of 51
JHEP11(2024)103
Published for SISSA by
Springer
Received:
August 19, 2024
Accepted:
October 2, 2024
Published:
November 19, 2024
Musings on SVD and pseudo entanglement entropies
Paweł Caputa
,
a,b
Souradeep Purkayastha
,
a
Abhigyan Saha
a
and Piotr Sułkowski
a,c
a
Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland
b
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
c
Walter Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, U.S.A.
E-mail:
pcaputa@fuw.edu.pl
, spurkayastha@fuw.edu.pl
, a.saha@uw.edu.pl
,
psulkows@fuw.edu.pl
Abstract:
Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy
that involve post-selection. In this work we analyze their properties as measures on the
spaces of quantum states and argue that their excess provides useful characterization of
a difference between two (i.e. pre-selected and post-selected) states, which shares certain
features and in certain cases can be identified as a metric. In particular, when applied to link
complement states that are associated to topological links via Chern-Simons theory, these
generalized entropies and their excess provide a novel quantification of a difference between
corresponding links. We discuss the dependence of such entropy measures on the level of
Chern-Simons theory and determine their asymptotic values for certain link states. We find
that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots.
We also consider properties of entropy measures for simpler quantum mechanical systems,
such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.
Keywords:
Chern-Simons Theories, Quantum Groups, Topological States of Matter
ArXiv ePrint:
2408.06791
Open Access
,
©
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP11(2024)103
JHEP11(2024)103
Contents
1 Introduction
1
2 Review of entropy measures
4
2.1 Entanglement entropy, pseudo-entropy and SVD entropy
4
2.2 Entropy measures for two-component states
6
3 Chern-Simons theory and link complement states
7
4 Quantum mechanical examples
11
4.1
SU(2)
coherent states
12
4.2
SU(1
,
1)
coherent states
16
4.3 Tripartite GHZ and W states
18
5 Entropy measures for link complement states
22
5.1 Two-component links in U(1) Chern-Simons theory
22
5.2
K
#2
2
1
links in non-abelian Chern-Simons theory
25
5.3
(
p,pn
)
torus links in non-abelian Chern-Simons theory
27
5.4 Example with Borromean links
31
6 Large
k
asymptotics for
T(
p, pn
)
links
32
7 Chirality and imaginary part
35
8 Conclusions and future directions
37
A Two-qubit states
39
B Pseudo-metric for two-component U(1) link states
40
C Details on large
k
calculations
41
C.1 Asymptotic eigenvalue numerics
41
C.2
T(2
,
4)
entanglement entropy integral form
43
1 Introduction
In recent years intriguing connections between high energy physics and quantum information
theory have been revealed. One link between these research areas is provided by the notion
of entanglement entropy and its generalizations (see e.g. review [
1
]). Apart from providing
means to describe complex systems, other motivations to study various incarnations of entropy
include their geometric interpretation via AdS/CFT correspondence [
2
5
], the potential to
characterize topological properties of various systems [
6
,
7
] and topological field theories in
particular, the capability to describe the process of post-selection [
8
,
9
], applications of these
– 1 –
JHEP11(2024)103
ideas in condensed matter physics [
10
], etc. Generalizations of the entanglement entropy
S
φ
E
(of a state
|
φ
) of our primary interest in this work are pseudo-entropy denoted
S
φ
|
ψ
P
[
11
], and
SVD entropy denoted
S
φ
|
ψ
SVD
[
12
]. Recall that entanglement entropy characterizes entanglement
between two subsets of a Hilbert space; they are often taken to be associated to two subregions
of the spatial domain on which a system under consideration is defined. Pseudo-entropy,
which arises naturally from the AdS/CFT perspective, is a generalization of the entanglement
entropy that involves post-selection and depends on two states, the initial one
|
φ
and the final
(post-selected) one
|
ψ
, as indicated in the notation above. Pseudo-entropy takes complex
values and can be also larger than the logarithm of the dimension of the Hilbert space, which
obscures its quantum-information interpretation. To remedy these issues, the SVD entropy
has been introduced in [
12
]. SVD entropy also depends on the initial and post-selected state,
however it takes real values, which in addition do not exceed the logarithm of the dimension
of the Hilbert space. Moreover, it admits an elegant operational meaning as a number of
Bell pairs in the intermediate states between
|
φ
and
|
ψ
.
In this work we argue that pseudo-entropy, SVD entropy, and their excess are useful
in quantifying a difference between (pre-selected and post-selected) quantum states. In
particular, for link states which are associated to topological links via Chern-Simons theory —
and which are of our main interest — these generalized entropies provide a novel quantification
of a difference between corresponding links. We analyze the dependence of such measures on
the level of Chern-Simons theory, and in particular determine their asymptotic values for large
level. Note that these results (as well as classes of links under our consideration) extend and
generalize earlier analysis of the entanglement and pseudo-entropy for link states in [
13
15
].
Furthermore, as a warm up, we also study these concepts for simpler quantum mechanical
systems, involving generalized
SU
(2)
and
SU
(1
,
1)
coherent states, as well as tripartite GHZ
and W states. The systems that we analyze are characterized by increasing dimension
of Hilbert spaces and increasing number of components; for link states the dimension of
the Hilbert space is determined by the level of Chern-Simons theory, while the number of
components is equal to the number of components of a link. These quantities may take
arbitrary values; in particular, we analyze the limit of infinite level, which is also of interest
in other contexts, such as the volume conjecture [
15
17
].
While we provide precise definitions of pseudo-entropy
S
φ
|
ψ
P
and SVD entropy
S
φ
|
ψ
SVD
in
section
2 , we note here that their excess is defined respectively as
S
φ
|
ψ
P
=
Re
S
φ
|
ψ
P

S
φ
E
+
S
ψ
E
2
,
S
φ
|
ψ
SVD
=
S
φ
|
ψ
SVD
S
φ
E
+
S
ψ
E
2
,
(1.1)
where
S
φ
E
is the entanglement entropy of a state
|
φ
. These excess functions have interesting
properties. For example, it was conjectured in [
19
,
20
] that the pseudo-entropy excess is
non-positive or positive if the two states are respectively in the same or different quantum
phases. In this work we analyze link states and other states from this perspective and argue
that they can be associated to the same or different phases, depending on particular choice
of parameters characterizing a given system we consider.
Furthermore, our main observation is that the entropy excess (
1.1) satisfies certain —
and in some cases all, depending on features of a given quantum system — axioms of the
– 2 –
JHEP11(2024)103
metric. In quantum information theory certain metrics have been introduced before (however
some of them only for pure states), which are referred to as Fisher metric, Fubini-Study
metric, Bures metric or Helstrom metric, and which provide a notion of distance on a space of
quantum states. We analyze for which systems and under which conditions the absolute value
of an excess function, i.e. either
|
S
φ
|
ψ
P
|
or
|
S
φ
|
ψ
SVD
|
, has analogous interpretation and thus
provides a proper notion of a distance between quantum states. The absolute value of either
pseudo-entropy’s or SVD entropy’s excess is clearly non-negative, equal to zero for
|
φ
=
|
ψ
,
and symmetric (with respect to the interchange of
|
φ
and
|
ψ
), which are a subset of the
axioms of a metric. In what follows we analyze for which systems of our interest the triangle
inequality holds (the space of states is called semi-metric if this inequality is violated), and
when the separation axiom (meaning that the distance cannot vanish for different states)
holds (the distance function is referred to as pseudo-metric when this axiom is violated).
A prototype example of a metric structure that we find is SVD entropy excess for
two-component link states in U(1) Chern-Simons theory. Consider two two-component links
with linking numbers
l
1
and
l
2
respectively. We show that for the corresponding pre-selected
and post-selected link states, in U(1) theory at level
k
, the SVD entropy takes the form
S
SVD
= log

k
gcd(
k,l
1
l
2
)

(1.2)
whenever the greatest common divisor (commonly denoted by
gcd
)
gcd
(
k,l
1
l
2
)
̸
=
np
2
for
n,p
N
(when
gcd
(
k,l
1
l
2
) =
np
2
, the expression is more complicated). It then follows that
the absolute value of the SVD entropy excess takes the form
|
S
SVD
|
=
1
2
log
(gcd(
k,l
1
l
2
))
2
gcd(
k,l
1
)
·
gcd(
k,l
2
)
!
.
(1.3)
We show that for this expression (and also more generally, for
gcd
(
k,l
1
l
2
) =
np
2
) the triangle
inequality holds and thus
|
S
SVD
|
provides a pseudo-metric on the space of two-component
links (it is a pseudo-metric, as in U(1) theory the entropy measures depend only on linking
numbers, so the distance between two different links with the same linking number vanishes).
Motivated by this example, we discuss for what other systems, including link states in
Chern-Simons theory with non-abelian gauge group as well as quantum mechanical examples,
the metric interpretation holds — this turns out to be the case for some specific ranges of
parameters specifying quantum states in a given system. We stress that whenever the SVD
entropy excess can be interpreted as a metric on the space of link states, it also provides a
measure on the space of links that may be of interest from the knot theory perspective.
Apart from the metric interpretation, we also identify other properties of entropy measures.
On one hand, we find classes of links states for which the SVD entropy take values between
the entanglement entropies of pre-selected and post-selected states, or exceeds the value of
one of these entanglement entropies. While the former case can be explained in terms of
Bell pairs exchanged between the two states under consideration, the latter phenomenon
is more surprising. Furthermore, we find that the imaginary part of the pseudo-entropy,
whose quantum information interpretation has been not so clear, detects chirality of link
states associated to topological links.
– 3 –
JHEP11(2024)103
The paper is organised as follows. In section
2 we introduce entropy measures of our
interest: entanglement entropy, pseudo-entropy and SVD entropy, and their excess. In
section
3 we review basics of Chern-Simons theory, knot invariants, and introduce the link
states that are of our main interest in what follows. In section
4 we analyze quantum
mechanical examples involving generalized
SU
(2)
and
SU
(1
,
1)
coherent states, as well as
tripartite GHZ and W states. In section
5 we determine entropy measures and discuss their
properties for various classes of link complement states: two-component links in U(1) Chern-
Simons theory, connected sums
K
#2
2
1
and
(
p,pn
)
torus links in non-abelian Chern-Simons
theory, and other examples involving in particular Borromean links. In section
6 we determine
asymptotic values of entropy measures for large
k
for various link states, and in section
7
we show that imaginary part of pseudo-entropy detects chirality of link states.
2 Review of entropy measures
In this section we introduce von Neumann entanglement entropy, pseudo-entropy [
11
], SVD
entropy [
12
], and the entropy excess. In the following sections we will employ these quantities
to characterize entanglement structure of quantum states in various models.
2.1 Entanglement entropy, pseudo-entropy and SVD entropy
To set up the stage, consider a Hilbert space
H
that admits a decomposition into two parts,
1
A
and its complement
B
H
=
H
A
⊗H
B
.
(2.1)
We denote dimensions of
H
A
and
H
B
by
d
A
and
d
B
respectively. In what follows we study
both finite and infinite dimensional spaces. Next, we pick a pure quantum state
|
ψ
in
H
and
define the (normalized) reduced density matrix of
A
by tracing over
B
ρ
A
=
Tr
B
(
ρ
)
,
Tr
(
ρ
A
) = 1
.
(2.2)
To characterize the entanglement between
A
and
B
, we will study von Neumann entanglement
entropy (denoted by the subscript E) of
ρ
A
S
ψ
E
=
S
(
ρ
A
)
≡−
Tr
(
ρ
A
log
ρ
A
) =
X
i
p
i
log
p
i
,
(2.3)
where
p
i
are eigenvalues of
ρ
A
. From the Schmidt decomposition of the pure state
|
ψ
in the
Hilbert spaces
H
A
⊗H
B
we have
S
(
ρ
A
) =
S
(
ρ
B
)
where
ρ
B
=
Tr
A
(
ρ
)
.
Next, we introduce two interesting generalisations of entanglement entropy. The first
one is the pseudo-entropy [
11
]. Its definition requires two pure states
|
φ
and
|
ψ
in
H
(2.1)
satisfying
φ
|
ψ
⟩ ̸
= 0
. In what follows we sometimes refer to them as the reference or pre-
selected state and the target or post-selected state respectively. Then, we define a transition
matrix for these two states
τ
φ
|
ψ
=
|
φ
⟩⟨
ψ
|
ψ
|
φ
.
(2.4)
1
A generalisation to multiple parts is analogous.
– 4 –
JHEP11(2024)103
Such objects are very natural not only in quantum information but also in physical studies of
post-selection or weak values and quantum measurements [
8
]. By analogy with the reduced
density matrix, we have the reduced transition matrix for
A
τ
φ
|
ψ
A
=
Tr
B
(
τ
φ
|
ψ
)
.
(2.5)
Since these transition matrices are not Hermitian, they will generally have complex eigenvalues
(see [
11
] for some classification), but one can still define a complex extension of the von
Neumann entropy, referred to as the pseudo-entropy (that we denote by subscript P) [
11
]
S
φ
|
ψ
P
=
Tr
A
(
τ
φ
|
ψ
A
log
τ
φ
|
ψ
A
)
.
(2.6)
Pseudo-entropy has several interesting properties and we only mention a few. Firstly, for
the specific case of
|
φ
=
|
ψ
,
(2.6)
reduces to the entanglement entropy (
2.3). It vanishes if
the states are product states
|
φ
1
A
|
φ
2
B
. Swapping the two states is equivalent to complex
conjugation of the pseudo-entropy. Its real part has various interesting properties, including
operational meaning for some classes of states and playing the role of an order parameter
for different quantum phases [
19
,
20
]. While imaginary part remains mysterious, in this
work we reveal some of its properties. Similar to von Neumann entropy, pseudo-entropy is
symmetric under exchanging
A
with its complement
B
. Nevertheless, it is still not clear
and very interesting open problem to determine conditions for obeying (perhaps saturation)
or violation of the famous entropy inequalities. For example violations of sub-additivity
were discussed in [
11
]. Further important developments on pseudo-entropy can be found
e.g. in [
21
32
].
Another generalization of entanglement entropy that involves post-selection is Singular
Value Decomposition entropy (SVD entropy for short) recently defined in [
12
]. To define it
we introduce analogous quantities as for the pseudo-entropy, up to the reduced transition
matrix
τ
φ
|
ψ
A
in
(2.5)
. Then we perform the SVD decomposition
τ
φ
|
ψ
A
=
U
Λ
V
,
(2.7)
with unitary matrices
U
and
V
and diagonal matrix with real and non-negative eigenvalues
Λ =
diag
(
λ
1
,
···
d
A
)
.
(2.8)
In general, these eigenvalues are not normalized, so we normalize them by introducing
ˆ
λ
i
=
λ
i
P
j
λ
j
,
d
A
X
i
=1
ˆ
λ
i
= 1
.
(2.9)
Note that it is useful to interpret them as eigenvalues of the following density matrix
constructed from the transition matrix
(2.5)
ρ
φ
|
ψ
A
=
r

τ
φ
|
ψ
A

τ
φ
|
ψ
A
Tr
r

τ
φ
|
ψ
A

τ
φ
|
ψ
A
!
.
(2.10)
– 5 –
JHEP11(2024)103
From this data, we finally define the SVD entropy as
S
φ
|
ψ
SVD
=
Tr
(
ρ
φ
|
ψ
A
log
ρ
φ
|
ψ
A
) =
X
i
ˆ
λ
i
log(
ˆ
λ
i
)
.
(2.11)
This quantity is manifestly real and has several interesting properties. It is positive and
bounded [
12
]
0
S
φ
|
ψ
SVD
log
d
A
,
(2.12)
and also vanishes if any of the states is a product
|
ψ
=
|
φ
A
|
φ
B
. Formally, it can be
defined for states that have a vanishing inner product (which cancels in the computation
with
ρ
φ
|
ψ
A
). However, in contrast to the previous two quantities above, it is not symmetric
under swapping
A
and
B
, i.e.
S
(
ρ
φ
|
ψ
A
)
̸
=
S
(
ρ
φ
|
ψ
B
)
. In fact, one can show that application
of a unitary operator on
B
, that we trace over, changes
S
(
ρ
φ
|
ψ
A
)
. In general, SVD entropy
violates Araki-Lieb inequality and (strong) sub-additivity. Nevertheless, it admits a very
elegant operational meaning as a number of Bell pairs in the intermediate states between
(arbitrary)
|
φ
and
|
ψ
. See [
33
38
] for further progress on this quantity.
Furthermore, following [
19
,
20
], we define the excess of the entropy measures introduced
above, i.e. the pseudo-entropy excess
S
φ
|
ψ
P
=
Re
S
φ
|
ψ
P

1
2
S
E
(
ρ
φ
A
) +
S
E
(
ρ
ψ
A
)

,
(2.13)
and analogously the excess of the SVD entropy
S
φ
|
ψ
SVD
=
S
φ
|
ψ
SVD
1
2
S
E
(
ρ
φ
A
) +
S
E
(
ρ
ψ
A
)

.
(2.14)
The entropy excess was conjectured to be a useful order parameter for detecting or distin-
guishing quantum phases in
|
φ
and
|
ψ
(see also [
18
]). In particular, this excess was observed
to be non-positive when the two states belong to the same quantum phase, while its positivity
was correlated with different phases of the two states under consideration. We will examine
this property for our quantum mechanical as well as link complement states below.
2.2 Entropy measures for two-component states
Before we proceed with specific models, let us analyze a general class of quantum states in
a product Hilbert space
H
=
H
A
⊗H
B
of the form
|
ψ
i
=
d
1
X
n
=0
c
(
i
)
n
|
n
A
⊗|
n
B
,
(2.15)
with equal dimensions of the two components
d
=
dim
H
A
=
dim
H
B
and complex coefficients
c
(
i
)
n
. These coefficients can be normalized as
P
d
1
n
=0
|
c
(
i
)
n
|
2
= 1
, however we do not necessarily
impose this condition, as the normalization cancels in the transition matrix
τ
1
|
2
=
|
ψ
1
⟩⟨
ψ
2
|
ψ
2
|
ψ
1
.
(2.16)
– 6 –
JHEP11(2024)103
We denote the overlap of our two states by
f
(1
|
2)
≡⟨
ψ
2
|
ψ
1
=
d
1
X
n
=0
c
(1)
n
̄
c
(2)
n
,
(2.17)
and compute the reduced transition matrix by tracing over
H
B
τ
1
|
2
A
=
1
f
(1
|
2)
d
1
X
n
=0
c
(1)
n
̄
c
(2)
n
|
n
A
n
|
A
.
(2.18)
This matrix is already diagonal and has complex eigenvalues. Moreover, its normalized
singular values are encoded in the density matrix
(2.10)
that becomes
ρ
1
|
2
A
=
1
̃
f
(1
|
2)
d
1
X
n
=0
|
c
(1)
n
̄
c
(2)
n
||
n
A
n
|
A
,
Tr
(
ρ
1
|
2
A
) = 1
,
(2.19)
where we denoted the real normalization by
̃
f
(1
|
2)
d
1
X
n
=0
|
c
(1)
n
̄
c
(2)
n
|
.
(2.20)
This way we derive the singular values
ˆ
λ
n
=
|
c
(1)
n
̄
c
(2)
n
|
̃
f
(1
|
2)
,
d
1
X
n
=0
ˆ
λ
n
= 1
.
(2.21)
Based on the above formulas, we obtain the pseudo-entropy
S
1
|
2
P
=
1
f
(1
|
2)
d
1
X
n
=0
c
(1)
n
̄
c
(2)
n
log
c
(1)
n
̄
c
(2)
n
f
(1
|
2)
!
=
= log(
f
(1
|
2)
)
1
f
(1
|
2)
d
1
X
n
=0
c
(1)
n
̄
c
(2)
n
log

c
(1)
n
̄
c
(2)
n

,
(2.22)
as well as the SVD entropy
S
1
|
2
SVD
=
1
̃
f
(1
|
2)
d
1
X
n
=0
|
c
(1)
n
̄
c
(2)
n
|
log
|
c
(1)
n
̄
c
(2)
n
|
̃
f
(1
|
2)
!
=
= log(
̃
f
(1
|
2)
)
1
̃
f
(1
|
2)
d
1
X
n
=0
|
c
(1)
n
̄
c
(2)
n
|
log

|
c
(1)
n
̄
c
(2)
n
|

.
(2.23)
In particular, we present explicit formulas for two qubits with
d
= 2
in appendix
A . We
take advantage of all these formulas in what follows.
3 Chern-Simons theory and link complement states
The main objects that we will examine using the quantum-information tools introduced
above will be the link complement states, which are defined using formalism of Chern-Simons
theory. Here we briefly review their construction and refer to [
39
43
] and [
13
,
14
,
44
46
]
for more details and applications.
– 7 –
JHEP11(2024)103
Chern-Simons theory is a 3-dimensional topological quantum field theory defined by
the action
S
=
k
4
π
Z
M
Tr

A
d
A
+
2
3
A
A
A

,
(3.1)
where
A
=
A
μ
dx
μ
is a gauge field, the coupling
k
(that takes integer values) is called the
level, and
M
is a 3-manifold on which the theory is defined. Various expectation values in
this theory are naturally expressed in terms of a parameter
q
= exp

2
πi
k
+
γ

,
(3.2)
where
γ
is the dual Coxeter number of the gauge group under consideration; in particular
γ
=
N
in
SU
(
N
)
theory. In what follows we also use the
q
-number,
q
-factorial and
q
-
Pochhammer symbol, defined respectively as
[
x
] =
q
x/
2
q
x/
2
q
1
/
2
q
1
/
2
,
[
x
]! = [
x
][
x
1]
···
[1]
,
(
z
;
q
)
k
=
k
1
Y
j
=0
(1
zq
j
)
.
(3.3)
An important role in Chern-Simons theory is played by modular matrices
S
and
T
. In
the
SU
(2)
case these matrices are related to the quantum representation of the modular
group
PSL
(2
,
Z
)
at level
k
; they take the form
S
lm
=
s
2
k
+ 2
sin

(
l
+ 1)(
m
+ 1)
π
k
+ 2

=
q
(
l
+1)(
m
+1)
2
q
(
l
+1)(
m
+1)
2
i
p
2(
k
+ 2)
,
T
lm
=
δ
lm
q
l
(
l
+2)
4
,
(3.4)
where
0
l,m
k
label integrable representations of
SU
(2)
. The above matrices satisfy
the relations [
41
,
47
]
S
2
= 1
,
(
ST
)
3
=
q
3
k
8
.
(3.5)
Interesting observables in Chern-Simons theory — which are also building blocks of the
link states that we are going to consider — are expectation values of Wilson loops associated
to knots
K
and (
n
-component) links
L
n
=
F
n
i
=1
K
i
(i.e. disjoint unions of knots
K
1
,...,
K
n
).
The simplest knot, i.e. unentangled loop, is called the unknot, denoted
0
1
. One infinite
family of knots and links that we consider are the torus knots (for relatively prime
p
and
q
)
and torus links (for
p
and
q
not relatively prime)
T(
p,q
)
, i.e. those that can be formed by
winding a piece of rope respectively
p
and
q
times along two cycles of a torus, see figure
1 .
A
T(
p,q
)
torus link has
gcd
(
p,q
)
components; any two of them weave around one another
with linking number
pq
gcd(
p,q
)
2
. The simplest non-trivial torus knot is the trefoil knot
T(2
,
3)
also denoted
3
1
, while the simplest torus link is the Hopf-link
T(2
,
2)
, also denoted
2
2
1
, which
is made of two interlacing unknots. Of our interest are also twist knots
K
p
, see figure
2 ,
which are an infinite family of knots constructed by taking a loop, making respectively
2
p
1
half-twists for positive
p
or
|
2
p
|
half-twists for negative
p
, and linking its ends together.
These include the unknot
K
0
= 0
1
, trefoil knot
K
1
= 3
1
, figure-8 knot
K
1
= 4
1
, as well as
– 8 –
JHEP11(2024)103
Figure 1.
Representative example of a generic torus link
T(
p,q
)
with
p
= 4
and
q
= 12
.
Figure 2.
Examples of twist knots
K
p
for
p
= 0
(unknot), 1 (trefoil),
1
(figure-8),
2
,
2
,
3
,
3
,
4
(in
anti-clockwise order, starting from the unknot at the top).
K
2
= 5
2
,
K
2
= 6
1
,
K
3
= 7
2
,
K
3
= 8
1
,
K
4
= 9
2
, etc. Another class of links that we consider
are connected sums of the form
K
#2
2
1
, which take the form of the Hopf-link whose one
component is replaced by the knot
K
; in our considerations we choose
K
to be a twist knot,
see figure
3 . In particular,
0
1
#2
2
1
= 2
2
1
is the Hopf-link.
For a link
L
n
(and in particular for a knot, for
n
= 1
), the expectation values of Wilson
loops in Chern-Simons theory reproduce colored (knot and) link invariants and take the form
C
L
n
m
1
,...,m
n
=
D
n
Y
i
=1
W
R
m
i
(
K
i
)
E
S
3
, W
R
m
i
(
K
i
) =
Tr
m
j
i
P
exp

i
I
K
i
A

,
(3.6)
where
W
R
m
i
(
K
i
)
involves an integral of the gauge field along
i
-th component of a link
K
i
,
P
denotes the path ordering, and
R
m
i
(also referred to as the color) for a given level
k
is an
integrable representation of the gauge group [
48
]. For
SU
(
N
)
gauge group, the labels
m
i
of
integrable representations are given by
(
k
+
N
1)!
(
N
1)!
k
!
Young diagrams that fit into the rectangle
of size
k
×
(
N
1)
; for
SU
(2)
they can be identified as integers
m
i
= 0
,...,k
that label
symmetric representations
S
m
i
. Furthermore, for
SU
(
N
)
gauge group, link invariants (
3.6) are
polynomials (or rational functions, depending on normalization) in
q
and
a
=
q
N
, referred to
as colored HOMFLY-PT polynomials; for
SU
(2)
and the specialization
a
=
q
2
they reduce to
colored Jones polynomials. The polynomials that we consider in what follows are normalized
so that for the unknot they are equal to 1. We denote HOMFLY-PT polynomials of a knot
K
colored by
m
-th symmetric representation
S
m
by
P
K
m
(
a,q
)
, while colored Jones polynomials
by
V
K
m
(
q
)
P
K
m
(
q
2
,q
)
. When we refer to an arbitrary knot, or it is clear to which knot
we refer to, we ignore the knot label, and we also often skip the representation label when
a knot polynomial is uncolored, i.e. when it is colored by the fundamental representation;
e.g.
V
(
q
)
V
1
(
q
)
. Let us provide some examples of knot invariants colored by symmetric
– 9 –
JHEP11(2024)103
Figure 3.
Links of the form
K
#
2
2
1
for
K
= 0
1
(unknot)
,
3
1
(trefoil knot)
and
4
1
(figure-8 knot) re-
spectively.
representations
S
m
(i.e. those represented by Young diagrams that consist of one row of
length
m
), which we also use in what follows. The colored HOMFLY-PT polynomials of
twist knots
K
p
take the form [
49
]
P
K
p
m
(
a,q
) =
X
k
=0
k
X
=0
q
k
aq
1
;
q

k
(
q
;
q
)
k

q
1
m
;
q

k

aq
m
1
;
q

k
×
(
1)
a
pℓ
q
(
p
+1
/
2)
(
1)
1
aq
2
1
(
aq
1
;
q
)
k
+1
(
q
;
q
)
k
(
q
;
q
)
(
q
;
q
)
k
.
(3.7)
For
N
= 2
(i.e.
a
=
q
2
) they reduce to colored Jones polynomials [
50
]
V
K
p
m
=
X
k
=0
k
X
=0
q
k
(
q
1
m
;
q
)
k
(
q
m
+1
;
q
)
k
(
1)
q
(
p
+1)+
p
(
1)
/
2
(1
q
2
+1
)
(
q
;
q
)
k
(
q
;
q
)
+
k
+1
(
q
;
q
)
k
,
(3.8)
and in particular, for trefoil
3
1
=
K
1
and figure-8 knot
4
1
=
K
1
, they take the form [
49
]
2
V
3
1
m
=
X
k
=0
q
k

q
1
m
;
q

k

q
1+
m
;
q

k
,
V
4
1
m
=
X
k
=0
(
1)
k
q
k
(
k
+1)
2

q
1
m
;
q

k

q
1+
m
;
q

k
.
(3.9)
Colored Jones polynomials for torus knots
T(
P,Q
)
(with relatively prime
P
and
Q
)
read [
52
]
V
T(
P,Q
)
m
=
q
PQ
(1
m
2
)
/
4
q
m/
2
q
m/
2
(
m
1)
/
2
X
r
=
(
m
1)
/
2

q
PQr
2
+(
P
+
Q
)
r
1
/
2
q
PQr
2
+(
P
Q
)
r
+1
/
2

.
(3.10)
For a Whitehead link shown in figure
18 , with its two components colored respectively
by
S
m
and
S
n
symmetric representations, Jones polynomial takes the form (
7.4). For the
three component Borromean link shown in figure
15 the colored Jones polynomial takes the
form (
5.25). Other examples of colored polynomials for various knots and links, and also
their generalizations to super-polynomials (i.e. deformations of HOMFLY-PT polynomials
that depend on an additional parameter
t
and capture some information about homological
invariants) can be found in [
53
,
54
].
2
Equivalent formulas can be found in [
51].
– 10 –
JHEP11(2024)103
The link states of our interest are associated to manifolds
M
n
=
S
3
\
N
(
L
n
)
, which
are obtained by removing a tubular neighbourhood
N
(
L
n
)
of a link
L
n
from
S
3
(a tubular
neighbourhood is obtained by thickening each component of a link to a solid torus). The
boundary of
M
n
takes the form of
n
copies of a torus,
∂M
n
=
n
i
=1
T
2
. The path integral
in Chern-Simons theory with gauge group
G
and level
k
produces then a link state, i.e.
quantum state in the
n
-fold tensor product Hilbert space
H
n
=
H
(
T
2
,G,k
)
n
, which
can be expanded as
|L
n
=
X
m
1
,...,m
n
C
L
n
m
1
,...,m
n
|
m
1
,...,m
n
,
(3.11)
where
|
m
are basis elements of the torus Hilbert space
H
labeled by integrable representations
m
of
G
, and the coefficients
C
L
n
m
1
,...,m
n
are the link invariants (
3.6). Note that these states
are not normalized, which however does not affect our considerations, as any normalization
factors cancel in (
2.4) and lead to the same values of entropy measures that we analyze.
In what follows we will take advantage of the unitarity of the
S
matrix (
3.4), which can
be used to implement a unitary basis transformation
|
m
⟩≡
X
n
S
mn
|
n
.
(3.12)
In our cases this transformation does not affect the entanglement properties of the link
states and can conveniently be used to diagonalize various transition matrices. For example,
in
SU
(2)
Chern-Simons theory at level
k
we can rewrite link states associated to links of
the form
K
#
2
2
1
as follows [
13
]
|K
#2
2
1
=
k
X
m,n
=0
C
K
m
S
0
m
S
mn
|
m,n
⟩≡
k
X
m
=0
̃
C
K
m
|
m,m
,
(3.13)
where
C
K
m
and
̃
C
K
m
=
C
K
m
/
S
0
m
are respectively unreduced and reduced colored invariants of
the knot
K
. The link states for
(
p,q
)
torus links with
d
=
gcd
(
p,q
)
components in
SU
(2)
Chern-Simons theory can be written as [
14
,
44
,
55
57
]
|
T(
p,q
)
=
k
X
m
=0
k
X
l
=0
1
S
d
1
0
m
S
ml
V
T
p/d,q/d
l
|
m,
···
,m
|
{z
}
d
entries
=
=
k
X
m
=0

S
X

p
d

T
q
p
S

m
0
1
S
d
1
0
m
|
m,...,m
,
(3.14)
where the matrix
X
(
t
)
, whose components
X
ab
(
t
)
are referred to as Adams coefficients, is
defined by [
55
]
Tr
a

U
t

=
k
X
b
=0
X
ab
(
t
)
Tr
b
(
U
)
,
for
U
SU(2)
.
(3.15)
4 Quantum mechanical examples
In this section we use entropy measures to characterize states in a few quantum mechanical
examples. We consider generalized coherent states for
SU
(2)
and
SU
(1
,
1)
Lie algebras in the
two-mode (bipartite) representation [
58
], as well as tripartite GHZ and W states.
– 11 –
JHEP11(2024)103
Note that the above systems have different numbers of modes and different sizes of a
Hilbert space (for each of those modes). In case of bipartite coherent states for
SU
(2)
, a
choice of a given representation determines the dimension of the Hilbert space for each mode,
which can be any (finite) integer number. On the other hand, for each of the two modes
of
SU
(1
,
1)
coherent states, the Hilbert space is infinite-dimensional. Finally, we consider
tripartite GHZ and W states associated to two-dimensional (qubit) Hilbert space. The link
states that we analyze in the next section can be thought of as generalizations of these
systems, in the sense they may involve an arbitrary number of components, and the size of
the Hilbert space is an arbitrary integer fixed by the choice of the level.
In this section, for the above quantum mechanical systems we evaluate their pseudo-
entropy and SVD entropy, and discuss how various states are distinguished by the en-
tropy excess.
4.1
SU(2)
coherent states
As the first example we consider coherent states associated to the
SU
(2)
Lie algebra. Its
generators
J
i
,
i
= 1
,
2
,
3
, satisfy commutation relations
[
J
i
,J
k
] =
ijk
J
k
, which can be written
in terms of the ladder operators
J
±
=
J
1
±
iJ
2
as
[
J
3
,J
±
] =
±
J
±
,
[
J
+
,J
] = 2
J
3
.
(4.1)
The lowest weight states
|−
j
are defined by
J
3
|−
j
=
j
|−
j
, J
|−
j
= 0
,
(4.2)
and generalized coherent states are conventionally defined by acting with a displacement
operator on
|−
j
|
z,j
=
e
ξJ
+
̄
ξJ
|−
j
.
(4.3)
Using the Baker-Campbell-Hausdorff (BCH) formula for
SU
(2)
we can expand this state as
|
z
i
,j
= (1 +
|
z
i
|
2
)
j
2
j
X
n
=0
z
n
i
s
Γ(2
j
+ 1)
n
!Γ(2
j
n
+ 1)
|
n
A
⊗|
n
B
,
(4.4)
where basis vectors
|
n
i
have
n
powers of
J
+
acting on them.
3
The Hilbert space
H
=
H
A
⊗H
B
is finite dimensional and each component
H
i
has dimension
d
i
= 2
j
+ 1
. Moreover, the
complex coordinate
z
i
admits a geometric interpretation as a point on the stereographic
projection of the unit sphere
4
z
i
= tan

θ
i
2

e
i
t
i
e
i
, θ
i
[0
]
, φ
i
[0
,
2
π
]
.
(4.5)
In our context, we consider the transition matrix between two coherent states labeled by
different
z
i
’s
τ
1
|
2
=
|
z
1
,j
⟩⟨
z
2
,j
|
z
2
,j
|
z
1
,j
.
(4.6)
3
Formally they can be constructed by using the two-mode representation of the algebra that we associate
with
A
and
B
.
4
Relation to coordinate
ξ
enters through
ξ
=
θ
2
e
.
– 12 –
JHEP11(2024)103
Since the coherent states form an over-complete basis, the overlap between them is non-trivial
and given by
z
2
,j
|
z
1
,j
=
(1 +
|
z
1
|
2
)
j
(1 +
|
z
2
|
2
)
j
(1 +
z
1
̄
z
2
)
2
j
.
(4.7)
This way, after tracing over the second Hilbert space, we obtain the reduced transition matrix
τ
1
|
2
A
=
Tr
H
B
τ
1
|
2

= (1 +
z
1
̄
z
2
)
2
j
2
j
X
n
=0
Γ(2
j
+ 1)
n
!Γ(2
j
n
+ 1)
(
z
1
̄
z
2
)
n
|
n
⟩⟨
n
|
,
(4.8)
and similarly the density matrix
(2.10)
ρ
1
|
2
A
= (1 +
|
z
1
̄
z
2
|
)
2
j
2
j
X
n
=0
Γ(2
j
+ 1)
n
!Γ(2
j
n
+ 1)
|
z
1
̄
z
2
|
n
|
n
⟩⟨
n
|
.
(4.9)
Clearly the
2
j
+ 1
complex eigenvalues of
τ
1
|
2
A
are parametrized by
z
1
̄
z
2
=
t
1
t
2
e
12
, φ
12
=
φ
1
φ
2
,
(4.10)
whereas those of
ρ
1
|
2
A
are real and parametrized by the absolute value of the expression above
which has the complex phase removed
ˆ
λ
n
=
Γ(2
j
+ 1)
n
!Γ(2
j
n
+ 1)
(1 +
t
1
t
2
)
2
j
(
t
1
t
2
)
n
,
2
j
X
n
=0
ˆ
λ
n
= 1
.
(4.11)
After closer examination of these eigenvalues, we can compute von Neumann, pseudo
and SVD entropies at once. Indeed,
ˆ
λ
simply follows the binomial distribution so it is
convenient to introduce
P
n
(
X
) =
2
j
n
!
p
n
(1
p
)
2
j
n
, p
=
X
1 +
X
,
(4.12)
and compute
S
(
j,X
) =
2
j
X
n
=0
P
n
(
X
) log(
P
n
(
X
))
,
(4.13)
where
X
=
t
2
i
for the computation of von Neumann entropies,
X
=
t
1
t
2
exp
(
12
)
for the
pseudo-entropy and
X
=
t
1
t
2
for the SVD entropy. While getting a closed form for arbitrary
j
is not possible, we can easily derive the answer for small
j
or perform the sums and plot
numerically. For instance, for the first three
j
=
m/
2
with
m
= 1
,
2
,
3
we simply get
S
(
m/
2
,X
) =
m
(
p
log(
p
)
(1
p
) log(1
p
))
p
(1
p
)
m
log(
m
)
.
(4.14)
For higher
j
this expression gets corrected by higher polynomials in
p
(1
p
)
. On the other
hand, we can use Stirling’s approximation, or equivalently the central limit or de Moivre-
Laplace theorem, to derive the asymptotic expression for large
j
. The answer diverges
logarithmically with
j
S
(
j,X
)
1
2
log (4
πep
(1
p
)
j
)
.
(4.15)
– 13 –
JHEP11(2024)103
0
5
10
15
20
25
30
j
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure 4.
Entanglement entropy as a function of
j
for
θ
=
π/
2
(dots) vs asymptotic formula
(4.15)
(solid blue curve).
Let us now analyze our quantities of interests. Firstly, the von Neumann entropies that
are parametrized by
t
=
tan
(
θ/
2)
have the maximum value (for any
j
) for
θ
=
π/
2
or
t
= 1
and
they vanish for
θ
=
{
0
}
. We will use these “maximally entangled states” as our target states
in the transition matrix. As an example, in figure
4 we present the von Neumann entropy as
a function of
j
for
θ
=
π/
2
(equivalent to
p
= 1
/
2
) vs. the asymptotic formula
(4.15)
.
Next, for concreteness and analytical control, let us focus on the
j
= 1
/
2
example where
entanglement entropies are
S
i
E
= log(1 +
t
2
i
)
2
t
2
i
log(
t
i
)
1 +
t
2
i
.
(4.16)
The pseudo-entropy becomes
S
1
|
2
P
= log

1 +
t
1
t
2
e
12

(
12
+ log(
t
1
t
2
))
t
1
t
2
e
12
1 +
t
1
t
2
e
12
,
(4.17)
and we can decompose it into real and imaginary parts as
S
1
|
2
P
=
1
2
log(∆
12
)
t
1
t
2
(
t
1
t
2
+cos(
φ
12
)) log(
t
1
t
2
)
φ
12
sin(
φ
12
)
12
i
"
t
1
t
2
(
t
1
t
2
+cos(
φ
12
))
φ
12
+sin(
φ
12
) log(
t
1
t
2
)
12
+
i
2
log
1+
t
1
t
2
e
12
1+
t
1
t
2
e
12
!#
,
(4.18)
where
12
=
|
1 +
t
1
t
2
e
12
|
2
= 1 +
t
2
1
t
2
2
+ 2
t
1
t
2
cos(
φ
12
)
.
(4.19)
Clearly, the imaginary part of the pseudo-entropy arises due to the non-trivial phases of
our two states that have different
φ
i
points on the sphere. Moreover, if we flip the phases,
φ
1
φ
2
the imaginary part changes the sign. We will elaborate on this property in the
context of link states in the last section.
Analogously, we can evaluate the SVD entropy that in our
j
= 1
/
2
example can be
simply obtained from pseudo-entropy by setting
φ
12
= 0
S
1
|
2
SVD
= log(1 +
t
1
t
2
)
t
1
t
2
log(
t
1
t
2
)
1 +
t
1
t
2
.
(4.20)
– 14 –