††institutetext: 1Department of Physics and Astronomy, University of Kentucky,
Lexington, KY 40506, U.S.A.††institutetext: 2Department of Theoretical
Physics, Tata Institute of Fundamental Research, Mumbai 400005, INDIA

# Gauge Invariant Target Space Entanglement in D-Brane Holography

Sumit R. Das1 Anurag Kaushal2 Sinong Liu1 Gautam Mandal2 Sandip P. Trivedi2
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###### Abstract

It has been suggested in https://arxiv.org/abs/2004.00613 that in Dp-brane
holography, entanglement in the target space of the D-brane Yang-Mills theory
provides a precise notion of bulk entanglement in the gravity dual. We expand
on this discussion by providing a gauge invariant characterization of operator
sub-algebras corresponding to such entanglement. This is achieved by finding a
projection operator which imposes a constraint characterizing the target space
region of interest. By considering probe branes in the Coloumb branch we
provide motivation for why the operator sub-algebras we consider are
appropriate for describing a class of measurements carried out with low-energy
probes in the corresponding bulk region of interest. We derive expressions for
the corresponding Renyi entropies in terms of path integrals which can be
directly used in numerical calculations.

TIFR-TH/20-48

##  1 Introduction

In quantum field theory on a fixed space-time background, entanglement between
two regions of space has a well defined meaning in the presence of a UV cutoff
and the corresponding entanglement entropy provides valuable information about
the nature of the quantum state. In quantum gravity this is a tricky issue
since space-time is dynamical. This becomes even more tricky in String Theory
where the fundamental degrees of freedom are extended objects. Nevertheless,
in a weakly coupled semi-classical regime there is an approximate notion which
comes from thinking of gravity as a field theory of gravitons in a background.
It is therefore interesting to ask if there is a precise notion of
entanglement in a complete theory of gravity which reduces to the above
approximate notion in the appropriate regime.

In Das:2020jhy  four of us proposed that in gravitational theories which have
holographic duals, such as the ones which arise in String Theory, such a
precise notion indeed exists. The proposal is that in Dp brane holography for
$p<3$ this notion is provided by entanglement in the target space of the Yang-
Mills theory on the brane. The idea is that a suitable target space constraint
can be associated with a co-dimension one spatial region in the bulk dual. In
the Yang-Mills theory the target space constraint then leads to a sub-algebra
of operators. The expectation values of operators in this sub-algebra can be
obtained correctly using a reduced density matrix lying in the sub-algebra
itself. The von-Neumann entropy for this reduced density matrix is the precise
notion, sought above, of a geometric entropy of the bulk sub-region. When the
entire system is in a pure state this entropy is entirely due to quantum
entanglement. When the system is in a mixed state, this contains a classical
piece.

We conjectured that when the state is the $N$ brane bound state or its
slightly heated up version, this von Neumann entropy is given by the
Bekenstein formula $A/4G$ where $A$ is the area of the entangling surface in
the dual black brane geometry 111For mixed states a spatial Bekenstein bound
is not generally valid. In such situations one may need to consider the fine
grained entropy of the degrees of freedom of a light sheet associated with a
boundary of area $A$ bousso  . In a complete theory of gravity one would
expect that in any definition of geometric entanglement entropy the UV cutoff
is automatic. Our conjecture therefore implies that the UV cutoff is provided
by Netwon’s constant, and not by e.g. the string length. Indeed, for simple
entangling surfaces for $p<3$ our conjecture yields answers which scale as
$N^{2}$ and are expressible purely in terms of the appropriate dimensionless
quantities of the Yang-Mills theory - as one would expect.

The bulk entanglement we are considering is across any codimension two
surface. This is distinct from the corrections to holographic entanglement
entropy Ryu:2006bv ; Hubeny:2007xt  due to entanglement across extremal
surfaces Faulkner:2013ana ; recentbulk  or what would become a quantum
extremal surface Engelhardt:2014gca . Note that the proposal discussed above,
for Dp branes with $p>0$, considers the same target space constraint holding
at all points on the Dp brane. The boundary of the corresponding codimension
one spatial surface in the bulk then includes the entire spatial boundary of
space-time. For $p>0$ it is also possible to discuss a more general notion of
entanglement which arises for a sub algebra of observables tied to a target
space constraint that applies only to a part of the base space along which the
Dp branes extend.

The fact that Newton’s constant is the natural cutoff is consistent with the
idea that the Bekenstein formula for black hole entropy involves a
renormalized Newton constant thooft ; susskind1 ; susskind2 . Furthermore it
has been argued in induced  that this naturally happens in theories of induced
gravity. In the past, jacobson  has argued that Einstein’s equations follow
from thermodynamics, provided the cutoff in the entanglement entropy in a
theory of gravity is Newton’s constant. myers  had also conjectured that the
entanglement entropy across an arbitrary surface in a theory of gravity
saturates the Bekenstein bound. The reasoning of Das:2020jhy  is initimately
tied with the identification of bulk entanglement with target space
entanglement and therefore differs from these other papers in an essential
way.

The appearance of Newton’s constant as the UV cutoff is also consistent with
the calculation of the bulk entanglement entropy in the c=1 Matrix Model / 2d
string theory duality Das:1995vj ; Das:1995jw ; Hartnoll:2015fca . The
holographic theory is now gauged quantum mechanics of a single $N\times N$
hermitian matrix. In this case the space of eigenvalues can be interpreted as
a bulk space and the only propagating mode of the two dimensional string is
related to the density of eigenvalues or the collective field Das:1990kaa
222The massless mode of two dimensional string theory is related to the
density of eigenvalues by an integral transform with a kernel whose scale is
the string scale polnat . Strictly speaking, the entanglement entropy
calculated here is in the eigenvalue space. However this would agree with a
bulk notion in terms of usual string theory upto an uncertainty of the order
of the string scale. . The Matrix Model is described exactly by $N$ free non-
relativistic fermions in an inverted harmonic oscillator potential which can
be rewritten as a second quantized field theory living in the eigenvalue space
Sengupta:1990bt . Entanglement of a region of the eigenvalue space can be then
defined in the usual way in this field theory. In fact, the c=1 theory
provides the simplest example of target space entanglement, since the emergent
space is the target space of the matrix model. In an approriate limit, the
entanglement entropy agrees with what one would expect from the low energy
effective field theory, but with the UV cutoff replaced by the position
dependent string coupling. After incorporating the appropriate factor of $N$,
the UV cutoff is identified with Newton’s constant.

The target space entanglement explored in Das:2020jhy  is in a gauge fixed
version of the holographic theory. This involves the temporal gauge for the
gauge field, and a further gauge choice. For the c=1 model the latter is the
gauge where the single matrix is diagonal. The remaining symmetries are Weyl
transformations which permute the eigenvalues. In the Dp brane theories we
have multiple matrices, and the remaining gauge freedom in the temporal gauge
can be used to diagonalize a single matrix which needs to be chosen to express
a desired target space constraint. The situation studied in detail in
Das:2020jhy  involves diagonalization of one of the scalar fields. The target
space constraint is then expressed in terms of an allowed range of the
eigenvalues, e.g. requiring the eigenvalue to be larger than some number. This
corresponds to a bulk region characterized by one of the transverse
coordinates being larger than some value and a spatial co-dimension one planar
entangling surface which bounds this region. The full Hilbert space breaks up
into a direct sum of superselection sectors characterized by the number of
eigenvalues which satisfy the constraint. In each sector, the smaller Hilbert
space is a direct product, which allows one to define a reduced density matrix
in the usual fashion. Two possible versions of the proposal for a
corresponding reduced density matrix were studied.

While Das:2020jhy  specified the general properties of the operators belonging
to the sub-algebra of operators associated with a given target space
constraint, a procedure to obtain such operators in terms of the operators of
the matrix theory was not specified. Furthermore, in this gauge fixed
formalism, it is difficult (though not impossible) to describe general
entangling surfaces.

In this paper we address both these issues by developing a gauge invariant
description of target space entanglement. This will be achieved by
constructing a projection operator appropriate for the desired target space
constraint. Starting with a gauge invariant operator which contains a string
of matrices, the subalgebra then consists of operators obtained by projecting
each of these matrices. We show that in the gauge used in Das:2020jhy  these
yield the correct class of operators in each superselection sector. Moreover,
the gauge invariant construction enables us to easily formulate other target
space constraints, e.g. those which correspond to entangling surfaces in the
bulk at a given value of the radial coordinate in the transverse space of the
D-branes - in this case we also show how the target space constraint can be
implemented explicitly by developing a formalism for a polar decomposition of
the matrices.

The proposed connection of a target space constraint with a bulk region is
based on several ingredients of gauge-gravity duality and closely tied to the
emergence of bulk locality. As is well known, the velocity dependent potential
between two stacks of D0 branes in supergravity follows from an effective
action calculation in the Coulomb branch of D0 brane quantum mechanics dkps ;
bfss ; kt ; tram ; rseven ; twoloop ; polbecker . For example, one may
consider a single D0 brane stripped off from a stack of $N$ D0 branes in their
bound state, corresponding to a point on the Coulomb branch of the D0 brane
matrix theory. The Higgs vev at this point is then the transverse location of
this probe D0 brane. This implies that a restriction to a region ${\cal{R}}$
of the bulk can be described as a restriction in the target space of the brane
theory. In the ’t Hooft limit, the gravity dual of the bound state of D0
branes is a non-trivial supergravity background imsy , and the velocity
dependent potential can be obtained from a DBI-CS action for the probe D0
brane moving in this background. It can also be calculated from the effective
action evaluated at the corresponding point in moduli space in the gauge
theory. Supersymmetry guarantees in fact that the leading terms in the
effective action may be calculated perturbatively. We argue that the potential
will also agree with the effective action for operators in the subalgebra,
${\cal A}_{\cal R}$, which we associate with the region ${\cal R}$, thereby
arguing that the subalgebra contains operators needed to describe bulk
measurements which can be carried out in ${\cal R}$.

A somewhat stronger connection comes from a point on the Coulomb branch
$SU(N)\rightarrow SU(N-2)\times U(1)\times U(1)$, which corresponds to two D
branes stripped off from the rest. This situation has been studied in detail
for D3 branes in branewaves . In this case, when the two individual branes are
excited, the lowest order terms in the effective action of the two $U(1)$’s
agree precisely with a supergravity calculation of the potential between the
two branes which follow from exchanges of supergravity modes propagating on
the $AdS_{5}\times S^{5}$ produced by the remaining $(N-1)$ branes. This
agreement is more detailed than the single D0 DBI+CS action since the
supergravity modes in this background mix non-trivially. There should be a
similar agreement for D0 branes.

Going beyond the ground state, in an excited state of the gauge theory which
corresponds to a modified supergravity background, also one expects that the
potential experiences by a probe brane can be obtained from the DBI+CS action,
and this potential should agree with an effective action calculation which can
be carried out in the gauge theory keeping operators in the subalgebra ${\cal
A}_{\cal R}$. Of course a perturbative calculation will no longer suffice to
demonstrate this333 unless the excited state preserves a high degree of
supersymmetryDouglas:1997pj . But one might hope to be able to check this as
numerical techniques improve further. In fact some progress has already been
made along these lines in obtaining the dynamics of probe branes at finite
temperatures hanadaprobe . In these calculations some evidence was found that
the supergravity fields couple to operators in the probe brane in a manner
consistent with the generalized AdS/CFT correspondence discussed in jevyoneya
; sekinoyoneya .

To summarize, the dynamics on the Coulomb branch should allow one to measure
the local background, at least at the level of one point functions, for
gravity and other supergravity modes, in a region ${\cal R}$. This dynamics we
argue can be obtained in the gauge theory by studying the effective action for
gauge invariant operators. If we are interested in measuring the supergravity
fields only in the region ${\cal{R}}$ of the bulk, we argue that it is
sufficient to only consider the operators in the subalgebra ${\cal A}_{\cal
R}$ associated with ${\cal R}$. As mentioned above, this subalgebra contains
gauge invariant operators obtained after carrying out a suitable projection
determined by the target space constraint which corresponds to the bulk region
${\cal R}$.

While the discussion above pertains to the Coulomb branch, the considerations
should be valid for a general configuration which appears in the wavefunction
of the $N$ D0 brane bound state. This motivates our identification of bulk
entanglement with target space entanglement.

We should mention that we expect the effective action and the related
correlation functions of the projected operators to provide only some and not
all of the detailed information about supergravity modes and the dual boundary
operators related to them via the BDHM-HKLL construction hkll . In particular,
the energy momentum tensor is not contained in the sub-algebra, only its
projected version is. We expect that this imposes important limitations on the
extent to which we can learn about the stress energy tensor’s correlation
functions from the sub-algebra. In fact, as was importantly argued in suvrat ;
suvrat2 , if the sub-algebra would allow all information pertaining to the
stress tensor to be obtained, then for an annular region adjacent to the
boundary, the entanglement entropy would be exactly zero 444The argument is as
follows: if the energy-momentum tensor at all points on the boundary is
included in the set of observables, so is the energy and the projector to the
ground state. In the vacuum, the latter is the density matrix of the whole
system. This would mean that the associated entanglement entropy must be
exactly zero. In our case the energy, and therefore also the ground state
projector, is not an element of the sub-algebra.. In this sense the
association of a target space constraint with a bulk region is approximate.

An analytic calculation of the target space entanglement entropy requires an
explicit expression for the wavefunction. Even in the simplest case of D0
branes, explicit expressions for the bound state wavefunction is not known,
though the the existence of bound states of D0 branes has been proven bound
555For bosonic BFSS models, the existence of bound states has been proved both
numerically Kawahara:2007fn  and analytically in the limit of large dimensions
Mandal:2009vz .. There are candidates for approximate wavefunctions which can
be in principle used to perform analytic computations of the von Neumann
entropy approx . However, there has been substantial progress in numerical
calculations of properties of D0 brane bound states at finite temperature:
these calculations provide precision tests of the AdS/CFT correspondence
catterall ; Hanada:2016zxj . These calculations deal with thermodynamic
quantities, correlation functions hanadayoneya  and investigations of probe
dynamics hanadaprobe . In this paper we derive path integral expressions for
target space Renyi entropies which can be directly used to perform numerical
calculations. Work in this direction is being developed currently ushanada :
these calculations should prove or disprove our conjecture about saturation of
the Bekenstein bound.

The formalism of target space entanglement entropy has been developed in
DMT:2018  and Mazenc:2019ety . Notions similar to target space entanglement
have been used to define entanglement in string theory in the worldsheet
formalism stringee  and in various explorations in holographic entanglement
internalextremal . Another notion of entanglement of internal degrees of
freedom (also combined with spatial degrees of freedom) called entwinement has
been discussed in entwinement . Notions of entanglement associated with other
kinds of partitions of large-N degrees of freedom have been explored in alet .
The proposal of Das:2020jhy  is distinct from these other works.

The paper vanrams  has explored general extremal surfaces in D brane
geometries (as distinct from RT surfaces) and speculated on possible meanings
of their areas with entanglement of degrees of freedom in the D0 brane quantum
mechanics. In particular, these authors have considered subsets of operators
consisting of linear combinations of traceless symmetric products of the
matrices in the D0 brane theory which would correspond to functions which have
support on some region of $S^{8}$ and speculated that an entropy can be
associated with such a subset. Our proposal is quite different from this: we
aim to describe bulk entanglement which involves the radial direction as well,
and we associate an entropy with a closed subalgebra.

This paper is organized as follows. In section 2 we introduce the gauge
invariant construction of operator algebras which define a target space
entanglement. We show how this construction leads to the gauge fixed version
discussed in Das:2020jhy  and review the proposed connection to bulk
entanglement and our conjecture about the saturation of Bekenstein bound. We
also discuss how to impose radial constraints in target space by developing a
polar decomposition of matrices. In section 3 we discuss the connection of
target space entanglement and bulk entanglement. In section 4 we recapitulate
the conjecture in Das:2020jhy  that the target space entanglement entropy
saturates the Bekenstein bound. In section 5 we derive path integral
expressions for target space Renyi entropies which can be directly used for
numerical calculations. Section 6 contains concluding remarks. The Appendix A
contains some details of the construction of projected operators. Appendix B
provides details of matrix polar decompositions for multiple matrices.
Appendix C deals with the DBI+CS action of a single D0 brane in the
supergravity background produced by $N$ other extremal branes and its
comparison with D0 brane quantum mechanics effective action.

##  2 Gauge Invariant Target Space Entanglement

In this section we will show how target space entanglement in a theory of
multiple matrices can be formulated in a gauge invariant fashion. A more
detailed description appears in Appendix A.

###  2.1 Review of the gauge-fixed formulation

In a previous paper Das:2020jhy , we considered the $D0$ brane theory and
discussed a bulk region specified by a condition on one of the spatial bulk
coordinates, say $x^{1}$. The condition took the form,

$x^{1}>a,$ (1)

for some real number $a$. We proposed that this condition mapped to a target
space constraint in the quantum mechanical dual theory that lives on the
boundary. And the bulk entanglement entropy maps to the entanglement entropy
associated with this target space constraint in the boundary theory. The
entanglement entropy defined in this way is manifestly finite when $N$ is
finite.

The action of D0 brane quantum mechanics is given by

$S=\frac{N}{2(g_{s}N)l_{s}}{\rm Tr}\int
dt\left[\sum_{I=1}^{9}(D_{t}X^{I})^{2}-\frac{1}{l_{s}^{4}}\sum_{I\neq
J=1}^{9}[X^{I},X^{J}]^{2}\right]+{\rm fermions}$ (2)

where $X^{I}$ are $N\times N$ hermitian matrices, and the covariant derivative
is defined by

$D_{t}X^{I}\equiv\partial_{t}X^{I}+i[A_{t},X^{I}]$ (3)

In the example above, the target space constraint involves the operator
$X^{1}$ in the boundary theory. To specify the target space constraint, we
worked in the gauge where $A_{t}=0$. The remaining gauge freedom consists of
time independent $SU(N)$ rotations, which we fixed by requiring $X^{1}$ to be
diagonal. The corresponding operators and their canonical conjuagte momenta
have the form

$\displaystyle\hat{X}^{1}$ $\displaystyle\rightarrow$ $\displaystyle{\rm
diag}\left(\hat{\lambda}_{1},\cdots\hat{\lambda}_{N}\right)$
$\displaystyle{\hat{\Pi}}_{1}$ $\displaystyle\rightarrow$ $\displaystyle{\rm
diag}\left({\hat{\pi}}_{1}\cdots{\hat{\pi}}_{N}\right)$ (4)

This does not fix the gauge completely: we are left with Weyl transformations,

$\displaystyle(\hat{\lambda}_{1},\hat{\lambda}_{2},\cdots,\hat{\lambda}_{N})$
$\displaystyle\mapsto$
$\displaystyle(\hat{\lambda}_{\sigma(1)},\hat{\lambda}_{\sigma(2)},\cdots,\hat{\lambda}_{\sigma(N)}),\,\sigma\in
S(N)$ $\displaystyle\hat{X}^{L}$ $\displaystyle\mapsto$
$\displaystyle\sigma(\hat{X}^{L}),\;\sigma(\hat{X}^{L}_{ij})=\hat{X}^{L}_{\sigma(i)\sigma(j)},\,L=2,\cdots,9.$
(5)

and $U(1)^{N}$ transformations which keep the diagonal matrix elements of all
the matrices invariant and multiplies the off-diagonal elements by phases
666In a previous version of this paper which appeared on the arXiv, we did not
consider these $U(1)^{N}$ transformations. Subsequently the paper lawrence
appeared, where these remaining symmetries were emphasized.

$\hat{X}^{L}_{ij}\mapsto\hat{X}^{L}_{ij}e^{i(\theta_{i}-\theta_{j})}$ (6)

where $\theta_{i}$ are angles. The physical state is constructed by adding
Weyl and $U(1)^{N}$ transforms. Let us work in a basis in the Hilbert space
comprised of eigenvectors of the operators
$\hat{\lambda}_{i},\hat{X}^{L}_{ij}$ with eigenvalues
$\lambda_{i},X^{L}_{ij}$. As is well known the transformation to the
eigenvalues of $X^{1}$ leads to a van der Monde factor in the measure of
integration. In the following we will absorb a square root of this factor in
the wavefunction so that the modified wavefunction is antisymmetric under an
interchange of the eigenvalues. Then a Weyl and $U(1)^{N}$ symmetrized state
is 777Our conventions for normalization of states is different from
Das:2020jhy .,888Since the Weyl group elements $g_{W}$ and the $U(1)^{N}$
group elements $g_{U}$ do not commute, the combined action on a given state
$|\psi\rangle$ depends on the order in which the group elements act. In (7) we
have applied $g_{U}$ followed by $g_{W}$. It is not difficult to see, however,
the ‘symmetrized’ state, which involves sum over the entire set of transforms,
does not depend on the order:
$\sum_{W,U}g_{W}g_{U}\psi=\sum_{W,U}g_{U}g_{W}\psi$.

$|\\{\lambda_{i}\\},\\{X^{L}_{ij}\\}\rangle_{W}=\frac{1}{N!}\int\prod_{I=1}^{N}\frac{d\theta_{i}}{2\pi}\sum_{\sigma\in
S_{N}}{\rm
sgn}(\sigma)|\\{\lambda_{\sigma(i)}\\},\\{X^{L}_{\sigma(i)\sigma(j)}e^{i(\theta_{\sigma(i)}-\theta_{\sigma(j)})}\\}\rangle$
(7)

Note that this symmetrized state is not an eigenstate of the individual
$\hat{\lambda}_{i},\hat{X}^{L}_{ij}$’s. They are eigenstates of gauge
invariant operators which are traces of products of
$\hat{X}^{I},{\hat{\Pi}}_{I}$ or products of these traces.

In this gauge, it was proposed that the required target space condition,
corresponding to (1), on an eigenvalue $\lambda_{i}$, was

$\lambda_{i}>a.$ (8)

The target space constraint can be generalized trivially to

$\lambda_{i}\in A$ (9)

where $A$ is some interval on the real line, with a corresponding change in
the bulk region (1).

Since there are $N$ eigenvalues the constraint gives rise to $N+1$ different
possibilities depending on whether $0,1,\cdots,N,$ of the eigenvalues meet the
constraint. These different possibilities actually can be thought of as giving
rise to different superselection sectors. The Hilbert space thus becomes a
direct sum of Hilbert spaces,

${\mathcal{H}}_{N}=\oplus_{k}{\mathcal{H}}_{k,N-k}$ (10)

where ${\mathcal{H}}_{k,N-k}$ denotes the sector where $k$ of the eigenvalues
of $\hat{X}^{1}$ are in the region of interest $A$ and the rest in its
complement ${\bar{A}}$.

The reduced density matrix in the $k^{\rm th}$ superselection sector,
${\tilde{\rho}}_{k,N-k}$, can be obtained by tracing out the degrees of
freedom corresponding to the remaining $(N-k)$ eigenvalues. The corresponding
target space entanglement entropy can then be obtained as the von Neumann
entropy for this density matrix and the full entanglement entropy for all
sectors can be obtained by adding the entropy from each sector 999The sector
with no eignvalues meeting the required condition is important to keep in
mind. It is taken to be one dimensional and the density matrix is then a
number corresponding to the probability of finding no eigenvalue meeting the
constraint.. Note that the density matrix in each individual sector is not
normalized. Rather the trace ${\rm tr}_{k}{\tilde{\rho}}_{k,N-k}$ is simply
the probability that $k$ of the eigenvalues are in the region of interest. The
full reduced density matrix is block diagonal, where each block corresponds to
a superselection sector

$\rho=\begin{pmatrix}{\tilde{\rho}}_{0,N}&{\bf 0}&{\bf 0}&\cdots&{\bf 0}\\\
{\bf 0}&{\tilde{\rho}}_{1,N-1}&{\bf 0}&\cdots&{\bf 0}\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\ {\bf 0}&{\bf 0}&{\bf 0}&{\bf
0}&{\tilde{\rho}}_{N,0}\end{pmatrix}$ (11)

does have unit trace, so that

$S=-{\rm tr}(\rho\log\rho)=-\sum_{k=0}^{N}{\rm
tr}_{k}({\tilde{\rho}}_{k,N-k}\log{\tilde{\rho}}_{k,N-k})$ (12)

is a legitimate von Neumann entropy.

Actually our proposal had two versions which arise when we think more
precisely about tracing out the degrees of freedom corresponding to the
remaining $N-k$ eigenvalues. By a suitable choice of gauge the eigenvalues of
$X^{1}$ meeting the constraint can be taken to be the first $k$ eigenvalues.

In the rest of the paper, the matrix indices $i,j,\cdots=1\cdots N$; the
indices $a,b=1\cdots k$ and $\alpha,\beta=k+1\cdots N$. In the rest of this
subsection, we denote $X^{2},X^{3},...,X^{9}$ by $X^{L}$, $L=2,3,...,9$.

  1. 1.

In the first version, one traces out the degrees of freedom corresponding to
the $(N-k)$ eigenvalues of $X^{1}$ which do not satisfy eq.(8),
$\lambda_{\alpha}$ and the degrees of freedom in $(N-k)\times(N-k)$ block for
the remaining spatial matrices,
$X_{\alpha\beta}^{2},X_{\alpha\beta}^{3},\cdots X_{\alpha\beta}^{9}$. In
addition, one also traces out the degrees of freedom corresponding to the off-
diagonal elements $(X^{2})_{a\alpha},(X^{2})_{\alpha a}$; and similarly for
$X^{3},X^{4},\cdots X^{9}$. As a result the only degrees of freedom we retain
are in the $k\times k$ block. In the basis we are using, the reduced density
matrix for the $k$’th sector is then given by

$\displaystyle{\tilde{\rho}}^{(1)}_{k,N-k}$
$\displaystyle\left(\lambda_{a},X^{L}_{ab};\lambda^{\prime}_{a},X^{\prime
L}_{ab}\right)=$ $\displaystyle{N\choose
k}\int[d\lambda_{\alpha}dX^{L}_{a\alpha}dX^{L}_{\alpha
a}dX^{L}_{\alpha\beta}]~{}\rho_{tot}\left(\lambda_{a},X^{L}_{ab},\lambda_{\alpha},X^{L}_{a\alpha},X^{L}_{\alpha
a},X^{L}_{\alpha\beta};\lambda^{\prime}_{a},,X^{\prime
L}_{ab},\lambda_{\alpha},X^{L}_{a\alpha}X^{L}_{\alpha
a}X^{L}_{\alpha\beta}\right)$ (13)

where $\rho_{tot}$ is the density matrix of the state of the entire system.

  2. 2.

In the second version, one traces out only the degrees of freedom which lie in
the $(N-k)\times(N-k)$ blocks for all matrices and retains the remaining
degrees of freedom. So for $X^{1}$ which is diagonal we retain the first $k$
eigenvalues which meet the constraint, but for $X^{2}$ we retain not only the
the elements $(X^{2})_{ab}$ but also the off-diagonal elements
$(X^{2})_{a\alpha},(X^{2})_{\alpha a}$ and similarly for $X^{3},\cdots X^{9}$,

$\displaystyle{\tilde{\rho}}^{(2)}_{k,N-k}$
$\displaystyle\left(\lambda_{a},X^{L}_{ab},dX^{L}_{a\alpha}dX^{L}_{\alpha
a};\lambda^{\prime}_{a},X^{\prime L}_{ab}dX^{\prime L}_{a\alpha}dX^{\prime
L}_{\alpha a}\right)=$ $\displaystyle{N\choose
k}\int[d\lambda_{\alpha}dX^{L}_{\alpha\beta}]~{}\rho_{tot}\left(\lambda_{a},X^{L}_{ab},\lambda_{\alpha},X^{L}_{a\alpha},X^{L}_{\alpha
a},X^{L}_{\alpha\beta};\lambda^{\prime}_{a},,X^{\prime
L}_{ab},\lambda_{\alpha},X^{\prime L}_{a\alpha}X^{\prime L}_{\alpha
a}X^{L}_{\alpha\beta}\right)$ (14)

In each sector labelled by $k$, the corresponding density matrix evaluates
expectation values of a closed subalgebra of operators which correspond to
measurements on the variables which are retained. In the first version, the
action of such an operator on a general Weyl and $U(1)^{N}$ symmetrized state
of the form (7) has the form

$\displaystyle{\hat{{\cal{O}}}}~{}$
$\displaystyle|\\{\lambda_{a},X^{L}_{ab},X^{L}_{a\alpha},X^{L}_{\alpha\beta},\lambda_{\alpha}\\}\rangle_{W}$
$\displaystyle=\int[d\lambda_{a}^{\prime}][dX^{\prime
L}_{ab}]~{}{\tilde{O}}(\\{\lambda_{a}^{\prime},X^{\prime
L}_{ab})\\},\\{\lambda_{a},X^{L}_{ab}\\})~{}|\\{\lambda_{a}^{\prime}X^{\prime
L}_{ab}\\};\\{\lambda_{\alpha},X^{L}_{a\alpha},X^{L}_{\alpha\beta}\\}\rangle_{W}$
(15)

Operators which satisfy this form a subalgebra:
${\tilde{O}}(\\{\lambda_{a},X^{L}_{ab}\\},\\{\lambda_{a}^{\prime},X^{\prime
L}_{ab})\\})$ then denote the matrix elements of an operator in the smaller
Hilbert space in this sector. The reduced density matrix which evaluates
expectation values of such operators is given by (13).

Similarly for the second version the action is given by

$\displaystyle{\hat{{\cal{O}}}}~{}$
$\displaystyle|\\{\lambda_{a},X^{L}_{ab},X^{L}_{a\alpha},X^{L}_{\alpha\beta},\lambda_{\alpha}\\}\rangle_{W}$
$\displaystyle=\int[d\lambda_{a}^{\prime}][dX^{\prime L}_{ab}][dX^{\prime
L}_{a\alpha}][dX^{\prime L}_{\alpha
a}]~{}{\tilde{O}}(\\{\lambda_{a}^{\prime},X^{\prime L}_{ab},X^{\prime
L}_{a\alpha},X^{\prime L}_{\alpha
a})\\},\\{\lambda_{a},X^{L}_{ab},X^{L}_{a\alpha},X^{L}_{\alpha a}\\})$
$\displaystyle\kern 200.0pt~{}|\\{\lambda_{a}^{\prime}X^{\prime
L}_{ab}X^{\prime L}_{a\alpha}X^{\prime L}_{\alpha
a}\\},\\{\lambda_{\alpha}X^{L}_{\alpha\beta}\\}\rangle_{W}$ (16)

It is clear that the density matrix is again of the form (11).

The associated entanglement entropy for a density matrix of the form eq.(11)
is expressible in terms of the normalized density matrices of the subsectors,
${\hat{\rho}}_{k,N-k}=\frac{1}{p_{k,N-k}}{\tilde{\rho}}_{k,N-k}$ as

$S=-\sum_{k=0}^{N}\left[p_{k,N-k}\log p_{k,N-k}+p_{k,N-k}{\rm
tr}_{k}({\hat{\rho}}_{k,N-k}\log{\hat{\rho}}_{k,N-k})\right]$ (17)

The distillable part of the entanglement is only the second term in (17),
while the first term is a classical piece which cannot be used as a quantum
resource for teleportation ST ; V .

Before closing this subsection let us note that while we have focussed on
bosonic operators above, a similar discussion also applies to fermionic
operators in the theory. Depending on which version of our proposal we
consider, the appropriate adjoint color degrees of freedom for fermonic
operators are also to be retained in the sub-algebra.

###  2.2 Gauge-invariant formulation

A drawback of the discussion in the previous paper Das:2020jhy , and our
discussion above, is that this description of the target space constraint and
the related entropy has been given in a particular gauge, e.g., for the
example above we worked in the gauge where $X^{1}$ is diagonal. Furthermore,
while (15) and (16) describe the properties satisfied by operators belonging
to the relevant subalgebra of observables, this does not tell us what these
operators are in terms of the basic operators of the theory. In this
subsection, we will address both these issues and give a gauge invariant
description of the target space constraint; this will also allow us to
generalise the discussion considerably to a much wider class of bulk regions.

In general, suppose we have a region in the bulk at time $t$ specified by one
condition among the $9$ spatial coordinates,

$f(x_{i})>0$ (18)

We would like to specify the target space constraint corresponding to this
bulk region in a gauge invariant manner. For this purpose, instead of starting
with a wave function, constructing the density matrix by a partial trace over
some degrees of freedom and calculating its entropy, it is useful to think of
the entanglement entropy as arising because one is dealing with a suitable
sub-algrebra of the set of all observables. The sub-algebra corresponds to the
operators whose expectation values can be obtained correctly from the reduced
density matrix obtained after tracing out the unwanted degrees of freedom.
Specifying the sub algebra is an equivalent way of specifying the tracing out
procedure and implementing the target space constraint.

Note that when we think in this way, starting from a sub-algebra of all
observables, the density matrix itself must lie in the sub-algebra of
observables and, as mentioned, must give the correct expectation values for
all operators in the sub-algebra. In addition the density matrix is
normalised, as usual, to meet the condition, ${\rm tr}\rho=1$. This specifies
the density matrix uniquely and the entanglement entropy is then the von-
Neumann entropy of this density matrix.101010The uniqueness can be easily
seen. Suppose there are two possible density matrices, $\rho$ and
$\tilde{\rho}$, which satisfy ${\rm Tr}(\rho O)$ =${\rm Tr}(\tilde{\rho}O)$ =
${\rm Tr}(\rho_{tot}O)$ for all operators $O$ in the sub-algebra. Hence ${\rm
Tr}[(\rho-\tilde{\rho})O]=0$ for all such operators; since both $\rho$ and
$\tilde{\rho}$ belong to the sub-algebra, this can only happen if
$\rho=\tilde{\rho}$. When there are superselection sectors, as in our current
discussion, we found above a corresponding density matrix in each sector.
However, as we will see, and this is one of the virtues of specifying a sub-
algebra to implement the target space constraint, the sub-algebra of interest
can in fact be specified once and for all in a gauge invariant manner
regardless of the sector we are working in.

Before proceeding to a gauge invariant formulation let us first address the
question; how do we determine the relevant subalgebra of operators even in a
fixed gauge. To illustrate the procedure it is useful to consider the simple
case of gauged quantum mechanics of a single matrix ${\hat{M}}$. In the gauge
where the matrix is diagonal, this reduces to a theory of $N$ fermions on a
line. The position and momentum operators of individual fermions are the
$\hat{\lambda}_{i}$ and ${\hat{\pi}}_{i}$. Consider a typical one body
operator in this theory

$\hat{\cal C}_{n,m}=\sum_{i=1}^{N}\hat{\lambda}_{i}^{n}{\hat{\pi}}_{i}^{m}$
(19)

We want to impose a target space constraint where the eigenvalues of
$\hat{\lambda}_{i}$ lie in a certain interval on the line denotes by $A$. This
corresponding subalegbra consists of operators which act only on the fermions
which lie in this interval. Such an operator can be constructed as follows.
Define a projection operator

$(\hat{P}_{A})_{i}=\int_{A}dx~{}\delta(x-\hat{\lambda}_{i})$ (20)

where $A\subset R$. By considering matrix elements between arbitrary states it
is clear that this operator indeed satisfies

$(\hat{P}_{A})_{i}^{2}=(\hat{P}_{A})_{i}$ (21)

Now, starting from an operator (19) construct an operator by replacing each of
the $\hat{\lambda}_{i},{\hat{\pi}}_{i}$ by
$(\hat{P}_{A})_{i}\hat{\lambda}_{i}(\hat{P}_{A})_{i}$ and
$(\hat{P}_{A})_{i}{\hat{\pi}}_{i}(\hat{P}_{A})_{i}$ to get

$(\hat{\cal
C}_{n,m})^{P}_{A}=\sum_{i=1}^{N}(\hat{P}_{A})_{i}\hat{\lambda}_{i}^{n}(\hat{P}_{A})_{i}{\hat{\pi}}_{i}(\hat{P}_{A})_{i}{\hat{\pi}}_{i}(\hat{P}_{A})_{i}\cdots(\hat{P}_{A})_{i}{\hat{\pi}}_{i}(\hat{P}_{A})_{i}$
(22)

where we have used (21) and the fact $[(\hat{P}_{A})_{i},\hat{\lambda}_{j}]=0$
to simplify the expression. It may be now easily checked that the expectation
value of $(\hat{\cal C}_{n,m})^{P}_{A}$ in a general many particle state
becomes a sum of terms: each term corresponds to a sector with $k$ particles
in the interval $A$. The k-th term contains the contribution only from the k
particles in A. This is discussed in more detail in Appendix A.

It is now straightforward to construct these operators in a gauge invariant
fashion. In terms of the matrix valued operator ${\hat{M}}$ the projector is
clearly given by

$(\hat{P}_{A})=\int_{A}dx~{}\delta(x{\bf I}-{\hat{M}})$ (23)

where ${\bf I}$ is the $N\times N$ identity operator.This procedure
generalizes to the D0 brane theory with multiple matrices as we now describe.

To obtain a sub-algebra which corresponds to the target space constraint
following from eq.(18) we consider its target space analogue,

$f({\hat{X}}^{I})>0$ (24)

and the following projection operator which follows from this constraint

$\hat{P}_{1}=\int_{x>0}dx\delta(x{\bf I}-f(\hat{X}^{I}))$ (25)

where $\hat{X}^{I}$ are the operators in D0 brane quantum mechanics. Note that
we have taken the function $f$ here to be the same as in eq. (18) but its
argument in eq.(25) are now operators 111111More generally the target space
constraint and bulk constraint could be related in a more complicated fashion,
see below for further discussion of this point.. The integral is over positive
values of $x$ which is a $c$ number. We will choose the operator $f(X^{I})$ to
be hermitian.

In general there will be ordering ambiguities which will arise in going from
the function $f$ in the bulk to the corresponding function $f$ of matrix
operators which appears in eq.(25); we will comment on this issue further
towards the end of this subsection.

By doing a unitary transformation and going to a basis in which $f(X^{I})$ is
diagonal one can easily check that $\hat{P}_{1}$ is a projection operator
satisfying the condition

$\hat{P}_{1}^{2}=\hat{P}_{1}$ (26)

Gauge invariant operators can now be obtained by conjugating with
$\hat{P}_{1}$ and taking a trace. For example, starting from
$\hat{X}^{I},I=1,\cdots 9,$ we construct the corresponding projected operators
$\hat{P}_{1}\hat{X}^{I}\hat{P}_{1},I=1,\cdots 9$, and then take a trace over
the color degrees to obtain gauge invariant operators from these projected
operators $Tr(\hat{P}_{1}\hat{X}^{1}),Tr(\hat{P}_{1}\hat{X}^{2}),\cdots
Tr(\hat{P}_{1}X^{9})$ (here we have used cyclicity of the trace and the facts
that $[\hat{P}_{1},\hat{X}^{I}]=0$ and $\hat{P}_{1}^{2}=1$ to drop one of the
two $\hat{P}_{1}$ factors) .

More generally let ${\cal O}$ be any operator obtained by multiplying a string
of $\hat{X}^{I}$’s and ${\hat{\Pi}}_{I}$’s where the ${\hat{\Pi}}_{I}$’s are
the momenta conjugate to the $\hat{X}^{I}$’s. Schematically we can write
${\cal O}=\cdots\hat{X}^{I}\cdots{\hat{\Pi}}_{J}\cdots$ to depict the string
of $X^{I}$’s and ${\hat{\Pi}}_{J}$’s in some order. We can obtain a gauge
invariant operator from ${\cal O}$ by taking the colour trace,

${\hat{{\cal{O}}}}=Tr({\cal
O})=Tr(\cdots\hat{X}^{I}\cdots{\hat{\Pi}}_{J}\cdots).$ (27)

Now to obtain elements of the desired sub-algebra we consider the projected
operators,

$\hat{X}^{I}\rightarrow(\hat{X}^{I})^{P_{1}}=\hat{P}_{1}X^{I}\hat{P}_{1}$ (28)

and

${\hat{\Pi}}_{J}\rightarrow({\hat{\Pi}}_{J})^{P_{1}}=\hat{P}_{1}{\hat{\Pi}}_{J}\hat{P}_{1}$
(29)

and construct the string

$\cdots(\hat{X}^{I})^{P_{1}}\cdots({\hat{\Pi}}_{J})^{P_{1}}\cdots$ (30)

by replacing every factor of $\hat{X}^{I},{\hat{\Pi}}_{J}$ in ${\cal O}$ above
with the projected counterpart. Then the projected operator corresponding to
${\hat{{\cal{O}}}}$ is given by taking the colour trace of eq.(30). We will
use the notation ${\hat{{\cal{O}}}}^{P_{1}}$ for this operator below, so we
have,

${\hat{{\cal{O}}}}^{P_{1}}=Tr(\cdots(\hat{X}^{I})^{P_{1}}\cdots({\hat{\Pi}}_{J})^{P_{1}}\cdots).$
(31)

It is important to note that the operator in eq.(31) is different from
$Tr(P_{1}{\cal O}P_{1})$, where ${\cal O}$ is given by eq.(27). E.g., when
${\cal O}$ above is $X^{I}X^{J}$,
${\hat{{\cal{O}}}}=Tr(\hat{X}^{I}\hat{X}^{J})$ and $Tr(P_{1}{\cal
O}P_{1})=Tr(P_{1}\hat{X}^{I}\hat{X}^{J}P_{1})$. However the operator
${\hat{{\cal{O}}}}^{P_{1}}=Tr(P_{1}\hat{X}^{I}P_{1}\hat{X}^{J})$ which is
different.

The full sub -algebra we consider associated with the constraint eq.(18)
involves all single trace operators obtained after projection in this manner
and the multi trace operators obtained from products of such single trace
projected operators.

Actually the projection operator $P_{1}$ above implements version 1) of the
proposal, for a constraint specified by the function $f(X^{I})$. To see this
consider the case $f(X^{I})=X^{1}-a$, discussed above. Working in the gauge
where $X^{1}$ is diagonal, let us consider the sector where the first $k$
eigenvalues $x^{1}_{i}>a,i=1,\cdots k$ are in the region of interest. Then it
is easy to see in this sector that the operator $P_{1}$ is the matrix

$P_{1}=\begin{pmatrix}{\bf I}_{k\times k}&{\bf 0}_{k\times(N-k)}\\\ {\bf
0}_{(N-k)\times k}&{\bf 0}_{(N-k)\times(N-k)}\end{pmatrix}$ (32)

where ${\bf I}_{k\times k}$ denotes the identity in the $k\times k$ block and
${\bf 0}$ denotes a matrix where all entries vanish. Projecting with this
operator we retain for all matrix operators their upper left hand $k\times k$
block, as shown in detail in the Appendix A. Gauge invariant operators made
out of such matrix operators are exactly the observables whose expectation
values can be calculated using the density matrix obtained from the tracing
out procedure described above for version 1) of the proposal.

More generally, for a constraint $f(x^{I})>0$ we can go to the gauge where
$f(\hat{X}^{I})$ is diagonal and in the sector where the first $k$ eigenvalues
satisfy the constraint find that multiplying with $P_{1}$ will retain
similarly the upper left hand $k\times k$ block for all operators and thus
give the correct sub-algebra associated with version 1).

Implementing the version 2) proposal in a gauge invariant manner is also
similarly doable. We first consider the orthogonal projector

${\tilde{P}}_{1}=\int_{x<0}dx\delta(x{\bf I}-f(\hat{X}^{I})),$ (33)

which involves the same argument for the delta function but with the range of
the $x$ integral now lying in the complementary region $x<0$. It follows that
${\tilde{P}}_{1}^{2}={\tilde{P}}_{1}$ To implement version 2) we consider the
operators, $\hat{X}^{I}$ and retain the elements corresponding to
$\hat{X}^{I}-{\tilde{P}}_{1}\hat{X}^{I}{\tilde{P}}_{1}$, so that

$\hat{X}^{I}\rightarrow(\hat{X}^{I})^{P_{2}}=\hat{X}^{I}-{\tilde{P}}_{1}\hat{X}^{I}{\tilde{P}}_{1}$
(34)

and similarly for the momentum operators ${\hat{\Pi}}_{I},I=1,\cdots 9$,

${\hat{\Pi}}_{I}\rightarrow({\hat{\Pi}}_{I})^{P_{2}}={\hat{\Pi}}_{I}-{\tilde{P}}_{1}{\hat{\Pi}}_{I}{\tilde{P}}_{1}$
(35)

Then taking the trace of a string of such operators we obtain gauge invariant
operators

${\hat{\cal{O}}}^{P_{2}}=Tr(\cdots(\hat{X}^{I})^{P_{2}}\cdots(\Pi_{J})^{P_{2}}\cdots)$
(36)

which should be compared with eq.(31) obtained above for the version 1) case.
It should be emphasised that the transformation
$\hat{X}^{I}\rightarrow(\hat{X}^{I})^{P_{2}},{\hat{\Pi}}_{J}\rightarrow({\hat{\Pi}}_{J})^{P_{2}}$
also squares to itself, since

$((\hat{X}^{I})^{P_{2}})^{P_{2}}=(\hat{X}^{I})^{P_{2}}-{\tilde{P}}_{1}(\hat{X}^{I})^{P_{2}}{\tilde{P}}_{1}=(\hat{X}^{I})^{P_{2}}$
(37)

where we used the property ${\tilde{P}}_{1}^{2}={\tilde{P}}_{1}$. This
transformation is therefore also a projection acting on the matrix operators
$\hat{X}^{i},{\hat{\Pi}}_{J}$. However, the transformation does not act by
conjugation, unlike $P_{1}$ for version 1).

The notation we have adopted referring to the gauge invariant operators
obtained in both cases, as ${\hat{\cal{O}}}^{P_{1}},{\hat{\cal{O}}}^{P_{2}}$,
allows for some simplification in the following discussion. We will often
refer to the operators obtained in both versions as ${\hat{O}}^{P}$ without
specifying which of the two cases $P_{1},P_{2}$ we have in mind; where needed
we will of course provide this clarification.

In the subsequent discussion we will also often denote the sub-algebra
associated with a bulk region ${\cal R}$ which is obtained after projection,
in either of the two versions as described above, as ${\cal A}_{R}$.

Let us end this subsection with two comments. First, in general, while passing
from the constraint in terms of bulk coordinates, $f(x^{I})$, eq.(18), to a
constraint in terms of matrix operators, $f(\hat{X}^{I})$, which appears in
the target space constraint, we will encounter ordering ambiguities as was
mentioned above. Note that in the matrix quantum mechanics, at any instant of
time, the different matrix elements of the matrix operators commute,
$[\hat{X}^{I}_{ij},\hat{X}^{J}_{kl}]=0$. However there are still matrix
ordering ambiguities which are present since as matrices $[X^{I},X^{J}]\neq
0$.

As we will soon discuss, the matrix model we are dealing with is formulated in
terms of matrix operators $X^{1},\cdots X^{9}$, which correspond to the
poincare coordinates in supergravity. For a linear constraint in the bulk
involving these coordinates, where $f(x^{I})=\sum_{I=1}^{9}c_{I}x^{I}-a$, it
is straightforward to obtain the operator constraint $f(\hat{X}^{I})$ to be
the corresponding function involving the matrix operators,
$f(\hat{X}^{i})=\sum_{I=1}^{9}c_{I}\hat{X}^{I}-a$. For some of the non-linear
constraints also there is a natural way to find the corresponding operator
constraints, for example $f(x^{I})=\sum_{I=1}^{9}c_{I}(x^{I})^{2}-a$, is
mapped in a straightforward manner to
$f(\hat{X}^{I})=\sum_{I=1}^{9}c_{I}(\hat{X}^{I})^{2}-a$. In fact, the last
example can be extended to more general constraints which involve terms
containing sums of monomials of individual coordinates, i.e.,

$f(x^{I})=\sum_{I}c_{I}(x^{I})^{p_{I}}-a.$ (38)

These are mapped to

$f(\hat{X}^{i})=\sum_{I}c_{I}(\hat{X}^{I})^{p_{I}}-a.$ (39)

However, more general non-linear constraints involving terms with multiple
coordinates cannot be mapped to a matrix constraint unambiguously, e.g. take
the case when $f(x^{I})=x^{1}x^{2}-a>0$, this could be mapped to either to
$\hat{X}^{1}\hat{X}^{2}-a>0$ or $\hat{X}^{2}\hat{X}^{1}-a>0$. Our discussion
below will primarily focus on cases like eq.(38) where the map to the target
space constraint eq.(39) is straightforward121212It could be that such
operator ordering ambiguities give rise to differences in entanglement entropy
which are subheading in $N$, we thank S. Minwalla for making this comment. .

Second, it is easy to see that the operators contained in the subalgebra
${\cal A}_{R}$ for both versions 1) and 2) do not include the Hamiltonian of
the system. Remaining in the temporal gauge, let us rescale the matrices in
(2) and their conjugate momenta as

$\hat{X}^{I}\rightarrow(g_{s}N)^{1/3}l_{s}\hat{X}^{I}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\hat{\Pi}}^{I}\rightarrow\frac{1}{(g_{s}N)^{1/3}l_{s}}{\hat{\Pi}}^{I}$
(40)

the hamiltonian becomes

$H=\frac{(g_{s}N)^{1/3}}{2l_{s}}{\rm
Tr}\left[\frac{1}{N}\sum_{I=1}^{9}({\hat{\Pi}}_{I})^{2}+N\sum_{I\neq
J=1}^{9}[\hat{X}^{I},\hat{X}^{J}]^{2}\right]+{\rm fermions}$ (41)

Instead ${\cal A}_{R}$ contains the operator

$H^{P}=\frac{(g_{s}N)^{1/3}}{2l_{s}}{\rm
Tr}\left[\frac{1}{N}\sum_{J}[({\hat{\Pi}}_{J})^{P}]^{2}+N\sum_{IJ}[(\hat{X}^{I})^{P},(\hat{X}^{J})^{P}]^{2}\right]+{\rm
fermions}$ (42)

which is different.

In this paper we will consider gauge theories which involve matter fields in
the adjoint representation. The main examples are gauged quantum mechanics of
a single matrix, a particular example of which is the dual description of two
dimensional strings, and $Dp$ brane field theories. The $D0$ brane quantum
mechanics is a particularly important example relevant for our discussion.

###  2.3 Implementing a non-linear target space constraint

As described above, the gauge invariant formulation of target space
entanglement applies to any constraint characterized by a hermitian operator
$f(\hat{X}^{I})$. In a practical calculation, however, one would need to fix a
gauge which diagonalizes this constraint. To perform a concrete calculation,
however, one needs to make a change of variables to a set of independent
variables which includes the eigenvalues. This is straightforward for a linear
constraint, but becomes complicated very soon when we consider nonlinear
constraints. In this subsection we explain how to do this for a constraint

$f(\hat{X}^{I})=\sum_{I=1}^{9}(\hat{X}^{I})^{2}\equiv\hat{R}^{2}$ (43)

The details of the procedure are given in the Appendix A. What we need is a
"polar" decomposition for matrices.

####  2.3.1 Two matrices

Let us begin with the simplest case of two matrices $\hat{X}^{1}$ and
$\hat{X}^{2}$. In the following all the matrices are operators in the Hilbert
space unless stated otherwise. We want to write this pair in terms of one
hermitian matrix $\hat{R}$ and a unitary matrix $\hat{Q}$, where

$\hat{R}^{2}=(\hat{X}^{1})^{2}+(\hat{X}^{2})^{2}$ (44)

Define the complex matrix

$\hat{Z}=\hat{X}^{1}+i\hat{X}^{2}$ (45)

Then it follows that

$2\hat{R}^{2}=\hat{Z}\hat{Z}^{\dagger}+\hat{Z}^{\dagger}\hat{Z}$ (46)

Now consider a singular value decomposition

$\hat{Z}=\hat{V}\hat{s}\hat{W}^{\dagger}$ (47)

where $\hat{V},\hat{W}$ are unitary matrices and $\hat{s}$ is a diagonal
matrix. Using (46) we then get

$\displaystyle 2(\hat{R}^{2})_{ij}$
$\displaystyle=(\hat{V}\hat{s}^{2}\hat{V}^{\dagger}+\hat{W}\hat{s}^{2}\hat{W}^{\dagger})_{ij}$
$\displaystyle=[\hat{V}^{\star}\otimes\hat{V}+\hat{W}^{\star}\otimes\hat{W}]_{ij,kl}(\hat{s}^{2})_{kl}$
(48)

where

$[\hat{V}^{\star}\otimes\hat{V}+\hat{W}^{\star}\otimes\hat{W}]_{ij,kl}\equiv\hat{V}_{ik}\hat{V}^{\star}_{jl}+\hat{W}_{ik}\hat{W}^{\star}_{jl}$
(49)

It is shown in the Appendix B that the direct product matrix appearing in (48)
is invertible in the sense

$[(\hat{V}^{\star}\otimes\hat{V}+\hat{W}^{\star}\otimes\hat{W})^{-1}]_{mn,ij}[(\hat{V}^{\star}\otimes\hat{V}+\hat{W}^{\star}\otimes\hat{W})]_{ij,kl}=\delta_{mk}\delta_{nl}$
(50)

An explicit expression for the inverse is

$[(\hat{V}^{\star}\otimes\hat{V}+\hat{W}^{\star}\otimes\hat{W})^{-1}]_{kl,rs}=\sum_{n=0}^{\infty}(-1)^{n}[\hat{V}^{\dagger}(\hat{Q}^{\dagger})^{n}]_{kr}[\hat{V}^{T}(\hat{Q}^{T})^{n}]_{ls}$
(51)

where we have defined the unitary matrix $\hat{Q}$

$\hat{Q}\equiv VW^{\dagger}$ (52)

We can now invert (48) to write

$\hat{s}^{2}=2\hat{V}^{\dagger}\left[\sum_{n=0}^{\infty}(-1)^{n}(\hat{Q}^{\dagger})^{n}\hat{R}^{2}\hat{Q}^{n}\right]\hat{V}$
(53)

In Appendix B we show that the matrix which appears in the square bracket in
(53) is positive semi-definite, so that we can take the square root of this
equation. Substituting this in (47) and using the definition of $\hat{Q}$ in
(52) we finally get

$\hat{Z}=({\mathfrak{L}}_{\hat{Q}}\hat{R})\hat{Q}$ (54)

where we have defined the hermitian matrix
${\mathfrak{L}}_{\hat{V}}{\hat{M}}$,

${\mathfrak{L}}_{\hat{V}}{\hat{M}}\equiv{\sqrt{2}}\left[\sum_{n=0}^{\infty}(-1)^{n}(\hat{V}^{\dagger})^{n}{\hat{M}}^{2}\hat{V}^{n}\right]^{1/2}$
(55)

where $\hat{V}$ is unitary and ${\hat{M}}$ is hermitian. This satisfies the
equation

$\hat{V}^{\dagger}({\mathfrak{L}}_{\hat{V}}{\hat{M}})^{2}\hat{V}+({\mathfrak{L}}_{\hat{V}}{\hat{M}})^{2}=2{\hat{M}}^{2}$
(56)

The matrices $\hat{X}^{1},\hat{X}^{2}$ can be then expressed as

$\displaystyle\hat{X}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[({\mathfrak{L}}_{\hat{Q}}\hat{R})\hat{Q}+\hat{Q}^{\dagger}({\mathfrak{L}}_{\hat{Q}}\hat{R})\right]$
$\displaystyle\hat{X}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left[({\mathfrak{L}}_{\hat{Q}}\hat{R})\hat{Q}-\hat{Q}^{\dagger}({\mathfrak{L}}_{\hat{Q}}\hat{R})\right]$
(57)

This is the matrix analog of a polar decomposition of cartesian coordinates in
two dimensions $x^{1}=r\cos\phi,x^{2}=r\sin\phi$. For matrices we have the
correspondence

$re^{i\phi}\rightarrow({\mathfrak{L}}_{\hat{Q}}\hat{R})\hat{Q}$ (58)

To construct the relevant subalgebra of operators we then need to use the
projector (25) with $f(X^{I})=\hat{R}^{2}$, replace $\hat{X}^{I},I=1,2$ using
(57) by their projected versions (28) and express the $\hat{X}^{I}$ in terms
of $\hat{Q}$ and $\hat{R}$ using (57).

An appropriate gauge-fixed version can be obtained by diagonalizing the matrix
$\hat{R}$,

$\hat{R}\rightarrow{\rm diag}[\hat{r}_{1},\hat{r}_{2},\cdots\hat{r}_{N}]$ (59)

We can then proceed to work in a Hilbert space basis which are eigenstates of
$\hat{r}_{i}$ and the $\hat{Q}_{ij}$ with eigenvalues $r_{i},Q_{ij}$. The
measure of integration then becomes

$[dX^{1}dX^{2}]=\mathbb{J}(r_{i},Q_{ij})\prod_{i}dr_{i}\prod_{ij}[dQ_{ij}]$
(60)

The jacobian $\mathbb{J}(r_{i},Q_{ij})$ can be obtained in principle by using
the explicit expressions (57). However this is rather complicated, and we have
not been able to obtain compact expressions for this. To proceed further, it
will be convenient to write the unitary matrix $Q$ in terms of a unitary
matrix $U$ and a set of angles $\phi_{i}$

$Q=Ue^{i\Phi}U^{\dagger},~{}~{}~{}~{}~{}~{}\Phi={\rm
diag}(\phi_{1},\phi_{2},\cdots\phi_{N})$ (61)

Defining

$dS\equiv U^{\dagger}dU$ (62)

the line element then becomes

${\rm
tr}(dQdQ^{\dagger})=\sum_{i}d\phi_{i}^{2}+8\sum_{i<j}\sin^{2}(\frac{\phi_{i}-\phi_{j}}{2})dS_{ij}dS^{\star}_{ij}$
(63)

which leads to the expression

$[dX^{1}dX^{2}]=\mathbb{J}(r_{i},\phi_{i},S_{ij})\prod_{i}dr_{i}\prod_{i}d\phi_{i}\prod_{i<j}[4\sin^{2}(\frac{\phi_{i}-\phi_{j}}{2})dS_{ij}dS^{\star}_{ij}]$
(64)

As in the case of simple linear constraints the projector leads to restriction
of the integration range of $r_{i}$.

The above construction is inspired by the work masuku  where the complex
matrix $\hat{Z}$ was written as $\hat{Z}={\tilde{R}}{\tilde{U}}$ where
${\tilde{R}}$ is a hermitian matrix operator and ${\tilde{U}}$ is a unitary
operator. However in this decomposition
${\tilde{R}}^{2}=(\hat{X}^{1})^{2}+(\hat{X}^{2})^{2}+i[\hat{X}^{1},\hat{X}^{2}]$
rather than (44).

####  2.3.2 Multiple Matrices

The above polar decomposition can be extended to an arbitary number of
matrices $\hat{X}^{I}$. To illustrate the procedure let us first consider the
case of three matrices $\hat{X}^{1},\hat{X}^{2},\hat{X}^{3}$. The idea is to
mimick the procedure to obtain spherical polar coordinates $(r,\theta,\phi)$
from usual cartesian coordinates $(x^{1},x^{2},x^{3})$,

$x^{1}=r\cos\phi_{1}\cos\phi_{2},~{}~{}~{}x^{2}=r\cos\phi_{1}\sin\phi_{2},~{}~{}~{}~{}x^{3}=r\sin\phi_{1}$
(65)

We want to make a change of variables from hermitian matrices
$\hat{X}^{I},I=1,2,3$ to a hermitian matrix $\hat{R}$ and two unitary matrices
$\hat{Q}_{1},\hat{Q}_{2}$. Here the matrix $\hat{Q}_{1}$ generalizes
$e^{i\theta}$, while $\hat{Q}_{2}$ generalizes $e^{i\phi}$. From (58) the
necessary replacements are

$\displaystyle re^{i\phi_{1}}$ $\displaystyle\rightarrow$
$\displaystyle({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}$ $\displaystyle
r\cos\phi_{1}e^{i\phi_{2}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}\left[{\mathfrak{L}}_{\hat{Q}_{2}}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}+\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)\right]\hat{Q}_{2}$
(66)

This leads to the final expressions

$\displaystyle\hat{X}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[{\mathfrak{L}}_{\hat{Q}_{2}}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}+\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)\right]\hat{Q}_{2}+\frac{1}{4}Q_{2}^{\dagger}\left[{\mathfrak{L}}_{\hat{Q}_{2}}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}+\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)\right]$
$\displaystyle\hat{X}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4i}\left[{\mathfrak{L}}_{\hat{Q}_{2}}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}+\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)\right]\hat{Q}_{2}-\frac{1}{4i}Q_{2}^{\dagger}\left[{\mathfrak{L}}_{\hat{Q}_{2}}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}+\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)\right]$
$\displaystyle\hat{X}^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\left(({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\hat{Q}_{1}-\hat{Q}_{1}^{\dagger}({\mathfrak{L}}_{\hat{Q}_{1}}\hat{R})\right)$
(67)

Using (56) repeatedly it is easy to see that

$(\hat{X}^{1})^{2}+(\hat{X}^{2})^{2}+(\hat{X}^{3})^{2}=\hat{R}^{2}$ (68)

However the domain of the unitary matrices need to be restricted. This is
because in (65) one has $-\pi/2<\phi_{1}<\pi/2$ while $-\pi<\phi_{2}<\pi$. To
obtain the corresponding restriction on the domain of the unitary matrices
$Q_{1},Q_{2}$, we resort to the decomposition in (61) for each of these
matrices.

$Q_{A}=U_{A}e^{i\Phi_{A}}U_{A}^{\dagger},~{}~{}~{}~{}~{}~{}\Phi_{A}={\rm
diag}[(\phi_{A})_{1},\cdots(\phi_{A})_{N}]~{}~{}~{}~{}~{}~{}A=1,2$ (69)

It is shown in the Appendix B that the requirement that the eigenvalues of
$X^{1}\cdots X^{3}$ should cover $\mathbb{R}^{3}$ once is equivalent to the
requirement that

$-\pi/2<(\phi_{1})_{i}<\pi/2~{}~{}~{}~{}~{}-\pi<(\phi_{2})_{i}<\pi$ (70)

The measure of integration is now

$[dX^{1}dX^{2}dX^{3}]=\mathbb{J}(r_{i},(\phi_{A})_{i},(S_{A})_{ij})\prod_{i}dr_{i}\prod_{A=1}^{2}\left[\prod_{i}d(\phi_{A})_{i}\prod_{i<j}[4\sin^{2}(\frac{(\phi_{A})_{i}-(\phi_{A})_{j}}{2})d(S_{A})_{ij}d(S_{A}^{\star})_{ij}]\right]$
(71)

where we have defined, in analogy with (62)

$dS_{A}\equiv U_{A}^{\dagger}dU_{A}$ (72)

It is now clear that this construction generalizes to arbitrary number of
matrices $\hat{X}^{I},I=1\cdots D$. Once again we start with the polar
coordinats of $\mathbb{R}^{D}$ and generalize to matrix polar decompositions
which generalize (66). Now we have a single hermitian matrix $\hat{R}$ and
$(D-1)$ unitary matrices $Q_{A},A=1,\cdots(D-1)$. Once each of the $Q_{A}$’s
are decomposed as in (69) we have the domains

$\displaystyle-\pi/2$ $\displaystyle<$
$\displaystyle(\phi_{A})_{i}<\pi/2~{}~{}~{}~{}~{}~{}~{}A=1\cdots D-2$
$\displaystyle-\pi$ $\displaystyle<$ $\displaystyle(\phi_{(D-1)})_{i}<\pi$
(73)

The integration measure is as in (71) with $A=1\cdots(D-1)$.

As discussed below we would like to identify the entanglement entropy
associated with a constraint which restricts the eigenvalues $r_{i}$ to be in
some range, e.g. $r_{i}>a$ in the dual supergravity background, at least for
sufficiently large values of $a$.

###  2.4 Dp Branes

The above considerations generalize for Dp branes for $p<3$. Now the matrices
are scalar fields $X^{I}(\xi)$ and gauge fields $A_{\mu}(\xi)$. The target
space restrictions are now on entire functions. The projector can be written
in terms of a functional integral

$\hat{P}_{A}=\int{\cal
D}x(\xi)~{}\prod_{\xi}\delta(x(\xi)-F[\hat{X}^{I}(\xi)])$ (74)

where the functional $F[\hat{X}^{I}(\xi)]$ needs to be chosen appropriately.
For example, for a "planar" constraint we have
$F[X^{I}(\xi)]=\hat{X}^{1}(\xi)$. Once again one can choose a gauge which is
tailored to the constraint, e.g. for the planar constraint we can pick a gauge
where $\hat{X}^{1}(\xi)$ is diagonal in matrix space. The discussion above can
be now repeated and it follows that in this gauge one recovers the results of
Das:2020jhy .

##  3 Target Space Entanglement as Bulk Entanglement

Some further remarks are called for at this stage. The purpose of our
investigation is to try and obtain a precise version of bulk entanglement by
mapping a bulk region to the target space in the boundary theory. Such an
investigation is of course closely tied to understanding how approximate bulk
locality arises in the boundary theory. In section 2 we showed how a sub-
algebra of observables can be associated with the corresponding target space
constraint. In this subsection we will discuss in what sense this subalgebra
can probe supergravity fields in the region ${\cal R}$.

Our main reasoning comes from the bulk meaning of the Coulomb branch of the
gauge theory. Let us start with the system being in the vacuum. For this case
the map we are using between the bulk and target space is in agreement with
what is well known about the system in the moduli space approximation. A
single $D0$ brane moving in the supergravity bulk dual to the D0 brane ground
state experiences a velocity dependent potential which depends on its bulk
location.This may be calculated by considering the DBI + Chern Simons action
in the non-trivial background of a large number of D0 branes (80). This
calculation is summarized in Appendix C. For small velocities, the coefficient
of $v^{2}$ is a constant, the next term goes like $v^{4}/r^{7}$, where $v$ is
its velocity and $r$ the distance from the origin. Exactly the same potential
follows from the gauge theory if the D brane location $(x^{1},\cdots x^{9})$
is mapped to a point in the Coulomb branch with one non-zero eigenvalue for
the matrices $X^{1},\cdots X^{9}$. For a D0 brane brane displaced along the
$x^{1}$ direction and lying at $x_{0}^{1}$ the matrix $X^{1}$ has one non-zero
eigenvalue,

$X^{1}=\begin{pmatrix}{\bf 0}_{(N-1)\times(N-1)}&0_{1\times(N-1)}\\\
0_{(N-1)\times 1}&x_{0}^{1}=r+vt\end{pmatrix}$ (75)

corresponding to $SU(N)\rightarrow SU(N-1)\times U(1)$. This represents
stripping off a single D0 brane from a bunch of $N-1$ D0 branes which form a
bound state. Non-renormalization theorems ensure the agreement, once this
identification is made between the bulk and moduli space, see e.g., rseven ;
twoloop ; polbecker  and references therein.

At next order a two loop calculation in D0 brane quantum mechanics yields a
term which behaves as $v^{6}/r^{14}$. This term can be also reproduced from
the DBI+CS action as discussed in Appendix C 131313A bulk calculation can also
be done in M theory with a compact null direction where $v$ is the relative
velocity between two eleven dimensional gravitons with momenta
$N_{1}/(g_{s}l_{s})$ and $N_{2}/g_{s}l_{s}$ in the M theory direction for
$N_{2}\ll N_{1}$. The effect of the graviton with momentum
$N_{1}/(g_{s}l_{s})$ is to produce an Aichelburg-Sexl metric and the other
graviton is considered as a probe in this background. The Aichelburg-Sexl
metric results from an infinite boost of a 11 dimensional Schwarzschild black
hole. We are interested in the extremal D0 brane background in 10 dimensions,
which is obtainable from 11 dimensions by infinitely boosting a 11 dimensional
black string along the string direction. While this looks like a different
limiting procedure, the expansion of the DBI+CS action (see Appendix C) in the
latter background is exactly identical to the particle action considered in
polbecker .

We can think of the calculations in the gauge theory as a computation of the
effective action for appropriate gauge invariant operators. In the example
above, eq.(75) we calculate the effective action for the operators,

$Tr(X^{1}),Tr(X^{1})^{2},\cdots Tr(X^{1})^{N-1}$ (76)

and then evaluate this effective action by setting
$Tr(X^{1})^{p}=(x_{0}^{1})^{p}$. The resulting value of the effective action
as a function of $x_{0}^{1}$ then gives the effective potential for the probe
$D0$ brane.

There is, conceptually speaking, another way to arrive at the same result in
the gauge theory. Consider a region ${\cal R}$ which includes the location of
the probe brane, i.e.,

$(x_{0}^{1},{\vec{0}})\in{\cal R}$ (77)

As discussed in Section 2.2, given some region ${\cal R}$ we can define a
target space constraint and an associated projector, leading to a sub-algebra
of operators ${\cal A}_{\cal R}$. This algebra also consists of gauge
invariant operators and we can also obtain an effective action for operators
in ${\cal A}_{R}$, i.e. obtain the Legendre transformation of the generating
function for operators in ${\cal A}_{R}$. In this effective action we now set
$Tr((X^{1})^{P}))^{m}=(x_{0}^{1})^{m}$, where the projector $P=P_{1}\ {\rm
or}\ P_{2},$ depending on whether we are considering version 1) or 2) of our
proposal. The result, as a function of $x_{0}^{1}$ will then agree with the
effective action for the set of operators eq.(76), and therefore will
correctly give the potential experienced by the probe brane in the bulk, as
long as eq.(77) is met. This is manifestly clear if we think of calculating
the effective action using the background field method in the gauge where
$X^{1}$ is diagonal with background value given by eq.(75), since we will then
be doing the same calculation in the two cases.

The force on the D0 brane in the bulk in the example above can be calculated
by using a DBI +CS action which is sensitive to the local values of the
metric, the 10 dimensional $U(1)$ RR gauge field and the dilaton. If the state
is changed from the vacuum to some other coherent state $|s>$ which leads to a
different background value of the metric and other bulk fields, we expect that
the force that the probe D0 brane experiences can continue to be obtained in
this way and will be sensitive to the local values of these bulk fields. For
concreteness consider the state $|s>$ to contain a gravity wave. Now, one way
to obtain the local value of the gravitational field can be to measure how the
potential for a probe brane at the location changes due to the presence of
this gravity wave. This should yield the same result as other methods which
may not involve a probe brane.

In the gauge theory we expect that the potential for the probe brane continues
to correspond to the effective action computed for suitable values of
operators, i.e. the set eq.(76), now in the state $|s>$ of the type we are
considering, and we also expect that this effective action is correctly
obtained from the effective action for operators in ${\cal A}_{\cal R}$ as in
the discussion above for the vacuum state. In this way we see that one expects
to be able to obtain the one point function for the graviton, and some other
supergravity modes, from operators contained in ${\cal A}_{R}$.

This reasoning above is in fact at the heart of our proposal for identifying
the bulk and boundary target space regions and also identifying the algebra
${\cal A}_{\cal R}$ in the manner we have done. In the sector where $k$ branes
are present in ${\cal R}$ we keep the $k\times k$ block $M_{ab}$ for all
matrix operators but not the complementary $(N-k)\times(N-k)$ block which
correspond to branes that are not present. In version 1) of our proposal we
also keep the off diagonal blocks $M_{a\alpha},M_{\alpha,a}$. The resulting
algebra of observables allows one to describe all measurements done on branes
present in ${\cal R}$ and this should then be sufficient to also detect low-
energy supergravity excitations in ${\cal R}$.

It is worth noting in this context that there is additional evidence, going
beyond the moduli space approximation, that the map between the bulk and
target space we are using continues to work. For example, one can consider two
stacks of D0 branes one at the origin and the other displaced from it and
excite open strings within branes in each set. This changes the potential
between the two stacks, but one still finds agreement in the bulk and in the
gauge theory for the resulting two-body interactions, after appropriately
identifying operators in the gauge theory with their counterpart currents in
the bulk, rseven . This suggests that the algebra ${\cal A}_{\cal R}$, which
retains the appropriate operators for all the superselection sectors where
different number of branes $k=0,1,\cdots N$ are present in the region of
interest, should suffice for describing the results of all measurements made
with local supergravity operators in ${\cal R}$.

Our intuition based on the above reasoning, can be extended to a given
configuration in the bound state wavefunction for $N$ D0 branes in the gauge
where the constraint function is diagonal. Consider a configuration where $k$
of these eigenvalues are in the region of interest ${\cal R}$. The degrees of
freedom $M_{a\alpha},M_{\alpha a}$ correspond to excitations in the bulk going
between ${\cal R}$ and ${\cal R}^{c}$ or vice versa. If the state $|s>$ has
some supergravity modes excited with support deep inside ${\cal R}$, by which
we mean the excitations are localised many string lengths away from the
boundary of ${\cal R}$, then neither the $M_{\alpha\beta}$ nor the
$M_{a\alpha},M_{\alpha a}$ degrees of freedom will be excited in $|s>$. The
$M_{\alpha\beta}$ degrees will not be excited because the excitations in $|s>$
are localised in ${\cal R}$. The $M_{a\alpha},M_{\alpha a}$ degrees will also
not be excited because they would correspond to open strings which would have
to stretch across the boundary across many string lengths and would therefore
be very heavy.

As a result, one might expect that the full change in expectation values of
any single trace operator ${\hat{O}}$ made out of a string of
$X^{I},\Pi_{J}$’s schematically depicted in eq.(27) will be obtained to good
approximation by the corresponding operator obtained after projection,
${\hat{O}}^{P}$, eq.(31), eq.(36) . Notice that at this level of admittedly
imprecise arguments we cannot distinguish between version 1) and 2) of our
proposal. Both contain the gauge invariant degrees of freedom coming from
$M_{ab}$. In version 1) there are extra degrees of freedom coming from
$M_{a\alpha},M_{\alpha a}$ as well but as per the intuitive argument above
they might not play an important role anyways.

However, there are reasons to believe that the sub-algebra we are considering
will not provide all details of bulk fields as defined e.g. by the BDHM-HKLL
map. In the low energy regime such a bulk field operator
$\phi(r,\theta_{i},t)$ is defined by

$\phi(r,\theta_{i},t)=\sum_{l_{i},\omega_{n}}[{\hat{O}}_{l_{i},\omega_{n}}f_{l_{i},\omega_{n}}(r)Y_{l_{i}}(\theta_{i})e^{-i\omega_{n}t}+cc]$
(78)

where ${\hat{O}}_{l_{i},\omega_{n}}$ are Fourier modes with frequency
$\omega_{n}$ obtained from the time dependent operators ${\hat{O}}_{l_{i}}(t)$
(we are being schematic here, $\omega_{n}$ need not be discrete). Consider
such a bulk operator with $(r,\theta_{i})\in{\cal R}$. The expectation value
of $\phi$ can be obtained if we know the expectation value of
${\hat{O}}_{l_{i}}(t)$ for all $l_{i}$ and all times $t$. Now we can regard
${\hat{O}}_{l_{i}}(t)$ as an operator acting on the Hilbert space of states at
$t=0$. Its expectation values can therefore be obtained, in principle, if the
expectation values of all operators are known at time $t=0$. In this way we
see that the expectation value of $\phi(r,\theta_{i},0)$ in any state can be
obtained once the expectation values of all operators in the corresponding
state in the boundary theory are known at $t=0$ . One such bulk operator is
the metric itself, for which the corresponding gauge theory operator is the
energy momentum tensor.

Consider now a region ${\cal R}$ which is a small annular region near the
boundary,

$r_{B}^{2}-\delta<r^{2}<r_{B}^{2},$ (79)

where $r_{B}$ is the boundary value of the radial variable
$r^{2}=\sum_{i}(x^{i})^{2}$, and $\delta$ is small. It has been argued in
suvrat , suvrat2 , that measurements carried out by observers in this region
will allow detailed information about the state in the bulk to be obtained. It
is crucial in these arguments that the observers close to the boundary have
access to the full Hamiltonian of the system. In fact, having access to the
Hamiltonian alone enables observers in ${\cal R}$ to reconstruct the full
density matrix of the vacuum, $|0><0|$. As a result any algebra, which
includes all operators corresponding to measurements bulk observers in the
region eq.(79) can make, in particular which includes the Hamiltonian, would
have a vanishing entanglement entropy for the vacuum state141414We are
grateful to Suvrat Raju for explaining this point to us.. The subalgebra we
are associating with the region ${\cal R}$ are the projected versions of the
gauge theory operators, ${\hat{\cal O}}_{l_{i}\omega_{n}}$, which appear in
(78). Thus, the bulk operators defined in eq.(78), with the restriction that
$(r,\theta_{i})\in{\cal R}$, are not contained in this subalgebra. In
particular the Hamiltonian is not an element of this subalgebra, only its
projected version is. We expect that this imposes significant restrictions on
the amount of information which can be obtained for the bulk operators. In
this sense our sub-algebra would only capture the notion of a local bulk
region in an approximate sense. A more detailed investigation of how
significant these restrictions are is left for the future.

Clearly, one would like a deeper understanding of the various issues discussed
in this section. In particular, one would like a better understanding of how
much information about the bulk region of information can actually be obtained
from the sub-algebras we propose, and also whether there are refinements to
our basic proposal, including an improved map between the constraint in the
bulk eq.(18) and the corresponding one in the boundary eq.(24), that are
needed. Since the issues at had are closely tied to how approximate bulk
locality arises, as was mentioned above at the outset, it is unlikely though
that we can make much progress through analytic methods alone . Numerical
calculations hold considerable promise in this regard. Roughly speaking one
wants to show that the change in the wave function which correspond to changes
in some bulk region ${\cal R}$, arises mainly in the target space region
associated with ${\cal R}$ and not its complement. This should also then shed
light on which operators would be needed to determine this change in the
boundary theory and whether a sub-algebra along the lines proposed here would
suffice.

##  4 Target Space Entanglement and Bekenstein Bound

In the ’t Hooft limit the usual ’t Hooft limit $g_{s}\rightarrow
0,N\rightarrow\infty$ with $(g_{s}N)$ held fixed, the bound state of $N$ Dp
branes is dual to a ten dimensional geometry. Our proposal implies that target
space constraints correspond to regions in the transverse space to these Dp
branes in this geometry and the target space entanglement entropy defined
above provides a notion of a bulk entanglement entropy associated with this
region. In Das:2020jhy  it was conjectured that for Dp brane matrix field
theories, the target space entanglement entropy saturates the Bekenstein bound
for this entangling surface. In this section we recapitulate the result for D0
branes.

We will be mostly interested in the bound state of $N$ D0 branes which are
slightly heated up to a temperature $T$. This is dual to the near-extremal
black D0 brane geometry in supergravity. The string frame metric, dilaton and
1-form gauge fields are

$\displaystyle ds_{string}^{2}$ $\displaystyle=$ $\displaystyle-
H_{0}(r)^{-1/2}g(r)dt^{2}+H_{0}(r)^{1/2}[\frac{dr^{2}}{g(r)}+r^{2}d\Omega_{8}^{2}]$
$\displaystyle e^{\phi}$ $\displaystyle=$ $\displaystyle
g_{s}H_{0}(r)^{3/4}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A_{0}=-\frac{1}{2}(H_{0}^{-1}-1)$
(80)

where

$g(r)=1-\left(\frac{r_{H}}{r}\right)^{7}~{}~{}~{}~{}H_{0}(r)=\frac{R^{7}}{r^{7}},~{}~{}~{}~{}~{}~{}r^{2}=x_{1}^{2}+\cdots
x_{9}^{2}.$ (81)

The horizon is at $r=r_{H}$. The Hawking temperature for this solution and the
length scale $R$ are given by

$T=\frac{7}{4\pi
R}\left(\frac{r_{H}}{R}\right)^{5/2}~{}~{}~{}~{}~{}R^{7}=60\pi^{3}l_{s}^{7}(g_{s}N).$
(82)

The supergravity solution above is valid in the regime

$g_{s}^{\frac{1}{3}}N^{1/7}<r<(g_{s}N)^{1/3}l_{s}~{}~{}~{}~{}~{}~{}~{}~{}T\frac{l_{s}}{(g_{s}N)^{1/3}}\ll
1$ (83)

In D0 brane quantum mechanics consider the simple linear constraint, e.g.
$f(X^{I})=X^{1}-a_{0}$. According to the proposal of Das:2020jhy  the bulk
region of interest is simply $x^{1}>a$. The relationship between the
dimensionless $a_{0}$ and the dimensionful $a$ can be read off from the
rescaling (40)

$a=a_{0}(g_{s}N)^{\frac{1}{3}}l_{s}$ (84)

This reflects the fact that in this holographic correspondence the transverse
distance becomes the energy scale of the D0 brane quantum mechanics which is
$\Lambda=(g_{s}N)^{1/3}/l_{s}$. Likewise the temperature appearing in (82) is
related to a dimensionless temperature $T_{0}$ by $T=T_{0}\Lambda$.

In Das:2020jhy  it was conjectured that this target space entanglement
saturates the Bekenstein bound

$S(a,T)=\frac{A_{a}(T)}{4G_{N}}$ (85)

where $A_{a}$ is the Einstein Frame area of the entangling surface $x^{1}=a$
in the geometry (80) and $G_{N}=8\pi^{6}g_{s}^{2}l_{s}^{8}$ is the ten
dimensional Newton constant. The quantity $S(a,T)$ is actually divergent, the
divergence coming from the large $r$ region. The large $r$ region is, however,
beyond the regime of validity of supergravity: thus one may consider using a
cutoff at $r=r_{0}$. However the difference $S(a,T)-S(a,T^{\prime})$ is
finite,

$S(a,T)-S(a,T^{\prime})=B_{0}~{}N^{2}a_{0}^{-5/2}\left[(T_{0})^{14/5}-(T_{0}^{\prime})^{14/5}\right]$
(86)

where $B_{0}$ is a number whose value is given in equation (29) of Das:2020jhy
.

Note that the expression (86) the dimensionless quantities which characterize
the state and the entangling region are those which are quantities which would
appear in D0 brane quantum mechanics. The only other number which appears is
$N$ : the answer is proportional to $N^{2}$. This is what one expects if our
proposal is correct. In particular all factors of $g_{s}$ nicely cancel. The
powers of $T_{0}$ and $d_{0}$ which appear in (86) does not follow from
general considerations of target space entanglement. If a numerical
calculation yields these powers we will have a very non-trivial evidence for
our proposal.

Let us make one comment before ending this section. We have emphasised above
that the discussion in this paper can be applied for constraints taking the
general form, eq.(18). Instead of the linear constraint considered above
suppose we take

$f(x^{1})=\sum_{i=1}^{9}(x^{i})^{2}>r_{0}^{2}$ (87)

where $r_{0}$ is some radius. In this case the Beckenstein- Hawking entropy
which is given by

$S=\Omega_{8}{R^{7/2}r_{0}^{9/2}\over 4G_{N}}$ (88)

is a function of $r_{0}$ but is independent of the temperature $T$. As per our
proposal we would like to equate this result with the entanglement entropy
associated with the target space constraint

$\sum_{I=1}^{9}({\hat{X}}^{I})^{2}>r_{0}^{2}.$ (89)

However it does seem rather strange then that the resulting entanglement
entropy is independent of the temperature $T$. One reason could be that
perhaps the map between a physical region in the bulk and the corresponding
target space constraint is more complicated at finite temperature, i.e. the
RHS in eq. (87) and eq.(89) are not equal but instead related by a temperature
dependent function. This might also help explain why when we take
$r_{0}=r_{H}$ in eq.(87), we get the entanglement entropy to be the full
entropy in the boundary theory and not a different value due to the additional
target space constraint eq.(89) being present. We leave a more detailed
investigation of such temperature dependent effects for the future.

##  5 Path Integral Expressions for Renyi Entropies

As discussed above, numerical calculations should be able to prove or disprove
our conjecture that the target space von Neumann entropy saturates the
Bekenstein bound. Recently there has been impressive advances in numerical
calculations for D0 branes Hanada:2016zxj . These calculations use euclidean
path integrals to calculate finite temperature partition functions as well as
some correlation functions. In this section we develop euclidean path integral
expressions for target space Renyi entropies which can be used directly for
numerical calculations. These expressions are in the gauge fixed formalism,
and we will develop them for planar constraints.

Consider the D0 brane theory at some finite temperature $T=1/\beta$. The
density operator is given by $\hat{\rho}_{0}={\rm exp}[-\beta H]$ where the
hamiltonian $H$ is given by (41). As in the previous sections, we will fix the
$A_{0}=0$ gauge, fix the time independent gauge transformations by
diagonalizing one of the matrices $X^{1}$, and impose the remaining Weyl and
$U(1)^{N}$ symmetries by explicitly summing over the corresponding
transformations. The basis states are given by (2.40). In the following we
will also ignore the fermions.

In the absence of any symmetrization the matrix elements of $\hat{\rho}$ can
be written as a path integral as follows

$\displaystyle\langle\lambda_{i},X^{L}_{ij}|\hat{\rho}_{0}|\lambda_{i}^{\prime},(X^{L}_{ij})^{\prime}\rangle=\int_{\lambda_{i}(0)=\lambda_{i}^{\prime}}^{\lambda_{i}(\beta)=\lambda_{i}}{\cal{D}}\lambda_{i}(\tau)\int_{X^{L}_{ij}(0)=(X^{L}_{ij})^{\prime}}^{X^{L}_{ij}(\beta)=X^{L}_{ij}}{\cal{D}}X^{L}_{ij}(\tau)~{}{\rm
exp}[-S_{\beta}]$ (90)

where the action $S_{\beta}$ is the euclidean action

$S=\frac{(g_{s}N)^{1/3}N}{2l_{s}}\int_{0}^{\beta}d\tau~{}{\rm
Tr}\left[\sum_{I=1}^{9}(\partial_{\tau}X^{I})^{2}+\sum_{I\neq
J=1}^{9}[{\tilde{X}}^{I},{\tilde{X}}^{J}]^{2}\right]$ (91)

Weyl and $U(1)^{N}$ symmetries are then imposed by explicitly summing over the
transformations, leading to braided boundary conditions. However, since the
action is symmetric under these transformations, we need to sum over
transforms of the boundary conditions at one of the ends of the euclidean time
interval. We therefore have

${}_{W}\langle\lambda_{i},X^{L}_{ij}|\hat{\rho}_{0}|\lambda_{i}^{\prime},(X^{L}_{ij})^{\prime}\rangle_{W}=\rho_{tot}(\lambda_{i},X^{L}_{ij};\lambda_{i}^{\prime},(X^{L}_{ij})^{\prime})$
(92) $\displaystyle=$
$\displaystyle~{}\frac{1}{N!}\int\prod_{i=1}^{N}\frac{d\theta_{i}}{2\pi}\sum_{\sigma\in
S(N)}(-)^{\sigma}\int_{\lambda_{i}(0)=\lambda_{i}^{\prime}}^{\lambda_{i}(\beta)=\lambda_{\sigma(i)}}{\cal{D}}\lambda_{i}(\tau)\int_{X^{L}_{ij}(0)=(X^{L}_{ij})^{\prime}}^{X^{L}_{ij}(\beta)=(X^{L}_{ij})^{W}}{\cal{D}}X^{L}_{ij}(\tau)~{}{\rm
exp}[-S_{\beta}]$

where we have introduced the notation

$(X^{L}_{ij})^{W}\equiv
X^{L}_{\sigma(i)\sigma(j)}e^{i(\theta_{\sigma(i)}-\theta_{\sigma(j)})}$ (93)

which we will use in the following equations as well.

The construction for $N=2$ and with two matrices $X^{1},X^{2}$ is illustrated
in Figure 1. Note that each term in the path integral is not a product of path
integrals over $\lambda_{i}$ and $X^{L}_{ij}$ since the interaction term in
the action couple them. These interactions are symbolically drawn as
rectangular boxes to emphasize this. The figure is meant to illustrate the
boundary conditions.

Figure 1: Path Integral Representation for the thermal density matrix for a
model of two $2\times 2$ matrices $X^{1}$ and $X^{2}$ in the gauge where
$X^{1}$ is diagonal with eigenvalues $\lambda_{1}$ and $\lambda_{2}$. The
blobs represent arbitrary number of interactions between the paths.

To obtain the reduced density matrix in some sector $(k,N-k)$ one needs to
integrate over the appropriate set of boundary values. Consider some interval
$A$ on the real line. As in the previous section we will split the matrix
indices into two sets, $a,b=1\cdots k$ and $\alpha,\beta=k+1\cdots N$ where
the eigenvalues $\lambda_{a}$ lie in $A$ while the remaining
$\lambda_{\alpha}$ lie in the complement ${\bar{A}}$. The boundary values of
the matrix elements of $X^{L}$ with $L=2\cdots 9$ are not constrained in any
fashion. Then the expressions for the two proposals are given in (13) and
(14).

In terms of the paths in the path integral this means the following. Along a
given path parametrized by $0<\tau<\beta$, the $\lambda_{a}(\tau)$ must begin
and end in distinct points in the interval $A$. The eigenvalues
$\lambda_{\alpha}(\tau)$ must begin and end at the same point in the
complement ${\bar{A}}$, and there is an integral over this point. It is
important to note that apart from these restrictions the paths are free to
wander around anywhere in the $\lambda$ space at intermediate times.

In proposal (1), the boundary values of
$X^{L}_{\alpha\beta},X^{L}_{a\alpha},X^{L}_{\alpha a}$ are the same at
$\tau=0$ and $\tau=\beta$ and are integrated over, while the boundary values
of $X^{L}_{ab}$ are different. This leads to the following expression for the
reduced density matrix:

$\begin{split}&\tilde{\rho}_{k,N-k}^{(1)}(\lambda_{a},X^{L}_{ab};\lambda_{a}^{\prime},(X^{L}_{ab})^{\prime})\\\
&=\frac{1}{N!}\binom{N}{k}\int_{\bar{A}}d\lambda_{\alpha}\int
dX^{L}_{a\alpha}dX^{L}_{\alpha
a}dX^{L}_{\alpha\gamma}\int\prod_{i=1}^{N}\frac{d\theta_{i}}{2\pi}\sum_{\sigma\in
S(N)}(-)^{\sigma}\int_{\mathcal{A}_{1}}\mathcal{D}\lambda_{i}(\tau)\int_{\mathcal{B}_{1}}\mathcal{D}X^{L}_{ij}(\tau)~{}{\rm
exp}[-S_{\beta}]\end{split}$ (94)

where the boundary conditions are denoted by

$\mathcal{A}_{1}=\begin{pmatrix}\lambda_{a}(0)=\lambda_{a}^{\prime},&\lambda_{\sigma(a)}(\beta)=\lambda_{a}\\\
\lambda_{\alpha}(0)=\lambda_{\alpha},&\lambda_{\sigma(\alpha)}(\beta)=\lambda_{\alpha}\end{pmatrix}$
(95)

$\mathcal{B}_{1}=\begin{pmatrix}X^{L}_{ab}(0)=(X^{L}_{ab})^{\prime},&X^{L}_{a\alpha}(0)=X^{L}_{a\alpha},&X^{L}_{\alpha
a}(0)=X^{L}_{\alpha a},&X^{L}_{\alpha\gamma}(0)=X^{L}_{\alpha\gamma}\\\
(X^{L}_{ab})^{W}(\beta)=X^{L}_{ab},&(X^{L}_{a\alpha})^{W}(\beta)=X^{L}_{a\alpha},&(X^{L}_{\alpha
a})^{W}(\beta)=X^{L}_{\alpha
a},&(X^{L}_{\alpha\gamma})^{W}(\beta)=X^{L}_{\alpha\gamma}\end{pmatrix}$ (96)

The Figures 2-4 show the paths for $N=2$ in the various sectors for our first
proposal, drawn as paths on a cylinder which is cut across the region $A$. In
each sector there are two terms. In these figures we have represented only the
boundary values of the eigenvalues of one of the matrices $X^{1}$. The other
matrix elements are braided in the manner indicated in Figure 1. As in Figure
1 these diagrams are illustrative of the boundary conditions: the rectangular
boxes represent interactions between the variables along the paths.

Figure 2: Path Integral Representation for the reduced density matrix in the
$(2,0)$ sector for a model of two $2\times 2$ matrices $X^{1}$ and $X^{2}$ in
the gauge where $X^{1}$ is diagonal. The red cut represents the region of
interest $A$. We have shown the end-point values only for the eigenvalues of
$X^{1}$. Figure 3: Path Integral Representation for the reduced density matrix
in the $(1,1)$ sector for a model of two $2\times 2$ matrices $X^{1}$ and
$X^{2}$ in the gauge where $X^{1}$ is diagonal. The red cut represents the
region of interest $A$. We have shown the end-point values only for the
eigenvalues of $X^{1}$. Figure 4: Path Integral Representation for the reduced
density matrix in the $(0,2)$ sector for a model of two $2\times 2$ matrices
$X^{1}$ and $X^{2}$ in the gauge where $X^{1}$ is diagonal. The red cut
represents the region of interest $A$. There are no specified boundary values.

In proposal (2), only the boundary values of $X^{L}_{\alpha\beta}$ are same
and integrated over.

$\begin{split}&\tilde{\rho}_{k,N-k}^{(2)}(\lambda_{a},X^{L}_{ab},X^{L}_{a\alpha},X^{L}_{\alpha
a};\lambda_{a}^{\prime},(X^{L}_{ab})^{\prime},(X^{L}_{a\alpha})^{\prime},(X^{L}_{\alpha
a})^{\prime})\\\ &=\frac{1}{N!}\binom{N}{k}\int_{\bar{A}}d\lambda_{\alpha}\int
dX^{L}_{\alpha\gamma}\int\prod_{i=1}^{N}\frac{d\theta_{i}}{2\pi}\sum_{\sigma\in
S(N)}(-)^{\sigma}\int_{\mathcal{A}}\mathcal{D}\lambda_{i}(\tau)\int_{\mathcal{B}_{2}}\mathcal{D}X^{L}_{ij}(\tau)~{}{\rm
exp}[-S_{\beta}]\end{split}$ (97)

The boundary conditions for the $\lambda_{i}$ remain the same as in (95) ,
while those for the $X^{L}_{ij}$ are denoted by

${\mathcal{B}}_{2}=\begin{pmatrix}X^{L}_{ab}(0)=(X^{L}_{ab})^{\prime},&X^{L}_{a\alpha}(0)=(X^{L})^{\prime}_{a\alpha},&(X^{L})_{\alpha
a}(0)=(X^{L})^{\prime}_{\alpha
a},&X^{L}_{\alpha\gamma}(0)=X^{L}_{\alpha\gamma}\\\
(X^{L}_{ab})^{W}(\beta)=X^{L}_{ab},&(X^{L}_{a\alpha})^{W}(\beta)=X^{L}_{a\alpha},&(X^{L}_{\alpha
a})^{W}(\beta)=X^{L}_{\alpha
a},&(X^{L}_{\alpha\gamma})^{W}(\beta)=X^{L}_{\alpha\gamma}\end{pmatrix}$ (98)

The figures for paths for the second proposal can be drawn as in the earlier
figures.

It is now straightforward to compute ${\rm tr}{\tilde{\rho}}^{n}_{k,N-k}$ by
taking powers of these expressions and tracing. For example, ${\rm
tr}{\tilde{\rho}}^{2}_{k,N-k}$ for Proposal (1) is

$\begin{split}&\operatorname{tr}(\tilde{\rho}_{k,N-k}^{(1)})^{2}\\\
=&\left[\frac{1}{N!}\binom{N}{k}\right]^{2}\int_{{A}}d\lambda_{a}d\lambda_{a}^{\prime}\int
dX^{L}_{ab}d(X^{L}_{ab})^{\prime}\times\int_{\bar{A}}d\lambda_{\alpha}\int
dX^{L}_{a\alpha}dX^{L}_{\alpha
a}dX^{L}_{\alpha\beta}\int_{\bar{A}}d\lambda_{\alpha}^{\prime}\int
d(X^{L}_{a\alpha})^{\prime}d(X^{L}_{\alpha
a})^{\prime}d(X^{L}_{\alpha\beta})^{\prime}\\\
&\times\int\prod_{i=1}^{N}d\theta_{i}\int\prod_{j=1}^{N}d\theta^{\prime}_{j}\sum_{\sigma,\sigma^{\prime}\in
S(N)}(-)^{\sigma+\sigma^{\prime}}\int_{{\cal
C}_{1}}\mathcal{D}\lambda_{i}(\tau)\int_{{\cal
C}_{2}}\mathcal{D}X^{L}_{ij}(\tau)~{}{\rm exp}[-S_{\beta}]\int_{{\cal
C}_{3}}\mathcal{D}\lambda_{i}(\tau)\int_{{\cal
C}_{4}}\mathcal{D}X^{L}_{ij}(\tau)~{}{\rm exp}[-S_{\beta}]\\\ \end{split}$
(99)

where the periodicity conditions are

${\cal
C}_{1}=\begin{pmatrix}\lambda_{a}(0)=\lambda_{a}^{\prime},&\lambda_{\alpha}(0)=\lambda_{\alpha}\\\
\lambda_{\sigma(a)}(\beta)=\lambda_{a},&\lambda_{\sigma(\alpha)}(\beta)=\lambda_{\alpha}\end{pmatrix}$
(100) ${\cal
C}_{2}=\begin{pmatrix}X^{L}_{ab}(0)=(X^{L}_{ab})^{\prime},&X^{L}_{a\alpha}(0)=X^{L}_{a\alpha},&X^{L}_{\alpha
a}(0)=X^{L}_{\alpha a},&X^{L}_{\alpha\gamma}(0)=X^{L}_{\alpha\gamma}\\\
(X^{L}_{ab})^{W}(\beta)=X^{L}_{ab},&(X^{L}_{a\alpha})^{W}(\beta)=X^{L}_{a\alpha},&(X^{L}_{\alpha
a})^{W}(\beta)=X^{L}_{\alpha
a},&(X^{L}_{\alpha\gamma})^{W}(\beta)=X^{L}_{\alpha\gamma}\end{pmatrix}$ (101)
${\cal
C}_{3}=\begin{pmatrix}\lambda_{a}(0)=\lambda_{a},&\lambda_{\alpha}(0)=\lambda_{\alpha}^{\prime}\\\
\lambda_{\sigma^{\prime}(a)}(\beta)=\lambda_{a}^{\prime},&\lambda_{\sigma^{\prime}(\alpha)}(\beta)=\lambda_{\alpha}^{\prime}\end{pmatrix}$
(102) ${\cal
C}_{4}=\begin{pmatrix}X^{L}_{ab}(0)=X^{L}_{ab},&X^{L}_{a\alpha}(0)=(X^{L}_{a\alpha})^{\prime},&X^{L}_{\alpha
a}(0)=(X^{L}_{\alpha
a})^{\prime},&X^{L}_{\alpha\gamma}(0)=(X^{L}_{\alpha\gamma})^{\prime}\\\
(X^{L}_{ab})^{W^{\prime}}(\beta)=(X^{L}_{ab})^{\prime},&(X^{L}_{a\alpha})^{W^{\prime}}(\beta)=(X^{L}_{a\alpha})^{\prime},&(X^{L}_{\alpha
a})^{W^{\prime}}(\beta)=(X^{L}_{\alpha
a})^{\prime},&(X^{L}_{\alpha\gamma})^{W^{\prime}}(\beta)=(X^{L}_{\alpha\gamma})^{\prime}\end{pmatrix}$
(103)

where in an obvious extension of the notation of (93)

$(X^{L}_{ij})^{W^{\prime}}\equiv
X^{L}_{\sigma^{\prime}(i)\sigma^{\prime}(j)}e^{i(\theta^{\prime}_{\sigma^{\prime}(i)}-\theta^{\prime}_{\sigma^{\prime}(j)})}$
(104)

The expression is invariant if we exchange
$\sigma\leftrightarrow\sigma^{\prime},\theta_{i}\leftrightarrow\theta_{i}^{\prime}$.
The expression for ${\rm tr}{\tilde{\rho}}^{n}_{k,N-k}$ in Proposal (2) can be
similarly written down. The Renyi entropies $S_{n}$ can be then computed using
these expressions,

$S_{n}=\frac{1}{n-1}\log\left[\sum_{k=0}^{N}{\rm
tr}_{k}{\tilde{\rho}}^{n}_{k,N-k}\right]$ (105)

These path integral expressions can be directly used in numerical
calculations. It is difficult to take the $\beta\rightarrow\infty$ limit to
recover a zero temperature answer. However it should be possible to compare
the difference of the entropies at two different temperatures with the
supergravity result.

##  6 Conclusions

In this paper we have have proposed that for near -horizon $Dp$ brane
backgrounds, target space entanglement in the boundary theory provides a
precise version of bulk entanglement in the gravity dual. We have described
how to obtain in a gauge invariant manner a sub -algebra related to a target
space constraint. A Von-Neumann entropy can be associated with this sub-
algebra in the standard manner and this then gives the target space
entanglement entropy. Our paper builds on Das:2020jhy  which dealt with linear
constraints in a gauge fixed formalism and we have provided here a general
gauge invariant description of the target space entanglement.

We have also provided some arguments here, based on comparisons between the
potential experienced by probe branes moving in some region of the bulk ${\cal
R}$ and the effective potential in the boundary theory evaluated in the
corresponding region of moduli space, to motivate why the sub-algebra of
operators ${\cal A}_{\cal R}$ that we identify is sufficient to describe some
local bulk measurements on gravitons and other supergravity modes, in addition
to D branes, carried out by observers in ${\cal R}$.

We should emphasise that our arguments are not completely precise. One
reflection of this is that there are in fact two versions of our proposal,
which give rise to two different sub-algebras related to a target space
constraint, and we cannot distinguish between them at our current level of
understanding.

One source of imprecision in our proposal could be that the map between the
bulk and target space constraints, eq.(18) and eq.(24), is more complicated
than we have assumed. This could happen due to operator ordering ambiguities
or for excited states, including at finite temperature, where some or all of
the supersymmetries are broken. The target space function $f({\hat{X}}^{i})$
which appears in eq.(24) in such situations could be more non-trivially
related to its bulk counterpart in eq.(18), with coupling constant and
temperature dependent corrections, as was also discussed in section 4 above.
By carrying out numerical calculations analogous to those in hanadaprobe  one
can try to determine the effective potential and equating these results to the
potential obtained in the bulk for a probe brane, one can further hope to
obtain the correct target space constraint corresponding to a bulk region. Our
proposal would then be that the sub-algebra for this possibly modified target
space constraint is the correct one to use for obtaining the bulk
entanglement. Hopefully, further developments, especially in numerical methods
will lead to concrete checks for our ideas and will allow them to be sharpened
further. In fact, connecting with some of these recent developments has been
one of our major motivations.

Our proposal for associating a target space constraint with a bulk region is
most straightforward when the bulk region is bounded by a surface with a
constant value of one of the cartesian coordinates. This is because the fields
in the Dp brane field theory as written in (2) directly relate to cartesian
coordinates in the geometry. For many physical questions, however, one would
like to consider subregions which are bounded by constant values of the radial
coordinate. A natural guess for the corresponding target space constraint is
to require that the eigenvalues of the hermitian operator $\sum(X^{I})^{2}$
are restricted to larger than some value. With this in mind, we have discussed
in section 2.3 in some detail how to implement a constraint in the radial
direction in the bulk. We have also derived path integral expressions for
Renyi entropies in a gauge fixed description which can be directly used in
numerical calculations.

There are several open questions which merit further study. As we have
discussed above we expect that the sub -algebra of operators we are
considering will allow one to determine the one- point function of the metric
and some other supergravity fields in the vacuum and in excited coherent
states. But we do not expect to be able to determine all correlators of
supergravity modes in the bulk region of interest from the sub-algebra, in
general. How much information can be extracted from the sub-algebra and how
does this contrast with the measurements which bulk observers can do using
supergravity probes restricted to the region of interest, is an important
issue which needs to be understood better. One would hope that bulk regions
whose boundary is given by an RT extremal surface should correspond to rather
special constraints in target space 151515We are thankful to Shiraz Minwalla
for emphasising this point.. Unfortunately we do not see any evidence for this
so far and leave it as an important question for further investigation. On a
related note, one would think that area of extremal surfaces not of the RT
type, as in D0 brane geometry vanrams  would also have some understanding in
terms of target space entanglement entropy. Finally, for usual $AdS/CFT$
duality there is evidence in favour of the conjecture that there is an
intimate connection between entanglement in base space and emergence of a
smooth AdS bulk with locality rams2  . One would expect that in models of
$AdS\times{\rm(Sphere)}/CFT$ there should be a similar connection between
entanglement in color space and locality in the sphere factor of the bulk.
Target space entanglement provides a concrete framework to study this
connection. In particular in D0 brane holography there is no base space of the
holographic theory: target space entanglement would entirely account for bulk
locality. This deep connection is well worth understanding further as well.

##  7 Acknowledgements

We thank Masanori Hanada, Antal Jevicki, Shiraz Minwalla, Mark van Raamsdonk
and especially Suvrat Raju, for discussions. S.R.D would like to thank Tata
Institute of Fundamental Research for hospitality during numerous extended
visits over the years. The works of S.R.D. and S.L. have been partially
supported by National Science Foundation grant NSF/PHY-1818878. The work of
S.L. is supported in part by a Macadam Fellowship. A.K., G. M. and S. P. T.
acknowledge the support of the Govt. Of India, Department of Atomic Energy,
under Project No. 12-R&D-TFR-5.02-0200 and support from the Quantum Space-Time
Endowment of the Infosys Science Foundation. S. P. T. acknowledges support
from a J. C. Bose Fellowship, Department of Science and Technology, Govt. of
India.

##  Appendix A Details of construction of sub-algebras

This appendix provides some details of the gauge invariant construction of the
subalgebra described in Section 2.

###  A.1 Single Matrix

First consider gauged matrix quantum mechanics of a single $N\times N$ matrix
$M$. Gauge invariant single trace operators are of the form

$\hat{C}={\rm Tr}\left({\hat{M}}^{m}{\hat{\Pi}_{M}}^{n}\right)_{\rm{order}}$
(106)

Here ${\hat{\Pi}_{M}}$ denotes the conjugate momentum to ${\hat{M}}$ and the
notation $\left(\right)_{\rm{order}}$ means that the ${\hat{M}}$ and
${\hat{\Pi}_{M}}$’s are sprinkled in all possible orders. However we can use
the commutation relations to bring e.g. all the ${\hat{M}}$’s together. The
trace Tr is over matrix indices.

We want to construct a sub-algebra of operators which can be used to make
measurements in a region $A$ of the space of eigenvalues of ${\hat{M}}$. This
is achieved by defining the projector (23). Then the operator which will
belong to this algebra is of the form

$\hat{C}_{A}={\rm
Tr}\left(\hat{P}_{A}M\hat{P}_{A}M\cdots\hat{P}_{A}M\hat{P}_{A}{\hat{\Pi}_{M}}\hat{P}_{A}\cdots{\hat{\Pi}_{M}}\hat{P}_{A}\right)$
(107)

In writing (107) we have used $\hat{P}_{A}^{2}=\hat{P}_{A}$ and the fact that
in this particular case $\left[\hat{P}_{A},{\hat{M}}\right]=0$. The operators
of the form (107) together with the identity operator form a sub-algebra of
operators of the theory.

To see that gives us the right sub-algebra, fix a gauge where ${\hat{M}}$ is
diagonal with its diagonal elements denoted by $\hat{\lambda}_{i},i=1\cdots
N$, while the diagonal elements of the conjugate momenta are denoted by
${\hat{\pi}}_{i}$. As explained in Section 2.1, the resulting constraint
requires the states to be singlets. The remaining gauge freedom of Weyl
transformations needs to be imposed by Weyl symmetrizing the states, and
absorbing the standard van der Monde factor then makes $\hat{\lambda}_{i}$
coordinate operators of $N$ fermions on a line. Let us begin by considering
operators which do not contain the conjugate momenta, i.e.

${\cal{O}}_{n}={\rm Tr}({\hat{M}}^{n})$ (108)

The expectation value of the operator ${\cal{O}}_{n}$ in some pure state
$|\Psi\rangle$ is given by

$\sum_{i=1}^{N}\langle\Psi|\hat{\lambda}_{i}^{n}|\Psi\rangle$ (109)

This measures the position of each of the fermions, takes its $n$-th power and
then sums over all the particles. Similarly, the expectation value of ${\rm
Tr}({\hat{\pi}}_{M}^{n})$ would measure the sum of $n$-th power of the momenta
of all the fermions. This is of course a standard measurement in a system of
many identical particles. The projected version of the operator (108) is

${\cal{O}}^{P}_{n}=\sum_{i=1}^{N}\int_{A}[\prod_{s=1}^{n}dx_{s}][\prod_{s=1}^{n}\delta(x_{s}-\hat{\lambda}_{i})]\hat{\lambda}_{i}^{n}$
(110)

We will use the basis

$|\lambda_{1},\lambda_{2},\cdots\lambda_{N}>_{a}=\frac{1}{N!}\sum_{\sigma\in
S_{N}}{\rm sgn}(\sigma)|\lambda_{\sigma(1)},\cdots\lambda_{\sigma(1)}\rangle$
(111)

where $|\rangle_{a}$ denotes an anti-symmetrized ket. Note that this is not an
eigenstate of each individual term in the sum in (110). However it is an
eigenstate of the sum. This follows from the fact that the sum is symmetric
under permutations.

Consider first the case $N=2$. The expectation value of the operator
${\cal{O}}^{P}_{3}$ in a state with a wavefunction
$\Psi(\lambda_{1},\lambda_{2})$ is given by

$\displaystyle\langle\Psi|{\cal{O}}^{P}_{3}|\Psi\rangle=\int_{A}[dx_{1}dx_{2}dx_{3}]\int_{R}d\lambda_{1}d\lambda_{2}~{}\Psi^{\star}($
$\displaystyle\lambda_{1},\lambda_{2})\\{\lambda_{1}^{3}~{}\delta(x_{1}-\lambda_{1})\delta(x_{2}-\lambda_{1})\delta(x_{3}-\lambda_{1})$
$\displaystyle+\lambda_{2}^{3}~{}\delta(x_{1}-\lambda_{2})\delta(x_{2}-\lambda_{2})\delta(x_{3}-\lambda_{2})\\}\Psi(\lambda_{1},\lambda_{2})$
(112)

where the wavefunction is
$\Psi[\lambda_{1}\cdots\lambda_{N}]=\langle\\{\lambda_{i}\\}|\Psi\rangle$.
where we have used antisymmetry of the wavefunctions. The integrals over
$\lambda_{1}$ and $\lambda_{2}$ are over the entire real line. Writing each of
these integrals as a sum over an integral over $A$ and an integral over the
complement ${\bar{A}}$, and noting that the delta functions ensure that only
the integrals over $A$ contribute, it is straightforward to see that the
result is

$\displaystyle\langle\Psi|{\cal{O}}^{P}_{3}|\Psi\rangle=$
$\displaystyle\int_{A}d\lambda_{1}\int_{A}d\lambda_{2}\Psi^{\star}(\lambda_{1},\lambda_{2})(\lambda_{1}^{3}+\lambda_{2}^{3})\Psi(\lambda_{1},\lambda_{2})$
$\displaystyle+2\int_{A}d\lambda_{1}\int_{{\bar{A}}}d\lambda_{2}\Psi^{\star}(\lambda_{1},\lambda_{2})(\lambda_{1}^{3})\Psi(\lambda_{1},\lambda_{2})$
(113)

The first term is the contribution from configurations when both the particles
are in the region of interest, while the second term from configurations where
one of the particles is in the region of interest. Clearly this expectation
value is equal to the expectation value of the operator without the projection
if the wavefunction is non-vanishing only when both the particles are in the
region of interest $A$.

This result can be easily generalized for arbitrary $N$. Then the expectation
value of the operator ${\cal{O}}^{P}_{n}$ becomes

$\langle\Psi|{\cal{O}}^{P}_{n}|\Psi\rangle=\sum_{k=1}^{N}~{}{N\choose
k}\sum_{a=1}^{k}\int_{A}\prod_{a=1}^{k}d\lambda_{a}\int_{{\bar{A}}}\prod_{\alpha=k+1}^{N}d\lambda_{\alpha}~{}\left(\Psi^{\star}[\\{\lambda_{i}\\}]~{}\lambda_{a}^{n}~{}\Psi[\\{\lambda_{i}\\}]\right)$
(114)

The expression (114) is a sum over sectors specified by the number of the
$\lambda_{i}$’s in the region of interest. The expectation value then measures
the sum of the $n$-th power of the position of all particles which are in the
region of interest. Equation (114) is simply a reflection of the decomposition
of the Hilbert space into sectors, as in (10).

The action of our projected operator on a basis state is

${\cal{O}}^{P}_{n}|\\{\lambda_{i}\\}\rangle_{a}=\sum_{i=1}^{N}\int[\prod_{s=1}^{n}dx_{s}][\prod_{s=1}^{n}\delta(x_{s}-\lambda_{i})]\lambda_{i}^{n}|\\{\lambda_{i}\\}\rangle_{a}$
(115)

Suppose the state is in the $(k,N-k)$ sector, i.e. $k$ of the $\lambda_{i}$’s
lie in the region of interest. We can choose these to be the
$\lambda_{a},a=1\cdots k$. Consider a term in the sum in (115). This contains
a product of delta functions, so this will be nonzero only when the
corresponding $\lambda_{i}$ lie in the region of interest $A$. This means that
the sum over $i$ is truncated to the first $k$ terms,

${\cal{O}}^{P}_{n}|\\{\lambda_{i}\\}\rangle_{a}=\sum_{i=1}^{k}\lambda_{i}^{m}|\\{\lambda_{i}\\}\rangle_{a}$
(116)

Thus this operator acting on a basis state in the $(k,N-k)$ sector has a
trivial action on the eigenvalues which are in the complement ${\bar{A}}$.
This is an example of an operator of the type $(k,N-k)$,

${\cal
O}_{k,N-k}~{}|\\{\lambda_{a}\\},\\{\lambda_{\alpha}\\}\rangle_{a}=\int\prod_{a=1}^{k}[d\lambda_{a}^{\prime}]~{}{\tilde{O}}\left(\\{\lambda_{a}^{\prime}\\},\\{\lambda_{a}\\}\right)~{}|\\{\lambda_{a}^{\prime}\\},\\{\lambda_{\alpha}\\}\rangle_{a}$
(117)

Let us define a smaller Hilbert space of $k$ particles which s spanned by

$|\\{\lambda_{a}\\}\rangle_{a}~{}~{}~{}~{}~{}~{}~{}~{}a=1\cdots
k~{}~{}\lambda_{a}\in A$ (118)

Then one can define an operator in this smaller Hilbert space,

${\tilde{{\cal{O}}}}_{k,N-k}=\int[\prod_{a=1}^{k}d\lambda_{a}d\lambda_{a}^{\prime}]~{}{\tilde{O}}_{k,N-k}\left(\\{\lambda_{a}\\},\\{\lambda_{a}^{\prime}\\}\right)~{}|\\{\lambda_{a}\\}\rangle_{a}~{}_{a}\langle\\{\lambda_{a}^{\prime}\\}|$
(119)

In the above discussion the sector $(0,N)$ did not enter in the expression
(114). This simply reflects the fact that if we measure any operator involving
the position and momenta in the region of interest $A$, we should get a non-
zero answer only if there are particles in $A$. However since the identity
operator is also a member of the sub-algebra, this sector needs to be
included. In fact the identity operator is the only operator which will
receive contributions from the $(0,N)$ sector.

Clearly the expression (114) can be written as a sum over traces in the
smaller Hilbert spaces,

$\langle\Psi|{\cal{O}}^{P}|\Psi\rangle={N\choose k}\sum_{k=1}^{N-1}{\rm
Tr}_{A}\left[{\tilde{\rho}}_{k,N-k}{\cal{O}}_{k,N-k}\right]$ (120)

where the density matrix ${\tilde{\rho}}_{k,N-k}$ is given by

${\tilde{\rho}}_{k,N-k}={N\choose
k}\int_{A}[\prod_{i=1}^{k}d\lambda_{a}d\lambda_{a}^{\prime}]\int_{{\bar{A}}}[\prod_{\alpha=k+1}^{N}d\lambda_{\alpha}]~{}\Psi^{\star}[\\{\lambda_{a}\\},\\{\lambda_{\alpha}\\}]~{}\Psi^{\star}[\\{\lambda_{a}^{\prime}\\},\\{\lambda_{\alpha}\\}]|\\{\lambda_{a}\\}\rangle_{a}~{}_{a}\langle\\{\lambda_{a}^{\prime}\\}|$
(121)

Note that this is an operator which lives in the $k$-particle sector of the
small Hilbert space defined in (118). This decomposition of the whole hilbert
space into sectors is exactly what appears in Das:2020jhy . The projected
operators therefore provide a gauge invariant formulation of the problem.

The above constructions easily generalize to the situation when the state of
the entire system is a mixed state. Let the density matrix of the whole system
be

$\rho_{tot}=\int[\prod_{i=1}^{N}d\lambda_{i}d\lambda_{i}^{\prime}]~{}\rho_{tot}\left(\\{\lambda_{i}\\},\\{\lambda_{i}^{\prime}\\}\right)~{}|\\{\lambda_{i}\\}\rangle_{a}~{}_{a}\langle\\{\lambda_{i}^{\prime}\\}|$
(122)

Then the reduced density matrix ${\tilde{\rho}}$ which evaluates expectation
values of operators belonging to this subalgebra is obtained by tracing over
${\bar{A}}$,

${\tilde{\rho}}_{k,N-k}={N\choose
k}\int[\prod_{i=1}^{k}d\lambda_{a}d\lambda_{a}^{\prime}][\prod_{\alpha=k+1}^{N}d\lambda_{\alpha}]~{}\rho_{tot}\left(\\{\lambda_{a}\\},\\{\lambda_{\alpha}\\};\\{\lambda_{a}^{\prime}\\},\\{\lambda_{\alpha}\\}\right)~{}|\\{\lambda_{a}\\}\rangle_{a}~{}_{a}\langle\\{\lambda_{a}^{\prime}\\}|$
(123)

This density matrix is not normalized. In fact the trace ${\rm
tr}_{k}{\tilde{\rho}}_{k}$ is the probability of $k$ particles to be in the
region of interest. Thus the density matrix in the Hilbert space which is a
direct sum of all the sector is properly normalized.

The von Neumann entropy associated with the reduced density matrix ${\rm
tr}{\tilde{\rho}}_{k}$ is given by

$S_{k,A}=-{\rm
tr}_{{\mathcal{H}}_{k,N-k}}({\tilde{\rho}}_{k,N-k}\log{\tilde{\rho}}_{k,N-k})$
(124)

This quantifies the entanglement between the target space region $A$ and its
complement ${\bar{A}}$ in this sector. Following the above steps, we can
easily see that a reduced density matrix $\rho_{{}_{RDM}}$ based on the gauge
invariant subalgebra ${\cal A}$, defined by

${\rm Tr}(\rho_{{}_{RDM}}{\cal{O}}):={\rm
Tr}(\rho_{tot}{\cal{O}})\quad\forall{\cal{O}}\in{\cal A}$

satisfies

$\rho_{{}_{RDM}}=\oplus_{k=0}^{N}{\tilde{\rho}}_{k,N-k}$

The total target space entanglement entropy is then a sum over all sectors

$S_{A}=\sum_{k=0}^{N}S_{k,A}$ (125)

As shown in Das:2020jhy  this quantity satisfies the usual positivity
properties and strong subadditivity. Similar sector-wise entanglement also
appears in discussions of entanglement entropy in gauge theories ST ; V .

We have used a first quantized description of the system. However there is an
equivalent second quantized description. In the latter, we have a conventional
nonrelativistic field theory of a fermion field $\psi(\lambda,t)$ : the space
of this theory is the space of eigenvalues. The target space entanglement we
discussed above now becomes a conventional geometric entanglement in this
field theory.

####  A.1.1 Momentum operators

Operators involving momenta are subtle and at a first sight, appear to require
introduction of other sectors. This can be illustrated by a calculation of the
expectation value of the projected version of an operator of the form (106)
with $m=0$ and $n=2$, with the region of interest $A$ being the positive real
line $\lambda\geq 0$. The result of a calculation analogous to (113) is, for
$N=2$,

$\begin{split}&\langle\Psi|\operatorname{tr}(\hat{P}\hat{\Pi}_{M}\hat{P}\hat{\Pi}_{M}\hat{P})|\Psi\rangle\\\
=&-\frac{1}{2}\int_{0}^{\infty}d\lambda_{1}\int_{0}^{\infty}d\lambda_{2}\left[\Psi^{*}(\lambda_{1},\lambda_{2})\left(\frac{\partial^{2}}{\partial\lambda_{1}^{2}}+\frac{\partial^{2}}{\partial\lambda_{2}^{2}}\right)\Psi(\lambda_{1},\lambda_{2})+\Psi(\lambda_{1},\lambda_{2})\left(\frac{\partial^{2}}{\partial\lambda_{1}^{2}}+\frac{\partial^{2}}{\partial\lambda_{2}^{2}}\right)\Psi^{*}(\lambda_{1},\lambda_{2})\right]\\\
&-\int_{0}^{\infty}d\lambda_{1}\int_{-\infty}^{0}d\lambda_{2}\left[\Psi^{*}(\lambda_{1},\lambda_{2})\frac{\partial^{2}}{\partial\lambda_{1}^{2}}\Psi(\lambda_{1},\lambda_{2})+\Psi(\lambda_{1},\lambda_{2})\frac{\partial^{2}}{\partial\lambda_{1}^{2}}\Psi^{*}(\lambda_{1},\lambda_{2})\right]\\\
&+\int_{-\infty}^{\infty}d\lambda\left(\frac{\partial}{\partial\lambda_{1}}|\Psi(\lambda_{1},\lambda)|^{2}\right)\bigg{|}_{\lambda_{1}=0}+2\delta(0)\int_{-\infty}^{\infty}d\lambda|\Psi(0,\lambda)|^{2}\end{split}$
(126)

The first and second lines of the RHS are analogous to what we got in (113);
the first line is the contribution from the $(2,0)$ sector, and the second
line is the contribution from the $(1,1)$ sector. However, we appear to also
have an extra line, the third line, which represents a sector that has one of
the particles exactly at $\lambda=0$.

Note that since these extra terms pertain to particles at the boundary, it is
tied to the question of how one defines the region $A$ precisely, e.g. as an
open or a closed set, or in terms of a target space lattice etc. We suggest
below an alternative treatment in terms of translation operators, rather than
momenta, which provide a proof of principle how these problems can be avoided.

To explain this, let us start with the case of $N=1$, that is, just the case
of a single $1\times 1$ matrix or equivalently the case of one particle. We
now have just two sectors $(1,0)$ and $(0,1)$, in the first one the particle
is in region $A$ (which we will again define as $x>0$) and in the second one
it is outside.

Consider the traslation operator

$O_{a}=\exp[-ia{\hat{\Pi}}]$ (127)

with the action

$O_{a}|x\rangle=|x+a\rangle$ (128)

Clearly such operators can take states in $(1,0)$ to $(0,1)$ (if $a<0$) or
vice versa (if $a>0$). In the following, we will take $a>0$ to be specific; a
similar analysis can be carried out with $a<0$. To obtain operators acting
within the sector $(1,0)$, let us use the projection $O_{a}\to O_{a}^{P}$. It
is useful to represent (127) in terms of the product of a large number $n$ of
exponentials (as in Feynman path integrals), with $a=n\epsilon$. In the limit
of $\epsilon\to 0$, each exponential can be approximated as
$\exp[-i\epsilon{\hat{\Pi}}]\approx 1+(-i\epsilon){\hat{\Pi}}$ whose projected
version is
$\hat{P}(1+(-i\epsilon){\hat{\Pi}})\hat{P}\approx\hat{P}(\exp[-i\epsilon{\hat{\Pi}}])\hat{P}$.
In short, we have

$\displaystyle O_{a}$
$\displaystyle\equiv\left(\exp[-i\epsilon{\hat{\Pi}}]\right)^{n}$
$\displaystyle O_{a}^{P}$ $\displaystyle=\lim_{\epsilon\to
0}\left(\hat{P}\exp[-i\epsilon{\hat{\Pi}}]\hat{P}\right)^{n}$

With the above expressions, it is easy to see that

$\displaystyle\langle x^{\prime}|O_{a}|x\rangle$ $\displaystyle=\langle
x^{\prime}|x+a\rangle=\delta(x^{\prime}-a-x)$ (129) $\displaystyle\langle
x^{\prime}|O_{a}^{P}|x\rangle$ $\displaystyle=\lim_{\epsilon\to 0}\langle
x^{\prime}|\left(\hat{P}\exp[-i\epsilon{\hat{\Pi}}]\hat{P}\right)^{n}|x\rangle$
$\displaystyle=\lim_{\epsilon\to 0}\theta(x^{\prime})\langle
x^{\prime}-\epsilon|\theta(x^{\prime}-\epsilon)\left(\hat{P}\exp[-i\epsilon{\hat{\Pi}}]\hat{P}\right)^{n-1}|x\rangle$
$\displaystyle=\lim_{\epsilon\to
0}\theta(x^{\prime})\theta(x^{\prime}-\epsilon)\theta(x^{\prime}-2\epsilon)...\theta(x^{\prime}-a)\delta(x^{\prime}-a-x)$
(130)

Note that unless $x$ and $x^{\prime}$ are both in $A$, the above matrix
element vanishes. This is because, say $x$ is not in $A$ while $x^{\prime}$ is
in $A$, then at least $\theta(x)=\theta(x^{\prime}-a)$ will vanish, making the
entire product (130) vanish.161616We avoid here the possibility $x=0$ by
assuming that the partition demarcating the region $A$ does not fall on $x=0$.
This is equivalent to assuming a lattice structure of the real line such that
none of the sites falls exactly on 0, which is possible for any lattice
separation, however small. Presumably a similar reasoning can get rid of the
boundary terms in (126) as well. On the other hand, if both $x$ and
$x^{\prime}$ are in $A$, then all the theta-functions in (130) evaluate to 1
(since the arguments of the theta-functions are all located on a straight
‘Feynman’ path joining $x$ and $x^{\prime}$ which are both in $A$ which is
convex). This leads to the original matrix element of $O$ (129) which
described the case with no restrictions. There are no extra terms
corresponding to particles located at $x=0$.

Now, an observable corresponds to the hermitian operator is ${\bf
O}_{a}=O_{a}+c.c.$, the projected operator being ${\bf
O}^{P}_{a}=O_{a}^{P}+c.c.$. Their expectation values are given as follows. For
a general wavefunction $\Psi(x)$

$\displaystyle\langle\Psi|{\bf O}_{a}|\Psi\rangle$
$\displaystyle=\int_{-\infty}^{\infty}dx\left[\Psi^{*}(x)\Psi(x+a)+\Psi^{*}(x+a)\Psi(x)\right]$
$\displaystyle\langle\Psi|{\bf O}_{a}^{P}|\Psi\rangle$
$\displaystyle=\int_{0}^{\infty}dx\left[\Psi^{*}(x)\Psi(x+a)+\Psi^{*}(x+a)\Psi(x)\right]$
(131)

Note that the projected operator merely restricts the range of the integral to
$x>0$ as it should, and does not introduce any unwarranted boundary terms,
unlike in (126), corresponding to particles located at $x=0$.

The generalization to $N>1$ can be done as follows. Consider the translation
operator $O_{a}=\exp[-ia{\rm
Tr}{\hat{\Pi}}]=\exp[-ia\sum_{m=0}^{N}{\hat{\Pi}}_{m}]$. This operator is
obviously gauge-invariant, since it involves ${\rm Tr}{\hat{\Pi}}$. In this
case the above argument for $N=1$ can be straightforwardly generalized. The
position space matrix elements of $O_{a}^{P}$ again involve a string of theta
functions all located along a straight line from $X=(x_{1},x_{2},...)$ to
$X^{\prime}=(x_{1}+a,x_{2}+a,...)$, which all evaluate to 1 if both $X$ and
$X^{\prime}$ are in region $A$, i.e. both $x_{1}$ and $x_{1}+a$ are positive.

Now the reader may justifiably point out that this is not the most general
translation operator, since the above operator translates the point
$(x_{1},x_{2},...)$ by the same amount. This can be remedied by considering an
operator $\exp[-i{\rm Tr}(A{\hat{\Pi}})]$ which evaluates to
$\exp[-i\sum_{m=0}^{N}a_{mm}{\hat{\Pi}}_{m}]$. This clearly describes a most
general translation. The operator is not gauge invariant, however, since $A$
is a fixed matrix. To make it gauge invariant, one can sum over terms with
Weyl-copies of $A$.171717In the previous paragraph, $A=a{\bf 1}$ which was
automatically Weyl-invariant.

We made these arguments in the context of a single matrix, but it is
generalizable to multiple matrices too.

The above considerations provide a proof of principle that if we replace the
momentum operators by appropriately defined ‘translation’ operators, then the
problem pointed out at the beginning of this subsubsection can be taken care
of.

###  A.2 Multiple Matrices

For multiple matrices we have two possible subalgebras which correspond to a
given target space constraint.

A projector leading to the first sub-algebra is defined by (25), and the
procedure to construct operators which belong to the sub-algebra is explained
in section 2.2.The gauge choice which makes the physics most transparent is
the one where the hermitian matrix $f(\hat{X}^{I})$ is chosen to be diagonal.
In this subsection we will discuss the simplest constraint where the function
which appears in (25) is

$F[\hat{X}]=X^{1}$ (132)

As in the single matrix example, we fix a $A_{t}=0$ gauge and fix the
remaining time independent gauge freedom by choosing $\hat{X}^{1}$ to be
diagonal with diagonal elements are $\hat{\lambda}_{i}$. The remaining
symmetries are Weyl transformations which permute the eigenvalues and the
matrix elements of the other matrices, and $U(1)^{N}$ transformations as in
(5) and (6). This symmetry is imposed by hand by adding the transforms in the
states, as in (7). As in the single matrix case, this is an eigenstate of the
traced operators of the form (31). Thus when ${\hat{{\cal{O}}}}^{P_{1}}$ acts
on such a state, we can replace the operators appearing in (31) by their
eigenvalues which we denote by the matrix without a hat.

Acting on a state where the $\lambda_{i}$ for $i=1\cdots k$ are in the region
A, the projected version of the operator $\hat{X}^{I}$ as defined in (28)

$(X^{I})^{P_{1}}_{ij}=\int dx_{1}\delta(x_{1}-\lambda_{i})X^{I}_{ij}\int
dx_{2}(x_{2}-\lambda_{j})=\begin{cases}X^{I}_{ij}&\text{if }i,j=1\cdots k\\\
0&\text{if }~{}~{}{\rm otherwise}\end{cases}$

Thus the projector projects each of the matrices to the $k\times k$ block, as
depicted in (32).

Consider, for example, an operator ${\hat{{\cal{O}}}}$ (as in (27) which is of
the form ${\cal{O}}={\rm Tr}(\hat{X}^{I}\hat{X}^{J}\hat{X}^{K})$. The action
of its projected version on a basis state is given by

$\displaystyle{\hat{{\cal{O}}}}^{P_{1}}$
$\displaystyle~{}|\\{\lambda_{a}\\},\\{\lambda_{\alpha}\\}\\{X^{L}_{ab}\\}\\{X^{L}_{a\alpha}\\}\\{X^{L}_{\alpha
a}\\}\\{X^{L}_{\alpha\beta}\\}\rangle_{W}$ $\displaystyle=\sum_{a_{1}\cdots
a_{3}=1}^{k}X^{I}_{a_{1}a_{2}}X^{J}_{a_{2}a_{3}}X^{K}_{a_{3}a_{1}}~{}|\\{\lambda_{a}\\},\\{\lambda_{\alpha}\\}\\{X^{L}_{ab}\\}\\{X^{L}_{a\alpha}\\}\\{X^{L}_{\alpha
a}\\}\\{X^{L}_{\alpha\beta}\\}\rangle_{W}$ (133)

This is an example of an operator of type $(k,N-k)$ in the first proposal for
a sub-algebra in Das:2020jhy . An operator belonging to this first sub-algebra
has a non-trivial action only on the $\lambda_{a},X^{L}_{ab},L\neq 1$ for
$a,b=1\cdots k$, as shown in (15). The reduced density matrix which evaluates
the expectation values of these operators is then obtained from the density
matrix $\rho_{tot}$ of the whole system by tracing over the variables
$\lambda_{\alpha},X^{I}_{a\alpha},X^{I}_{\alpha\beta}$. This expression is
given in (13)

A second sub-algebra was also defined which retains the off diagonal matrices
of the type $X^{L}_{a\alpha}$. This is defined in equations (33) - (36) Acting
on a state of the form (7) where $\lambda_{i},i=1\cdots k$ we then have

$(X^{I})^{P_{2}}_{ij}=\begin{cases}X^{I}_{ij}&\text{if}~{}~{}~{}i=1\cdots
k,j=1\cdots N\\\ X^{I}_{ij}&\text{if}~{}~{}~{}i=1\cdots N,j=1\cdots k\\\
0&\text{otherwise}\end{cases}$

It is now straightforward to see that a projected operator has a non-trivial
action only on $\lambda_{a},X^{I}_{ab},X^{I}_{a\alpha}$ and $X^{I}_{\alpha
a}$. This is an operator of type $(k,N-k)$ in the second subalgebra defined in
Das:2020jhy , whose action is given in (16) The corresponding reduced density
matrix is given in (14)

##  Appendix B Polar Decomposition of Matrices

In this appendix we provide the details of the polar decomposition of multiple
matrices.

###  B.1 Two Matrices

For two matrices, the positive semi-definite matrix $\hat{R}$ given by (44) is
expressed in terms of unitary matrices $\hat{V},\hat{W}$ by (48). The inverse
of the direct product matrix
$\left[\hat{V}\otimes\hat{V}^{*}+\hat{W}\otimes\hat{W}^{*}\right]$ in terms of
a infinite series as follows

$\left(\hat{V}^{*}\otimes\hat{V}+\hat{W}^{*}\otimes\hat{W}\right)^{-1}=\hat{V}^{T}\otimes\hat{V}^{\dagger}\left(\mathbb{I}\otimes\mathbb{I}+(\hat{W}\hat{V}^{\dagger})^{*}\otimes\hat{W}\hat{V}^{\dagger}\right)^{-1}=\hat{V}^{T}\otimes\hat{V}^{\dagger}\sum_{n=0}^{\infty}(-1)^{n}\left[(\hat{W}\hat{V}^{\dagger})^{*}\otimes\hat{W}\hat{V}^{\dagger}\right]^{n}$
(134)

which proves (51) with the matrix $\hat{Q}$ defined in (52).

Thus $\left(\hat{s}^{2}\right)_{kl}$ can be solved by applying the relation
(51)

$\begin{split}\left(\hat{s}^{2}\right)_{ij}=&\left[\left(\hat{V}^{*}\otimes\hat{V}+\hat{W}^{*}\otimes\hat{W}\right)^{-1}\right]_{ij,kl}2\left(\hat{R}^{2}\right)_{kl}\\\
=&2\left[\sum_{n=0}^{\infty}(-1)^{n}\hat{V}^{T}\left[(\hat{W}\hat{V}^{\dagger})^{*}\right]^{n}\otimes\hat{V}^{\dagger}\left[\hat{W}\hat{V}^{\dagger}\right]^{n}\right]_{ij,kl}\left(\hat{R}^{2}\right)_{kl}\\\
=&2\sum_{n=0}^{\infty}(-1)^{n}\left[\hat{V}^{\dagger}\left[\hat{W}\hat{V}^{\dagger}\right]^{n}\right]_{ik}\left(\hat{R}^{2}\right)_{kl}\left[\left[(\hat{W}\hat{V}^{\dagger})^{\dagger}\right]^{n}\hat{V}\right]_{lj}\\\
\end{split}$ (135)

which proves (53).

We now prove that the right hand side of (53) is positive semi-definite. Let
$\\{v_{i}\\}$ be the set of eigenvectors of $\hat{Q}$. Because $\hat{Q}$ is
unitary, the eigenvalues of $\hat{Q}$ take the form

$\hat{Q}v_{i}=e^{i\phi_{i}}v_{i}$ (136)

then we find the inner product

$\begin{split}\langle
v_{i},2\left\\{\sum_{n=0}^{\infty}(-1)^{n}\left[\hat{Q}^{\dagger}\right]^{n}\hat{R}^{2}\hat{Q}^{n}\right\\}v_{i}\rangle=&2\sum_{n=0}^{\infty}(-1)^{n}\langle
v_{i},\left(\hat{Q}^{\dagger}\right)^{n}\hat{R}^{2}\hat{Q}^{n}v_{i}\rangle=2\sum_{n=0}^{\infty}(-1)^{n}\langle\hat{Q}^{n}v_{i},\hat{R}^{2}\hat{Q}^{n}v_{i}\rangle\\\
=&2\sum_{n=0}^{\infty}(-1)^{n}\langle v_{i},\hat{R}^{2}v_{i}\rangle=\langle
v_{i},\hat{R}^{2}v_{i}\rangle>0\end{split}$ (137)

since $\hat{R}^{2}$ is positive semi-definite according to (44).

Since $\hat{s}$ is positive semi-definite diagonal matrix,
$\left(\hat{s}^{2}\right)_{ij}=\hat{s}_{i}^{2}\delta_{ij}$, we can take the
square root of both sides of (53) to get

$\hat{s}=\sqrt{2}\hat{V}^{\dagger}\left\\{\sum_{n=0}^{\infty}(-1)^{n}\left[\hat{Q}^{\dagger}\right]^{n}\hat{R}^{2}\hat{Q}^{n}\right\\}^{1/2}\hat{V}$
(53)

Plugging (53) back into (47) we obtain $\hat{Z}$,

$\hat{Z}=\hat{V}\hat{s}\hat{W}^{\dagger}=\sqrt{2}\left\\{\sum_{n=0}^{\infty}(-1)^{n}\left[\hat{Q}^{\dagger}\right]^{n}\hat{R}^{2}\hat{Q}^{n}\right\\}^{1/2}\hat{V}\hat{W}^{\dagger}=\sqrt{2}\left\\{\sum_{n=0}^{\infty}(-1)^{n}\left[\hat{Q}^{\dagger}\right]^{n}\hat{R}^{2}\hat{Q}^{n}\right\\}^{1/2}\hat{Q}$
(138)

This proves (54), where the operation ${\mathfrak{L}}_{\hat{V}}$ is defined in
(55). In terms of matrix elements, the equation (54) can be written as

$\hat{Z}_{ij}=\sqrt{2}\left\\{\left[\left(\mathbb{I}\otimes\mathbb{I}+\hat{Q}^{T}\otimes\hat{Q}^{\dagger}\right)^{-1}\right]_{il,mn}\left(\hat{R}^{2}\right)_{mn}\right\\}^{1/2}\hat{Q}_{lj}.$
(139)

The identity (56) follows from the definition (55),

$\hat{V}^{\dagger}\left(\mathfrak{L}_{\hat{V}}\hat{M}\right)^{2}\hat{V}+\left(\mathfrak{L}_{\hat{V}}\hat{M}\right)^{2}=-2\sum_{n=1}^{\infty}(-1)^{n}\left(\hat{V}^{\dagger}\right)^{n}\hat{M}^{2}\hat{V}^{n}+2\sum_{n=0}^{\infty}(-1)^{n}\left(\hat{V}^{\dagger}\right)^{n}\hat{M}^{2}\hat{V}^{n}=2\hat{M}^{2},$
(56)

we obtain (54).

We now explain the derivation of the integration measure (61). In the gauge
where $\hat{R}$ is diagonal, as in (59), the expression for the complex matrix
$\hat{Z}$ is given by (53) with $\hat{R}^{2}$ replaced by $\hat{r}^{2}$. In a
Hilbert space basis where $\hat{r}_{i}$ and $\hat{Q}_{ij}$ are diagonal, the
measure is given by (60). Parametrizing $Q$ as in (61) we have

$\begin{split}dQ=U\left(de^{i\Phi}-\left[e^{i\Phi},U^{\dagger}dU\right]\right)U^{\dagger}\end{split}$
(140)

where we have used the identity $dU^{\dagger}=-U^{\dagger}dUU^{\dagger}$. Then
define

$dS\equiv U^{\dagger}dU$ (62)

We can see that $dS^{\dagger}=-dS$. Then the line element becomes

$\begin{split}\operatorname{tr}\left(dQdQ^{\dagger}\right)=&\operatorname{tr}\left\\{de^{i\Phi}de^{-i\Phi}+\left[e^{i\Phi},dS\right]\left[e^{-i\Phi},dS\right]\right\\}\\\
=&\sum_{i}d\phi_{i}^{2}+2\sum_{ij}e^{i\phi_{i}}dS_{ij}e^{-i\phi_{j}}dS_{ji}-2\sum_{ij}dS_{ij}dS_{ji}\\\
=&\sum_{i}d\phi_{i}^{2}+8\sum_{i<j}\sin^{2}\frac{\phi_{i}-\phi_{j}}{2}dS_{ij}dS_{ij}^{*}\\\
\end{split}$ (63)

This implies that we can choose $dS_{ii}=0$. Now we have metric
$ds^{2}=g_{AB}d\bar{x}^{A}dx^{B}$ with
$dx^{A}=\left(d\phi_{i},dS_{ij(i<j)},dS_{ij(i<j)}^{*}\right)$, where

$g_{AB}=\left(\begin{matrix}1&0&0\\\ 0&4\sin^{2}\frac{\phi_{i}-\phi_{j}}{2}\\\
0&0&4\sin^{2}\frac{\phi_{i}-\phi_{j}}{2}\\\ \end{matrix}\right)$ (141)

The determinant of $g$ is

$\det g_{AB}=\prod_{i<j}\left(4\sin^{2}\frac{\phi_{i}-\phi_{j}}{2}\right)^{2}$
(142)

Thus

$\prod_{ij}[dQ_{ij}]=\sqrt{\det
g_{AB}}\prod_{i}d\phi_{i}\prod_{i<j}dS_{ij}dS_{ij}^{*}$ (143)

The final expression (64) follows when we use this in (60). To ensure that the
variables $r_{i},\phi_{i}$ cover the $\mathbb{R}^{2}$ formed by $X^{1},X^{2}$
once we see that the ranges of the angles $\phi_{i}$ are

$-\pi<\phi_{i}<\pi,~{}~{}~{}i=1,\cdots,N$ (144)

###  B.2 Three Matrices

Now consider three matrices $\hat{X}^{1},\hat{X}^{2},\hat{X}^{3}$, with
$\hat{R}$ defined by (68). To obtain a polar decomposition we first form a
complex matrix as follows

$\hat{Y}\equiv\sqrt{(\hat{X}^{1})^{2}+(\hat{X}^{2})^{2}}+i\hat{X}^{3}$ (145)