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