Topological Entanglement Entropy
Alexei Kitaev
1,2
and John Preskill
1
1
Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA
2
Microsoft Research, One Microsoft Way, Redmond, Washington 98052, USA
(Received 13 October 2005; published 24 March 2006)
We formulate a universal characterization of the many-particle quantum entanglement in the ground
state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the
plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state,
by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator
for the degrees of freedom in the interior. The von Neumann entropy of , a measure of the entanglement
of the interior and exterior variables, has the form SL , where the ellipsis represents
terms that vanish in the limit L !1. We show that is a universal constant characterizing a global
feature of the entanglement in the ground state. Using topological quantum field theory methods, we
derive a formula for in terms of properties of the superselection sectors of the medium.
DOI: 10.1103/PhysRevLett.96.110404 PACS numbers: 03.65.Ud, 03.67.Mn, 71.10.Pm, 73.43.Nq
In a quantum many-body system at zero temperature, a
quantum phase transition may occur as a parameter varies
in the Hamiltonian of the system. The two phases on either
side of a quantum critical point may be characterized by
different types of quantum order; the quantum correlations
among the microscopic degrees of freedom have qualita-
tively different properties in the two phases. Yet in some
cases, the phases cannot be distinguished by any local
order parameter.
For example, in two spatial dimensions a system with a
mass gap can exhibit topological order [1]. The quantum
entanglement in the ground state of a topologically ordered
medium has global properties with remarkable consequen-
ces. For one thing, the quasiparticle excitations of the
system (anyons) exhibit an exotic variant of indistinguish-
able particle statistics. Furthermore, in the infinite-volume
limit the ground-state degeneracy depends on the genus
(number of handles) of the closed surface on which the
system resides.
While it is clear that these unusual properties emerge
because the ground state is profoundly entangled, up until
now no firm connection has been established between
topological order and any quantitative measure of entan-
glement. In this Letter we provide such a connection by
relating topological order to von Neumann entropy, which
quantifies the entanglement of a bipartite pure state.
Specifically, we consider a disk in the plane, with a
smooth boundary of length L, large compared to the cor-
relation length. In the ground state, by tracing out all
degrees of freedom in the exterior of the disk, we obtain
a marginal density operator for the degrees of freedom in
the interior. The von Neumann entropy Str log
of this density operator, a measure of the entanglement of
the interior and exterior variables, has the form
SL ; (1)
where the ellipsis represents terms that vanish in the limit
L !1. The coefficient , arising from short wavelength
modes localized near the boundary, is nonuniversal and
ultraviolet divergent [2], but (where is nonnegative)
is a universal additive constant characterizing a global
feature of the entanglement in the ground state. We call
the topological entanglement entropy.
This universal quantity reflects topological properties of
the entanglement that survive at arbitrarily long distances,
and therefore can be studied using an effective field theory
that captures the far-infrared behavior of the medium,
namely, a topological quantum field theory (TQFT) that
describes the long-range Aharonov-Bohm interactions of
the medium’s massive quasiparticle excitations. We find
logD; (2)
where D 1 is the total quantum dimension of the me-
dium, given by
D
X
a
d
2
a
s
; (3)
here the sum is over all the superselection sectors of the
medium, and d
a
is the quantum dimension of a particle
with charge a.
Any abelian anyon has quantum dimension d 1;
therefore, for a model of Abelian anyons, D
2
is simply
the number of superselection sectors. Thus for a Laughlin
state [3] realized in a fractional quantum Hall system with
filling factor 1=q where q is an odd integer, we have
D
q
p
. For the toric code [4], which has four sectors, the
topological entropy is log2, as has already been noted
in [5].
However, non-Abelian anyons have quantum dimension
greater than one. The significance of d
a
(which need not be
a rational number) is that the dimension N
aaaa
of the
fusion vector space spanned by all the distinguishable ways
in which n anyons of type a can be glued together to yield a
trivial total charge grows asymptotically like the nth power
of d
a
. For example, in the SU2
k
Chern-Simons theory, we
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have
D
1
2
k 2
s
sin
k 2
: (4)
To justify computing the entropy using effective field
theory, we require that the boundary have no ‘‘sharp’’
features that might be sensitive to short-distance physics;
yet for a boundary drawn on a lattice, sharp corners are
unavoidable. Furthermore, there is an inherent ambiguity
in separating the term that scales with the length L from the
constant term. We can circumvent these difficulties by the
following construction. We now divide the plane into four
regions, all large compared to the correlation length,
labeled A; B; C; D as in Fig. 1(a). Let S
A
denote the von
Neumann entropy of the density operator
A
that is ob-
tained from the ground state by tracing out the degrees of
freedom outside region A, let S
AB
denote the von Neumann
entropy of the density operator
AB
obtained by tracing out
the degrees of freedom outside region AB A [ B, etc.
Then we define the topological entropy S
topo
as
S
topo
S
A
S
B
S
C
S
AB
S
BC
S
AC
S
ABC
: (5)
This linear combination of entropies has been strategically
chosen to ensure that the dependence on the length of the
boundaries of the regions cancels out. For example, the
term proportional to the length of the double intersection of
A and D appears in S
A
and S
ABC
with a sign, and in
S
AB
and S
AC
with a minus sign. Similarly, the double
intersection of A and B appears in S
A
and S
B
with a sign,
and in S
AC
and in S
BC
with a minus sign. [The obser-
vation that the ultraviolet divergent terms cancel in a
suitably constructed linear combination of entropies has
also been exploited in [6] and applied there to 1
1-dimensional systems.]
Assuming the behavior Eq. (1) in each term, we find
S
topo
. But the advantage of defining S
topo
using a
division into four regions is that we can then argue persua-
sively that S
topo
is a topological invariant (dependent only
on the topology of how the regions join and not on their
geometry) and a universal quantity (unchanged by smooth
deformations of the Hamiltonian unless a quantum critical
point is encountered).
To see that S
topo
is topologically invariant, first consider
deforming the boundary between two regions, far from any
triple point where three regions meet. Deforming the
boundary between C and D, say, has no effect on regions
A, B, and AB; therefore, if all regions are large compared to
the correlation length, we expect the changes in S
A
, S
B
, and
S
AB
to all be negligible. Thus the change in S
topo
can be
expressed as
S
topo
S
ABC
S
BC
S
AC
S
C
: (6)
We expect, though, that if the regions are large compared to
the correlation length, then appending region A to BC
should have a negligible effect on the change in the en-
tropy, since A is far away from where the deformation is
occurring; similarly, appending A to C should not affect the
change in the entropy. Thus both terms on the right-hand
side of Eq. (6) vanish, and S
topo
is unchanged. The same
reasoning applies to the deformation of any other boundary
between two regions.
Next consider deforming the position of a triple point,
such as the point where B, C, and D meet as in Fig. 1(b).
Again we may argue that S
A
is unchanged by the deforma-
tion. We recall that for a bipartite pure state (like the
ground state), the marginal density operators for both sub-
systems have the same nonzero eigenvalues and therefore
the same entropy; thus S
ABC
S
D
and S
BC
S
AD
. We see
that the change in S
topo
can be expressed as
S
topo
S
B
S
AB
S
C
S
AC
S
D
S
AD
: (7)
All three terms on the right-hand side of Eq. (7) vanish
because appending A does not affect the change in the
entropy. The same reasoning applies when any other triple
point moves; we conclude that S
topo
is unchanged by any
deformation of the geometry of the regions that preserves
their topology, as long as all regions remain large com-
pared to the correlation length.
Now, what happens to S
topo
as the Hamiltonian of the
system is smoothly deformed? We assume that the
Hamiltonian is a sum of local terms, and that the correla-
tion length remains finite during the deformation (and in
fact that the correlation length stays small compared to the
size of regions A; B; C; D). If the Hamiltonian changes
locally in a region far from any boundary, then this change
has a negligible effect on the ground state in the vicinity of
the boundary, and therefore does not affect S
topo
. If the
Hamiltonian changes locally close to a boundary, we can
exploit the topological invariance of S
topo
to first move the
boundary far away, then deform the Hamiltonian, and
finally return the boundary to its original location. Thus
we see that S
topo
is a universal quantity characteristic of a
particular kind of topological order, which remains invari-
FIG. 1 (color online). (a) The plane is divided into four re-
gions, labeled A; B; C; D, that meet at double and triple inter-
sections. (b) Moving the triple intersection where B; C; D meet
deforms the regions as shown.
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ant if no quantum critical point is encountered as the
Hamiltonian varies.
To facilitate the computation of S
topo
, it is convenient to
imagine joining together the planar medium we wish to
study with its time-reversal conjugate. We glue together the
medium and its conjugate at spatial infinity, and then attach
‘‘wormholes’’ that connect the two planes at the positions
of the four triple intersections, as indicated in Fig. 2. The
resulting closed surface has the topology of a sphere with
four handles. If an isolated planar medium is punctured,
then massless chiral modes propagate around the edge of
the puncture, but the edge states of the medium and its
conjugate have opposite chirality, so the edge states ac-
quire masses when the two surfaces are coupled; therefore,
the wormholes can be created adiabatically without de-
stroying the mass gap. No anyons are produced during this
adiabatic process, so that the mouth of each wormhole
carries trivial anyonic charge.
The boundaries that separate regions in the plane and in
its double can be joined through the wormholes as in Fig. 2;
then each region of the doubled surface has the topology of
a sphere with three punctures, and each union of two
adjacent regions becomes a sphere with four punctures.
The topological entanglement entropy of the medium and
its conjugate are both equal to S
topo
, so that the topological
entanglement entropy of the doubled surface is twice S
topo
.
The entropy of a region depends only on its topology, so for
the doubled surface we have
2S
topo
4S
3
3S
4
; (8)
where S
3
denotes the entropy for the sphere with three
punctures and S
4
denotes the entropy for the sphere with
four punctures.
The quantities S
3
and S
4
can be computed using the
appropriate effective field theory, a TQFT [7]. We use the
property that no charge is detected by an anyon that winds
around the throat of a wormhole. A cycle that encloses a
puncture in the double of (say) region A is complementary
to a cycle that winds around the wormhole throat; it follows
that the puncture carries charge a with probability
p
a
jS
a
1
j
2
d
2
a
=D
2
; (9)
where S
a
b
is the topological S matrix of the TQFT, and 1
denotes the trivial charge. To find the joint probability
distribution p
abc
governing the charges a; b; c on the punc-
tures of the sphere with three punctures, we may use
standard TQFT methods to compute the probability
p
ab!
c
that when charges a and b fuse the total charge is
c. The result is
p
ab!
c
N
abc
d
c
=d
a
d
b
; (10)
where N
abc
is the dimension of the fusion vector space
spanned by all the distinguishable ways in which charges
a, b, and c can fuse to yield trivial total charge; it follows
that
p
abc
p
a
p
b
p
ab!
c
N
abc
d
a
d
b
d
c
=D
4
: (11)
Evaluating the entropy in the basis in which each puncture
has a definite charge, and summing over all the distinguish-
able fusion states that occur for specified values of the
charges, we find
S
3
X
a;b;c
X
N
abc
1
p
abc
N
abc
log
p
abc
N
abc
4 logD
X
a;b;c
p
abc
logd
a
d
b
d
c
4 logD 3
X
a
p
a
logd
a
: (12)
For the sphere with four punctures, a similar calculation
yields
p
abcd
p
a
p
b
p
c
p
abc!
d
N
abcd
d
a
d
b
d
c
d
d
=D
6
(13)
and
S
4
6 logD 4
X
a
p
a
logd
a
: (14)
Plugging into Eq. (8), we find
S
topo
2S
3
3
2
S
4
logD : (15)
Equation (15) is our main result. Note that it follows if
we use Eq. (1) to evaluate the entropy of each region, since
appears 4 times in the expression for S
topo
with a negative
FIG. 2 (color online). The planar medium is glued at spatial
infinity to its time-reversal conjugate, and wormholes are at-
tached that connect the two conjugate media at the locations of
the triple intersections, creating a sphere with four handles. Each
region, together with its image, becomes a sphere with three
punctures, and each union of two regions, together with its
image, becomes a sphere with four punctures. The punctures
carry charges labeled a; b; c; d. Anyons that wind around a cycle
enclosing a wormhole throat as shown detect a trivial charge.
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sign and 3 times with a positive sign. We also observe that
S
topo
actually depends on the topology of the regions
A; B; C. For example, consider the arrangement shown in
Fig. 3, in which B and AC both have two connected
components, and ABC is not simply connected. Since
regions B, AC, and ABC each have boundaries with two
components, now appears 6 times with a negative sign
and 4 times with a positive sign, so that S
topo
2.
Using a different approach, we can formulate a simpler
but more heuristic derivation of the formula for . First we
write the marginal density operator for the disk as
e
H
. This is just a definition of H and has no other content
in itself; furthermore, the parameter T
1
is arbi-
trary—so we are free to choose it to be small compared
to the bulk energy gap of the two-dimensional me-
dium. Now we make a natural but nontrivial assumption:
that H can be regarded as the Hamiltonian of a 1
1-dimensional conformal field theory (CFT). This CFT
ignores short-distance properties of the bulk medium, and
therefore will not account correctly for the term in the
entropy proportional to L, but it should reproduce correctly
the universal constant term.
To compute the entropy for the case of a disk that
contains an anyon with charge a (far from the boundary),
we evaluate the partition function Z
a
tr
a
e
H
for the
associated conformal block of the CFT. Z
a
can be ex-
pressed as a path integral on a torus of length in the
Euclidean time direction and length L in the spatial direc-
tion, in the presence of a Wilson loop carrying anyon
charge a that winds through the interior of the torus in
the timelike direction. After a modular transformation, we
have
Z
a
X
b
S
b
a
~
Z
b
; (16)
where
~
Z
b
is the partition function for the b block on a torus
of length L in the Euclidean time direction and length in
the spatial direction, and S is the modular S matrix of the
CFT, which matches the topological S matrix of the anyon
model. In the limit L !1, the sum is dominated by the
trivial block
~
Z
1
, and we find
logZ
a
logS
1
a
~
Z
1
logS
1
a
12
c
cL=; (17)
where c and
c are the holomorphic and antiholomorphic
central charges of the CFT, and S
1
a
d
a
=D is a topologi-
cal S matrix element. Applying the thermodynamic iden-
tity S @F=@T (where F T logZ is the free
energy), we then find
S
@
@T
T logZL logD=d
a
: (18)
Thus when a is the trivial charge and d
a
1, we recover
the result of Eqs. (1) and (2). While this derivation is not on
so firm a footing as the derivation leading to Eq. (15), it is
more transparent and it generalizes readily to the case
where the disk contains an anyon.
We have found an intriguing connection between en-
tanglement entropy and topological order in two dimen-
sions. We note that there are close mathematical ties be-
tween the topological entanglement entropy and the
1 1-dimensional boundary entropy discussed in [8],
and we expect that further insights can be derived from
studying higher-dimensional analogs of S
topo
. We also
hope that our results can provide guidance for the impor-
tant task of constructing explicit microscopic models that
realize topological order.
Results similar to ours have been obtained indepen-
dently by Levin and Wen [9].
We thank Anton Kapustin for discussions. This work has
been supported in part by: the Department of Energy under
Grant No. DE-FG03-92-ER40701, the National Science
Foundation under Grant No. PHY-0456720, the Army
Research Office under Grants No. W911NF-04-1-0236
and No. W911NF-05-1-0294, and the Caltech MURI
Center for Quantum Networks under ARO Grant
No. DAAD19-00-1-0374.
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110405 (2006).
FIG. 3 (color online). If the regions A; B; C have the topology
shown, then S
topo
2.
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