Entanglement in Gauge Theories,
Matrices and Space-Time
Sandip Trivedi
Tata Institute of Fundamental Research,
Mumbai, India.

Outline• Introduction to Entanglement•

Gauge Theories: a) Definitionb) Operational Meaning• Matrix Theory and Space-Time
Entanglement: a) Definition b) Proposal• Conclusions

The Entanglement Frontier• We are in the midst of exploring a newfrontier of science today.
• It does Not pertain to:i) The very small, deep within the atomii) The very large, way beyond our galaxyiii) Or the very many: Atoms, Molecules or
Neurons.

The Entanglement FrontierInstead, it pertains to our growing understanding of
Quantum Mechanics. Its true meaning andsignificance.
Since quantum mechanics provides the bedrock for
our understanding of science, this new frontier
promises to have important implications for manybranches of sciences:
Physics, Chemistry, Computer Science, Other
aspects of Engineering, Mathematics.

The Entanglement Frontier
Quantum Mechanics is Strange.
Much of this strangeness has to do with aproperty called Entanglement which has noclassical analogue.
This new frontier is therefore often referred toas the ``Entanglement Frontier’’ (Preskill) Of course we all know that the underlying description in
Quantum Mechanics is in terms of a wave function(probability amplitude).

 Different paths can interfere with each other as they

evolve, like the different ripples of a wave interfere as itpropagates.

Probability to findthe particle atpoint xWave function

Einstein,
Podolsky, Rosen

 Actually there is something else which is very

strange about Quantum Mechanics. It was Einstein
who first put his finger on this “weirdness” and it
caused him great disquiet.

 A kind of ``spooky action at a distance”.
 The wave function which describes a system is nonlocal, even though all consequences are local.

 It is this additional weirdness which Entanglement
refers to.

Quantum systems have non-local
correlations. These have no ClassicalAnalogue.
Even though all dynamics is local.
These non-local correlations arecalled entanglement
A quantitative measure is provided
by the Entanglement Entropy.

Entangled SpinsEntangled Spins
The first electron does not have a definite spin inany direction.
Regardless, once one find out its spin along any
direction, x,y,z, by doing a measurement one
knows the spin of the second electron as well.
This kind of Quantum Correlation is called
Entanglement.

Of course correlations can arise inclassical systems as well.
Looking at
one sock you
can tell thecolour of theother one.

Entangled Spins

But the quantum correlations in the two electron
system are different.
The first electron does not have a definite spin in any
direction.
Regardless, once one find out its spin along any
direction, by doing a measurement, one knows thespin of the second electron with certainty.
Entanglement characterises this kind of certainty
present in the midst of the inherent uncertainty ofquantum mechanics.

Einstein,
Podolsky,
Rosen• All physical effects are local. No signal
travels faster than light. Neverthelessour description of a state is inherently
non-local.
• Lead Einstein to suggest that there
was a more complete description
which was classical, local and withadditional hidden variables•John Bell - brilliant paper in 1964.

•

Pointed out that Entanglement would leadto experimental consequences which could
be used to distinguish quantum theory fromany classical theory with extra local hidden
variable.

The differences are summarised by
Bell’s inequalities.
No classical system can violate them.
The essential point is that probabilities
cannot be negative.
Quantum systems do violate themfor entangled states.

Alain Aspect
Experiments have brilliantly confirmedthat Quantum Mechanics is correct
and ruled out the hidden variable
theories.

Quantifying Entanglement

The two electrons in the state above
are entangled.
The electrons in the state :are also correlated. But they are notentangled. This state does not leadto violations of Bell’s inequalities.

How do we quantify the quantum
nature of these correlations?
Entanglement Entropy has emerged
as an important measure.

Operator acting
on 1st electron;
Called Density
MatrixSum over all possible
states for 2nd electron

Entanglement Entropy

For the state:

Entanglement Entropy

Essentially the two states |+>, |-> areequally probable with probability p=1/2

Entanglement Entropy is the von
Neumann Entropy of the Density Matrix.

• For the state• The two electrons are stillcorrelated. But not entangled.
• This state will not lead toviolations of Bell’s inequalities.

• More generally, it is necessary thatthe state has a non-vanishingentanglement entropy to get aviolation of Bell’s inequalities.
• This makes entanglement entropy
a good measure of ``quantum
weirdness”

Entanglement Entropy in Gauge
Theories•

Three of the four fundamental forces of nature
are described by gauge theories.

•

Gravity is also a kind of gauge theory.

• Gauge theories often emerge at low energies
in condensed matter systems.
•

Thus it is important to understand how to
precisely define entanglement entropy in
Gauge theories.

Four Fundamental Forces
StrongInteractions

Weak
Interactions

Gravity

ElectroMagnetism

Entanglement in Gauge Theories• Turns out to be subtle to define.
• In a spin system the degrees of
freedom are local.
• This makes it straightforward to
define a measure of the non-local
quantum correlations.

Entanglement in Gauge Theories
In contrast, for a gauge theory the degrees of
freedom are non-local.
Electric and Magnetic lines cannot abruptly
end at the boundary of the region of interest,and must satisfy continuity conditions.
This means the flux cannot be varied inside
the region and outside in an independent
manner.

Gauss’ Law:

(Electromagnetism without charge)

Lines of Force cross from on side to another.
The inside and outside are not independentIn the quantum theory (of pure
electromagnetism without any charges)
Gauss’ Law:
Ensures that the physical states aregauge invariant.
(gauge).

Note that this is different from the twoelectron system.

The particular state with S=0 was
correlated. But in general the state of one
electron could be changed independentof the other.

For example:

The full Hilbert space of the twoelectrons is therefore a tensor
product

The tensor product nature of H thenmakes it straightforward to define theEntanglement Entropy for a spin system.
In contrast, since the electric andmagnetic fluxes cannot be varied
independently inside and outside theHilbert Space of Gauge Invariant states ina Gauge Theory does not admit a tensor
product decomposition.
This is the essential complication.

Ronak Soni Sudip
Ghosh

Upamanyu
MoitraJHEP 1509, 069 (2015), JHEP 1601, 136 (2016),
JHEP 1702,101 (2017), JHEP 08 (2019) 059

Important Contributions
Casini, Huerta, Rosabal (PRD 89, 2014)
Donnelley (PRD 85, 2012, Class Quant.
Grav. 31, 2014,…)
Casini, …
.
.

``On The Entanglement Entropy For
Gauge Theories”,
Sudip Ghosh, Ronak Soni and SPTJHEP 1509, 069 (2015)``Aspects of Entanglement Entropy For
Gauge Theories”,
Ronak Soni, SPT, JHEP 1601, 136 (2016)``Entanglement Entropy in U(1) Gauge”
Theory, Ronak Soni, SPT, JHEP 1702,101(2017)

``On The Entanglement Entropy For
Gauge Theories”,
Sudip Ghosh, Ronak Soni and SPTJHEP 1509, 069 (2015)``Aspects of Entanglement Entropy For
Gauge Theories”,
Ronak Soni, SPT, JHEP 1601, 136 (2016)``Entanglement Entropy in U(1) Gauge”
Theory, Ronak Soni, SPT, JHEP 1702,101(2017)

The approach we took was to embed
the physical states in an Extended
Hilbert space which include states that
do not satisfy the Gauss law
constraint.
These additional states are not Gauge
Invariant.
The extended Hilbert space now has atensor product decomposition giving
rise to a Entanglement Entropy.

The result for the Entanglement
Entropy is gauge invariant.
Method works for gauge theories ingeneral, including Abelian and NonAbelian ones, with and without
matter.
Agrees with the replica trick.

The algebra of gauge invariant operators
has a non-trivial center.
This means the Hilbert space is a sum
over tensor products.
This structure generalizes for the nonAbelian case too.

(Casini, Huerta, Rosabal )For a Gauge Theory:A

B

All theentanglement
cannot be
distilled

Soni, ST,
Acoleyen et.al.,

N entangled Bel
pairs

Entanglement Distillation :AB
2N unentangled
qubitsThe von Neumann entropy for the density
matrix in gauge theories has a distillable
and non-distillable piece.
The distillable amount tells you how many
Bell pairs you can extract.
The non-distillable amount is alsointeresting. In some cases ( e.g.Toric
Code) it is a measure of `topological”
properties of the state.
(Topological Entanglememnt Entropy)

Matrix Theory Entanglement+ fermions: 9 NXN matrices: degrees of freedom.

Matrix Theory
Gauge Theory: Gauge Field
Action Invariant under
Colour degrees of freedom

Roughly speaking, we want to ask how
entangled are some of the degrees of
freedom of the matrices with others?

Gauge Symmetry makes this question
complicated!

m
m

Keep
block for all matrices
and ask what is the entanglement entropyof the resulting density matrix?

NXN Matrix
U(N) gauge transformation will mix theblue block with the rest.
So what is the gauge invariant way to ask
this question?
This is the question we will address.

Some References:
1) Mazenc and Ranard ( hep-th/1910.07449 )2) S. Das, A. Kashyap, G. Mandal, ST, (hepth/2004.00613)
3) S. Das, S. Liu, A. Kashyap, G. Mandal, ST,(hep-th/2011.13857)
4) H. Hampapura, J. Harper, A. Lawrence, (hep-th/
2012.15683)

Context:
Why is this kind of Matrix Model interesting?
The AdS/CFT correspondence and related
developments in the past two decades haveshown that such models (and their higher
dimensional analogues) describe spacetimes
which are smooth in appropriate limits.

Juan Maldacena, …

A well known example of AdS/CFT:The N=4 supersymmetric Yang Mills Theory(CFT) in 3+1 dimensions is equivalent toString Theory in 5 Dimensional Anti-de Sitter
Spacetime(In the matrix theory, the boundary theory
only lives in time, not along any spatial
directions, so this is a higher dimensional
analogue).

CFT (Conformal Field Theory)

How did this correspondence come about?
• D-branes, which are soliton- like objects instring theory, give rise to 10 dimensional
solutions of (super) gravity.
• Their low energy excitations can be
described in two ways, as a gauge theory, or
as excitations in a gravity theory.
• The equivalence between these two
descriptions gives rise to thecorrespondence.

For D0 branes (Near Horizon limit)

Coulomb Branch

D0 branes in Bulk (in Coulomb branch) One to
one correspondence with the vacuua in Matrix
Model.

Coulomb Branch in Matrix Theory
Moduli space of vacua (Coulomb branch): All 9 matrices commute.

brane etc

: location of first D0

Matrix Theory Entanglement
This definition is actually motivated fromthe bulk (the gravity side).
Suppose we are interested in the bulk
regionand want to ascribe an
entropy to it.
Look at corresponding constraint inboundaryEntanglement Entropy in Matrix Theory
Let us go to a gauge whereis diagonal

Arrange eigenvalues in descending order

Let’s say first m eigenvalues meet thecondition

m
m

Keep
block for all matrices andcompute the density matrix for these
degrees of freedom.
Note: in general the other matrices,
will not be diagonal in this gauge.

But the Hilbert space (in this sector)
still admits a tensor productdecomposition for all matrix degrees
of freedom.
By tracing over the remaining matrix
elements we can then obtain adensity matrix(This is version 1 of the proposal)

Entanglement in the m th sector:

Full entanglement given by a sum overall sectors
Target spaceentanglement
Include possibility that no eigenvalues
lies in region of interestmmVersion 1

Keepblock for all matrices
and compute its entanglement entropy.

m N-m
m

A

B

N-m

C

D

Version 2
N-m

N-m
Keep blocks A, B, C and remove D –which is of size ( N-m)X(N-m)- for all
matrices.

For either version we now have to sum
over the various sectors: m = 0,1, …, N
Full Target Space Entanglement is thesum over the contributions from thevarious sectors.

Matrix Theory Entanglement
We have used a particular gauge todefine the entropy.
But in fact it can be given a gauge
invariant meaning using suitableprojection operators and the resulting
gauge invariant subalgebra.
(There are two versions of our definition
as we will see shortly)Matrix Entanglement
Definition can be generalised for any
constraint involving the matrices
As operators
commute, any constraintof the form
can be dealt with bydiagonalising the constraint and then obtaining
the entanglement sector by sector.
E.g.

Target Space Entanglement
This definition is actually motivated fromthe bulk (the gravity side).
Suppose we are interested in the bulk
regionand want to ascribe an
entropy to it.
Look at corresponding target spaceconstraint in boundary

In the bulk it is difficult to make theidea of entanglement precise.
But we do have a precise definitionof a related quantity in the Matrix
theory.

Physical Picture:

4 D0 branes to meet constraint.
(m=4 sector)

Version 1:

4 D0 branes meet constraint. Keep
all degrees of freedom associatedwith them denoted by blue strings.

Version 2:

In addition also keep black strings running
between the 4 D0 branes and the rest.

On the bulk side the entanglement
corresponds to strings stretching
between the D branes.
On the boundary side it corresponds
to ``colour” degrees of freedom of.
In fact there are two versions of ourdefinition as the bulk pictures show.

Version 1
m values

N-m values

Wave function

m th sector density matrix
Complement of the region of interest,

It is tempting to speculate that theanswer in the matrix theory willagree with the bulk area (in Planck
units)

Connection with bulk area is a proposal.
Remains to be checked.
Impressive advances in numerical work on
D0 brane system make one hopeful…

Boundary Quantum Mechanics+ fermions: NXN matrices: degrees of freedom.

At finite temperature: Black Hole

Agrees with Numerics:
Berkowiz, Rinaldi, Hanada, Ishiki,
Shimasaki, Vranas, 2016

(slide taken from J. Maldacena, ICTS lecture
2018)Boundary Quantum Mechanics
Characterised by one energy scale:Ryu Takayanagi Formula
For the higher dimensional cases, where
the boundary theory lives along somespatial directions (eg along the 3 spatial
directions for 3+1 dimensional CFT):One can ask how entangled are thedegrees of freedom in some region,
along these spatial directions, with therest?

Entanglement of region B on boundary with A
is calculated from the Area of the Minimal
surface.

Remarkable Result of Ryu andTakayanagi!
This is ``base space” entanglement.

We have been interested here inentanglement along the Target Space, inwhich the boundary theory does notextend.
The D0 branes (or Matrix Mechanics)
only lives in time and no spatial
directions. So the 9 matrices,
are target space directions.
Connection to the RT formula remains to
be understood.

Summary• Entanglement is a key tounderstanding the strangeness inquantum mechanics, and unlocking
the full potential of quantum
techniques and technology.
• We discussed how to define andmake precise the meaning ofentanglement in gauge theories

Summary• And also for Matrix models of
the kind which are promising in
our attempts to understand how
space and time arise.
Much excitement lies ahead!

Thank You!

Banyan Tree In TIFR Mumbai

Thank you!

Entanglement Distillation :AB
2N unentangled
qubits

To finally arrive at the situation:

A

B

N entangled Bellpairs

Only this term can be extracted

Target Space Entanglement
Often for a field theory in
One considers a subregion of
And asks how entangled are the degrees
of freedom inside with those outside.

Spatialregion in

Ryu Takayanagi Formula
For the higher dimensional cases, where
the boundary theory lives along somespatial directions (eg along the 3 spatial
directions for 3+1 dimensional CFT):One can ask how entangled are thedegrees of freedom in some region
along these spatial directions with the
rest?

A

How entangled is A with.

?

We will be interested here in theentanglement along the target
space.
E.g AdS_5 X S^5 has 9 spatial
directions. The field theory lives in 3of them. The remaining 6 directions
are target space directions.
(Experts: Adjoint scalar fields takevalues along the target space
directions)

The matrix model we started with has
no spatial directions. Its gravity dual isa solution in IIA string theory in 9+1
dimensions.
All of the 9 spatial directions then are
target space directions in this case,with no base space directions.
This makes the matrix model aconvenient system for our purposes.

For target space entanglement
Define projector:

For target space constraint

Version 1:
Project all operators by acting with thisprojector.
Then take a trace to obtain gauge
invariant operators.

Version 1:
This gives rise to a sub algebra.
Entangelment entropy is associated withthis sub algebra.
Similarly for version 2.
Subalgebra defined With an appropriately
chosen projector.

Bulk Entangelment : Proposal
Bulk Entangelment Entropy for region
Is given by Target space entanglement
associated with constraint
And this bulk entanglement saturates
the Beckenstein-Hawking bound (for ageneral surface).

Proposal for Bulk Entanglement
And
Target spaceentanglement

Boundary witharea
(In one of the two versions of our
proposal for)