar{A}}|\right)$.
Note that the connected correlation functions of local operators exponentially
decay with distance in this massive free lattice theory. Actually, we use the
stronger fact that the mutual information $I(B_{R}:\bar{A})$ in the vacuum
tends to zero as the size $R$ of the buffer region increases, which can be
shown with the methods of [1]. Then the connected correlation of any bounded
operators on $B_{R}$ and $\bar{A}$ must tend to zero for large $R$, using the
fact that mutual information upper bounds connected correlations, by Pinsker’s
inequality. Thus for any operator $O_{A}$ on $A$ with operator norm 1,

$\displaystyle\operatorname{Tr}(\sigma O_{A})$
$\displaystyle=\langle\Omega|O_{A}\left(\sum_{x_{i}\in\bar{A}}f(x_{i})\phi(x_{i})\right)^{2}|\Omega\rangle$
(100)
$\displaystyle\to\langle\Omega|O_{A}|\Omega\rangle\langle\Omega|\left(\sum_{x_{i}\in\bar{A}}f(x_{i})\phi(x_{i})\right)^{2}|\Omega\rangle$
(101) $\displaystyle\to\langle\Omega|O_{A}|\Omega\rangle(1-p_{A}),$ (102)

where again all limits are taken as described above.777Because the operators
$\phi(x_{i})$ are not bounded, and the mutual information only upper bounds
connected correlations of bounded operators, we cannot directly apply the
upper bound as stated. However, it turns out the action of $\phi(x_{i})$ on
the vacuum can be sufficiently well-approximated by the action of a bounded
operator for our purpose. Thus the state $\sigma/(1-p_{A})$ approaches the
reduced state of the vacuum.

Likewise, we have
$\operatorname{Tr}_{A}\left(|\psi_{B_{R}}\rangle\langle\psi_{B_{R}}|\right)/p_{A}$
approaching the reduced state of the vacuum as well. (Note the previous
expression traces out $A$ rather than $\bar{A}$, but the entanglement entropy
will be the same the same whether we trace out $A$ or $\bar{A}$.)

Now we analyze the second and third terms of Eqn. 99. These terms are actually
equal by the Hermiticiy of $\phi$; call this term
$\tau\equiv\operatorname{Tr}_{\bar{A}}(|\psi_{\bar{A}}\rangle\langle\psi_{B_{R}}|)$.
Similar to the above calculation, we have

$\displaystyle\operatorname{Tr}(\tau
O_{A})\to\langle\Omega|O_{A}|\Omega\rangle\langle\Omega|\sum_{x_{i}\in\bar{A}}f(x_{i})\phi(x_{i})|\Omega\rangle=0$
(103)

for any fixed norm operator $O_{A}$ local to $A$, again using the fact that
$I(B_{R}:\bar{A})\to 0$ in the vacuum as $R$ increases. The RHS is zero above
simply because $\phi$ has zero vacuum expectation. So $\tau\to 0$, and we can
discard these terms.

Finally, the first and fourth terms of Eqn. 99 tend to orthogonal operators,
so the entropy of their sum is the average of their entropies, plus the
Shannon entropy associated to their traces. We conclude

$\displaystyle S(\rho_{A})\to S_{0}+H(\\{p_{A},1-p_{A}\\})$ (104)

in the given limit, as desired.

##  8 Target space entanglement entropy for field theories

Above we considered the spatial entanglement in first-quantized many-particle
systems, alternatively interpreted as target space entanglement in a
0+1-dimensional theory. Now we consider target space entanglement for a more
general $d+1$-dimensional theory, i.e. with a higher-dimensional base space.
This quantity is more akin to what might be desired in worldsheet string
theory, if one desires to partition the target spacetime. However, worldsheet
string theory will offer further complications due to re-parameterization
invariance, as further discussed in Section 10.

Consider a $d+1$-dimensional field theory defined on the base space $B$ (for
example, $B\cong\mathbb{R}^{d}$), with its field $\phi$ taking values in the
target space $T$. Referring again to the ordinary base space factorization
described in Eqn. 68, we have the formal expression

$\displaystyle\mathcal{H}_{QFT}$ $\displaystyle=\bigotimes_{\vec{x}\in
B}\mathcal{P}_{\vec{x}}$ $\displaystyle\mathcal{P}_{\vec{x}}$
$\displaystyle\equiv\textrm{span}\\{|\phi\rangle_{\vec{x}}:\phi\in T\\}\cong
L^{2}(T).$

Consider a partition of target space $T$ into complementary regions
$A,\bar{A}\subset T$. We associate the following algebra to the target space
region $A$,

$\displaystyle\mathcal{A}\equiv\bigg{\langle}\\{\ket{\phi}_{\vec{x}}\bra{\phi^{\prime}}_{\vec{x}}\otimes_{\vec{y}\neq\vec{x}}\mathds{1}_{\mathcal{P}_{\vec{y}}}:\phi,\phi^{\prime}\in
A,\forall\vec{x}\in B\\}\cup\mathds{1}_{\mathcal{H}_{QFT}.}\bigg{\rangle}$
(105)

Defining the projector $\Pi_{A}^{\vec{x}}$ on local Hilbert space
$\mathcal{P}_{\vec{x}}$ at $\vec{x}$ as

$\displaystyle\Pi_{A}^{\vec{x}}\equiv\int_{\phi\in
A}d\phi\,|\phi\rangle_{\vec{x}}\langle\phi|_{\vec{x}}.$ (106)

we can immediately write down the center of $\mathcal{A}$

$\displaystyle
Z(\mathcal{A})=\bigg{\langle}\\{\Pi^{\vec{x}}_{A}\otimes_{\vec{y}\neq\vec{x}}\mathds{1}_{\mathcal{P}_{\vec{y}}}:\forall\vec{x}\\}\cup\mathds{1}_{\mathcal{H}_{QFT}.}\bigg{\rangle}$
(107)

These algebras resemble those of Section 4.3.1 because we may view our QFT as
the first-quantized theory of many distinguishable particles, labeled by
$\vec{x}$, moving in target $T$ with coordinates $\phi$.

With this definition in hand, we can calculate the target space entanglement
entropy of general field theories defined on the lattice. Unfortunately, the
calculation does not appear straightforward. 888We find this quite reminiscent
of the early days of base space entanglement entropy, where analytical
progress appeared similarly difficult. Both [22, 23] ultimately resorted to
numerical methods to discover the area law in the ground state of scalar field
theory. What we need is a suitable analog of the powerful path integral
replica trick. We settle for the simplest non-trivial example: a massive
scalar “field theory” on two spatial lattice points, i.e. two coupled harmonic
oscillators.

Consider two field degrees of freedom $\phi_{1}$ and $\phi_{2}$ at lattice
points 1 and 2, with Hilbert space $\mathcal{H}=L^{2}(\mathbb{R})\otimes
L^{2}(\mathbb{R})$ and Hamiltonian

$\displaystyle
H=\frac{1}{2}(\pi_{1}^{2}+\pi_{2}^{2}+(\phi_{1}-\phi_{2})^{2}+m^{2}(\phi_{1}^{2}+\phi_{2}^{2}))$
(108)

where $\pi_{1},\pi_{2}$ are the conjugate momentum operators and $m$ the mass.
The interaction term $(\phi_{1}-\phi_{2})^{2}$ comes from the lattice
discretized spatial gradient of the field.

We choose to compute the entanglement entropy of the ground state,
partitioning the target space into the positive and negative half-lines,
$A\equiv\\{\phi\in\mathbb{R}:\phi>0\\}\subset\mathbb{R}$. In this case, the
algebra $\mathcal{A}$ of Eqn. 105 has four sectors. Defining the projectors
$\Pi^{1}_{A},\Pi^{2}_{A},\Pi^{1}_{\bar{A}},\Pi^{2}_{\bar{A}}$ as in Eqn. 105,
the four sectors of $\mathcal{A}$ are simply the images of the four projectors
$\Pi^{1}_{A}\Pi^{2}_{A},\;\Pi^{1}_{A}\Pi^{2}_{\bar{A}},\;\Pi^{1}_{\bar{A}}\Pi^{1}_{A},\;\Pi^{1}_{\bar{A}}\Pi^{2}_{\bar{A}}$.
The sector projected onto by $\Pi^{1}_{A}\Pi^{2}_{A}$ indicates “the field on
both lattice points takes values in $A$,” while the sector projected onto by
$\Pi^{1}_{A}\Pi^{2}_{\bar{A}}$ indicates $\phi_{1}\in A,\phi_{2}\in\bar{A}$,
and so on. Note there are four rather than three sectors because the lattice
sites are distinguishable.

The normalized ground state wavefunction for the Hamiltonian (108) is given by

$\displaystyle\psi(\phi_{1},\phi_{2})=\left(\frac{\omega_{+}\omega_{-}}{\pi^{2}}\right)^{1/4}e^{-\frac{1}{2}(\omega_{+}\phi_{+}^{2}\,+\,\omega_{-}\phi_{-})^{2}}$
(109)

where $\phi_{\pm}=(\phi_{1}\pm\phi_{2})/\sqrt{2}$, $\omega_{+}=m$,
$\omega_{-}=\sqrt{m^{2}+2}$; see for instance the similar example in [23].

Figure 3: Base (Left) vs. Target (Right) Space Partition & Associated Algebras
for scalar field on two lattice sites.

We consider the state $\rho=|\psi\rangle\langle\psi|$ projected separately
onto the four sectors described above. In the sector where
$\phi_{1},\phi_{2}\in A$, the Hilbert space factorizes in a trivial way, as in
the discussion surrounding Eqn. 27 for the particle on a line. Hence the
projection of the state onto this sector yields a product state, with no
contribution to the quantum piece of the entanglement entropy. The same holds
for the sector associated to $\phi_{1},\phi_{2}\in\bar{A}$. The only
contribution to the quantum part of the entanglement entropy thus comes from
from the two sectors where $\phi_{1},\phi_{2}$ are in different regions of the
target space. Since the groundstate is symmetric under the exchange of
$\phi_{1}\leftrightarrow\phi_{2}$ the contribution in each such sector will be
identical. Thus we need only consider one sector, say the image of
$\Pi^{1}_{A}\Pi^{2}_{\bar{A}}$.

The sector factorizes as
$\mathcal{V}^{1}_{A}\otimes\mathcal{V}^{2}_{\bar{A}}$, where
$\mathcal{V}^{1}_{A}\equiv\textrm{span}\\{|\phi_{1}\rangle:\phi_{1}\in A\\}$
and
$\mathcal{V}^{2}_{\bar{A}}\equiv\textrm{span}\\{|\phi_{2}\rangle:\phi_{2}\in\bar{A}\\}$.
We need to take the state projected on this sector,
$\Pi^{1}_{A}\Pi^{2}_{\bar{A}}|\psi\rangle\langle\psi|\Pi^{1}_{A}\Pi^{2}_{\bar{A}}$,
and trace out the second factor $\mathcal{V}^{2}_{\bar{A}}$. We obtain the
(non-normalized) density matrix $\sigma$ on $\mathcal{V}^{1}_{A}$ given by

$\displaystyle\sigma(x_{1},y_{1})=\int_{x_{2}\in\bar{A}}dx_{2}\,\psi(x_{1},x_{2})\psi(y_{1},x_{2})^{*}.$
(110)

The integral above can be expressed in terms of error functions. To calculate
the entanglement entropy, it remains to diagonalize the above density matrix
$\sigma$. Returning attention to the full reduced state $\rho_{\mathcal{A}}$,
the classical part of the entanglement entropy is then given by

$\displaystyle S(\rho,\mathcal{A})_{classical}=H(\\{p,p,1-p,1-p\\})$ (111)

where $H(\\{\cdot\\})$ is the classical (Shannon) entropy of the probability
distribution, and $p=\operatorname{Tr}(\sigma)$, with $\sigma$ given above.
Meanwhile, the quantum piece of the entanglement entropy is given by

$\displaystyle
S(\rho,\mathcal{A})_{quantum}=2pS\left(\sigma/\operatorname{Tr}(\sigma)\right).$
(112)

In lieu of an analytic method, we discretize the $x_{1},y_{1}$ coordinates of
Eqn. 110 and numerically diagonalize the resulting finite matrix. We ensure
that the discretization is at sufficient resolution that the results converge
when decreasing the spacing or increasing the total number of discretized
points. Section 9 guarantees convergence, the end result being finite.
Ultimately we produce a numerical answer for the quantum and classical piece
of the entanglement entropy of the ground state, as a function of the mass in
the Hamiltonian. The results are depicted in Figure 4. Numerical error due to
discretization appears to be somewhat smaller than $10^{-3}$, but we do not
include a rigorous analysis.

Figure 4: Entanglement entropy with respect to a partition of target space,
for two coupled oscillators governed by the Hamiltonian in Eqn. 108.

At high mass, the two harmonic oscillators approximately de-couple. The
wavefunction spreads equally between the four sectors, so that the classical
piece of the entanglement entropy gives two bits. Meanwhile, the quantum piece
of the entanglement entropy tends to zero, because the only sectors that can
contribute must have $\phi_{1}$ and $\phi_{2}$ in different regions $A$ and
$\bar{A}$, and in these sectors, the wavefunction approximately factorizes due
to the de-coupling of the oscillators.

Figure 4 also illustrates that the quantum term of the target space
entanglement entropy is not monotonic with respect to the mass parameter. The
non-monotonicity is associated with the fact that in Eqn. 112, the first
factor $p$ increases monotonically with mass, whereas the second factor
$S(\sigma/\operatorname{Tr}(\sigma))$ decreases monotonically.

##  9 Finiteness of the entanglement entropy

The entanglement entropies discussed in this paper involve infinite-
dimensional Hilbert spaces and algebras. In infinite dimensions, we must take
care that density matrices and entropies remain well-defined. Fortunately, we
will see that most of the infinities present here are of a relatively tame
variety.

In this section, we will take more mathematical care, recalling for instance
that the “position eigenstate” $|\psi\rangle$ is not a true state in the
Hilbert space $L^{2}(\mathbb{R})$ as traditionally defined.

The algebra associated to a region in multi-particle quantum mechanics (like
Eqn. 43) is a finite direct sum of factors, where each factor is an infinite-
dimensional “Type $I$” factor, according to the type theory of von Neumann
algebras [24]. Type $I$ factors are algebras which are isomorphic to the full
algebra of bounded operators on some Hilbert space. The Type $I$ property of
this algebra is therefore apparent from the schematic form of the algebra in
Eqn. 48.

Similarly, an algebra associated to a region in the target space of a lattice
field theory – like the algebra in Eqn. 105 on a finite lattice, or the
algebra in Figure 3 – is also a direct sum of Type $I$ sectors, even when the
target space is infinite-dimensional.

For the general algebra decomposition of Eqn. 3 to make sense as written, it
is indeed essential the algebra is a direct sum of Type $I$ factors. Otherwise
the use of the tensor product there is incorrect.

Even for these Type $I$ algebras, we must take care with the entanglement
entropy. The formula for the algebraic entanglement entropy in Eqn. 13
requires defining the von Neumann entropy of the partial trace of a pure state
in a bipartite Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, where
the factors $\mathcal{H}_{1},\mathcal{H}_{2}$ may be countably infinite-
dimensional. (The full algebraic entanglement entropy was then a sum of such
entropies in each sector of the algebra.) We therefore focus on the question
of ordinary von Neumann entanglement entropies of pure states in bipartite
Hilbert spaces. For any (mathematically legitimate, i.e. normalizable) state
$|\psi\rangle\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, the partial trace
$\rho=\operatorname{Tr}_{2}(|\psi\rangle\langle\psi|)$ can be taken using any
(legitimate) orthonormal basis. The result will be a trace-class Hermitian
operator $\rho$. (To see that $\rho$ is trace class, we can take its trace in
any orthonormal basis, and the resulting sum will be convergent by the
normalizability of $|\psi\rangle$.) Recall that a trace-class Hermitian
operator $\rho$ has an eigen-decomposition

$\displaystyle\rho=\sum_{i=1}^{\infty}\lambda_{i}|v_{i}\rangle\langle v_{i}|$
(113)

for some countably infinite set of eigenvectors $\\{|v_{i}\rangle\\}$ and
eigenvalues $\lambda_{i}$. Thus we are in the position to define the
entanglement entropy

$\displaystyle S(\rho)\equiv\sum_{i=1}^{\infty}-\lambda_{i}\log(\lambda_{i}).$
(114)

However, the above sum may be infinite, even though a normalized state
$|\psi\rangle$ guarantees $\sum_{i}\lambda_{i}=1$. In fact, the set of states
$|\psi\rangle$ with infinite entanglement entropy is dense in the total
Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, so in some sense the
divergence is generic.

Yet, for states of interest, the sum is often finite. For instance, a finite
energy condition may imply finiteness. The authors of [21] prove that, for any
non-interacting Hamiltonian
$H=H_{1}\otimes\mathds{1}_{2}+\mathds{1}_{1}\otimes H_{2}$ on
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ with discrete spectrum such that
$\operatorname{Tr}(e^{-\beta H})$ is finite for all $\beta>0$, any state
$|\psi\rangle\in\mathcal{H}$ that has finite expected energy
$\langle\psi|H|\psi\rangle<\infty$ with respect to this Hamiltonian will have
finite entanglement entropy. Note the state $|\psi\rangle$ may have nonzero
overlap with energy eigenstates of arbitrarily high energy; as long as the
expected energy is finite, the theorem applies.

Although the theorem of [21] requires one to find a non-interacting
Hamiltonian with respect to which $|\psi\rangle$ has finite energy, this
reference Hamiltonian need not bear any relation to the dynamics of the system
of interest. Rather, the assumption of finite energy with respect to the
reference merely ensures that $\rho$, which might have infinite nonzero
eigenvalues, nonetheless has sufficiently accurate low-rank approximations.
For instance, if we have
$\mathcal{H}_{1},\mathcal{H}_{2}=L^{2}(\mathbb{R}^{d})$ and one chooses $H$ to
be the Hamiltonian of two independent harmonic oscillators,

$\displaystyle
H=\vec{p}_{1}^{2}+\vec{p}_{2}^{2}+\vec{x}_{1}^{2}+\vec{x}_{2}^{2},$ (115)

then any state $|\psi\rangle$ with a smooth spatial wavefunction that decays
at spatial infinity at least as fast as $1/r^{(d+3)/2}$ will have finite
energy with respect to $H$, and hence the theorem of [21] implies this large
class of wavefunctions has finite entanglement entropy. (If one tries to
weaken this condition to include wavefunctions that are not smooth but decay,
or decay but are not smooth, counterexamples exist with infinite entanglement
entropy in both cases.) In particular, entanglement entropy of the density
matrix in Eqn. 110 will be finite, as corroborated by the convergence of the
numerics used for Fig. 4.

Similarly, the algebraic entanglement entropies of Section 5 will be finite
for states with smooth, decaying wavefunctions. The finiteness highlights the
difference between the two notions of locality discussed in Section 6. Our
first-quantized algebraic approach uses Type $I$ algebras and gives finite
entanglement entropies, whereas the ordinary “factorization” of field theory
gives area-law divergences, associated to the Type $III$ sub-algebras present
in field theory.

##  10 Discussion

###  10.1 Worldlines, worldsheets & reparametrization invariance

This work has highlighted the largely unexplored realm of target space
partitions and their relevance in the quantum gravitational context. However,
we remain far from our original hope of using an algebraic approach to define
target space entanglement entropy in worldsheet string theory.

In ordinary field theory, our algebraic definition successfully captured the
entanglement entropy with respect to a certain factorization. However, as
discussed in Section 6, the field theory admits at least two seemingly natural
factorizations, which we called the “Fock-based” and “field-based”
factorizations. It is the entanglement with respect to the former that is
captured by our algebraic definition, whereas only the latter factorization
exhibits the divergent area law contribution. On one hand, the calculation
outlined in Section 6 demonstrates that the additional entanglement of
particle excitations atop the vacuum can be meaningfully compared between the
two factorizations. On the other hand, the first-quantized algebraic approach
remains unable to analyze the area law contribution itself.

In fact, it might appear senseless to imagine a first-quantized particle
description teaching us anything about the spatial structure of the vacuum.
The wordline framing of QFT suggests otherwise. Consider the relativistic free
massive scalar field. (Interactions can be incorporated but are not the focus
of the argument.) The logarithm of its partition function can be recast as the
path integral of a point particle coupled to $1$-dimensional gravity on its
worldline 999This might be more familiar under the guise of the “Schwinger
paramterization” of Feynman diagrams. [25] :

$\displaystyle\log Z_{QFT}$ $\displaystyle=$ $\displaystyle\log\int D\phi
e^{-\frac{1}{2}\int
d^{d}x\sqrt{g}\left(g^{\mu\nu}\partial_{\mu}\phi(x)\partial_{\mu}\phi(x)+m^{2}\phi(x)^{2}\right)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\operatorname{Tr}\left[\log\left(-g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+m^{2}\right)\right]$
$\displaystyle=$ $\displaystyle\int
d^{d}x\bra{x^{\mu}}\int_{\epsilon}^{\infty}\frac{ds}{2s}e^{-s(p_{\mu}p^{\mu}+m^{2})}\ket{x^{\mu}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x\int_{y^{\mu}(0)=x^{\mu}}^{y^{\mu}(s)=x^{\mu}}\frac{Dy^{\mu}(\tau)De(\tau)}{Vol(Diff)}e^{-\frac{1}{2}\int_{0}^{s}d\tau
e\left(\frac{1}{e^{2}}\partial_{\tau}y^{\mu}(\tau)\partial_{\tau}y^{\nu}(\tau)g_{\mu\nu}(y(\tau))+m^{2}\right)}$

This worldline approach to field theory is the most immediate field-theoretic
analog of worldsheet string theory. In the worldline setting, we know we can
access the area law entanglement pattern of the QFT via a replica trick
Euclidean path integral. Schematically, we can compute it as

$S_{EE}=\left(1-n\partial_{n})\log Z_{QFT}[n]\right){\bigg{|}}_{n=1}$ (116)

The right hand side of this equation, including the the necessary field-
theoretic $UV$-regulator, may be completely recast in terms of worldline
quantities. The euclidean path integral immediately gives us the entropy. Its
Lorentzian interpretation on the other hand, remains elusive.

Indeed, while the area law manifests itself as above in the worldline
formalism, it is unclear there exists any partition of the point particle
Hilbert space that yields this entropy. An algebraic approach would require
such a partition. However, hope remains. Two salient features deserve further
notice. Firstly, we see the spatial arguments of the fields, the $x^{\mu}$ in
$\phi(x^{\mu})$ appear as boundary conditions on the worldline trajectories.
This is the familiar statement that, in string theory, D-branes help us probe
target space locality [26, 27]. Note the states $\ket{x^{\mu}}$ do not belong
to the physical subspace of the point particle Hilbert space, as they do not
satisfy the constraint $\hat{p}^{2}+m^{2}\ket{\psi}=0$. In the language of
[25], they do not reside in the BRST coholomogy of $Q_{BRST}=c(p^{2}+m^{2})$.
This simply reflects the fact that reparametrization invariance breaks down at
the endpoints of the worldline. Secondly, from a more algebraic perspective,
we know that any reduced density matrix reproducing all two-point correlation
functions $\braket{\phi(x^{\mu})\phi(y^{\nu})}$,
$\braket{\phi(x^{\mu})\Pi(y^{\nu})}$ and $\braket{\Pi(x^{\mu})\Pi(y^{\nu})}$
for $x^{\mu},y^{\nu}\in A$ will gives us the field theory entanglement entropy
relative to the $\phi(x)$ tensor product factorization. These correlators can
also be rewritten purely in terms of worldline variables:

$\braket{\phi(x^{\mu})\phi(y^{\nu})}=\int_{\epsilon}^{\infty}ds\bra{x^{\mu}}e^{-s(p^{2}+m^{2})}\ket{y^{\nu}}$
(117)

We would therefore need to consider some sort of restriction on the set of
allowable “D-branes” for the worldline. While we have not succeeded in
defining an associated reduced density matrix, it is at least clearly a
Lorentzian setup.

The single particle Hamiltonian $\sqrt{\vec{p}^{2}+m^{2}}$ considered in
Section 6 arises via gauge fixing the relativistic point particle action. More
precisely, it is the canonical Hamiltonian after choosing static gauge
$x^{0}(\tau)=\tau$. We feel there is something important about
reparametization invariance we have yet to pinpoint, and hope to explore this
avenue in future work.

###  10.2 Different factorizations and the c=1 matrix quantum mechanics

The discussion in Section 6, teasing out the subtle differences in our notion
of “spatial locality,” could appear somewhat artificial. Yet such competing
notions of locality might, in fact, be rather generic within the emergent
spacetime paradigm. The $c=1$ matrix quantum mechanics provides a sharp
holographic example. As highlighted in [5], there exists at least two
seemingly natural emergent spatial dimensions. On one hand, the matrix quantum
mechanics in the singlet sector can be recast as a local fermionic field
theory on matrix eigenvalue space. On the other hand, the low-energy target
space dynamics, derived from the worldsheet Liouville string theory, is most
naturally formulated in terms of the string embedding coordinates $X^{0}$ (the
$c=1$ boson) and $\phi$ (the Liouville field).101010 [28] points out important
subtleties in viewing the Liouville direction as spatial coordinate. They
consider instead yet another space on which they define a string field theory
of loop operators, parametrized by the length of the strings they create.
Section 11 of [29] shows precisely how bosonization of the matrix model’s
fermionic field theory maps onto the closed string tachyon dynamics in the
target spacetime. In momentum space, a simple multiplicative phase factor
relates the two - the celebrated “leg-pole factor” - in close parallel to the
$\phi(p)\sim(2E_{p})^{-1/2}a^{\dagger}_{p}$ example discussed in Section 7. In
position space, this gives a non-local map. Natsuume and Polchinski argued all
the (admittedly very simple) gravitational dynamics on the 2d target space
were encoded in the matrix model via this non-local map [30]. Reference [31]
reproduced the entanglement entropy of the bulk 2d tachyon by partitioning the
matrix eigenvalue space. It failed, however, in capturing any
$\mathcal{O}(1/g_{st}^{2})$ contribution - the closest 2d relative of
Area/4$G$. One might blame this on choosing a notion of locality similar to
the Fock space factorization discussed above, thereby capturing only
excitations around the background. Making this precise might help guide future
attempts at diagnosing emergent locality from matrix degrees of freedom.

##  11 Acknowledgements

It is a pleasure to thank Sean Hartnoll for stressing over the years the
importance of defining spatial entanglement entropy in first-quantized
(matrix) quantum mechanics. We thank him, Tom Hartman and Jordan Cotler for
many fruitful discussions and early collaborations on this topic. We also
gratefully acknowledge helpful conversations with Eva Silverstein and Ronak
Soni. Finally, we would like to thank the Yukawa Institute for Theoretical
Physics in Kyoto, where part of this work was completed during the workshop
“Quantum Information and String Theory.”

##  12 Appendix: The algebra for bosons

Here we provide more detail justifying Eqn. 43 using the definition of the
algebra $\mathcal{A}$ in Eqn. 48.

First off, we can write down the commutant $\mathcal{A}^{\prime}$ as

$\mathcal{A}=\bigg{\langle}\\{P_{S_{N}}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}_{1}\otimes\mathds{1}_{2}\otimes...\otimes\mathds{1}_{N}\right)P_{S_{N}}:\vec{x},\vec{x}^{\prime}\in\bar{A}\\}\cup\mathds{1}_{\mathcal{H}_{N}}\bigg{\rangle}$
(118)

so that the center $Z(\mathcal{A})$ is generated by

$\mathcal{Z}=\bigg{\langle}\\{P_{S_{N}}\left(\int_{A}d\vec{x}\ket{\vec{x}}_{1}\bra{\vec{x}}_{1}\otimes\mathds{1}_{2}\otimes...\otimes\mathds{1}_{N}\right)P_{S_{N}}\\}\cup\mathds{1}_{\mathcal{H}_{N}}\bigg{\rangle}$
(119)

At this point we wish to identify the minimal projectors which span the center
$\mathcal{Z}$, as in Eqn. 7. Here there are $N+1$ such projectors, which we
can write as

$\Pi_{k}=\binom{N}{k}P_{S_{N}}\left(\underbrace{\Pi_{A}\otimes...\otimes\Pi_{A}}_{k\>\text{times}}\otimes\underbrace{\Pi_{\bar{A}}\otimes...\otimes\Pi_{\bar{A}}}_{(N-k)\>\text{times}}\right)P_{S_{N}}$
(120)

Physically, $\Pi_{k}$ is the projector onto the subspace with $k$ particles in
$A$ and $N-k$ particles in $\bar{A}$.

The algebra $\Pi_{k}\mathcal{A}\Pi_{k}$ projected onto this subspace takes the
form

$\Pi_{k}\mathcal{A}\Pi_{k}=\bigg{\langle}P_{S_{N}}\left(\ket{\vec{x}}\bra{\vec{x}^{\prime}}\otimes\Pi_{A}...\otimes\Pi_{A}\otimes\Pi_{\bar{A}}\otimes...\otimes\Pi_{\bar{A}}\right)P_{S_{N}}:\vec{x},\vec{x}^{\prime}\in
A\bigg{\rangle}$ (121)

has trivial center on $\Pi_{k}\mathcal{H}_{N}$. To see this, we first write
its commutant restricted to the subspace

$\mathcal{A}^{\prime}\big{|}_{\Pi_{k}\mathcal{H}_{N}}=\bigg{\langle}P_{S_{N}}\left(\ket{\vec{x}}\bra{\vec{x}^{\prime}}\otimes\Pi_{\bar{A}}...\otimes\Pi_{\bar{A}}\otimes\Pi_{A}\otimes...\otimes\Pi_{A}\right)P_{S_{N}}:\vec{x},\vec{x}^{\prime}\in\bar{A}\bigg{\rangle}$
(122)

so that indeed the center on this subspace is trivial

$\mathcal{A}\cap\mathcal{A}^{\prime}\big{|}_{\Pi_{k}\mathcal{H}_{N}}=P_{S_{N}}\left(\underbrace{\Pi_{A}\otimes...\otimes\Pi_{A}}_{k\>\text{times}}\underbrace{\Pi_{\bar{A}}\otimes...\otimes\Pi_{\bar{A}}}_{N-k\>\text{times}}\right)P_{S_{N}}=\mathds{1}\big{|}_{\Pi_{k}\mathcal{H}_{N}}$
(123)

Since the algebra restricted to each subspace is a factor, we know there
exists a tensor product factorization in each block such that all
$\mathcal{O}\in\mathcal{A}$ take the form

$\mathcal{O}=\oplus_{k=0}^{N}\mathcal{O}_{A_{k}}\otimes\mathds{1}_{\bar{A}_{k}}$
(124)

What is this tensor product factorization? It is nothing but the decomposition

$\frac{(\mathcal{H}_{A}\oplus\mathcal{H}_{\bar{A}})^{\otimes
N}}{S_{N}}=\oplus^{N}_{k=0}\left(\frac{\mathcal{H}_{A}^{\otimes
k}}{S_{k}}\otimes\frac{\mathcal{H}_{\bar{A}}^{\otimes N-k}}{S_{N-k}}\right)$
(125)

where we define $\mathcal{H}_{A}^{0}=\mathcal{H}_{\bar{A}}^{0}=\mathbb{C}$. In
particular

$\Pi_{k}\mathcal{H}_{N}=\frac{\mathcal{H}_{A}^{\otimes
k}}{S_{k}}\otimes\frac{\mathcal{H}_{\bar{A}}^{\otimes N-k}}{S_{N-k}}$ (126)

Finally, we stress the symmetric projectors $P_{S_{N}}$ are crucial for
$\mathcal{A}$ to contain multi-particle operators. For example, in the case of
$N=2$, multiplying two basis algebra elements can generate all symmetric
2-particle operators:

$\displaystyle
P_{S_{2}}\left(\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}\otimes\mathds{1}_{2}\right)P_{S_{2}}P_{S_{2}}\left(\ket{\vec{y}}_{1}\bra{\vec{y}^{\prime}}\otimes\mathds{1}_{2}\right)P_{S_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2!}P_{S_{2}}\ket{\vec{x}}_{1}\bra{\vec{x}^{\prime}}\otimes\ket{\vec{y}}_{2}\bra{\vec{y}^{\prime}}P_{S_{2}}$
(128)
$\displaystyle+\frac{1}{2!}\delta(x^{\prime}-y)P_{S_{2}}\left(\ket{\vec{x}}_{1}\bra{\vec{y}^{\prime}}\otimes\mathds{1}_{2}\right)P_{S_{2}}$

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