††institutetext: Department of Physics, National Tsing Hua University, Hsinchu
30013, Taiwan

# Topological entanglement entropy for torus knot bipartitions and the
Verlinde-like formulas

Chih-Yu Lo  Po-Yao Chang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The topological Rényi and entanglement entropies depend on the bipartition of
the manifold and the choice of the ground states. However, these entanglement
quantities remain invariant under a coordinate transformation when the
bipartition also undergoes the same transformation. In the context of
topological quantum field theories, these coordinate transformations reduce to
representations of the mapping class group on the manifold of the Hilbert
space. We employ this invariant property of the Rényi and entanglement
entropies under coordinate transformations for TQFTs in (2 + 1) dimensions on
a torus with various bipartitions. By utilizing the replica trick and the
surgery method to compute the topological Rényi and entanglement entropies,
the invariant property results in Verlinde-like formulas. Furthermore, for the
bipartition with interfaces as two non-intersecting torus knots, an
$SL(2,\mathbb{Z})$ transformation can untwist the torus knots, leading to a
simple bipartition with an effective ground state. This invariant property
allows us to demonstrate that the topological entanglement entropy has a lower
bound $-2\ln D$, where $D$ is the total quantum dimensions of the system.

###### Contents

  1. 1 Introduction
  2. 2 Coordinate transformation
  3. 3 Partition functions, replica trick, and the surgery method
  4. 4 Topological entanglement entropy with different bipartitions, and the applications of the coordinate transformation
    1. 4.1 Single $S^{1}$ interface
    2. 4.2 Two $S^{1}$ interfaces related by modular $S$ transformation
      1. 4.2.1 Two $S^{1}$ interfaces being the longitudes of the torus
      2. 4.2.2 Two $S^{1}$ interfaces being the meridians of the torus
    3. 4.3 General Torus knot bipartitions
  5. 5 Conclusion
  6. 6 Acknowledgments
  7. A The Verlinde-like formula
  8. B The bounds on ground state TEE

##  1 Introduction

After the publication of seminal papers on Chern-Simon Theory TQFT1 ; TQFT2 ,
topological quantum field theories (TQFTs) have found applications in studying
the properties of topologically ordered systems with a mass gap. Notable
examples include the fractional quantum Hall effect FQH ; FQH2 ; FQH3 ; FQH4 ,
gapped quantum spin liquids SpinLiquid , quantum dimer models Dimer , and
$p_{x}+ip_{y}$ superconductors px+ipy . TQFTs can be regarded as effective
field theories for these systems, treating the mass gap as infinite. Under
this stringent condition, all excitations become irrelevant, allowing us to
focus solely on the ground states, which exhibit long-range entanglement. Due
to these long-range entangled properties, the number of ground states depends
on the topology of the manifold, leading to these systems being referred to as
having topological orders TO .

The concept of topological entanglement entropy (TEE) was later introduced by
TEE ; TEE2  as a diagnostic tool for discerning topological orders in two
dimensions. The TEE of these systems can be written as $\alpha
L+S_{\text{TEE}}$, where $\alpha L$ is a term proportional to the area of the
entanglement interface Area  and the $S_{\text{TEE}}$ is a negative term due
to the topological constraints of these systems. Extensive computations of the
TEE in various lattice models have substantiated its efficacy in
characterizing topological orders TO1 ; TO2 ; TO3 ; TO4 ; TO5 .

Alternatively, the TEE can be directly computed using the Von-Neumann entropy
within the framework of TQFTs through the replica method TEEinCS ,
successfully reproducing the negative TEE. Furthermore, by employing the
replica method in TQFTs, several entanglement quantities including the
negativity negativity ; diagTEE ; EdgeTEE , pseudo entropy psuedoTEE , mutual
information negativity ; EdgeTEE  and the reflective entropy diagTEE  are
investigated. Moreover, the TEE can be derived using the diagrammatic
approach, which captures the TEE generated by the braiding of anyons anyonTEE
; diagTEE ; multiTEE . Additionally, the area term of the entanglement entropy
in the context of TQFTs can be recovered by considering a regularization of
the entanglement interface EdgeTEE ; interfaceTEE ; diagTEE .

Consider a spatial bipartition where subsystem $A$ forms an annulus on a torus
in $(2+1)$ dimensions. The topological entanglement entropy (TEE) associated
with this bipartition is proposed TEE ; TEE2  to be given by $-2\ln D$.
However, for the same bipartition, the TEE may vary for different ground
states depending on the presence of Wilson lines TEEinCS ; gsTEE ; EdgeTEE .
In this paper, we demonstrate that if one fixes a ground state, the TEE can
still differ depending on the twist of the annulus. Specifically, we explore
the nuanced scenario where, even with a fixed ground state, the TEE can vary
based on the twist of the bipartition. In particular, we examine bipartitions
characterized by two non-intersecting torus knots interfaces between subsystem
$A$ and its complement. We will refer to these bipartitions as torus knot
bipartitions (or twisted annulus bipartitions). We demonstrate that the TEE of
these different torus knots is given by

$S_{\mathrm{TEE}}=-2\ln D+S_{\mathrm{gs}},$ (1.1)

where $0\leq S_{\mathrm{gs}}\leq 2\ln D$ represents a quantity arising from
the effective ground states induced by the twists of bipartitions.. A
comparable property has been explored in TEEinCS ; gsTEE ; EdgeTEE , where the
authors examine untwisted bipartitions with different ground states.

To compute the TEE for general torus knot bipartitions, we leverage coordinate
transformations on the torus, specifically the mapping class group
$SL(2,\mathbb{Z})$ in TQFTs. When the bipartition of the manifold respect the
coordinate transformation, the TEE and other entanglement quantities must
remain invariant under this transformation. Exploiting this invariant property
of entropies, we map torus knot bipartitions to a simpler annulus bipartition,
whose TEE has been previously computed EdgeTEE . As a byproduct of our
approach, for the torus knot bipartition with two $S^{1}$ interfaces forming
the meridians of the torus, the invariance of the R’enyi entropies under
modular $S$ transformation yields Verlinde-like formulas. This establishes a
relationship between quantum dimensions and modular data.

This paper is organized as follows. In Sec. 2, we provide the definition of
coordinate transformations, interpreted as relabelings of sites on the
manifold. This expression establishes the invariant property of entanglement
quantities. Sec. 3 review the replica trick and the surgery method for
computing Rényi entropies in TQFTs. In Sec. 4, we apply the invariant property
of Rényi entropies under coordinate transformations with various annulus
bipartitions, including those with meridian interfaces, longitude interfaces,
and general torus knot interfaces. The invariance of Rényi entropies is
demonstrated to lead to Verlinde-like formulas. Additionally, torus knot
bipartitions give rise to an effective ground state. Finally, we summarize our
findings and conclude the paper in Section 5.

##  2 Coordinate transformation

Before delving into the calculation of the TEE, let’s begin by explaining the
invariance property under coordinate transformations. Suppose we have a system
of sites indexed by a set $C$ placed on a surface $S$ through a bijective
morphism111For example, it preserves topology for topological manifolds or
smoothness for smooth manifolds, respectively. $\phi:C\rightarrow S$. The
Hilbert space of the system is given by

$\mathcal{H}_{C}=\otimes_{c\in C}\mathcal{H}^{c}_{\phi(c)}=\otimes_{x\in
S}\mathcal{H}^{\phi^{-1}(x)}_{x}=\mathcal{H}_{S}.$

In the context of quantum field theories, $\mathcal{H}^{c}_{\phi(c)}$
represents the same Hilbert space, independent of site indices. Let
${\ket{i}^{c}_{\phi(c)}}{i\in J}$ be an orthonormal basis for
$\mathcal{H}^{c}{\phi(c)}$ (possibly infinite-dimensional in the case of
bosons). Then, a basis for $\mathcal{H}_{S}$ can be expressed as

$\\{\ket{f}_{\phi}^{C}=\otimes_{c}\ket{f(c)}^{c}_{\phi(c)}=\otimes_{x}\ket{f\circ\phi^{-1}(x)}^{\phi^{-1}(x)}_{x}|f\in
F(C,J)\\},$ (2.1)

where $F(C,J)$ denotes the set of functions from $C$ to $J$. The label $\phi$
indicates the position of sites, and the label $C$ indicates that it is the
functional basis ranged in $C$. We define our states as intrinsic objects
attached to sites, independent of spatial labeling [See Fig. 1]. That is, for
any given labelings $\phi_{1},\phi_{2}:C\rightarrow S$,

$\ket{i}^{c}_{\phi_{1}(c)}=\ket{i}^{c}_{\phi_{2}(c)}\quad\text{and}\,\,\quad\ket{f}^{C}_{\phi_{1}}=\ket{f}^{C}_{\phi_{2}}.$
(2.2)

Figure 1: Two different maps $\phi_{1},\phi_{2}$ from sites to space
resembling the same state.

Now, a state in $\mathcal{H}^{C}$ can be expressed as

$\ket{\psi^{C}}_{\phi}=\sum_{f\in F(C,J)}\psi^{C}(f)\ket{f}^{C}_{\phi},$

where the functionals $\psi^{C}:F(C,J)\rightarrow\mathbb{C}$ satisfy the
equation of motion for the underlying system.

Let us consider a bipartition on sites $C=A\cup B$ and let
$\\{\ket{f^{A}}^{C}_{\phi}\\}_{f^{A}\in F(A,I)}$,
$\\{\ket{f^{B}}^{C}_{\phi}\\}_{f^{B}\in F(B,I)}$ be the bases for
$\mathcal{H}^{A},\mathcal{H}^{B}$ respectively. A continuous function $f$ can
be bijectively mapped to continuous functions $f^{A},f^{B}$ on $A$ and $B$ by
restriction. Therefore, we write
$\ket{f}^{C}_{\phi}=\ket{f^{A}}^{C}_{\phi}\otimes\ket{f^{B}}^{C}_{\phi}$. With
this identification, we define $\psi^{AB}:F(A,J)\times
F(B,J)\rightarrow\mathbb{C}$ by

$\psi^{AB}(f^{A},f^{B})=\psi^{C}(f).$

Now, we have

$\ket{\psi^{C}}_{\phi}=\sum_{f}\psi^{C}(f)\ket{f}^{C}_{\phi}=\sum_{f^{A},f^{B}}\psi^{AB}(f_{A},f_{B})\ket{f^{A}}^{A}_{\phi}\otimes\ket{f^{B}}^{B}_{\phi},$
(2.3)

and

$\rho_{A}=\sum_{f^{A},f^{A^{\prime}}}\Big{(}\sum_{f^{B}}\psi^{AB}(f_{A},f_{B})\bar{\psi}^{A^{\prime}B}(f_{A^{\prime}},f_{B})\Big{)}\ket{f^{A}}^{A}\langle
f^{A^{\prime}}|^{A^{\prime}}.$ (2.4)

Suppose we perform a coordinate transformation on $S$, which is an
automorphism $K:S\rightarrow S$ on the manifold222For example, we can change
the Cartesian coordinate to the polar coordinate. A more exotic example is a
Dehn twist on a torus, where the coordinate transformation can be considered
as the twisting map $[e^{i\theta},t]\to[e^{i(\theta+2\pi t)},t]$ of the
annulus with $\theta\in[0,2\pi]$ and $t\in[0,1]$.. The Hilbert space can now
be written as

$\mathcal{H}^{C}=\otimes_{c\in C}\mathcal{H}^{c}_{K\circ\phi(c)}=\otimes_{x\in
S}\mathcal{H}^{(K\circ\phi)^{-1}(x)}_{x}=\mathcal{H}_{S}.$

Under such an automorphism, everything discussed above remains the same with
the substitution $\phi\rightarrow K\circ\phi$. Essentially, we are changing
nothing but shuffling the position of sites around. Therefore, if we perform a
partial trace with respect to the relabeling of the site label, the reduced
density matrix will remain the same.

On the other hand, we can also look at the state in the position $x$. By
definition Eq. (2.1) we have the relation

$\ket{f}^{C}_{\phi}=\ket{f\circ\phi^{-1}}^{\phi}_{S},$ (2.5)

One should note that there can be ambiguity in defining states in positions in
quantum field theories333There is no unique way to discretize continuous
theories, and the states can depend on the regularization scheme., and in
general, $\ket{i}_{x}^{\phi^{-1}_{1}(x)}\neq\ket{i}^{\phi^{-1}_{2}(x)}_{x}$
and $\ket{f}^{\phi_{1}}_{S}\neq\ket{f}^{\phi_{2}}_{S}$. In fact, we can define
an automorphism on $S$ by $\phi_{2}\circ\phi_{1}^{-1}$, then

$\ket{f}^{\phi_{2}}_{S}=\sum_{f^{\prime}}U_{\phi_{2}\circ\phi_{1}^{-1}}(f,f^{\prime})\ket{f^{\prime}}^{\phi_{1}}_{S},$
(2.6)

where $U$ is an unitary operator on the functional space. This can be think of
as performing a relabeling on sites while keeping the spatial wave functional
basis fixed. On the other hand, from Eq. (2.2) and Eq. (2.5) we have
$\ket{f}^{\phi}_{S}=\ket{f\circ K^{-1}}^{K\circ\phi}_{S}$, thus

$\ket{f\circ
K}^{\phi}_{S}=\sum_{f^{\prime}}U_{K}(f,f^{\prime})\ket{f^{\prime}}^{\phi}_{S}.$
(2.7)

This is essentially changing the spatial functional while keeping the site
labels fixed. A change of labeling of sites can be thought of as a two-step
process. The unitary transformation cancels out on each other, leaving the
state itself invariant. More explicitly, define
$K=\phi_{1}\circ\phi^{-1}_{2}$, then

$\ket{f}^{C}_{\phi_{1}}=\ket{f}^{C}_{K\circ\phi_{2}}=\ket{f\circ
K^{-1}}^{\phi_{1}}_{S}=\sum_{f^{\prime},f^{\prime\prime}}U_{K^{-1}}(f,f^{\prime})U_{K}(f^{\prime},f^{\prime\prime})\ket{f^{\prime\prime}}^{\phi_{2}}_{S}=\ket{f}^{C}_{\phi_{2}}.$
(2.8)

It’s crucial to note that while the bipartition with respect to the sites is
fixed, the bipartition with respect to the positions will also change
according to the relabeling of sites $S=\phi_{1}(A)\cup\phi_{1}(B)\rightarrow
S=\phi_{2}(A)\cup\phi_{2}(B)$ [See Fig. 2 for example].

Figure 2: Same bipartition on sites can gives different bipartitions on space

In the case of TQFTs, the Hilbert space of functionals is reduced to a finite-
dimensional Hilbert space of anyons if the gauge group and the manifold are
compact TQFT1 ; TQFT2 . The coordinate transformations are also replaced by
the mapping class group, which consists of topologically equivalent classes of
transformations.

##  3 Partition functions, replica trick, and the surgery method

Here, we briefly introduce the method of computing TEE in the context of TQFT
following TEEinCS . Consider the Chern-Simon action on a three manifold $M$

$S_{\mathrm{CS}}=\frac{k}{4\pi}\int_{M}\mathrm{Tr}(A\wedge dA+A\wedge A\wedge
A),$ (3.1)

where $A$ is the connection one-form of the principal bundle of the underlying
gauge group, and $k$ is the level of the theory. Doing a path integral over
$M$ will give rise to a state in the Hilbert space of its boundary,

$\ket{M}=\int\mathcal{D}A\,e^{-iS_{\mathrm{CS}}}\in\mathcal{H}_{\partial M}.$
(3.2)

In general, one can also insert Wilson loops into the bulk to obtain different
boundary states

$\ket{M,C_{a}}=\int\mathcal{D}A\,e^{-iS_{\mathrm{CS}}[A]}W_{a}^{C}[A],$ (3.3)

where the Wilson loop operator

$W_{a}^{C}[A]=\mathrm{Tr}_{a}\mathcal{P}\\{e^{\oint_{C}A}\\}$ (3.4)

traces the holonomy on a closed curve $C$ in representation $a$.

In particular, $\mathcal{H}_{T^{2}}$ is spanned by $\ket{a}=\ket{D^{2}\times
S^{1},C_{a}}$, with $a$ ranging over the highest weight representations of the
gauge group at level $k$ Ver ; TQFT1 ; TQFT2 . One can also construct the dual
vectors by reversing the orientation of the manifold

$\bra{a}=|\overline{D^{2}\times
S^{1}},\overline{C_{a}}\rangle\in\mathcal{H}^{*}_{T^{2}}=\mathcal{H}_{\overline{T^{2}}}.$
(3.5)

Partition function on $S^{2}\times S^{1}$ can be obtained by gluing two
$D^{2}\times S^{1}$ with opposite orientation by identity map

$Z(S^{2}\times S^{1},C_{\bar{a}},C_{b})=\bra{a}\ket{b}=\delta_{ab}.$ (3.6)

On the other hand, one can also obtain the partition function of $S^{3}$ by
gluing two $D^{2}\times S^{1}$ with opposite orientation related by the
modular $S$ transformation

$Z(S^{3},L_{ab})=\bra{a}S\ket{b}=S_{ab},$ (3.7)

where $L_{ab}$ is the configuration of $C_{\bar{a}}$ and $C_{b}$ being linked
once  . In particular, if $a=b=0$, we obtain $Z(S^{2}\times S^{1})=1$ and
$Z(S^{3})=S_{00}=D^{-1}$. Throughout this paper, we will refer to
$\\{\ket{a}\\}_{a}$ as the inside basis and $\\{S\ket{a}\\}_{a}$ as the
outside basis.444One can imagine a torus embedded in
$\mathbb{R}^{3}\cup{\infty}\simeq S^{3}$ divide the ambient space into inside
and outside solid tori. We then have the basis transformation

$S\ket{a}=\sum_{b}S_{ab}\ket{b}$ (3.8)

Given a state $\ket{M}$, one can obtain the corresponding density matrix
$\rho$ by the disjoint union of $M$ with $\overline{M}$. Given a bipartition
$\partial M=A\cup B$, one can obtain the reduced density matrix
$\rho_{A}=\mathrm{Tr}_{B}\rho$ by gluing $B$ of $M$ to the corresponding
$B^{\prime}$ of $\overline{M}$ by the identity map. $\rho_{A}$ is then a
manifold with boundary $A\cup A^{\prime}$, where $A^{\prime}$ is the
corresponding region to $A$ in $\overline{M}$. One can then obtain
$\mathrm{Tr}\rho_{A}^{2}$ by making two copies of $\rho_{A}$ and then perform
two gluings from $A$ to $A^{\prime}$, which is a manifold without boundary.
Similarly, $\mathrm{Tr}\rho_{A}^{n}$ can be constructed by $n$ copies of $M$
with corresponding gluings of the boundaries. One can then obtain the Rényi
entropies

$S_{n}=\frac{1}{1-n}\ln\mathrm{Tr}(\rho_{A})^{n}.$ (3.9)

The entanglement entropy is then the limit $n\rightarrow 1$ by performing
analytical continuation

$S_{\mathrm{TEE}}=\lim_{n\rightarrow 1}S_{n}.$ (3.10)

To evaluate the partition function of the glued manifold, one then applies the
method of surgeryTQFT1  to decompose the manifold into simple ones whose
partition functions are known, for example, Eq. (3.6) and Eq. (3.7).

Suppose one has a manifold $M$ without a boundary, which can be obtained by
gluing $\overline{M_{1}},M_{2}$ along their common $S^{2}$ boundary

$Z(M)=\bra{M_{1}}\ket{M_{2}}.$ (3.11)

Due to the one dimensional property of $\mathcal{H}_{S^{2}}$, one has

<EMAIL_ADDRESS>2.84544pt=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (3.12)

Let $\tilde{M_{1}},\tilde{M_{2}}$ be the manifolds obtained by capping off the
$S^{2}$ boundary with $D^{3}$ of $M_{1},M_{2}$ respectively, then

$Z(M)Z(S^{3})=Z(\tilde{M_{1}})Z(\tilde{M_{2}}).$ (3.13)

For example

(3.14)

In general, one can also perform surgery with a Wilson loop insert due to the
one dimensional property of $\mathcal{H}_{S^{2},a,\bar{a}}$ TQFT1 ; negativity

<EMAIL_ADDRESS>2.84544pt=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (3.15)

##  4 Topological entanglement entropy with different bipartitions, and the
applications of the coordinate transformation

In this section, we apply the invariant property discussed in Sec. 2 to
compute entropies using different coordinates. In Sec. 4.1, we will discuss
bipartitions with a single $S^{1}$ interface as a consistency check of the
method. In Sec. 4.2, we will discuss bipartitions with two $S^{1}$ interfaces
being the meridians or the longitudes and obtain Verlinde-like formulas. In
Sec. 4.3, we will discuss the case where the two $S^{1}$ interfaces are
generic torus knots, which is made possible to compute by means of coordinate
transformations.

###  4.1 Single $S^{1}$ interface

Consider the vacuum state $\ket{0}$ on $T^{2}$ generated by the empty solid
doughnut $D^{2}\times S^{1}$ with a bipartition which has a single $S^{1}$
interface

$\ket{0}=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.1)

We consider the reduced density matrix

$\rho_{A}=\mathrm{Tr}_{B}\rho=\mathrm{Tr}_{B}\outerproduct{0}{0}=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.2)

We can compute the $n$-th Rényi entropy by making $n$ copies of $\rho_{A}$ and
gluing them together according to the boundary orientation. In the end, we
will obtain that $\mathrm{Tr}{(\rho_{A}}^{n})$ is given by the connected sum
of $n$ copies of $S^{2}\times S^{1}$.

$\mathrm{Tr}{(\rho_{A}}^{n})=\hskip
<EMAIL_ADDRESS>(4.3)

The partition function of such a configuration can be computed by the method
of surgery similar to Eq. (3.14). By inserting $(n-1)$ copies of $S^{3}$ the
configuration can be separated into $n$ copies of $S^{2}\times S^{1}$. The
result is given by

$\mathrm{Tr}{(\rho_{A}}^{n})=\frac{Z(S^{2}\times
S^{1})^{n}}{Z(S^{3})^{n-1}}=D^{n-1}.$ (4.4)

On the other hand, we can perform the same calculation using the outside basis
instead. We know from Eq. (3.8) that the inside torus is related to the
outside torus by the transformation

<EMAIL_ADDRESS>2.84544pt=\sum_{b}S_{ab}\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.5)

where we use vertical torus to indicate the outside basis and the $S_{ab}$ is
the modular $S$ matrix. Using the outside basis, the reduced density matrix is
given by

$\rho_{A}=\sum_{a,b}S_{0a}S_{0\bar{b}}\hskip
<EMAIL_ADDRESS>2.84544pt=\sum_{a,b}\frac{d_{a}}{D}\frac{d_{b}}{D}\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.6)

where $d_{a}=d_{\bar{a}}=S_{0a}D$ is the quantum dimension of anyon $a$ and
$D=\sqrt{\sum_{a}d_{a}^{2}}$ is the total quantum dimension. For simplicity,
we use a single black line to indicate a copy of $D^{2}\times S^{1}$, we use
the dot to indicate that they are being connected summed and we use blue lines
to indicate the anyon loops circling the corresponding non-contractible loops
of $D^{2}\times S^{1}$. Using this notation, we have

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{a_{1},b_{1}}\sum_{a_{2},b_{2}}\dots\sum_{a_{n},b_{n}}\Big{(}\prod_{k=1}^{n}\frac{d_{a_{k}}d_{b_{k}}}{D^{2}}\Big{)}\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.7)

where the double lines indicates that the two copies of $D^{2}\times S^{1}$
are glued together along their torus boundary to form a $S^{2}\times S^{1}$,
and the giant dot indicates that these $S^{2}\times S^{1}$ are connected
summed together. We should also mentioned that all the lines are all in the
outside basis even if we do not put them in the vertical manner. To evaluate
the partition function, we apply the method of surgery again to cut out all
the separate $S^{2}\times S^{1}$’s with Wilson lines by inserting $S^{3}$’s.
We then obtain

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{a_{1},b_{1}}\sum_{a_{2},b_{2}}\dots\sum_{a_{n},b_{n}}\prod_{k=1}^{n}\frac{d_{a_{k}}d_{b_{k}}}{D^{2}}\hskip
<EMAIL_ADDRESS><EMAIL_ADDRESS>2.84544pt\frac{1}{Z(S^{3})^{n-1}}$ (4.8)

Since the partition function of each $S^{2}\times S^{1}$ with Wilson lines
$\bar{a},b$ is given by $\delta_{ab}$, and the partition function of $S^{3}$
is simply $D^{-1}$, we then have

$\mathrm{Tr}{(\rho_{A}}^{n})=\Big{(}\sum_{a}\frac{d_{a}^{2}}{D^{2}}\Big{)}^{n}D^{n-1}=D^{n-1}.$
(4.9)

As expected, we obtain the same result as using the inside basis. This serves
as a consistency check of our method.

###  4.2 Two $S^{1}$ interfaces related by modular $S$ transformation

Next, we consider the case where the two sub-regions meet at two $S^{1}$
interfaces. If both $S^{1}$ interfaces are contractible, then the result will
be similar to the case of a single interface. If only one $S^{1}$ is
contractible, then the bipartition is ill-defined. The next possible
bipartition is when the two interfaces are given by the same torus knots. In
this subsection, we will consider the case where the torus knots are simply
given by the longitudes and meridians.

####  4.2.1 Two $S^{1}$ interfaces being the longitudes of the torus

Consider the vacuum state in the solid torus and with the bipartition given by
two longitudes which can be viewed as cutting the doughnut horizontally

$\ket{0}=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.10)

In this case, the reduced density matrix after tracing out $B$ is given by

$\rho_{A}=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.11)

We have

$\mathrm{Tr}({\rho_{A}}^{n})=Z(S^{2}\times S^{1})=1.$ (4.12)

On the other hand, the calculation in the outside basis is rather non-trivial.
Now we have

$\ket{0}=\hskip
<EMAIL_ADDRESS>2.84544pt=\sum_{a}S_{0a}\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.13)

where the bipartition also transforms with respect to the rewiring of sites as
discussed in Sec. 2. We obtain the reduced density matrix $\rho_{A}$ by gluing
$B$ to $B^{\prime}$,

$\rho_{A}=\sum_{a,b}S_{0a}S_{0\bar{b}}\hskip
5.69046pt\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[height=113.81102pt]{rho_A_out_3.png}}}}}}}$}=\sum_{a,b}S_{0a}S_{0b}\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.14)

Here the first graph in the left-hand side of Eq. (4.14), we flatten our solid
torus to be an annulus for simplicity. The glued region indicated by color
gray is $S^{2}\times D^{1}$ with two copies of $S^{2}$ boundaries. Also, each
half solid torus ($A$ and $A^{\prime}$) has boundary being two copies of
$D^{2}$. Therefore, two $D^{2}$ of separate $A$ and $A^{\prime}$ then be glued
to form $S^{2}$ which matches each side of the boundary of the $S^{2}\times
D^{1}$. The right-hand side of Eq. (4.14) is the view at the angle parallel to
the radial direction of the circles (side view). For example,

$\mathrm{Tr}{\rho_{A}}^{2}=\hskip
<EMAIL_ADDRESS>2.84544pt=\hskip
<EMAIL_ADDRESS>2.84544pt$ (4.15)

With this notation, we have

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{\begin{subarray}{c}a_{1},\dots,a_{n}\\\
b_{1},\dots,b_{n}\end{subarray}}\big{(}\prod_{i=1}^{n}\frac{d_{a_{i}}}{D}\frac{d_{b_{i}}}{D}\big{)}\hskip
<EMAIL_ADDRESS>2.84544pt$ (4.16)

We should note that each double line now represent $S^{2}\times D^{1}$ which
is different from we have seen in Eq. (4.7). Each $S^{2}\times D^{1}$ has two
disconnected $S^{2}$ boundaries and each $S^{2}$ boundary is separately
connected summed to other corresponding $S^{2}$ boundaries. Next, we flatten
the configuration by pushing all the handles ($S^{2}\times D^{1}$) in to the
same plane in the following fashion

<EMAIL_ADDRESS>2.84544pt=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.17)

The right-land side of Eq. (4.17) is a connected sum of $S^{2}\times S^{1}$,
where each hole represent a copy of $S^{2}\times S^{1}$. The anyon lines are
given from left to right in the order
$b_{1},a_{2},...,b_{k-1},a_{k},...b_{n-1},a_{n},b_{n}$ running in the same
direction and $a_{1}$ winds all the holes. For example, one can refer to the
left-hand side of Eq. (4.15) and deform it to the right-land side of Eq.
(4.17) with three holes. We then apply the method of surgery by inserting
$2n-2$ copies of $S^{3}$ with $a_{1}$ anyon loop as in Eq. (3.15). Therefore,
we can obtain that

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{\begin{subarray}{c}a_{1},\dots,a_{n}\\\
b_{1},\dots,b_{n}\end{subarray}}\prod_{i=1}^{n}\frac{d_{a_{i}}}{D}\frac{d_{b_{i}}}{D}\delta_{\bar{a_{1}}a_{i}}\delta_{\bar{a_{1}}b_{i}}\Big{(}\frac{D}{d_{a_{1}}}\Big{)}^{2n-2}=\sum_{a_{1}}\frac{d_{a_{1}}^{2}}{D^{2}}=1$
(4.18)

Although the intermediate steps are less trivial, the final results using the
outside basis and inside basis match.

####  4.2.2 Two $S^{1}$ interfaces being the meridians of the torus

Next, let us consider the vacuum state whose bipartition contain two meridians
of the torus

$\ket{0}=\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.19)

In this case, we have the reduced density matrix

$\rho_{A}=\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.20)

where we flatten the tori similar to previous section. Following the same
procedure, we then obtain the same configuration as equation Eq. (4.17) but
without the anyon lines. The final result is then a connected sum of $2n-1$
copies of $S^{2}\times S^{1}$. Therefore, we apply the method of surgery by
inserting $2n-2$ copies of $S^{3}$ and get

$\mathrm{Tr}{(\rho_{A}}^{n})=\frac{Z(S^{2}\times
S^{1})^{2n-1}}{Z(S^{3})^{2n-2}}=D^{2n-2}.$ (4.21)

Now, we do the same calculation but using the outside basis instead. The
change of basis is

$\ket{0}=\hskip
<EMAIL_ADDRESS>2.84544pt=\sum_{a}S_{0a}\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.22)

Now, we have

$\rho_{A}=\sum_{a,b}S_{0a}S_{0b}\hskip
<EMAIL_ADDRESS>2.84544pt,$ (4.23)

and

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{a_{1},b_{1}}\sum_{a_{2},b_{2}}\dots\sum_{a_{n},b_{n}}\prod_{k=1}^{n}\frac{d_{a_{k}}d_{b_{k}}}{D^{2}}\hskip
<EMAIL_ADDRESS>2.84544pt$ (4.24)

The partition function of $S^{2}\times S^{1}$ with anyon lines $a_{k}$,
$b_{k}$ in its non-contractible loop is given by
$\mathrm{dim}V_{a_{1},b_{1},\dots a_{n},b_{n}}$. Equating this to Eq. (4.21),
we then obtain

$\sum_{a_{1},b_{1}}\sum_{a_{2},b_{2}}\dots\sum_{a_{n},b_{n}}d_{a_{1}}d_{b_{1}}\dots,d_{a_{n}}d_{b_{n}}\mathrm{dim}V_{a_{1},b_{1},\dots
a_{n},b_{n}}=D^{4n-2}.$ (4.25)

This formula tells us that if we sum over the product of quantum dimensions of
all the possible arrangements of $2n$ anyons weighted by the dimensions of the
fusion trees $\mathrm{dim}V_{a_{1},b_{1},\dots,a_{n},b_{n}}$, then we obtain
the total quantum dimension $D^{4n-2}$. Eq. (4.25) reminds us of the Verlinde
formula Ver ; genVer , which also relates the quantum dimensions to the
modular data. For example, in the case where $n=1$,
$\mathrm{dim}V_{a_{1},b_{1}}=\delta_{a_{1},b_{1}}$, and thus we have the
definition of quantum dimension $\sum_{a}d_{a}^{2}=D^{2}$. Although not quite
obvious, one can also derive Eq. (4.25) using the Verlinde formula, which we
will discuss in appendix A.

###  4.3 General Torus knot bipartitions

Now that we have established the consistency of entropy calculation using
different coordinates, we can now compute the entanglement for more
complicated configurations. In particular, we will consider bipartitions with
two $S^{1}$ interfaces, both being torus knots. We will refer to such
bipartitions as torus knot bipartitions throughout this paper. To visualize a
torus knot bipartition, one can imagine a single torus knot on a torus, as in
Fig. 3, and widen the line to become a ribbon that defines the subregion.

A general torus knot can be written as $K(k_{a},k_{b})$, where $k_{a}$ and
$k_{b}$ are integers with $\mathrm{gcd}(k_{a},k_{b})=1$. Let $K(1,0)$ and
$K(0,1)$ denote the meridians and the longitudes, then $K(k_{a},k_{b})$ is a
torus knot that winds $k_{a}$ and $k_{b}$ times in the meridian and longitude
direction, respectively. For example, Fig. 3 shows a torus knot $K(8,3)$.

Figure 3: Torus knot $K(8,3)$ embedded in $\mathbb{R}^{3}$ and on its
fundamental domain.

Suppose we have a bipartition $T^{2}=A\amalg B$, with interfaces $\partial
A=\partial B$ composed of two copies of $S^{1}$, then the two $S^{1}$ should
be the same torus knot to avoid intersecting with each other. Therefore, we
can uniquely determine a torus knot bipartitions by the type of torus knots
its interfaces are made of.

Let us consider a vacuum state with no Wilson lines inserted and a torus knot
bipartition with the interface being the torus knot $K(k_{a},k_{b})$ with
$\mathrm{gcd}(k_{a},k_{b})=1$. By applying modular $S$ and $T$
transformations, one can transform $K(k_{a},k_{b})$ into $K(k_{b},k_{a})$ and
$K(k_{a},k_{b}+k_{a})$, respectively. Therefore, using the Euclidean
algorithm, one can transform any torus knot into the meridian $K(1,0)$ by a
series of modular $S$ and $T$ transformations.

Figure 4: The modular $S$ and $T$ transformation generate the whole modular
group $SL(2,\mathbb{Z})$.

That is, all the possible torus knot bipartitions can be mapped back to the
canonical bipartition consist of meridian interfaces by some $\mathcal{O}\in
SL(2,\mathbb{Z})$. For example, $K(8,3)$ can be turned into the canonical
bipartition by the sequentially application of $ST^{2}S^{-1},T,ST^{2}S^{-1},S$
on the torus. The combined transformation is then given by
$\mathcal{O}=SST^{2}S^{-1}TST^{2}S^{-1}=(ST)^{-1}$. With the expression of
$\mathcal{O}$, we rewrite

$\ket{0}=\mathcal{O}\mathcal{O}^{-1}\hskip
<EMAIL_ADDRESS>2.84544pt=\sum_{a}\mathcal{O}_{0a}\hskip
<EMAIL_ADDRESS>2.84544pt.$ (4.26)

The gray line in the left figure of Eq. (4.26) indicate the subregion $A$ and
the solid torus has no Wilson lines insert. In the right figure of Eq. (4.26),
the coordinate grid is twisted compared to the left figure. However, we can
perform exactly the same replica method in the new coordinate system as long
as all the copies are in the same coordinate. Following similar calculation in
the previous sections, we then obtain that

$\mathrm{Tr}{(\rho_{A}}^{n})=\sum_{\begin{subarray}{c}a_{1},\dots,a_{n}\\\
b_{1},\dots,b_{n}\end{subarray}}\prod_{i=1}^{n}\mathcal{O}_{0a_{i}}\mathcal{O}^{*}_{0b_{i}}\delta_{a_{1}a_{i}}\delta_{a_{1}b_{i}}\Big{(}\frac{D}{d_{a_{1}}}\Big{)}^{2n-2}=\sum_{a}\frac{|\mathcal{O}_{0a}|^{2n}}{(S_{0a})^{2n-2}}$
(4.27)

Taking the limit $n\rightarrow 1$, we then obtain the topological entanglement
entropy. One interesting observation is that if we define
$\psi_{a}=\mathcal{O}_{0a}$, then the TEE is given by

$S_{A}=\sum_{a}2|\psi_{a}|^{2}(\ln d_{a}-\ln|\psi_{a}|)-2\ln D.$ (4.28)

This is exactly the same as computed in EdgeTEE  for a generic ground state in
the edge theory approach. That is, although we start with a vacuum state, the
twisted bipartition induces an effective state which is no longer vacuum. In
general, if we start with a state $\ket{\psi}=\sum_{a}\psi_{a}\ket{a}$ with a
torus knot bipartition, which can be rotated back to the canonical bipartition
by an operator $\mathcal{O}$, the effective state is then given by
$\ket{\psi^{\prime}}=\sum_{a,b}\mathcal{O}_{ab}\psi_{b}\ket{a}$. For our
$K(8,3)$ example where the initial state has no Wilson lines insert, the
effective state is given by
$\ket{\psi^{\prime}}=\sum_{a,b}(ST)^{-1}_{ab}\delta_{0b}\ket{a}=\sum_{a}T^{*}_{aa}S_{0a}\ket{a}$.
The topological entanglement entropy is then given by

$S_{A}=\sum_{a}2(S_{0a})^{2}(\ln d_{a}-\ln S_{0a})-2\ln D=0.$ (4.29)

We have shown that the entropy for an arbitrary ground state and an arbitrary
torus knot bipartition can be written as Eq. (1.1), where the ground state TEE
is given by

$S_{\mathrm{gs}}=\sum_{a}2|\psi_{a}|^{2}(\ln d_{a}-\ln|\psi_{a}|),$ (4.30)

for a given state $\ket{\psi}=\sum_{a}\psi_{a}\ket{a}$ depending on the twists
of bipartition. This ground state TEE is bounded by555This fact has also been
mentioned in EdgeTEE  for a generic ground states. We extend their result to
generic ground states in generic bipartitions. A proof can be seen in appendix
B.

$0\leq S_{\mathrm{gs}}\leq 2\ln D.$ (4.31)

That is, in the context of TQFT, $-2\ln D$ shown in TEE ; TEE2  is the lower
bound for the TEE of any bipartition independent of the anyon type, ground
state, and the bipartition as long as the number of interfaces is fixed. The
TEE correction due to the ground state and the Dehn twist of bipartition will
always be non-negative.

##  5 Conclusion

In this paper, we apply the invariant property of the entanglement quantities
under coordinate transformations for generic torus knots bipartitions. We
demonstrate a Verlinde-like formula can be derived. We find the TEE of torus
knots bipartitions with twists can be decomposed into a state independent part
which depends only on the interface and a non-negative correction caused by
effective Wilson lines inserted into the system. As a final remark, in the
follow-up paper CYL , we will discuss a general decomposition of the TEE

$S_{\mathrm{TEE}}(A,\psi)=S_{\mathrm{min}}(A)+S_{\mathrm{gs}}(A,\psi),$ (5.1)

where

$S_{\mathrm{min}}(A)=\min_{\ket{\psi}}S_{\mathrm{TEE}}(A,\ket{\psi})\leq 0,$
(5.2)

is a universal topological quantity that depends only on the number of
interfaces of the bipartition, and $S_{\mathrm{gs}}\geq 0$ is a quantity that
detects effective ground states caused by the twisted bipartition or the
Wilson loop inserted in the system. Overall, the value of this correction term
will never exceed the absolute value of the universal term, ensuring that the
total TEE, i.e., $S_{\mathrm{TEE}}$ is always non-positive. Furthermore, we
can also show that $S_{\mathrm{gs}}$ behaves just like the usual entanglement
entropy, as it is non-negative and satisfies both the strong subadditivity and
the subadditivity. $S_{\mathrm{TEE}}$, on the other hand, is always non-
positive and only satisfies the strong subadditivity.

##  6 Acknowledgments

We are grateful to Xueda Wen for useful discussions. P.-Y.C. acknowledges
support from the National Science and Technology Council of Taiwan under
Grants No. NSTC 112-2636-M-007-007 and No. 112-2112-M-007-043. Both P.-Y.C and
C.-Y. L. thank the National Center for Theoretical Sciences, Physics Division
for their support.

##  Appendix A The Verlinde-like formula

Here, we derive Eq. (4.25) by using the Verlinde formula. The Verlinde formula
states that the fusion rules can be simultaneously diagonalized by the modular
$S$ matrices Ver

$\mathrm{dim}V^{a}_{bc}=(N_{c})^{a}_{b}=(N_{b})^{a}_{c}=\sum_{d}\frac{S_{bd}S_{cd}S^{*}_{da}}{S_{0d}}.$
(A.1)

One can generalize this to $n$ anyons by factorizing the fusion trees

$V^{0}_{a_{1},\dots a_{n}}=\sum_{c_{2},\dots
c_{n-1}}\prod_{i=2}^{n}V^{c_{i}}_{c_{i-1}a_{i}},$ (A.2)

where $c_{n-1}=0$ and $c_{1}=a_{1}$. Substituting the Verlinde formula into
the above expression, we then obtain

$\mathrm{dim}V^{0}_{a_{1},\dots
a_{n}}=\sum_{\begin{subarray}{c}c_{2},\dots,c_{n-1}\\\
d_{2},\dots,d_{n}\end{subarray}}\prod_{i=2}^{n}\frac{S_{a_{i}d_{i}}S_{c_{i-1}d_{i}}S^{*}_{d_{i}c_{i}}}{S_{0d_{i}}}.$
(A.3)

Sum over the internal $c_{i}$’s for $i=2,\dots,n-1$ with
$\sum_{c_{i}}S^{*}_{d_{i}c_{i}}S_{c_{i}d_{i+1}}=\delta_{d_{i}d_{i+1}}$, one
has

$\displaystyle\mathrm{dim}V^{0}_{a_{1},\dots a_{n}}$
$\displaystyle=\sum_{d_{2},\dots,d_{n}}S_{c_{1}d_{2}}\Big{(}\prod_{i=2}^{n-1}\frac{S_{a_{i}d_{i}}\delta_{d_{i}d_{i+1}}}{S_{0d_{i}}}\Big{)}\frac{S_{a_{n}d_{n}}S^{*}_{c_{n-1}d_{n}}}{S_{0d_{n}}}$
(A.4)
$\displaystyle=\sum_{d}\Big{(}\prod_{i=1}^{n}\frac{S_{a_{i}d}}{S_{0d}}\Big{)}(S_{0d})^{2}.$

This is the Verlinde-like formula discussed in genVer . Now, substituting in
$S_{0a}=S^{*}_{0a}=d_{a}$, $S_{00}^{-1}=D$ and $S^{2}=C$, the charge
conjugation operator, one arrive at

$\sum_{a_{1},\dots,a_{n}}\big{(}\prod_{i=1}^{n}d_{a_{i}}\big{)}\mathrm{dim}V^{0}_{a_{1},\dots
a_{n}}=\sum_{d}\Big{(}\prod_{i=1}^{n}\frac{C_{0d}}{S_{00}S_{0d}}\Big{)}(S_{0d})^{2}=D^{2n-2}.$
(A.5)

This is exactly equation Eq. (4.25) by setting $n$ to $2n$ due to the presence
of both $a_{k}$ and $b_{k}$.

##  Appendix B The bounds on ground state TEE

Here, we give a brief proof of Eq. (4.31). Since we are extremizing under the
constraint $\sum_{a}|\psi_{a}|^{2}=1$, we can employ the method of Lagrange
multiplier by defining

$\tilde{S}_{\mathrm{gs}}=\sum_{a}2|\psi_{a}|^{2}(\ln
d_{a}-\ln|\psi_{a}|)-\lambda(\sum_{a}|\psi_{a}|^{2}-1).$ (B.1)

Differentiating with respect to $|\psi_{a}|$ yields

$2(\ln d_{a}-\ln|\psi_{a}|)-1=2\lambda,$ (B.2)

which normalizes to $|\psi_{a}|=S_{0a}$. That is, the extremum of
$S_{\mathrm{gs}}$ happens only if $|\psi_{a}|=S_{0a}$ or $|\psi_{a}|$ takes
boundary values . Therefore, one then verify that $S_{\mathrm{gs}}$ takes
minimum $S_{\mathrm{gs}}=0$ when $|\psi_{a}|=\delta_{a0}$ and takes maximum
$S_{\mathrm{gs}}=2\ln D$ when $|\psi_{a}|=S_{0a}$ for all $a$.

## References

  * (1) E. Witten, _Quantum Field Theory and the Jones Polynomial_ , _Commun.Math.Phys._ 121 (1989) .
  * (2) S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, _Remarks on the canonical quantization of the Chern-Simons-Witten theory_ , _Nucl.Phys.B_ 326 (1989) .
  * (3) D. Tsui, H. Stormer and A. Gossard, _Two-dimensional magnetotransport in the extreme quantum limit_ , _Phys. Rev. Lett._ 48 (1982) .
  * (4) R. Laughlin, _Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations_ , _Phys. Rev. Lett._ 50 (1983) .
  * (5) X.-G. Wen, _Topological orders and Edge excitations in FQH states_ , _Advances in Physics_ 44 (1995) .
  * (6) H. Li and F. D. M. Haldane, _Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States_ , _Phys. Rev. Lett._ 101 (2008) .
  * (7) X.-G. Wen, _Quantum Orders and Symmetric Spin Liquids_ , _Phys. Rev. B_ 65 (2002) .
  * (8) D. S. Rokhsar and S. A. Kivelson, _Superconductivity and the Quantum Hard-Core Dimer Gas_ , _Phys. Rev. Lett._ 61 (1988) .
  * (9) M. Stone and S. B. Chung, _Fusion rules and vortices in px+ipy superconductors_ , _Phys. Rev. B_ 73 (2006) .
  * (10) X. Weng, _Topological Order in Rigid States_ , _Int.J.Mod.Phys.B_ 4 (1990) .
  * (11) A. Kitaev and J. Preskill, _Topological entanglement entropy_ , _Phys.Rev.Lett._ 96 (2006) .
  * (12) M. Levin and X.-G. Wen, _Detecting topological order in a ground state wave function_ , _Phys.Rev.Lett._ 96 (2006) .
  * (13) M. Srednicki, _Entropy and Area_ , _Phys.Rev.Lett._ 71 (1993) .
  * (14) M. Haque, O. Zozulya and K. Schoutens, _Entanglement entropy in fermionic Laughlin states_ , _Nature Physics_ 7 (2007) .
  * (15) H.-C. Jiang, Z. Wang and L. Balents, _Identifying Topological Order by Entanglement Entropy_ , _Nature Physics_ 8 (2012) .
  * (16) H. Li and F. D. M. Haldane, _Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States_ , _Phys. Rev. Lett._ 101 (2008) .
  * (17) H. Li and F. D. M. Haldane, _Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States_ , _Phys. Rev. Lett._ 101 (2008) .
  * (18) S. V. Isakov, M. B. Hastings and R. G. Melko, _Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid_ , _Nature Physics_ 7 (2011) .
  * (19) S. Dong, E. Fradkin, R. G. Leigh and S. Nowling, _Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids_ , _JHEP_ 016 (2008) .
  * (20) X. Wen, P.-Y. Chang and S. Ryu, _Topological entanglement negativity in Chern-Simons theories_ , _JHEP_ 09 (2016) .
  * (21) R. Sohal and S. Ryu, _Entanglement in tripartitions of topological orders: a diagrammatic approach_ , _Phys. Rev. B_ 108 (2023) .
  * (22) X. Wen, S. Matsuura and S. Ryu, _Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories_ , _Phys. Rev. B_ 93 (2016) .
  * (23) T. Nishioka, T. Takayanagi and Y. Taki, _Topological pseudo entropy_ , _JHEP_ 09 (2021) .
  * (24) P. Bonderson, C. Knapp and K. Patel, _Anyonic Entanglement and Topological Entanglement Entropy_ , _Annals of Physics_ 385 (2017) .
  * (25) S. Dwivedi, V. K. Singh, P. Ramadevi, Y. Zhou and S. Dhara, _Entanglement on multiple S2 boundaries in Chern-Simons theory_ , _JHEP_ 08 (2019) .
  * (26) J. R. Fliss, X. Wen, O. Parrikar, C.-T. Hsieh, T. L. H. Bo Han and R. G. Leigh, _Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory_ , _JHEP_ 09 (2017) .
  * (27) Y. Zhang, T. Grover, A. Turner, M. Oshikawa and A. Vishwanath, _Quasi-particle Statistics and Braiding from Ground State Entanglement_ , _Phys. Rev. B_ 85 (2012) .
  * (28) E. P. Verlinde, _Fusion Rules and Modular Transformations in 2D Conformal Field Theory_ , _Nucl.Phys.B_ 300 (1988) .
  * (29) G. W. Moore and N. Seiberg, _Classical and Quantum Conformal Field Theory_ , _Commun.Math.Phys._ 123 (1989) .
  * (30) C.-Y. Lo and P.-Y. Chang, _Strong subadditivity for topological entanglement entropy_ , _in preparation_ .


# Generalized Uncertainty Principle from the Regularized Self-Energy

Kimet Jusufi⊙<EMAIL_ADDRESS>Ahmed Farag Ali△∇<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>⊙ Physics Department, State University of Tetovo,
Ilinden Street nn, 1200, Tetovo, North Macedonia  △Essex County College, 303
University Ave, Newark, NJ 07102, United States.  ∇Department of Physics,
Faculty of Science, Benha University, Benha, 13518, Egypt.

###### Abstract

We use the Schrödinger–Newton equation to calculate the regularized self-
energy of the particle using a regular self-gravitational and electrostatic
potential derived in the string T-duality. The particle mass $M$ is no longer
concentrated into a point but it is diluted and described by a quantum-
corrected smeared energy density resulting in corrections to the energy of the
particle which is interpreted as a regularized self-energy. We extend our
results and find corrections to the relativistic particles using the Klein-
Gordon, Proca, and Dirac equations. An important finding is that we extract a
form of generalized uncertainty principle (GUP) from the corrected energy. The
form of GUP is shown to depend on the nature of particles; namely, for bosons
(spin $0$ and spin $1$) we obtain a quadratic form of GUP, while for fermions
(spin $1/2$) we obtain a linear form of GUP. The correlation we found between
spin and GUP may offer insights into investigating quantum gravity.

.

##  I Introduction

A significant challenge in the field of physics remains unresolved, which is
how to integrate the principles of quantum mechanics and gravity into a
quantum theory of gravity. The fundamental equation of gravity is Einstein’s
field equation Einstein (1916). The fundamental equation in quantum mechanics
is Schrödinger’s equation Schrödinger (1926), while Klein-Gordon equation
Gordon (1926); Klein (1926), and Dirac equation Dirac (1928), are viewed as
relativistic field equations. A modification to Schrödinger’s equation was
suggested by Diósi and Penrose Diósi (1984); Diosi (1987); Penrose (1996);
Diosi (1989); Penrose (1998, 2014) to include the force of gravity. It is
known as the Schrödinger-Newton equation. This new term takes into account the
gravitational potential generated by the distribution of mass in the system.
The effect can be generalized for the Dirac equation, Klein-Gordon equation,
and Proca equation Proca (1936).

In this work, we would like to study a specific form of gravitational
potential that arises from string T-duality Sathiapalan (1987); Strominger
_et al._ (1996) with these equations. On the other hand, a new spin forward
was made by Padmanabhan who related the concept of duality and zero point
length as a tool to obtain quantum gravity effect Padmanabhan (1997) and was
applied to black hole physics to cure the black hole singularity for regular
and charged solutions Nicolini  _et al._ (2019); Gaete  _et al._ (2022);
Nicolini (2022). In Ref. Jusufi (2023) one of the authors of the present work
argued how the zero-point length which modifies the black hole spacetime can
lead to a non-singular solution during the gravitational collapse. In the
context of cosmology, however, it was found a modified Friedmann’s equation
Jusufi and Sheykhi (2023) and in particular, it was found how the zero-point
length corrections can lead to a bouncing scenario (see, Jusufi and Sheykhi
(2023); Millano  _et al._ (2023)). Such a potential is regular at $r=0$ and
modifies the energy of the system. In this picture, the particle is no longer
concentrated into a point but it is diluted along with smeared energy density.
The gravitational potential implied by T-duality suggests a modification of
energy that agrees with the modifications implied by generalized uncertainty
principle models Amati  _et al._ (1989); Ali  _et al._ (2009); Kempf  _et al._
(1995); Brau (1999); Maggiore (1993); Adler (1994), see also Carr  _et al._
(2015); Carr (2022). We realize a new connection between the spin concept of
particles and these energy modifications.

Phenomenological and experimental implications of the GUP have been
investigated in low and high-energy regimes. These include atomic systems Ali
_et al._ (2011); Das and Vagenas (2008), quantum optical systems Pikovski  _et
al._ (2012), gravitational bar detectors Marin  _et al._ (2013), gravitational
decoherence Petruzziello and Illuminati (2021), gravitational waves Das  _et
al._ (2021); Moussa  _et al._ (2021), composite particles Kumar and Plenio
(2020), astrophysical systems Moradpour  _et al._ (2019), condensed matter
systems Iorio  _et al._ (2018), cold atoms Gao and Zhan (2016), macroscopic
harmonic oscillators Bawaj  _et al._ (2015), gauge theories Kober (2010),
quantum noise Girdhar and Doherty (2020) and cosmological models Ashoorioon
_et al._ (2005); Easther  _et al._ (2003); Dabrowski and Wagner (2019); Ali
and Majumder (2014); Easther  _et al._ (2001); Ali  _et al._ (2015). Reviews
of the GUP, its phenomenology, and its experimental implications can be found
in Addazi  _et al._ (2021); Hossenfelder (2013). Our work establishes a
connection between T-duality in string theory and the generalized uncertainty
principle. Among other things, in this work, we would like to understand if
the form of GUP depends on the nature of particles, for example, the spin. It
is possible to interpret that the experimental and phenomenological research
on GUP provides support in favor of T-duality and string theory.

The paper is organized as follows. In Sec. II, we study the modified
Shrodinger-Newton equation to compute the corrections to the energy of the
particle. In Sec. III, we compute the self-regularized energy due to the
electric field. In Sec. IV, we study the modified Klein-Gordon equation and
the relativistic energy corrections. In Sec. V, we investigate the modified
Dirac equation and its energy corrections. In Sec. VI, we study the modified
Proca equation. In Sec. VII, we show a link between the GUP and self-
regularized energy. Importantly, here we shall elaborate on how the nature of
particles, say the spin of the particle, can lead to a different form of GUP.
Finally in Sec. VIII, we comment on our results.

##  II Modified Schrödinger–Newton equation

We shall review the quantum-corrected static interaction potential according
to the field theory enjoying the path integral duality. We recall that the
momentum space propagator induced by the path integral duality for the
massless case is given by Nicolini  _et al._ (2019)

$G(k)=-\frac{l_{0}}{\sqrt{k^{2}}}\,K_{1}\\!\left(l_{0}\sqrt{k^{2}}\right)$ (1)

with $l_{0}$ being the zero-point length and $K_{1}\\!\left(x\right)$ is a
modified Bessel function of the second kind. One has two cases: at small
momenta, i.e. for ${(l_{0}k)}^{2}\to 0$, we obtain the conventional massless
propagator $G(k)=-k^{-2}$, at large momenta, interestingly, the exponential
suppression is responsible for curing UV divergences Nicolini  _et al._ (2019)

$G(k)\sim-l_{0}^{1/2}\,{\left(k^{2}\right)}^{-3/4}\,e^{-l_{0}\sqrt{k^{2}}}.$
(2)

Consider a static external source $J$ which consists of two point-like masses,
$m$ and $M$, at relative distance $\vec{r}$, then for the potential it was
found Nicolini  _et al._ (2019)

$\displaystyle V(r)$ $\displaystyle=-\frac{1}{m}\,\frac{W[J]}{T}$ (3)
$\displaystyle=-M\,\int\\!\frac{d^{3}k}{{\left(2\pi\right)}^{3}}\;{G_{F}(k)|}_{k^{0}=0}\;\exp\\!\left(i\vec{k}\vec{r}\right)$
(4) $\displaystyle=-\frac{M}{\sqrt{r^{2}+l_{0}^{2}}},$ (5)

In this section, we will use the Schrödinger–Newton equation to obtain the
modified energy-time uncertainty relation. The Schrödinger–Newton equation for
a free particle with the self-gravitational potential $V$ produced by a
quantum source in state $\Psi$ can be written as Diósi (1984); Diosi (1987);
Penrose (1996); Diosi (1989); Penrose (1998, 2014)

$i\hbar\frac{\partial\Psi(t,\vec{r})}{\partial
t}=\left(-\frac{\hbar^{2}}{2M}\nabla^{2}-M^{2}G\int\frac{|\Psi(t,\vec{r}^{\prime})|^{2}d^{3}\vec{r}^{\prime}}{\vec{r}-\vec{r}^{\prime}}\right)\Psi(t,\vec{r})\,,$
(6)

studied by Diósi and Penrose Diósi (1984); Diosi (1987); Penrose (1996); Diosi
(1989); Penrose (1998, 2014). Along with the semiclassical Poisson equation

$\nabla^{2}V=4\pi GM|\Psi|^{2}$ (7)

The mass density of the source is related to the quantum probability density
via

$\displaystyle\rho(r,t)=M|\Psi(t,\vec{r}^{\prime})|^{2}.$ (8)

Our aim is to use a well-behaved expression without a singularity at $r=0$.
Such regularised expression was found in the string T-duality with the
gravitational potential given by Nicolini  _et al._ (2019)

$\displaystyle V(r)\to-\frac{GM}{\sqrt{r^{2}+l_{0}^{2}}},$ (9)

where $l_{0}$ is the zero point length. The modified potential suggests a
modification to the Schrödinger–Newton as follows

$i\hbar\frac{\partial\Psi(t,\vec{r})}{\partial
t}=\left(-\frac{\hbar^{2}}{2M}\nabla^{2}-M^{2}G\int\frac{|\Psi(t,\vec{r})|^{2}d^{3}\vec{r}}{\sqrt{r^{2}+l_{0}^{2}}}\right)\Psi(t,\vec{r}).$
(10)

On one hand, we see that when $l_{0}=0$, the standard relation is obtained. On
the other hand, we can use a coupling of gravity to matter by means of the
semi-classical Einstein equations Bahrami  _et al._ (2014)

$R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}R=\frac{8\pi
G}{c^{4}}\,\bra{\Psi}T_{\mu\nu}\ket{\Psi}.$ (11)

In the Newtonian limit, the components of the energy-momentum tensor become
the Poisson equation Bahrami  _et al._ (2014)

$\nabla^{2}V=\frac{4\pi G}{c^{2}}\bra{\Psi}\hat{T}_{00}\ket{\Psi}.$ (12)

In the Newtonian limit, where $T_{00}$ is the dominant term of the energy-
momentum tensor, the interaction Hamiltonian then becomes Bahrami  _et al._
(2014)

$\displaystyle H_{\text{int}}$ $\displaystyle=$ $\displaystyle\int
d^{3}r\,V\,T^{00}$ (13) $\displaystyle=$ $\displaystyle-
MG\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}d^{3}r\frac{\rho(\vec{r})}{\sqrt{r^{2}+l_{0}^{2}}}\,,$
(14)

whereby using $V(r)$ and Poisson equation where $\rho(r)$ is given by Nicolini
_et al._ (2019)

$\rho(r)=\frac{1}{4\pi}\nabla^{2}V(r)=\frac{3l_{0}^{2}M}{4\pi\left(r^{2}+l_{0}^{2}\right)^{5/2}}.$
(15)

Using spherical coordinates

$d^{3}r=r^{2}\sin^{2}\theta\,d\theta\,d\phi\,dr,$ (16)

and integrating this equation we obtain a finite contribution

$H_{\text{int}}=-\frac{3\pi GM^{2}}{16\,l_{0}}.$ (17)

Therefore the apparently nonlinear form of the Schrödinger–Newton equation
reduces to the linear form

$i\hbar\frac{\partial}{\partial
t}\Psi(t,\vec{r})=-\frac{\hbar^{2}}{2M}\nabla^{2}\Psi(t,\vec{r})+\hat{H}_{int}\Psi(t,\vec{r}),$
(18)

or in terms of operators

$\displaystyle\hat{H}\Psi(t,r)$ $\displaystyle=$
$\displaystyle\left(-\frac{\hbar^{2}}{2M}\nabla^{2}+\hat{H}_{int}\right)\Psi(t,r),$
(19)

and if we define $\hat{H}_{tot}=\hat{H}-\hat{H}_{int}$, we can write

$\displaystyle\hat{H}_{tot}\Psi(t,r)$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2M}\nabla^{2}\Psi(t,r),$ (20)

yielding

$\displaystyle E_{tot}\Psi(t,r)$ $\displaystyle=$
$\displaystyle\left(E+\frac{3\pi GM^{2}}{16\,l_{0}}\right)\Psi(t,r),$ (21)

here $E$ can be interpreted as the bare energy of the free particle and the
wave function of the particle is now

$\Psi=A\exp(-\frac{i}{\hbar}(E_{tot}t-\vec{p}\cdot\vec{r})),$ (22)

meaning that the total energy in the weave function is modified.

###  II.1 Significance of effects in the SN equation

The Shrodinger-Newton equation reduces to
$E_{tot}\Psi(t,r)=\frac{p^{2}}{2M}\Psi(t,r)$. Consider a free particle
solution described by the initial Gaussian wave function

$\Psi(r,t)=\left(\pi\,a^{2}\right)^{-3/4}\,\left(1+\frac{i\,\hbar\,t}{M\,a^{2}}\right)^{-3/2}\,e^{-\frac{r^{2}}{2a^{2}\left(1+\frac{i\,\hbar\,t}{M\,a^{2}}\right)}},$
(23)

and note that the peak of $r^{2}\,\Psi^{*}(r,t)\,\Psi(r,t)$ is located at

$r_{p}(t)=\sqrt{a^{2}+\left(\frac{\hbar\,t}{a\,M}\right)^{2}}.$ (24)

Now let us set the acceleration

$\ddot{r}_{p}(0)=\frac{\hbar^{2}}{a^{3}\,M^{2}},$ (25)

while the peak probability equal to the acceleration due to Newtonian gravity

$\ddot{r}_{p}(0)=-\frac{d}{dr}\,V(r_{p}(0))=\frac{M\,a}{(l_{0}^{2}+a^{2})^{3/2}}.$
(26)

From the last two equations, we get the mass

$\displaystyle
M^{3}\,a=\frac{\hbar^{2}}{G}\left(1+\frac{l_{0}^{2}}{a^{2}}\right)^{3/2}.$
(27)

From the last equation, we get the standard result when $l_{0}=0$, i.e.,
$M^{3}a=\hbar^{2}/G$. This equation allows us to determine a critical width
for a given mass value and conversely. In this way, we can estimate the regime
where the effects of the Schrodinger-Newton are important.

###  II.2 Curved spacetime

We can consider further corrections by considering a curved spacetime
background. In particular, the general solution in the case of a static,
spherically symmetric source reads

$ds^{2}=g_{00}\,dt^{2}+g_{rr}\,dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (28)

with $g_{00}(r)=-f(r)$, and

$\begin{split}f(r)=\left(1-\frac{2Gm(r)}{r}\right)=\left(1-2\Phi(r)\right).\end{split}$
(29)

Using the energy density (15) it was found that

$m(r)=\frac{Mr^{3}}{{\left(r^{2}+l_{0}^{2}\right)}^{3/2}},$ (30)

along with

$\begin{split}f(r)=1-\frac{2GMr^{2}}{{\left(r^{2}+l_{0}^{2}\right)}^{3/2}}.\end{split}$
(31)

In this case, the Schrödinger–Newton equation will be

$i\hbar\frac{\partial\Psi(t,\vec{r})}{\partial
t}=\left(-\frac{\hbar^{2}}{2M}\nabla_{LB}^{2}+H_{int}\right)\Psi(t,\vec{r}).$
(32)

where
$\nabla^{2}_{LB}=g^{-1/2}\partial_{\mu}(g^{\mu\nu}g^{1/2}\partial_{\nu})$ is
the Laplace-Beltrami operator in the curved spacetime geometry, with $g=\det
g_{\mu\nu}$. For the self-gravitational interaction, we get

$\displaystyle H_{\text{int}}$ $\displaystyle=$ $\displaystyle-
MG\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}\Phi(r)\rho(r)d^{3}r$ (33)
$\displaystyle=$ $\displaystyle-\frac{3\pi GM^{2}}{32l_{0}}.$

Compared to Eq. (17) this only suggests an improved correction by a factor
$1/2$.

##  III Regularised self-energy due to the electric field

In T-duality, it was shown that for the electric potential of the system which
consists of two point-like masses, at relative distance $\vec{r}$, one can
write Gaete and Nicolini (2022)

$\displaystyle V_{em}(r)\to-\frac{Q}{4\pi\epsilon_{0}\sqrt{r^{2}+l_{0}^{2}}}.$
(34)

In addition, we have Maxwell’s equation which is written as

$\nabla_{\mu}F^{\mu\nu}=J^{\nu},$ (35)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$J^{\mu}=(c\rho_{em},\vec{j})$ is the four-current. One can check that the
above equations lead to the charge density

$\rho_{em}(r)=\frac{3l_{0}^{2}Q}{4\pi\epsilon_{0}{(r^{2}+l_{0}^{2})}^{5/2}}.$
(36)

The interaction Hamiltonian then becomes

$\displaystyle H^{em}_{\text{int}}$ $\displaystyle=$
$\displaystyle-Q\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}d^{3}r\frac{\rho_{em}(\vec{r})}{4\pi\epsilon_{0}\sqrt{r^{2}+l_{0}^{2}}}\,,$
(37)

Solving the integral we get

$H^{em}_{\text{int}}=-\frac{3\pi Q^{2}}{16\epsilon_{0}l_{0}}.$ (38)

###  III.1 Curved spacetime

If we consider again the effect of spacetime curvature, by using the mass
profile with corrections due to the charge density (36) we obtain

$f(r)=1-\frac{2Qr^{2}}{{\left(r^{2}+l_{0}^{2}\right)}^{3/2}}=\left(1-2\Phi_{em}(r)\right),$
(39)

which does not represent a new solution since the total mass is simply scaled
to $M\to M+Q$. The Schrödinger–Newton equation will be

$i\hbar\frac{\partial\Psi(t,\vec{r})}{\partial
t}=\left(-\frac{\hbar^{2}}{2M}\nabla_{LB}^{2}+H_{int}+H_{int}^{em}\right)\Psi(t,\vec{r}).$
(40)

The interaction Hamiltonian due to the charge density we get

$\displaystyle H^{em}_{\text{int}}$ $\displaystyle=$
$\displaystyle-Q\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\infty}d^{3}r\,\Phi_{em}(r){\rho}_{em},$
(41)

yielding

$H^{em}_{\text{int}}=-\frac{3\pi Q^{2}}{32\epsilon_{0}l_{0}}$ (42)

Compared to (38), we obtain an improved correction by a factor of $1/2$. The
total energy of the particle is

$E_{tot}=E+\frac{3\pi GM^{2}}{32l_{0}}+\frac{3\pi
Q^{2}}{32\epsilon_{0}l_{0}}.$ (43)

This equation shows that the electrostatic field also modifies the energy of
the particle. Here we found the energy stored in the charge via the charged
density and the curved spacetime. It is very interesting to note that the
corrections to the energy due to the electromagnetic field can be found in an
alternative way using the stress-energy tensor for the electromagnetic field.
We will assume that the form of the energy-tensor the electromagnetic field
has the form

$T_{\mu\nu}^{em}=F_{\mu\sigma}{F_{\nu}}^{\sigma}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma},$
(44)

where $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, with

$\displaystyle A_{\mu}=\left(\frac{V_{em}}{c},0,0,0\right).$ (45)

It is easy to show that the ${T^{0}}_{0}$ component of the energy-momentum
tensor for the electromagnetic field gives the energy density

$\rho_{em}=\frac{Q^{2}r^{2}}{8\pi(r^{2}+l_{0}^{2})^{3}}.$ (46)

This means that the energy has to be

$\displaystyle H_{int}^{em}$ $\displaystyle=-$ $\displaystyle\int
d^{3}r\frac{Q^{2}r^{2}}{8\pi(r^{2}+l_{0}^{2})^{3}}$ (47) $\displaystyle=-$
$\displaystyle\frac{3\pi Q^{2}}{32\epsilon_{0}l_{0}},$ (48)

which precisely matches Eq. (42). Such a correction was found also for a
charged black hole in T-duality Gaete  _et al._ (2022).

##  IV Modified Klein-Gordon equation

We can obtain the corrected relativistic energy for the particle with a
different spin. Let us consider here the Klein-Gordon described by the scalar
field $\Phi(x,t)$

$\displaystyle\square\Phi(x,t)-\frac{M^{2}c^{2}}{\hbar^{2}}\Phi(x,t)-\frac{H^{2}_{int}}{\hbar^{2}c^{2}}\Phi(x,t)=0,$
(49)

which is modified due to the self-gravitational interaction term. This
equation can be further written as

$\displaystyle\left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}-\frac{M^{2}c^{2}}{\hbar^{2}}\right)\Phi(x,t)-\frac{H^{2}_{int}}{\hbar^{2}c^{2}}\Phi(x,t)=0.$
(50)

Taking a plane weave solution for the scalar field

$\Phi(x,t)=A\exp(-\frac{i}{\hbar}x^{\mu}p_{\mu}),$ (51)

we get the energy relation given by

$\displaystyle
E^{2}_{tot}=M^{2}c^{4}+p^{2}c^{2}+\frac{9\pi^{2}G^{2}M^{4}}{16^{2}l_{0}^{2}}.$
(52)

We can also rewrite this equation in a more compact form

$\displaystyle E^{2}_{tot}=M_{tot}^{2}c^{4}+p^{2}c^{2}.$ (53)

in terms of the new definition of the mass

$\displaystyle
M^{2}_{tot}=M^{2}\left(1+\frac{9\pi^{2}G^{2}M^{2}}{16^{2}c^{4}l_{0}^{2}}\right).$
(54)

By considering a series expansion, we get

$\displaystyle M_{tot}\simeq
M\left(1+\frac{1}{2}\frac{9\pi^{2}G^{2}M^{2}}{16^{2}c^{4}l_{0}^{2}}\right).$
(55)

Using the fact that $l_{0}\sim l_{Pl}=\sqrt{\frac{\hbar G}{c^{3}}}$, we can
also write

$\displaystyle M_{tot}\simeq M\left(1+\alpha\frac{M^{2}}{M_{Pl}^{2}}\right).$
(56)

where $\alpha=(9/2)\,\pi^{2}/16^{2}$ and $M_{Pl}=\sqrt{\frac{\hbar c}{G}}$ is
the Planck mass. This could be related to gravitational self-completeness with
quantum mechanical mass limits. In particular, we may have two cases:
1) When $M<<M_{Pl}$, i.e., in the particle section then Eq. (56) is valid and
the corrections are small.
2) When $M>M_{Pl}$, we have a connection between elementary particles and
black holes. As was shown Carr  _et al._ (2015); Carr (2022) the Compton-
Schwarzschild correspondence posits a smooth transition between the Compton
wavelength $(R_{C}\sim 1/M)$ below the Planck mass and the Schwarzschild
radius $(R_{S}\sim M)$ above it. The Compton and Schwarzschild lines transform
into one another under the transformation $M\to M^{2}_{Pl}/M$, which suggests
the following form for the mass

$\displaystyle M_{tot}\simeq M\left(1+\alpha\frac{M_{Pl}^{2}}{M^{2}}\right).$
(57)

This relation was recently obtained in Refs. Carr  _et al._ (2015); Carr
(2022).

##  V Modified Dirac equation

Let us consider also here the Dirac equation described by the spinor field
$\psi(x,t)$ which is modified due to the self-gravitational interaction term

$\displaystyle
i\gamma^{\mu}\nabla_{\mu}\psi(x,t)-\frac{Mc}{\hbar}\psi(x,t)+\frac{H_{int}}{\hbar
c}\psi(x,t)=0,$ (58)

where for the $\gamma^{\mu}$ matrices in terms of $2\times 2$ from the Pauli
matrices sub-matrices we can write

$\displaystyle\gamma^{t}=\left(\begin{matrix}I_{2}&0\\\
0&-I_{2}\end{matrix}\right),\,\,\,\gamma^{1}=\left(\begin{matrix}0&\sigma_{x}\\\
-\sigma_{x}&0\end{matrix}\right),$ (59)
$\displaystyle\gamma^{2}=\left(\begin{matrix}0&\sigma_{y}\\\
-\sigma_{y}&0\end{matrix}\right),\,\,\,\gamma^{3}=\left(\begin{matrix}0&\sigma_{z}\\\
-\sigma_{z}&0\end{matrix}\right).$ (60)

If we take the solution for the spinor field

$\displaystyle\psi(x,t)=\left(\begin{matrix}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\\\ \psi_{4}\end{matrix}\right),$ (61)

or, in a different notation

$\displaystyle\psi(x,t)=\left(\begin{matrix}\psi_{A}\\\ \psi_{B}\\\
\end{matrix}\right),\,\,\,\psi_{A}=\left(\begin{matrix}\psi_{1}\\\ \psi_{2}\\\
\end{matrix}\right),\,\,\,\psi_{B}=\left(\begin{matrix}\psi_{3}\\\ \psi_{4}\\\
\end{matrix}\right).$ (62)

It is well known that the solution can be written as

$\displaystyle\psi(x^{\mu})=A\exp\left(-\frac{i}{\hbar}x^{\mu}p_{\mu}\right)\left(\begin{matrix}u_{A}\\\
u_{B}\\\ \end{matrix}\right)$ (63)

then the Dirac equation reduces to

$\displaystyle\left(\gamma^{\mu}p_{\mu}-Mc-\frac{H_{int}}{c}\right)\left(\begin{matrix}u_{A}\\\
u_{B}\\\ \end{matrix}\right)=0,$ (64)

From this equation, we get a system of two equations

$\displaystyle\left(\begin{matrix}\frac{E}{c}-Mc+\frac{H_{int}}{c}&-\vec{p}\cdot\vec{\sigma}\\\
\vec{p}\cdot\vec{\sigma}&\frac{E}{c}-Mc+\frac{H_{int}}{c}\end{matrix}\right)\left(\begin{matrix}u_{A}\\\
u_{B}\\\ \end{matrix}\right)=0.$ (65)

By solving the determinant of this system we get two solutions for the energy

$\displaystyle E_{tot}=\pm\sqrt{M_{tot}^{2}c^{4}+p^{2}c^{2}},$ (66)

where it has been defined

$\displaystyle M^{2}_{tot}c^{4}=\left(Mc^{2}+\frac{3\pi
GM^{2}}{16l_{0}}\right)^{2}.$ (67)

Finally, the last equation can be further written as

$\displaystyle M_{tot}=M\left(1+\frac{3\pi GM}{16c^{2}l_{0}}\right),$ (68)

or

$\displaystyle M_{tot}=M\left(1+\alpha\frac{M}{M_{Pl}}\right),$ (69)

where $\alpha=3\pi/16$. It is rather amazing that the correction to the mass
for fermions (spin 1/2) is linear as can be seen from the last equation. On
the other hand bosons (spin zero) is quadratic as was found in Eq. (56). This
is a surprising result. Again we have two cases:
1) When $M<<M_{Pl}$, then Eq. (69) is valid for the particle sector.
2) When $M>M_{Pl}$, we have a connection between elementary particles and
black holes and we have to use the transformation $M\to M^{2}_{Pl}/M$ Carr
_et al._ (2015); Carr (2022), which suggests the following form for the mass

$\displaystyle M_{tot}\simeq M\left(1+\alpha\frac{M_{Pl}}{M}\right).$ (70)

##  VI Modified Proca equation

Here we shall consider the motion of a massive vector particle of mass $M$,
with spin $1$ described by the vector field $\psi^{\mu}$, which be studied by
the Proca equation (PE), which reads

$\nabla_{\mu}\nabla^{[\mu}\psi^{\nu]}-\frac{M^{2}c^{2}}{\hbar^{2}}\psi^{\nu}-\frac{H_{int}^{2}}{\hbar^{2}c^{2}}\psi^{\nu}=0$
(71)

where in the last term we introduced the corrections due to the self-
regularized energy. Note also that

$\nabla_{[\mu}\psi_{\nu]}=\frac{1}{2}(\nabla_{\mu}\psi_{\nu}-\nabla_{\nu}\psi_{\mu}):=\psi_{\mu\nu}~{}.$
(72)

The solution of the equation can be written as

$\psi_{\nu}(x^{\mu})=C_{\nu}\exp\left(-\frac{i}{\hbar}x^{\mu}p_{\mu}\right)$
(73)

These equations $p^{\mu}=(E/c,\vec{p})$ and can help us to construct a
$4\times 4$ matrix $\aleph$, which satisfies the following matrix equation

$\aleph(C_{0},C_{1},C_{2},C_{3})^{T}=0~{}.$ (74)

Taking for simplicity only one component of the momentum
$\vec{p}=(p_{x},0,0)$, the matrix $\aleph$ has the components

$\displaystyle\aleph=\left(\begin{matrix}\aleph_{00}&0&0&\aleph_{03}\\\
\aleph_{10}&0&0&\aleph_{13}\\\ 0&\aleph_{21}&0&0\\\ 0&0&\aleph_{32}&0\\\
\end{matrix}\right)$ (75)

where

$\displaystyle\aleph_{00}$ $\displaystyle=$
$\displaystyle\frac{Ep}{c^{2}\hbar^{2}},$ (76) $\displaystyle\aleph_{03}$
$\displaystyle=$
$\displaystyle-\frac{M^{2}c^{4}+c^{2}p^{2}-H_{int}^{2}}{c^{2}},$ (77)
$\displaystyle\aleph_{10}$ $\displaystyle=$
$\displaystyle\frac{M^{2}c^{4}-E^{2}+H_{int}^{2}}{c^{2}},$ (78)
$\displaystyle\aleph_{13}$ $\displaystyle=$ $\displaystyle-\frac{Ep}{c^{2}},$
(79) $\displaystyle\aleph_{21}$ $\displaystyle=$
$\displaystyle\aleph_{32}=\frac{M^{2}c^{4}+c^{2}p^{2}-E^{2}+H_{int}^{2}}{c^{2}}.$
(80)

Taking the determinant of this matrix and solving for the energy we get

$\displaystyle E_{tot}=\pm\sqrt{M_{tot}^{2}c^{2}+p^{2}c^{2}}$ (81)

where

$\displaystyle
M^{2}_{tot}=M^{2}\left(1+\frac{9\pi^{2}G^{2}M^{2}}{16^{2}c^{4}l_{0}^{2}}\right),$
(82)

and therefore

$\displaystyle M_{tot}\simeq M\left(1+\alpha\frac{M^{2}}{M_{Pl}^{2}}\right).$
(83)

In other words, the corrections to the energy and mass of the particle with
spin 1 particles described by the Proca equation are similar to to scalar
particles with spin 0 described by the Klein-Gordon equation.

##  VII GUP from regularized self-energy

According to the non-commutative relation between position and momentum, we
have

$\displaystyle\left[x_{i},p_{j}\right]=i~{}\delta_{ij}~{}\hbar,$ (84)

meaning that the position and momentum can not have real eigenvalue for the
same eigenstate. In quantum gravity theories, such as string theory Amati  _et
al._ (1989) the quadratic form of GUP has been suggested and has the following
form Kempf  _et al._ (1995); Brau (1999); Maggiore (1993):

$\displaystyle\Delta x\Delta p$ $\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}(1+\beta\Delta p^{2}),$
$\displaystyle\left[x_{i},p_{j}\right]$ $\displaystyle=$ $\displaystyle
i\hbar\left[\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta p_{i}p_{j}\right]$ (85)

where $\beta=\beta_{0}l_{p}^{2}/\hbar^{2}$, $\beta_{0}$ is a dimensionless
constant, and $l_{p}=1.6162\times 10^{-35}\text{m}$ is the Planck length.
Besides, the form of linear GUP is also motivated by doubly special relativity
Magueijo and Smolin (2002, 2003). The linear GUP has the following formAli
_et al._ (2009); Cortes and Gamboa (2005); Ali  _et al._ (2011)

$\displaystyle[x_{i},p_{j}]=i\hbar\left[\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)\right],~{}~{}~{}$
(86)

where $\alpha=\alpha_{0}l_{p}/\hbar$, and $\alpha_{0}$ is a dimensionless
constant. In what follows we shall argue how these two types of GUP appear
naturally from regularized self-energy.

###  VII.1 Quadratic GUP and bosons particles

Let us now show a very interesting result where one can obtain the GUP
principle using the modified energy. Let us assume $p\sim 0$, we can rewrite
Eqs. (56) and (83) as follows

$\displaystyle E_{tot}\simeq
E\left(1+\alpha\frac{E^{2}}{c^{4}M_{Pl}^{2}}\right).$ (87)

where $E_{corr}=M_{corr}c^{2}$ and $E=Mc^{2}$. By differentiating with respect
to the energy we get

$\displaystyle
dE_{tot}=dE\left(1+3\alpha\frac{E^{2}}{c^{4}M_{Pl}^{2}}\right),$ (88)

assuming

$\displaystyle dE_{tot}dt\sim\hbar,$ (89)

we get

$\displaystyle
dEdt\sim\hbar\left(1-3\alpha\frac{E^{2}}{c^{4}M_{Pl}^{2}}\right),$ (90)

but since

$\displaystyle l_{0}\sim l_{Pl}=\sqrt{\frac{\hbar G}{c^{3}}}$ (91)

we can write further as

$\displaystyle
dEdt\sim\hbar\left(1-\frac{\beta\,l^{2}_{Pl}E^{2}}{c^{2}\hbar^{2}}\right),$
(92)

where $\beta=3\alpha$. This shows that the GUP is a consequence of the
modified energy of the particle due to the regularized self-energy. We can
rewrite the last equation in different forms, say we take $dE=dp\,c$, along
with $dt=dx/c$, and we get

$\displaystyle
dp\,dx\sim\hbar\left(1-\frac{\beta\,l^{2}_{Pl}\,p^{2}}{\hbar^{2}}\right).$
(93)

This GUP expression is consistent with the quadratic form of GUP given by Eq.
(85).

###  VII.2 Linear GUP and Dirac particles

Consider the corrections to the mass given by Eq. (69) and assume $p\sim 0$,
we get

$\displaystyle E_{tot}\simeq E\left(1+\alpha\frac{E}{c^{2}M_{Pl}}\right).$
(94)

By differentiating with respect to the energy we get

$\displaystyle dE_{tot}=dE\left(1+2\alpha\frac{E}{c^{2}M_{Pl}}\right),$ (95)

assuming again that

$\displaystyle dE_{tot}dt\sim\hbar,$ (96)

we get

$\displaystyle dEdt\sim\hbar\left(1-2\alpha\frac{E}{c^{2}M_{Pl}}\right),$ (97)

we can write further this as

$\displaystyle dEdt\sim\hbar\left(1-\frac{\beta\,l_{Pl}E}{c\hbar}\right),$
(98)

where $\beta=2\alpha$. This shows that the GUP is a consequence of the
modified energy of the particle due to the regularized self-energy. If we
further take $dE=dp\,c$, along with $dt=dx/c$, we get

$\displaystyle dp\,dx\sim\hbar\left(1-\frac{\beta\,l_{Pl}\,p}{\hbar}\right).$
(99)

this is the linear GUP in agreement with Eq. (86). We, therefore, conclude
that the type of GUP depends on the nature of the particles being considered.
In particular, for scalar particles with spin zero, we obtained a quadratic
form of GUP, while for fermion particles with spin 1/2, we got a linear GUP.
This is consistent with the connection between spin and linear GUP that was
found in Ali and Majumder (2021) that have implications with explaining the
quantum entanglement Ali (2022).

##  VIII Conclusions

In the present paper, we have computed the regularized self-energy of the
particle using the Schrödinger–Newton equation. We used regular and well-
defined gravitational and electric potentials obtained in T-duality. In this
picture, the particle with mass $M$ is no longer concentrated into a point but
it is diluted and can be described by a quantum-corrected stress-energy tensor
with a smeared energy density resulting in corrections to the energy of the
particle which can be interpreted as a regularized self-energy of the
particle. To this end, we have found the corrections to the energy due to the
electrostatic field or the energy stored in the charge density of the
particle.

In the second part of this work, we have extended the corrections to energy by
incorporating the relativistic effects using the relativistic field equations
such as the Klein-Gordon, Proca, and Dirac equations. We found that the
corrections to the energy can be linked to the GUP. Quite surprisingly, the
form of GUP is shown to depend on the spin of the particles; namely, for
bosons having spin $0$ and $1$ we obtain a quadratic form of GUP, on the other
hand for fermions having spin $1/2$ we obtain a linear form of GUP. In the
near future, we would like to study more about the phenomenological aspects of
GUP which is linked to regularized self-energy as was argued in this work. We
would like to see how the regularized self-energy affects other types of
particles such as the spin-2 particles. Like spin-0 and spin-1 particles, one
can of course assume also for gravitons a similar expression for the energy
correction or modified dispersion relation

$\displaystyle E_{tot}^{2}=(p^{g})^{2}c^{2}+(M^{g})^{2}_{tot}c^{4},$ (100)

where $M^{g}_{tot}\simeq M^{g}\left(1+\alpha\,(M^{g}/M_{Pl})^{2}\right)$, with
$M^{g}$ and $p^{g}$ being the bare graviton mass and the graviton momentum,
respectively. Using the expression for the gravitons three-velocity
$v^{g}=p^{g}\,c^{2}/E_{tot}$, we can obtain

$\displaystyle\frac{v^{g}}{c}=\sqrt{1-\frac{(M^{g}_{tot})^{2}c^{4}}{E_{tot}^{2}}}.$
(101)

For the energy we may take $E_{tot}=hf$, with $f$ being the graviton’s
frequency, then using the gravitational waves it might be interesting to see
whether one can constrain the graviton mass along with the zero-point length.
There is also this possibility that we cannot constrain the zero-point length
from observations. This has to do with the fact that $M_{tot}^{g}$ might be
the true mass being measured by observations and, it simply shifts by some
constant compared to the bare mass $M^{g}$, meaning that we cannot distinguish
these quantities with observations. We should like to study more about the
implication of GUP on the graviton mass in the near future.

## References

  * Einstein (1916) A. Einstein, Annalen Phys. 49, 769 (1916).
  * Schrödinger (1926) E. Schrödinger, Annalen Phys. 384, 361 (1926).
  * Gordon (1926) W. Gordon, Zeitschrift für Physik 40, 117 (1926).
  * Klein (1926) O. Klein, Zeitschrift für Physik 37, 895 (1926).
  * Dirac (1928) P. A. M. Dirac, Proc. Roy. Soc. Lond. A 117, 610 (1928).
  * Diósi (1984) L. Diósi, Phys. Lett. A 105, 199 (1984), arXiv:1412.0201 [quant-ph] .
  * Diosi (1987) L. Diosi, Phys. Lett. A 120, 377 (1987).
  * Penrose (1996) R. Penrose, Gen. Rel. Grav. 28, 581 (1996).
  * Diosi (1989) L. Diosi, Phys. Rev. A 40, 1165 (1989).
  * Penrose (1998) R. Penrose, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 356, 1927 (1998).
  * Penrose (2014) R. Penrose, Found. Phys. 44, 557 (2014).
  * Proca (1936) A. Proca, J. Phys. Radium 7, 347 (1936).
  * Sathiapalan (1987) B. Sathiapalan, Phys. Rev. Lett. 58, 1597 (1987).
  * Strominger  _et al._ (1996) A. Strominger, S.-T. Yau, and E. Zaslow, Nucl. Phys. B 479, 243 (1996), arXiv:hep-th/9606040 .
  * Padmanabhan (1997) T. Padmanabhan, Phys. Rev. Lett. 78, 1854 (1997), arXiv:hep-th/9608182 .
  * Nicolini  _et al._ (2019) P. Nicolini, E. Spallucci, and M. F. Wondrak, Phys. Lett. B 797, 134888 (2019), arXiv:1902.11242 [gr-qc] .
  * Gaete  _et al._ (2022) P. Gaete, K. Jusufi, and P. Nicolini, Phys. Lett. B 835, 137546 (2022), arXiv:2205.15441 [hep-th] .
  * Nicolini (2022) P. Nicolini, Gen. Rel. Grav. 54, 106 (2022), arXiv:2208.05390 [hep-th] .
  * Jusufi (2023) K. Jusufi, Universe 9, 41 (2023), arXiv:2301.03590 [gr-qc] .
  * Jusufi and Sheykhi (2023) K. Jusufi and A. Sheykhi, Phys. Lett. B 836, 137621 (2023), arXiv:2210.01584 [gr-qc] .
  * Millano  _et al._ (2023) A. D. Millano, K. Jusufi, and G. Leon, (2023), arXiv:2302.00223 [gr-qc] .
  * Amati  _et al._ (1989) D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216, 41 (1989).
  * Ali  _et al._ (2009) A. F. Ali, S. Das, and E. C. Vagenas, Phys. Lett. B 678, 497 (2009), arXiv:0906.5396 [hep-th] .
  * Kempf  _et al._ (1995) A. Kempf, G. Mangano, and R. B. Mann, Phys. Rev. D 52, 1108 (1995), arXiv:hep-th/9412167 .
  * Brau (1999) F. Brau, J. Phys. A 32, 7691 (1999), arXiv:quant-ph/9905033 .
  * Maggiore (1993) M. Maggiore, Phys. Lett. B 304, 65 (1993), arXiv:hep-th/9301067 .
  * Adler (1994) S. L. Adler, Nucl. Phys. B 415, 195 (1994), arXiv:hep-th/9306009 .
  * Carr  _et al._ (2015) B. J. Carr, J. Mureika, and P. Nicolini, JHEP 07, 052 (2015), arXiv:1504.07637 [gr-qc] .
  * Carr (2022) B. J. Carr, Front. Astron. Space Sci. 9, 1008221 (2022), arXiv:2302.12609 [gr-qc] .
  * Ali  _et al._ (2011) A. F. Ali, S. Das, and E. C. Vagenas, Phys. Rev. D 84, 044013 (2011), arXiv:1107.3164 [hep-th] .
  * Das and Vagenas (2008) S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008), arXiv:0810.5333 [hep-th] .
  * Pikovski  _et al._ (2012) I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, Nature Phys. 8, 393 (2012), arXiv:1111.1979 [quant-ph] .
  * Marin  _et al._ (2013) F. Marin _et al._ , Nature Phys. 9, 71 (2013).
  * Petruzziello and Illuminati (2021) L. Petruzziello and F. Illuminati, Nature Commun. 12, 4449 (2021), arXiv:2011.01255 [gr-qc] .
  * Das  _et al._ (2021) A. Das, S. Das, N. R. Mansour, and E. C. Vagenas, Physics Letters B 819, 136429 (2021).
  * Moussa  _et al._ (2021) M. Moussa, H. Shababi, and A. Farag Ali, Phys. Lett. B 814, 136071 (2021), arXiv:2101.04747 [gr-qc] .
  * Kumar and Plenio (2020) S. P. Kumar and M. B. Plenio, Nature Commun. 11, 3900 (2020), arXiv:1908.11164 [quant-ph] .
  * Moradpour  _et al._ (2019) H. Moradpour, A. H. Ziaie, S. Ghaffari, and F. Feleppa, Mon. Not. Roy. Astron. Soc. 488, L69 (2019), arXiv:1907.12940 [gr-qc] .
  * Iorio  _et al._ (2018) A. Iorio, P. Pais, I. A. Elmashad, A. F. Ali, M. Faizal, and L. I. Abou-Salem, Int. J. Mod. Phys. D 27, 1850080 (2018), arXiv:1706.01332 [physics.gen-ph] .
  * Gao and Zhan (2016) D. Gao and M. Zhan, Physical Review A 94, 013607 (2016).
  * Bawaj  _et al._ (2015) M. Bawaj _et al._ , Nature Commun. 6, 7503 (2015), arXiv:1411.6410 [gr-qc] .
  * Kober (2010) M. Kober, Phys. Rev. D 82, 085017 (2010), arXiv:1008.0154 [physics.gen-ph] .
  * Girdhar and Doherty (2020) P. Girdhar and A. C. Doherty, New J. Phys. 22, 093073 (2020), arXiv:2005.08984 [quant-ph] .
  * Ashoorioon  _et al._ (2005) A. Ashoorioon, A. Kempf, and R. B. Mann, Phys. Rev. D 71, 023503 (2005), arXiv:astro-ph/0410139 .
  * Easther  _et al._ (2003) R. Easther, B. R. Greene, W. H. Kinney, and G. Shiu, Phys. Rev. D 67, 063508 (2003), arXiv:hep-th/0110226 .
  * Dabrowski and Wagner (2019) M. P. Dabrowski and F. Wagner, The European Physical Journal C 79, 1 (2019).
  * Ali and Majumder (2014) A. F. Ali and B. Majumder, Class. Quant. Grav. 31, 215007 (2014), arXiv:1402.5104 [gr-qc] .
  * Easther  _et al._ (2001) R. Easther, B. R. Greene, W. H. Kinney, and G. Shiu, Phys. Rev. D 64, 103502 (2001), arXiv:hep-th/0104102 .
  * Ali  _et al._ (2015) A. F. Ali, M. Faizal, and M. M. Khalil, JCAP 09, 025 (2015), arXiv:1505.06963 [gr-qc] .
  * Addazi  _et al._ (2021) A. Addazi _et al._ , (2021), arXiv:2111.05659 [hep-ph] .
  * Hossenfelder (2013) S. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc] .
  * Bahrami  _et al._ (2014) M. Bahrami, A. Großardt, S. Donadi, and A. Bassi, New J. Phys. 16, 115007 (2014), arXiv:1407.4370 [quant-ph] .
  * Gaete and Nicolini (2022) P. Gaete and P. Nicolini, Phys. Lett. B 829, 137100 (2022), arXiv:2202.09311 [hep-th] .
  * Magueijo and Smolin (2002) J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002), arXiv:hep-th/0112090 .
  * Magueijo and Smolin (2003) J. Magueijo and L. Smolin, Phys. Rev. D 67, 044017 (2003), arXiv:gr-qc/0207085 .
  * Cortes and Gamboa (2005) J. L. Cortes and J. Gamboa, Phys. Rev. D 71, 065015 (2005), arXiv:hep-th/0405285 .
  * Ali and Majumder (2021) A. F. Ali and B. Majumder, Eur. Phys. J. C 81, 360 (2021), arXiv:2104.14563 [gr-qc] .
  * Ali (2022) A. F. Ali, (2022), arXiv:2210.13974 [quant-ph] .


$\cdot\vdash\mathcal{C}_{i}:\mathbf{A};\textnormal{{qbit}}^{k}$ such that
$c_{i}{\blacktriangleleft_{\mathbf{A}}^{k}}\mathcal{C}_{i}$ for $i\in F$.
Assume further that for each $i\in F$, we are given a reduction path
$\pi_{i}\in\mathrm{Paths}(\mathcal{C},\mathcal{C}_{i})$, such that all paths
$\pi_{i}$ are distinct. Then

$\sum_{i\in
F}P(\pi_{i})c_{i}{\blacktriangleleft_{\mathbf{A}}^{k}}\mathcal{C}.$

###### Proof.

Fully analogous to Lemma D.9.45. ∎

######  Lemma D.9.59.

If
$q_{1}\blacktriangleleft_{\mathbf{\Gamma}_{1}\vdash\mathbf{B}_{1}}\mathbf{q}_{1}$
and
$q_{2}\blacktriangleleft_{\mathbf{\Gamma}_{2}\vdash\mathbf{B}_{2}}\mathbf{q}_{2}$,
then $q_{1}\otimes
q_{2}\blacktriangleleft_{\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}\vdash\mathbf{B}_{1}\otimes\mathbf{B}_{2}}\mathbf{q}_{1}\otimes\mathbf{q}_{2}$.

######  Lemma D.9.60.

If
$q\blacktriangleleft_{\mathbf{\Gamma}_{1}\vdash\mathbf{A}_{1}\otimes\mathbf{A}_{2}}\mathbf{q}$
and
$r\blacktriangleleft_{\mathbf{\Gamma}_{2},\mathbf{x}:\mathbf{A}_{1},\mathbf{y}:\mathbf{A}_{2}\vdash\mathbf{B}}\mathbf{r}$,
then

$r\circ(\mathrm{id}\otimes
q)\circ\mathrm{swap}\blacktriangleleft_{\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}\vdash\mathbf{B}}\mathbf{let\
x\otimes y=q\ in\ r}.$

######  Lemma D.9.61.

For $i\in\\{1,2\\},$ if
$q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{A}}\mathbf{q}$ then
$\mathbf{in}_{i}\circ
q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{A}\oplus\mathbf{B}}\mathbf{in}_{i}\
\mathbf{q}$.

######  Lemma D.9.62.

If
$q\blacktriangleleft_{\mathbf{\Gamma}_{1}\vdash\mathbf{A}_{1}\oplus\mathbf{A}_{2}}\mathbf{q}$
and
$r_{1}\blacktriangleleft_{\mathbf{\Gamma}_{2},\mathbf{x}:\mathbf{A}_{1}\vdash\mathbf{B}}\mathbf{r}_{1}$
and
$r_{2}\blacktriangleleft_{\mathbf{\Gamma}_{2},\mathbf{y}:\mathbf{A}_{2}\vdash\mathbf{B}}\mathbf{r}_{2}$,
then

$[r_{1},r_{2}]\circ d\circ(\mathrm{id}\otimes
q)\circ\mathrm{swap}\blacktriangleleft_{\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}\vdash\mathbf{B}}\mathbf{(case\
q\ of\ in_{1}x\Rightarrow r_{1}\ |\ in_{2}y\Rightarrow r_{2})}.$

######  Lemma D.9.63.

If $m\ \overline{\vartriangleleft_{Q(\mathbf{A},\mathbf{B})}}\ M$ and
$q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{A}}\mathbf{q},$ then
$\beta(m(*))\circ
q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{B}}M\mathbf{q}$.

######  Lemma D.9.64.

If $c\blacktriangleleft_{\mathbf{O}}^{k}\mathcal{C}$, then $(*\mapsto
r_{\mathbf{O}}(\mathrm{drop}_{k}\circ c))\
\overline{\vartriangleleft_{|\mathbf{O}|}}\ \text{run}\ \mathcal{C}.$

######  Lemma D.9.65.

If $m\ \overline{\vartriangleleft_{|\mathbf{O}|}}\ M$, then
$r^{-1}_{\mathbf{O}}(m_{*})\blacktriangleleft_{\cdot\vdash\mathbf{O}}\textbf{init}\
M.$

######  Lemma D.9.66.

Let $q\blacktriangleleft_{\mathbf{\Gamma}_{1}\vdash\mathbf{O}}\mathbf{q}$. Let
$x:|\mathbf{O}|;\mathbf{\Gamma}_{2}\vdash\mathbf{r}:\mathbf{A}$ be a term and
$r:{\llbracket|\mathbf{O}|\rrbracket}\to\mathbf{Q}({\llbracket\mathbf{\Gamma}_{2}\rrbracket},{\llbracket\mathbf{A}\rrbracket})$
a Scott-continuous function such that for any observable value $\cdot\vdash
V:|\mathbf{O}|$ we have that $r_{{\llbracket
V\rrbracket}}\blacktriangleleft_{\mathbf{\Gamma}_{2}\vdash\mathbf{A}}\mathbf{r}[V/x].$
Then
$\widehat{r}_{*}\circ(q\otimes\mathrm{id})\blacktriangleleft_{\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}\vdash\mathbf{A}}\mathbf{let}\
x=\text{lift}\ \mathbf{q}\ \mathbf{in\ r}.$

######  Lemma D.9.67.

If $q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{P[\mu
X.P/X]}}\mathbf{q}$ then $\mathrm{fold}\circ
q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{\mu X.P}}\mathbf{fold}\
\mathbf{q}$.

######  Lemma D.9.68.

If $q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{\mu X.P}}\mathbf{q}$
then $\mathrm{unfold}\circ
q\blacktriangleleft_{\mathbf{\Gamma}\vdash\mathbf{P[\mu
X.P/X]}}\mathbf{unfold}\ \mathbf{q}$.

######  Lemma D.9.69.

If $q\blacktriangleleft_{\mathbf{\Gamma}_{1}\vdash\mathbf{I}}\mathbf{q}$ and
$r\blacktriangleleft_{\mathbf{\Gamma}_{2}\vdash\mathbf{A}}\mathbf{r}$ then
$\cong\circ(q\otimes
r)\blacktriangleleft_{\mathbf{\Gamma}_{1},\mathbf{\Gamma}_{2}\vdash\mathbf{A}}\mathbf{q};\mathbf{r}$.

###  D.10. Fundamental Lemma and Strong Adequacy

We extend the definition of the logical relations to cover _all terms_ ,
including those whose non-linear context may be non-empty.

###### Definition D.10.70.

For any classical term $x_{1}:P_{1},\ldots,x_{n}:P_{n}\vdash M:R,$ and
morphism $m\in\mathbf{DCPO}_{\mathcal{M}}({\llbracket
P_{1}\rrbracket}\times\cdots\times{\llbracket P_{n}\rrbracket},{\llbracket
R\rrbracket})$ we shall write $m\
\overline{\vartriangleleft_{x_{1}:P_{1},\ldots,x_{n}:P_{n}\vdash R}}\ M$ iff
$\forall(v_{1}\vartriangleleft_{P_{1}}V_{1}),\ldots,(v_{n}\vartriangleleft_{P_{n}}V_{n})$
we have that
$m\mathbin{\raisebox{0.9pt}{\scalebox{0.7}{$\varodot$}}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\
\vec{v}\
\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\
\overline{\vartriangleleft_{R}}\ M[\vec{V}/\vec{x}].$

###### Definition D.10.71.

For any quantum term
$x_{1}:P_{1},\ldots,x_{n}:P_{n};\mathbf{y}_{1}:\mathbf{A}_{1},\ldots,\mathbf{y}_{m}:\mathbf{A}_{m}\vdash\mathbf{q}:\mathbf{B}$
and morphism $q\in\mathbf{DCPO}({\llbracket
P_{1}\rrbracket}\times\cdots\times{\llbracket
P_{n}\rrbracket},\mathbf{Q}({\llbracket\mathbf{A}_{1}\rrbracket}\otimes\cdots\otimes{\llbracket\mathbf{A}_{m}\rrbracket},{\llbracket\mathbf{B}\rrbracket}))$
we shall write
$q\blacktriangleleft_{x_{1}:P_{1},\ldots,x_{n}:P_{n};\mathbf{y}_{1}:\mathbf{A}_{1},\ldots,\mathbf{y}_{m}:\mathbf{A}_{m}\vdash\mathbf{B}}\mathbf{q}$
iff
$\forall(v_{1}\vartriangleleft_{P_{1}}V_{1}),\ldots,(v_{n}\vartriangleleft_{P_{n}}V_{n})$
we have that
$q_{\vec{v}}{\blacktriangleleft}_{\mathbf{y}_{1}:\mathbf{A}_{1},\ldots,\mathbf{y}_{m}:\mathbf{A}_{m}\vdash\mathbf{B}}\mathbf{q}[\vec{V}/\vec{x}].$

We may now prove the Fundamental Lemma which then easily implies our adequacy
result.

######  Lemma D.10.72 (Fundamental).

For any classical term $\Phi\vdash M:R$ and any quantum term
$\Phi;\mathbf{\Gamma}\vdash\mathbf{q}:\mathbf{B}$ we have that

${\llbracket M\rrbracket}\ \overline{\vartriangleleft_{\Phi\vdash R}}\
M\qquad\text{ and
}\qquad{\llbracket\mathbf{q}\rrbracket}{\blacktriangleleft}_{\Phi;\mathbf{\Gamma}\vdash\mathbf{B}}\mathbf{q}$

###### Proof.

By induction on the derivation of the term.

Classical Lambda Abstractions. The case for classical lambda abstractions
follows using exactly the same arguments as (Jia et al., 2021, Lemma 117).

Quantum Lambda Abstractions. The case for quantum lambda abstractions follows
using similar arguments which we now present.

Let us assume that the term of the induction hypothesis is

$x_{1}:P_{1},\ldots,x_{n}:P_{n};\mathbf{y}_{1}:\mathbf{A}_{1},\ldots,\mathbf{y}_{m}:\mathbf{A}_{m}\vdash\mathbf{q}:\mathbf{B}.$

Let
$(v_{1}\vartriangleleft_{P_{1}}V_{1}),\ldots,(v_{n}\vartriangleleft_{P_{n}}V_{n})$
be arbitrary. Let us write $l\stackrel{{\scriptstyle\textrm{{\scriptsize
def}}}}{{=}}{\llbracket{\lambda}\vec{\mathbf{y}}.\mathbf{q}\rrbracket}\mathbin{\raisebox{0.9pt}{\scalebox{0.7}{$\varodot$}}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\
\vec{v}\
\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
and $R\stackrel{{\scriptstyle\textrm{{\scriptsize
def}}}}{{=}}{\lambda}\vec{\mathbf{y}}.\mathbf{q}[\vec{V}/\vec{x}]$. Observe
that $l\in\mathbf{TD}$ and therefore by Theorem D.8.44 (C5), we may
equivalently show that

$l\vartriangleleft_{Q(\mathbf{A}_{1}\otimes\cdots\otimes\mathbf{A}_{m},\mathbf{B})}R.$

By Theorem D.8.44 (A5), this is in turn equivalent to showing that

$l\leq{\llbracket R\rrbracket}\text{ and }\forall
k\in\mathbb{N}.\forall(\cdot\vdash[\lvert\psi\rangle,\ell,\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m}]:\mathbf{A}_{1}\otimes\cdots\otimes\mathbf{A}_{m};\textnormal{{qbit}}^{k}).$
$(\beta(l(*))\otimes\mathrm{id})\circ{\llbracket[\lvert\psi\rangle,\ell,\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m}]\rrbracket}\blacktriangleleft_{\mathbf{B}}^{k}[\lvert\psi\rangle,\ell,R(\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m})]$

The inequality is satisfied, because

$\displaystyle l$
$\displaystyle={\llbracket{\lambda}\vec{\mathbf{y}}.\mathbf{q}\rrbracket}\mathbin{\raisebox{0.9pt}{\scalebox{0.7}{$\varodot$}}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\
\vec{v}\
\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
$\displaystyle\leq{\llbracket{\lambda}\vec{\mathbf{y}}.\mathbf{q}\rrbracket}\mathbin{\raisebox{0.9pt}{\scalebox{0.7}{$\varodot$}}}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}\
\vec{{\llbracket V\rrbracket}}\
\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
$\displaystyle(\text{Theorem \ref{thm:formal-relations} (C1)})$
$\displaystyle={\llbracket R\rrbracket}.$ $\displaystyle(\text{Lemma
\ref{lem:term-substitution}})$

For the other requirement, we reason as follows

$\displaystyle\beta(l(*))$
$\displaystyle=\beta({\llbracket{\lambda}\vec{\mathbf{y}}.\mathbf{q}\rrbracket}_{\vec{v}})$
$\displaystyle(\text{Definition})$
$\displaystyle=\beta(\mathcal{J}{\llbracket\mathbf{q}\rrbracket}_{\vec{v}})$
$\displaystyle={\llbracket\mathbf{q}\rrbracket}_{\vec{v}}$
$\displaystyle{\blacktriangleleft_{\mathbf{y}_{1}:\mathbf{A}_{1},\ldots,\mathbf{y}_{m}:\mathbf{A}_{m}\vdash\mathbf{B}}}\mathbf{q}[\vec{V}/\vec{x}].$
(Induction Hypothesis)

It now follows (by definition) that for any $k\in\mathbb{N}$ and any
$\cdot\vdash[\lvert\psi\rangle,\ell,\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m}]:\mathbf{A}_{1}\otimes\cdots\otimes\mathbf{A}_{m};\textnormal{{qbit}}^{k}$
we have that

$(\beta(l(*))\otimes\mathrm{id})\circ{\llbracket[\lvert\psi\rangle,\ell,\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m}]\rrbracket}\blacktriangleleft_{\mathbf{B}}^{k}[\lvert\psi\rangle,\ell,\mathbf{q}[\vec{\mathbf{v}}/\vec{\mathbf{y}},\vec{V}/\vec{x}]].$

Finally, observe that
$[\lvert\psi\rangle,\ell,R(\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m})]\xrightarrow{1}[\lvert\psi\rangle,\ell,\mathbf{q}[\vec{\mathbf{v}}/\vec{\mathbf{y}},\vec{V}/\vec{x}]]$
and then by Lemma D.9.58 it follows that

$(\beta(l(*))\otimes\mathrm{id})\circ{\llbracket[\lvert\psi\rangle,\ell,\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m}]\rrbracket}\blacktriangleleft_{\mathbf{B}}^{k}[\lvert\psi\rangle,\ell,R(\mathbf{v}_{1}\otimes\cdots\otimes\mathbf{v}_{m})],$

as required.

The cases for terms whose formation rules do not have a premise follow by
straightforward verification. All other cases follow by induction using the
relevant closure lemma from Section D.9. ∎

######  Corollary D.10.73.

For any closed quantum configuration
$\cdot\vdash\mathcal{C}:\mathbf{A};\textnormal{{qbit}}^{k}$, we have that
${\llbracket\mathcal{C}\rrbracket}\blacktriangleleft_{\mathbf{A}}^{k}\mathcal{C}.$

###### Proof.

Let $\mathcal{C}=[\lvert\psi\rangle,\ell,\mathbf{q}]$. It follows that for
$n=\dim(\lvert\psi\rangle)-k$ we have
$\mathbf{x}_{1}:\textnormal{{qbit}},\ldots,\mathbf{x}_{n}:\textnormal{{qbit}}\vdash\mathbf{q}:\mathbf{A}$
and then by the Fundamental Lemma we know that

${\llbracket\mathbf{q}\rrbracket}\blacktriangleleft_{\mathbf{x}_{1}:\textnormal{{qbit}},\ldots,\mathbf{x}_{n}:\textnormal{{qbit}}\vdash\mathbf{A}}\mathbf{q}.$

By definition it now follows that for
$\cdot\vdash[\lvert\psi\rangle,\ell,\mathbf{x}_{1}\otimes\cdots\otimes\mathbf{x}_{n}]:\textnormal{{qbit}}\otimes\cdots\otimes\textnormal{{qbit}};\textnormal{{qbit}}^{k}$
we have

$({\llbracket\mathbf{q}\rrbracket}\otimes\mathrm{id}_{\textnormal{{qbit}}^{k}})\circ{\llbracket[\lvert\psi\rangle,\ell,\mathbf{x}_{1}\otimes\cdots\otimes\mathbf{x}_{n}]\rrbracket}\blacktriangleleft_{\mathbf{A}}^{k}[\mathbf{q}\twoheadleftarrow[\lvert\psi\rangle,\ell,\mathbf{x}_{1}\otimes\cdots\otimes\mathbf{x}_{n}]].$

The LHS is precisely ${\llbracket\mathcal{C}\rrbracket}$ and the RHS is
exactly $\mathcal{C}$ which completes the proof. ∎

Adequacy now follows as a consequence of the Fundamental Lemma.

######  Theorem D.10.74 (Strong Adequacy).

Let $\cdot\vdash m:P$ be a closed classical term and
$\cdot\vdash\mathcal{C}:\mathbf{A};\textnormal{{qbit}}^{k}$ a closed quantum
configuration. Then:

$\displaystyle{\llbracket
m\rrbracket}=\sum_{v\in\text{Val}(m)}P(m\xrightarrow{}_{*}v){\llbracket
v\rrbracket}\qquad\qquad{\llbracket\mathcal{C}\rrbracket}=\sum_{\mathcal{V}\in\mathrm{ValC}(\mathcal{C})}P(\mathcal{C}\xrightarrow{}_{*}\mathcal{V}){\llbracket\mathcal{V}\rrbracket}.$

###### Proof.

The equation for classical terms may be established using exactly the same
arguments as (Jia et al., 2021, Theorem 118). The equation for quantum
configurations also follows using the same arguments which we repeat now. Let

$u\stackrel{{\scriptstyle\textrm{{\scriptsize
def}}}}{{=}}\sum_{\mathcal{V}\in\mathrm{ValC}(\mathcal{C})}P(\mathcal{C}\xrightarrow{}_{*}\mathcal{V}){\llbracket\mathcal{V}\rrbracket}.$

From Corollary D.0.1, we know that ${\llbracket\mathcal{C}\rrbracket}\geq u.$
It remains to show the converse inequality. Towards this end, observe that
$\mathcal{S}_{0}(\blacktriangleleft_{\mathbf{A}}^{k};\mathcal{C})\subseteq\mathop{\downarrow}u$.
To establish this, we reason as follows. Taking an arbitrary element of
$\mathcal{S}_{0}(\blacktriangleleft_{\mathbf{A}}^{k};\mathcal{C})$ as in
Definition D.3.7:

$\displaystyle\sum_{\pi\in F}P(\pi){\llbracket\mathcal{V}_{\pi}\rrbracket}$
$\displaystyle=\sum_{\mathcal{V}\in\cup\\{\mathcal{V}_{\pi}|\pi\in
F\\}}\left(\sum_{\begin{subarray}{c}\pi\in F\\\
\mathcal{V}_{\pi}=\mathcal{V}\end{subarray}}P(\pi)\right){\llbracket\mathcal{V}\rrbracket}$
$\displaystyle\leq\sum_{\mathcal{V}\in\cup\\{\mathcal{V}_{\pi}|\pi\in
F\\}}\left(\sum_{\pi\in\mathrm{Paths}(\mathcal{C},\mathcal{V})}P(\pi)\right){\llbracket\mathcal{V}\rrbracket}$
$\displaystyle=\sum_{\mathcal{V}\in\cup\\{\mathcal{V}_{\pi}|\pi\in
F\\}}P(\mathcal{C}\xrightarrow{}_{*}\mathcal{V}){\llbracket\mathcal{V}\rrbracket}$
$\displaystyle\leq\sum_{\mathcal{V}\in\mathrm{ValC}(\mathcal{C})}P(\mathcal{C}\xrightarrow{}_{*}\mathcal{V}){\llbracket\mathcal{V}\rrbracket}.$

The set $\mathop{\downarrow}u$ is Scott-closed and therefore
$\mathcal{S}(\blacktriangleleft_{\mathbf{A}}^{k};\mathcal{C})\subseteq\mathop{\downarrow}u$.
By Corollary D.10.73, we know that
${\llbracket\mathcal{C}\rrbracket}{\blacktriangleleft_{\mathbf{A}}^{k}}\mathcal{C}.$
By definition of ${\blacktriangleleft_{\mathbf{A}}^{k}}$ it follows
${\llbracket\mathcal{C}\rrbracket}\in\mathcal{S}(\blacktriangleleft_{\mathbf{A}}^{k};\mathcal{C})$
and therefore ${\llbracket\mathcal{C}\rrbracket}\leq u,$ thus finishing the
proof. ∎