# Essential Surfaces in the Exterior of $K13n586$

Chaeryn Lee

###### Abstract

We count the number of isotopy classes of closed, connected, orientable,
essential surfaces embedded in the exterior $B$ of the knot K13n586. The main
result is that the count of surfaces by genus is equal to the Euler totent
function. This is the first manifold for which we know the number of surfaces
for any genus. The main argument is to show when normal surfaces in $B$ are
connected by counting their number of components. We implement tools from
Agol, Hass and Thurston to convert the problem of counting components of
surfaces into counting the number of orbits in a set of integers under a
collection of bijections defined on its subsets.

##  1 Introduction

Listing and counting the number of essential surfaces is one interesting
approach to the study of essential surfaces in 3-manifolds. It has been shown
in [3] that the number of closed, orientable, essential surfaces in a given
3-manifold counted by Euler characteristic have a particular quasi-polynomial
structure. For a 3-manifold $M$, let $b_{M}(n)$ denote the number of isotopy
classes of closed, orientable, essential surfaces with Euler characteristic
$n$. If $M$ is compact, orientable, irreducible, $\partial$-irreducible,
atoroidal, acylindrical and does not contain any closed, nonorientable,
essential surfaces then the corresponding generating function
$B_{M}(x)=\sum_{n=1}^{\infty}b_{M}(-2n)x^{n}$ is short (Theorem 1.3 of [3]).

Figure 1: The knot $K13n586$.

No general results are known when we turn to counting connected essential
surfaces. Define $a_{M}(g)$ as the number of isotopy classes of closed,
connected, orientable surfaces of genus $g$ in $M$. To relate this index to
Euler characteristic $n=g-1$, we may adjust it and define
$\tilde{a}_{M}(n)=a_{M}(n+1)$. Section 8 of [3] provides data of $a_{M}(g)$
for $g\leq 200$ of certain manifolds but there are very few manifolds for
which we actually know the full list of essential surfaces embedded inside.
The only known cases are where $a_{M}(g)=0$ for all large $g$. In this paper
we study a specific manifold, the exterior of the knot $K13n586$, and present
a complete description of all of its essential connected surfaces counted by
Euler characteristic. This is the first manifold for which we know $a_{M}(g)$
for all $g$. In particular, we will prove the following result which was
asserted as a conjecture in Section 8.1 of [3].

######  Theorem 1.

For the exterior $B$ of the knot $K13n586$, $\tilde{a}_{B}(n)=\phi(n)$, $n>1$
where $\phi(n)$ is the Euler totent function.

In Section 2, we give a detailed description of the triangulation of $B$ and
the normal surfaces embedded in $B$ with respect to this triangulation. In
Section 3, we look at combinatorial tools that enable us to count the number
of components of surfaces. We then prove our theorem in Section 4.

This work was partially supported by U.S. National Science Foundation grant
DMS-1811156.

##  2 The structure of $B$ and its normal surfaces

We first look at some normal surface theory introduced in the work by [4]. Let
$M$ be a manifold with a triangulation $\mathcal{T}$. A surface is said to be
normal if it is in general position with the 1-skeleton $\mathcal{T}^{(1)}$ of
$\mathcal{T}$ and meets each tetrahedron only in triangles and quadrilaterals
that we call elementary disks. An elementary disk $E$ in a tetrahedron
$\Delta$ of $\mathcal{T}$ is a properly embedded disk that meets each edge of
$\Delta$ in at most one point and each face of $\Delta$ in at most one line.
We will follow the convention that if $E\cap\mathcal{T}^{(1)}$ is a planar set
then $E$ is planar and if not then $E$ is the cone $b\ast\partial E$ where $b$
is the centroid of the 3-simplex $\Delta$ spanned by $E\cap\mathcal{T}^{(1)}$.
Elementary disks and hence normal surfaces are uniquely determined by their
intersection points with $\mathcal{T}^{(1)}$. A normal isotopy of $M$ is an
isotopy that leaves every simplex of $\mathcal{T}$ invariant. There are
exactly 7 normal isotopy classes of elementary disks, four triangles that each
cut off a vertex and three quadrilaterals that separate the three pairs of
disjoint edges. We call such normal isotopy classes the disk types of a
tetrahedron. Suppose we have two normal surfaces $S$ and $T$ that intersect
transversely and have the same quad types in each tetrahedron of
$\mathcal{T}$. Then there is a unique normal surface determined by $(S\cup
T)\cap\mathcal{T}^{(1)}$ which we call the normal sum of $S$ and $T$ and write
$S+T$.

$0$$1$$2$$3$E01E02E03E12E13E23tri0tri1tri2tri3quadF0F1F2F3 Figure 2: Standard tetrahedron with elementary disk types, labelled with names of vertices, edges and faces. | F0 | F1 | F2 | F3
---|---|---|---|---
tet0 | tet2 | tet1 | tet7 | tet5
| (0, 1, 3, 2) | (0, 1, 3, 2) | (0, 2, 1, 3) | (2, 1, 0, 3)
tet1 | tet4 | tet0 | tet6 | tet3
| (3, 0, 1, 2) | (0, 1, 3, 2) | (3, 1, 2, 0) | (0, 1, 3, 2)
tet2 | tet0 | tet5 | tet3 | tet9
| (0, 1, 3, 2) | (0, 2, 1, 3) | (1, 3, 0, 2) | (3, 1, 2, 0)
tet3 | tet2 | tet9 | tet1 | tet8
| (2, 0, 3, 1) | (0, 1, 3, 2) | (0, 1, 3, 2) | (2, 1, 0, 3)
tet4 | tet7 | tet8 | tet9 | tet1
| (3, 1, 2, 0) | (0, 1, 3, 2) | (3, 1, 2, 0) | (1, 2, 3, 0)
tet5 | tet7 | tet6 | tet2 | tet0
| (0, 1, 3, 2) | (0, 1, 3, 2) | (0, 2, 1, 3) | (2, 1, 0, 3)
tet6 | tet9 | tet5 | tet1 | tet8
| (3, 0, 1, 2) | (0, 1, 3, 2) | (3, 1, 2, 0) | (0, 1, 3, 2)
tet7 | tet5 | tet0 | tet8 | tet4
| (0, 1, 3, 2) | (0, 2, 1, 3) | (1, 3, 0, 2) | (3, 1, 2, 0)
tet8 | tet7 | tet4 | tet6 | tet3
| (2, 0, 3, 1) | (0, 1, 3, 2) | (0, 1, 3, 2) | (2, 1, 0, 3)
tet9 | tet2 | tet3 | tet4 | tet6
| (3, 1, 2, 0) | (0, 1, 3, 2) | (3, 1, 2, 0) | (1, 2, 3, 0)
Table 1: Triangulation of $\mathcal{S}$ of $B$. The $(i,j)$-th entry shows which tetrahedron is glued to face F$j-1$ of tetrahedron tet$i-1$. The permutations below show the order with which their vertices are glued. | Edges Identified
---|---
e0 | E10 : tet0, E23 : tet4, E12 : tet5, E20 : tet7, E13 : tet7, E32: tet8
e1 | E20 : tet0, E30 : tet1, E02 : tet5, E03 : tet6
e2 | E12 : tet0, E20 : tet2, E13 : tet2, E32 : tet3, E10 : tet5, E23 : tet9
e3 | E30 : tet0, E01 : tet1, E20 : tet1, E01 : tet3, E30 : tet3
E31 : tet6, E30 : tet7, E21 : tet8, E20 : tet9
e4 | E31 : tet0, E21 : tet2, E23 : tet5, E32 : tet6, E32 : tet7, E21 : tet9
e5 | E23 : tet0, E32 : tet1, E32 : tet2, E21 : tet4, E31 : tet5, E21 : tet7
e6 | E12 : tet1, E01 : tet2, E13 : tet3, E01 : tet4, E31 : tet9
e7 | E31 : tet1, E03 : tet2, E21 : tet3, E20 : tet4, E30 : tet5
E01 : tet6, E20 : tet6, E01 : tet8, E30 : tet8
e8 | E20 : tet3, E03 : tet4, E02 : tet8, E30 : tet9
e9 | E31 : tet4, E12 : tet6, E01 : tet7, E13 : tet8, E01 : tet9
Table 2: List of edges in each tetrahedron that is glued to the edge e$i$ of
$\mathcal{S}$.

| $F$ | $G$
---|---|---
tri0 | tri1 | tri2 | tri3 | quad | tri0 | tri1 | tri2 | tri3 | quad
tet0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0
tet1 | 3 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 2 | 1
tet2 | 1 | 1 | 0 | 0 | 1 | 2 | 0 | 0 | 2 | 0
tet3 | 3 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 0 | 1
tet4 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1
tet5 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2
tet6 | 1 | 1 | 0 | 2 | 1 | 3 | 1 | 0 | 0 | 1
tet7 | 2 | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 0 | 1
tet8 | 1 | 1 | 2 | 0 | 1 | 3 | 1 | 1 | 1 | 0
tet9 | 2 | 0 | 1 | 1 | 1 | 2 | 0 | 0 | 2 | 0

| $uF+vG$
---|---
tri0 | tri1 | tri2 | tri3 | quad
tet0 | $2u+2v$ | 0 | $2v$ | 0 | $2u$
tet1 | $3u+v$ | $u+v$ | 0 | $2v$ | $u+v$
tet2 | $u+2v$ | $u$ | 0 | $2v$ | $u$
tet3 | $3u+v$ | $u+v$ | $u+2v$ | $u$ | $v$
tet4 | $2u+2v$ | 0 | $v$ | $2u+v$ | $v$
tet5 | $2u+2v$ | 0 | $2u$ | 0 | $2v$
tet6 | $u+3v$ | $u+v$ | 0 | $2u$ | $u+v$
tet7 | $2u+v$ | $v$ | 0 | $2u$ | $v$
tet8 | $u+3v$ | $u+v$ | $2u+v$ | $v$ | $u$
tet9 | $2u+2v$ | 0 | $u$ | $u+2v$ | $u$

Table 3: Number of each disk type that forms the surfaces $F$, $G$ and
$uF+vG$. Disk types are depicted in Figure 2.

Throughout this paper we will fix a specific triangulation $\mathcal{S}$ of
$B$ that is provided by snappy.HTLinkExteriors on SnapPy [2] (the current
labelling on the tetrahedra differs). Tables 1 and 2 give a description of
this triangulation. The triangulation $\mathcal{S}$ consists of a single
vertex, 10 edges, named e$i$ ($0\leq i\leq 9$), and 10 tetrahedra, named
tet$j$ $0\leq i\leq 9$, each with an ordering $0,1,2,3$ on their vertices as
in Figure 2. In each tetrahedron E$ij$ denotes the edge between vertices $i$
and $j$ while F$k$ denotes the face opposite to vertex $k$, that is the
2-simplex spanned by all vertices but $k$. Note that an orientation is fixed
on edges E$ij$ from $i$ to $j$ so that E$ij$ and E$ji$ represent the same edge
but with opposite orientation. The entry in Table 1 in the row labeled tet$i$
and column labeled F$j$ shows which tetrahedron is glued to face F$(j-1)$ of
tetrahedron tet$(i-1)$ (the discrepancy in the indices comes from the fact
that labels in our triangulation start at 0) and the order with which their
vertices are glued. For instance, the vertices $(0,1,3)$ in F$2$ of tet$2$ are
glued in order to the vertices $(1,3,2)$ which is F$0$ of tet$3$. Note that we
give the full permutations of size 4 in Table 1 but one value in each
permutation is redundant since every face is made up of 3 vertices. Table 2
gives a list of the edges of the 10 tetrahedra that are identified to each
$e_{i}$. The orientation with which they are identified are taken into account
by the indices E$ij$.

Section 8.1 of [3] shows that there is a bijection from the isotopy classes of
essential surfaces in $B$ to surfaces of the form $uF+vG$
($u,v\in\mathbb{N}$). In our triangulation $\mathcal{S}$ the only disk types
we consider are the triangles, ordered by the vertex they are positioned at,
and the quadrilateral that separates the edges E01 and E23 as in Figure 2.
Note that $F$ and $G$ have the same disk types, hence it is possible to define
their normal sum. Table 3 gives a description of the normal surfaces $F$, $G$
and how they are embedded in $B$ with respect to $\mathcal{S}$. Table 3 gives
the number of each disc type that forms the surfaces $F$, $G$ and $uF+vG$.
Both $F$ and $G$ are surfaces of genus 2.

Our goal is to count $\tilde{a}_{B}(n)$, the number of connected surfaces
$uF+vG$ with genus $n+1$ or Euler characteristic $-2n$. Following the work of
[3] we can in fact reduce Theorem 1 to Lemma 2 below. Since $F$ and $G$ are
surfaces of genus 2, surfaces $uF+vG$ with Euler characteristic $-2n$
correspond to lattice points in $\mathbb{N}^{2}$ on the line $x+y=n$. We want
to show that $\tilde{a}_{B}(n)$ is exactly the number of all primitive such
lattice points. If $uF+vG$ is connected then $gcd(u,v)=1$, otherwise $uF+vG$
would consist of disjoint copies of the surface $\frac{1}{gcd(u,v)}(uF+vG)$.
Hence it suffices to show the converse to prove Theorem 1.

######  Lemma 2.

If $gcd(u,v)=1$ then $uF+vG$ is a connected surface.

The rest of this paper will focus on proving Lemma 2.

##  3 Counting orbits of integer intervals

In this section we follow the work of Section 4 in [1] and introduce some
techniques used to count the number of orbits in a set of integers under a
particular pseudogroup action. Fixing an order and orientation on the edges of
our triangulation $\mathcal{S}$, we may correspond an integer interval
$[1,N]\subseteq\mathbb{N}$ to the intersection points of $uF+vG$ with the
edges of $\mathcal{S}$. The intersection arcs of $uF+vG$ with the faces of
$\mathcal{S}$ then give rise to bijections defined on subintervals of $[1,N]$.
Hence we can change the problem of counting the number of components of
$uF+vG$ to counting the number of orbits in the interval $[1,N]$ under these
bijections.

Let $[1,N]\subseteq\mathbb{N}$ be a set of integers. A bijection
$g\colon[a,b]\to[c,d]$ defined on subintervals $[a,b],[c,d]\subseteq[1,N]$ is
called a pairing. A pairing is orientation preserving if it is increasing and
orientation reversing if it is decreasing. The width of a pairing
$g\colon[a,b]\to[c,d]$ is the number of integers in its domain or range, and
denoted $|g|=b-a+1=d-c+1$. Note that all sets and maps we work with henceforth
are discrete.

Assume that we have the interval $[1,N]$ and a collection of pairings
$\\{g_{i}\\},1\leq i\leq k$ defined on subintervals of $[1,N]$. These pairings
naturally generate a pseudogroup on $[1,N]$. Any two integers are said to be
in the same orbit if some pairing in the pseudogroup generated by
$\\{g_{i}\\}$ sends one to the other. Note that orbits form a partition of
$[1,N]$. Our goal is to count the number of such orbits.

We now describe some modifications we can perform on the interval $[1,N]$ and
pairings $\\{g_{i}\\}$. An important trait of these modifications is that they
do not change the number of orbits (Lemma 11 of [1]).

Trimming

Trimming modifies an orientation reversing pairing so that its domain and
range are disjoint. Let $g\colon[a,b]\to[c,d]$ be an orientation reversing
pairing so that its domain and range overlap, $g(a)=d,g(b)=c,c\leq b$. Define
the pairing $g^{\prime}\colon[a,\frac{a+d}{2})\to(\frac{a+d}{2},d]$ and
replace $g$ with $g^{\prime}$. We say that $g^{\prime}$ is obtained by
trimming $g$. $g^{\prime}$ is also an orientation reversing pairing.

Truncation

Suppose there is some rightmost subinterval $[N^{\prime}+1,N]$ of $[1,N]$ that
lies in the range of only one pairing $g\colon[a,b]\to[c,N]$, $c\leq
N^{\prime}+1\leq N$. Truncation "peels off" this subinterval from $g$ and
shortens the entire interval $[1,N]$ to $[1,N^{\prime}]$ without changing the
orbit structure. If $g$ is orientation preserving define
$g^{\prime}\colon[a,b-(N-N^{\prime})]\to[c,N^{\prime}]$. If $g$ is orientation
reversing first make sure that its domain and range are disjoint, trim $g$ if
necessary. Then define
$g^{\prime}\colon[a+(N-N^{\prime}),b]\to[c,N^{\prime}]$. Replacing $g$ with
$g^{\prime}$ is called truncating $g$.

We can similarly truncate a leftmost subinterval $[1,N^{\prime\prime}]$ of
$[1,N]$ that lies in the domain and range of a single pairing. After this
modification is applied the interval $[1,N]$ is shortened to
$[N^{\prime\prime}+1,N]$. To ensure our interval starts at 1 we may translate
the domain and range of every pairing in $\\{g_{i}\\}$ by $-N^{\prime\prime}$.

Transmission

Transmission composes two pairings to shift the domain and range of one by the
other. Assume $g_{1}$ has disjoint domain and range and that the range of
$g_{2}$ lies in the range of $g_{1}$. If the domain of $g_{2}$ does not lie in
the range of $g_{1}$, define $g_{2}^{\prime}={g_{1}}^{-1}\circ g_{2}$. If the
domain of $g_{2}$ does lie in the range of $g_{1}$, define
$g_{2}^{\prime}={g_{1}}^{-1}\circ g_{2}\circ g_{1}$. Replacing $g_{2}$ with
$g_{2}^{\prime}$ we transmit $g_{2}$ by $g_{1}$.

##  4 Proof of Lemma 2

We will order the edges of $\mathcal{S}$ as $e1,e8,e6,e9,e0,e2,e3,e4,e5,e7$
(this order is intentionally chosen in this way) then label each of the
intersection points by the integers $1,2,...,24u+24v$ to obtain an interval
$[1,24u+24v]\subseteq\mathbb{N}$. Faces of $\mathcal{S}$ intersect the surface
$uF+vG$ in a collection of arcs joining edges of each face. A collection of
arcs that join the same edges of a face will be called an arc type. Every arc
type defines a pairing $p_{i}\colon[a_{i},b_{i}]\to[c_{i},d_{i}]$ where
$[a_{i},b_{i}]$ and $[c_{i},d_{i}]$ are subintervals of $[1,24u+24v]$. To
prove Lemma 2 our goal is to show that there is one orbit of $[1,24u+24v]$
under the collection of pairings $\\{p_{i}\\}$.

######  Claim 1.

The number of orbits of interval $[1,24u+24v]$ under pairings $\\{p_{i}\\}$ is
at most the number of orbits of the interval $[1,2u+2v]$ under the orientation
reversing pairings

$\displaystyle\\{f\colon[1,u]\to[u+1,2u],$ $\displaystyle
g\colon[2u+1,2u+v]\to[2u+v+1,2u+2v],$ $\displaystyle
h\colon[1,u+v]\to[u+v+1,2u+2v]\\}.$

$f$$g$$h$ Figure 3: The thick interval in the middle represents the interval
$[1,2u+2v]$. Pairings $f,g$ and $h$ are depicted as the bands attached to this
interval.

###### Proof.

The first step will be to reduce the interval $[1,24u+24v]$ to its subinterval
corresponding to $e1$ which is $[1,4u+4v]$. We start from the rightmost
subinterval corresponding to $e7$. Observe that all pairings that have their
ranges on the edge $e7$ (these include the pairings that have both their
domain and range on $e7$) can be transmitted by the pairing $t_{7}$ depicted
as in Figure 4 so that their ranges are shifted into $e1$. This is because the
domain of $t_{7}$ lies on $e1$ and its range is the entire $e7$. For any
pairing whose domain lies on $e7$ and range lies on a distinct edge, simply
swap the domain and range and apply transmission. Now $e7$ only lies in the
range of $t_{7}$ so we can truncate $t_{7}$ and remove the subinterval
corresponding to $e7$ from $[1,24u+24v]$. Similarly, all pairings that have
their range on the edges $e0,e2,e3,e4,e5$ can be transmitted by the pairings
$t_{0},t_{2},t_{3},t_{4},t_{5}$ into $e_{1}$ and
$t_{0},t_{2},t_{3},t_{4},t_{5}$ can be truncated. All pairings defined on
$e6,e9$ can be transmitted by pairings $t_{6},t_{9}$ into $e8$ and
$t_{6},t_{9}$ are truncated. Finally, all pairings defined on $e8$ can be
transmitted by $t_{8a},t_{8b}$ into $e1$ where pairings whose range lie in
both the range of $t_{8a},t_{8b}$ can be cut into two pairings before
transmission.

$1$$2$$0$e1e0F3 of tet5$t_{0}(2u+2v)$$1$$2$$0$e1e2F3 of
tet0$t_{2}(2u+2v)$$2$$3$$0$e1e3e5F1 of
tet0$t_{3}$$(4u+2v)$$t_{5}(2v)$$2$$3$$0$e1e7e4F1 of
tet5$t_{4}(2u)$$t_{7}$$(2u+4v)$$1$$3$$0$e8e6F2 of
tet9$t_{6}(2u+2v)$$1$$3$$0$e8e9F2 of tet4$t_{9}(2u+2v)$$1$$2$$0$e3e8e7F3 of
tet3$t_{8b}$$(u+3v)$$t_{8a}$$(3u+v)$ Figure 4: Faces with vertices, edges, and
pairings $t_{i}$ used for transmission. The numbers in parentheses are the
widths of each pairing. Details about pairings other than $t_{i}$ are omitted.

We now have $[1,4u+4v]$ and pairings $\\{p^{\prime}_{i}\\}$ where the domains
and ranges of $p^{\prime}_{i}$ are all subintervals of $[1,4u+4v]$. Let
$\\{p^{\prime}_{i1},p^{\prime}_{i2},...,p^{\prime}_{in}\\}$ be a subset of
$\\{p^{\prime}_{i}\\}$. If there are $n$ orbits of $[1,4u+4v]$ under
$\\{p^{\prime}_{i1},p^{\prime}_{i2},...,p^{\prime}_{in}\\}$ there are at most
$n$ orbits of $[1,4u+4v]$ under $\\{p^{\prime}_{i}\\}$.

$2$$3$$0$e1e7F1 of tet5$2u+4v$$2u$$1$$2$$0$e7e7F3 of
tet6$u+3v$$u+v$$u+v$e1e7e7e1$4u+4v$$1$$u+3v$$2u+4v$$u+v+1$$f_{1}\colon[1,u+3v]\to[u+v+1,2u+4v]$
Figure 5: The triangles above show the faces and pairings that give rise to
$f_{1}\colon[1,u+3v]\to[u+v+1,2u+4v]$. The arrows on the edges describe the
orientation on their edges while the numbers on the pairings show their
widths. The poles below are the edges aligned according to the integer
intervals they correspond to. The strip in between the edges shows the pairing
$f_{1}$. In particular, $f_{1}$ is orientation reversing.

Henceforth we will only consider a subset of $\\{p^{\prime}_{i}\\}$,
$\\{f_{1},g_{1},h_{1}\\}$ that consists of three special pairings. Let
$f_{1}\colon[1,u+3v]\to[u+v+1,2u+4v]$,
$g_{1}\colon[2v+1,3u+3v]\to[u+3v+1,4u+4v]$,
$h_{1}\colon[u+3v+1,2u+4v]\to[3u+3v+1,4u+4v]$ be pairings obtained as in
Figures 5, 6 and 7. $f_{1}$ and $g_{1}$ are orientation reversing and $h_{1}$
is orientation preserving. Trim $f_{1}$ and $g_{1}$ to obtain
$f_{2}\colon[1,u+2v]\to[u+2v+1,2u+4v]$,
$g_{2}\colon[2v,2u+3v]\to[2u+3v+1,4u+4v]$. By composing $h_{1}$ with $g_{2}$
we can replace $h_{1}$ with $h_{2}\colon[u+3v+1,2u+4v]\to[2v+1,u+3v]$. $h_{2}$
is orientation reversing. Now $[1,2v]$ and $[2u+4v+1,4u+4v]$ each lie in the
domain and range of exactly one pairing so we can truncate $f_{2}$, $g_{2}$ to
$f_{3}\colon[2v+1,u+2v]\to[u+2v+1,2u+2v]$,
$g_{3}\colon[2u+2v+1,2u+3v]\to[2u+3v+1,2u+4v]$. Translating the domain and
range of all pairings by $-2v$ we reach the interval $[1,2u+2v]$ with a
collection of pairings
$\\{f\colon[1,u]\to[u+1,2u],g\colon[2u+1,2u+v]\to[2u+v+1,2u+2v],h\colon[1,u+v]\to[u+v+1,2u+2v]\\}$.

$2$$3$$0$e1e3F1 of tet0$4u+2v$$2v$$1$$3$$0$e1e3F2 of
tet6$u+3v$$u+3v$$3u+v$e1e1e3$1$$2v+1$$3u+3v$$4u+2v$$3u+v$$4u+4v$$u+3v+1$$g_{1}\colon[2v+1,3u+3v]\to[u+3v+1,4u+4v]$
Figure 6: Pairings that give rise to
$g_{1}\colon[2v+1,3u+3v]\to[u+3v+1,4u+4v]$. $g_{1}$ is orientation reversing.
$2$$3$$0$e1e7F1 of tet5$2u+4v$$2u$$2$$3$$0$e1e3F1 of
tet0$4u+2v$$2v$$1$$2$$0$e3e7F3 of
tet3$3u+v$$u+v$$u+3v$e1e1e7e3$1$$u+3v+1$$2u+4v$$3u+v+1$$4u+2v$$4u+4v$$3u+3v+1$$h_{1}\colon[u+3v+1,2u+4v]\to[3u+3v+1,4u+4v]$
Figure 7: Pairings that give rise to
$h_{1}\colon[u+3v+1,2u+4v]\to[3u+3v+1,4u+4v]$. Unlike $f_{1}$ and $g_{1}$,
$h_{1}$ is orientation preserving.

∎

Now it remains to count the number of orbits.

######  Claim 2.

There are $gcd(u,v)$ orbits of $[1,2u+2v]$ under $\\{f,g,h\\}$.

###### Proof.

The three pairings $f,g,h$ are orientation reversing and
$|f|=u,|g|=v,|h|=u+v$. $1$ is in the domain of $f$ whilst $2u+2v$ is in the
range of $g$ meaning $f$ lies on the left side of the interval and $g$ lies on
the right side of the interval. Without loss of generality we assume $u\geq
v$, if not we may simply reverse the order on the entire interval and apply
the following proof. We will now describe a step by step orbit counting
procedure with a sequence of pairings
$\\{\\{f_{1},g_{1},h_{1}\\},\\{f_{2},g_{2},h_{2}\\},...,\\{f_{N},g_{N},h_{N}\\}\\}$
that starts with $\\{f_{1},g_{1},h_{1}\\}=\\{f,g,h\\}$ and ends with a set of
pairings $\\{f_{N},g_{N},h_{N}\\}$ where $|f_{N}|=|g_{N}|$. Geometrically,
this process can be illustrated as in Figure 8. The bands attached to the
thickened interval in the middle illustrate the pairings.

$f_{i}$$g_{i}$$h_{i}$$f_{i}$$g_{i}^{\prime}$$h_{i}^{\prime}$$f_{i+1}$$g_{i+1}$$h_{i+1}$$\Longleftrightarrow$
Figure 8: Picture description of gcd calculation. The thick interval in the
middle illustrates our interval while the bands attached are the pairings
defined on subintervals.

  1. 1.

Transmit $g_{i}$ by $h_{i}$ to obtain $g_{i}^{\prime}$.

  2. 2.

Truncate $h_{i}$ and peel off the domain and range of $g_{i}$ from $h_{i}$ to
obtain $h_{i}^{\prime}$.

  3. 3.

Assign $h_{i+1}=f_{i}$.

  4. 4.

Compare the widths of $g_{i}^{\prime}$ and $h_{i}^{\prime}$ then assign
$f_{i+1}$ with the pairing of larger width and $g_{i+1}$ with the pairing of
smaller width,

In each step the pairings are of the form

$\displaystyle
f_{i}\colon\left[a_{i},\frac{a_{i}+b_{i}}{2}\right)\to\left(\frac{a_{i}+b_{i}}{2},b_{i}\right],$
$\displaystyle
g_{i}\colon\left[b_{i},\frac{b_{i}+c_{i}}{2}\right)\to\left(\frac{b_{i}+c_{i}}{2},c_{i}\right],$
$\displaystyle
h_{i}\colon\left[a_{i},\frac{a_{i}+c_{i}}{2}\right)\to\left(\frac{a_{i}+c_{i}}{2},c_{i}\right],\quad
a_{i}\leq b_{i}\leq c_{i}.$

The number of orbits are unchanged in this procedure and $|f_{N}|=|g_{N}|$
will give the number of orbits of $[1,2u+2v]$ under $\\{f,g,h\\}$ since we
have the pairings

$\displaystyle
f_{N}\colon\left[a_{N},\frac{a_{N}+c_{N}}{4}\right)\to\left(\frac{a_{N}+c_{N}}{4},\frac{a_{N}+c_{N}}{2}\right],$
$\displaystyle
g_{N}\colon\left[\frac{a_{N}+c_{N}}{2},\frac{3(a_{N}+c_{N})}{4}\right)\to\left(\frac{3(a_{N}+c_{N})}{4},c_{N}\right]$
$\displaystyle
h_{N}\colon\left[a_{N},\frac{a_{N}+c_{N}}{2}\right)\to\left(\frac{a_{N}+c_{N}}{2},c_{N}\right].$

Observe that in each step $|h_{i}|=|f_{i}|+|g_{i}|$, $|f_{i}|\geq|g_{i}|$.
Moreover, $|h_{i+1}|=|f_{i}|$ and
$\\{|f_{i+1}|,|g_{i+1}|\\}=\\{|g_{i}|,|f_{i}|-|g_{i}|\\}$. The pairs of
integers $(|f_{i}|,|g_{i}|)$ represent the sequence of integers that computes
$gcd(|f_{1}|,|g_{1}|)=gcd(u,v)$ by Euclid’s algorithm. Hence the number of
orbits $|f_{N}|=|g_{N}|=gcd(u,v)$.

∎

By assumption $gcd(u,v)=1$ so we are done by Claim $1$.

## References

  * [1] I. Agol, J. Hass, and W. Thurston, The computational complexity of knot genus and spanning area, Trans.Amer.Math.Soc 358 (2006), 3821-3850.
  * [2] M. Culler, N. M. Dunfield, M. Goerner and J. R. Weeks, SnapPy, a computer program for studying the topology and geometry of 3-manifolds, Version 3.0.1, https://snappy.computop.org/
  * [3] N. M. Dunfield, S. Garoufalidis, J. H. Rubinstein, Counting essential surfaces in 3-manifolds, arXiv:2007.10053.
  * [4] J. Tollefson, Isotopy classes of incompressible surfaces in irreducible 3-manifolds, Osaka J.Math 32 (1995), 1087-1111.


# Experimental demonstration of robustness of Gaussian quantum coherence

Haijun Kang1,2 Dongmei Han1,2 Na Wang1,2 Yang Liu1,2 Shuhong Hao3
<EMAIL_ADDRESS>Xiaolong Su1,2<EMAIL_ADDRESS>1State Key Laboratory of
Quantum Optics and Quantum Optics Devices,
Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006, People’s
Republic of China
2Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan,Shanxi 030006, People’s Republic of China
3School of Mathematics and Physics, Anhui University of Technology, Maanshan,
243000, People’s Republic of China

###### Abstract

Besides quantum entanglement and steering, quantum coherence has also been
identified as a useful quantum resource in quantum information. It is
important to investigate the evolution of quantum coherence in practical
quantum channels. In this paper, we experimentally quantify the quantum
coherence of a squeezed state and a Gaussian Einstein-Podolsky-Rosen (EPR)
entangled state transmitted in Gaussian thermal noise channel, respectively.
By reconstructing the covariance matrix of the transmitted states, quantum
coherence of these Gaussian states is quantified by calculating the relative
entropy. We show that quantum coherence of the squeezed state and the Gaussian
EPR entangled state is robust against loss and noise in a quantum channel,
which is different from the properties of squeezing and Gaussian entanglement.
Our experimental results pave the way for application of Gaussian quantum
coherence in lossy and noisy environments.

##  I I. Introduction

The principle of coherent superposition of waves plays important roles in many
well-known phenomena, such as interference and diffraction. In quantum
mechanics, the superposition principle which is one of the fundamental
nonclassical characteristics of quantum states, underlies many nonclassical
properties of quantum mechanics including entanglement or coherence
Baumgratz2014 . Recently, resource theories of coherence have attracted a lot
of attention Winter2016 ; Streltsov2017 ; Chitambar2019 . Quantum coherence,
which characterizes the quantumness and underpins quantum correlations in
quantum systems, plays a key role in many novel quantum phenomena and has been
identified as a quantum resource for quantum information processing Aberg2014
; Mark2016 ; Li2016 ; Shi2017 ; Shi2020 . Quantum coherence also plays a
strong role in biology systems Plenio2013 , such as photosynthetic energy
transport, the avian compass and sense of smell.

To quantify coherence, Baumgratz et al. established a framework by referring
to the method of quantifying entanglement Baumgratz2014 . The quantum
coherence of a quantum state is defined as the minimum distance between the
quantum state and an incoherent state in the Hilbert space Baumgratz2014 . In
addition to relative entropy and $l_{1}$-norm Baumgratz2014 , it has been
shown that quantum coherence can also be quantified by Fisher information
Feng2017 , skew information entropy Yu2017 , Tsallis relative $\alpha$ entropy
Rastegin2016 , robustness Napoli2016 , and so on. The frozen Bromley2015 ,
distillation Chitambar2016116 , catalytic Bu2016 , and erasure Singh2015  of
quantum coherence and the relationships between quantum coherence and
complementarity relationCheng2015 , uncertainty relationYuan2017U , quantum
entanglement or other types of quantum correlation Xi2015 ; Chitambar2016117
have also been investigated. With the rapid development in quantum coherence
theory, the experimental demonstration related to quantum coherence is in
progress Yuan2017D ; Wu2017 ; Gao2018 ; Wu2018 ; Zhang2019 ; Xu2020 . Gaussian
states, such as the squeezed state and the Einstein-Podolsky-Rosen (EPR)
entangled state, play essential roles in continuous variable (CV) quantum
information Braunstein2005 ; Wang2007 ; Weedbrook2012 , where Gaussian states
are generated deterministically and information is encoded in the position or
momentum quadrature of photonic harmonic oscillators. For example, Gaussian
states has been applied in quantum computation Ukai2011 ; Su2013 , quantum key
distribution Su2009 ; Gehring2015 ; Diamanti2016 , quantum teleportation
Furusawa1998 ; Huo2018 , quantum entanglement swapping Takei2005 ; Jia2004 ;
Su2016 , quantum dense coding Li2002 ; Jun2005 , and verification of the
error-disturbance uncertainty relation Liu2019npj ; Liu2019pr . Recently, it
has been shown that quantum coherence with infinite-dimensional systems can be
quantified by relative entropy Fan2016 . Then, the investigations of Gaussian
quantum coherence attracted lots of attention Xu2016 ; Buono2016 ;
Albarelli2017 .

In practical quantum computation and quantum information, decoherence coming
from the inevitable interaction between a quantum resource and the environment
is a main obstacle Siena2005 . Up to now, the decoherence of squeezing,
entanglement and quantum steering in the thermal noise channel have been
experimentally demonstrated Barbosa2010 ; Deng2016 ; Deng2017 ; liu2020 ;
Deng2021 . Toward the application of quantum coherence, it is necessary to
investigate the evolution of quantum coherence in lossy and noisy environment
Suciu2016 ; Croitoru2020 . Very recently, it has been shown that quantum
coherence can be robust against noise theoretically Buruaga2017  and quantum
coherence of optical cat states can be robust against loss Zhang2021 .

In this paper, we experimentally quantify quantum coherence of Gaussian
states, for example, a squeezed state and a Gaussian EPR entangled state, by
measuring their covariance matrices. Then we investigate quantum coherence of
these Gaussian states through lossy and noisy channels. We show that quantum
coherence of these Gaussian states is robust against loss in a lossy channel,
which is similar to the case of squeezing and entanglement of Gaussian state.
The most interesting thing is that the quantum coherence of these Gaussian
states is still robust in a noisy channel even if the squeezing and
entanglement of Gaussian state disappear. The presented results provide useful
reference for applying quantum coherence of a Gaussian state in practical
quantum information processing.

##  II II. The principle of experiment

Figure 1: (a) Schematic of transmitting the quantum coherence of a Gaussian
state in a thermal noise channel. (b) Experimental setup. The 1080 nm and the
540 nm laser output from the Nd:YAP/LBO laser, pass through two mode cleaners
and are injected into the nondegenerate-optical-parametric-amplifier (NOPA) as
signal light and pump light, respectively. The output modes of the
polarization beam splitter (PBS) behind the NOPA are an amplitude squeezed
state (transmitted mode) and a phase squeezed state (reflected mode) or EPR
entanglement state, when the half wave plate (HWP) behind the NOPA is set to
$22.5^{\textup{o}}$ or $0^{\textup{o}}$, respectively. We use homodyne
detectors to measure the output modes and a digital storage oscilloscope to
record the experimental data. The interference efficiencies of homodyne
detectors are 99% and the quantum efficiencies of photodiodes (LASER
COMPONENTS, InGaAs-PD-500um) are 99.6%. AM, amplitude modulator; PM, phase
modulator.

As shown in Fig. 1(a), the quantum coherence of a Gaussian state is
distributed through a Gaussian thermal noise channel, and the quantum
coherence of the output state is measured. Here we consider two kinds of
Gaussian channel: one is a lossy channel where only vacuum noise is involved,
and the other is a noisy channel where the noise higher than vacuum noise
exists. The squeezed state and the EPR entangled state, which are generated
from a nondegenerate-optical-parametric-amplifier (NOPA) as shown in Fig.
1(b), are used as two examples to investigate the quantum coherence of a
Gaussian state in our experiment. The NOPA cavity is in a semi-monolithic
structure, which is composed by an $\alpha$-cut type-II potassium titanyl
phosphate (KTP) crystal ($3\times 3\times 10$ mm), whose front surface is used
as input mirror, and a concave mirror with curvature radius of 50 mm. The NOPA
is operating in the case of deamplification, i.e., the relative phase between
the signal and the pump light is locked to $(2n+1)\pi$. The lossy channel is
simulated by combination of a half-wave plate (HWP) and a polarization beam
splitter (PBS) and the noisy channel is simulated by combination of a HWP, two
PBSs, and an ancillary coherent beam carrying Gaussian noise.

At first, we distribute the amplitude squeezed state through a lossy channel,
where the output state is measured by the homemade homodyne detector at Bob’s
side in time domain. Then we investigate quantum coherence of an EPR entangled
state in lossy channel, where output states are measured by Alice’s and Bob’s
homodyne detectors simultaneously in the time domain. Finally, we quantify the
quantum coherence of the amplitude squeezed state and EPR entangled state in a
noisy channel, where the Gaussian noise is added through amplitude and phase
modulators (LINOS, LM0202 P and LM0202 PHAS) on an ancillary coherent beam and
coupled into the lossy channel by the PBS. In the measurement of the
covariance matrix of output states in the time domain, the electrical signal
of each homodyne detector is mixed with a 3 MHz reference signal (SRS, DS345),
then pass through a low-pass filter and a low-noise preamplifier (SRS, SR560),
and finally recorded in a digital storage oscilloscope (TELEDYNE LECROY,
WaveRunner 640Zi). The bandwidth of the homodyne detectors we used is 8 MHz,
and common-mode rejection ratio is 30 dB (at 3 MHz). The sampling rate of the
digital storage oscilloscope is 500 KS/s, and there are $5\times 10^{5}$ data
for each sampling space.

##  III III. Quantum coherence

Quantum coherence of a quantum state $\hat{\rho}$ in Fock space can be
calculated by Baumgratz2014

$\mathcal{C}_{rel.~{}ent.}(\hat{\rho})=S\left(\hat{\rho}_{diag}\right)-S\left(\hat{\rho}\right),$
(1)

where $S$ is van Neumann entropy and $\hat{\rho}_{diag}$ is a diagonal matrix
which removing all off-diagonal elements of the density matrix $\hat{\rho}$.
In the case of Gaussian quantum information, a Gaussian state
$\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ can be completely represented by the
displacement $\bar{\mathbf{x}}$ and the covariance matrix $\mathbf{V}$ in
phase space, which correspond to the first and second statistical moments of
the quadrature operators respectively Wang2007 ; Weedbrook2012 . The
displacement $\mathbf{\bar{x}}=\langle\mathbf{\hat{x}}\rangle$, where
$\mathbf{\hat{x}=(}\mathit{\hat{X}}_{\mathit{1}}\mathit{,\mathit{\hat{Y}}_{\mathit{1}},...,\mathit{\hat{X}}}_{\mathit{\mathit{N}}}\mathit{,\hat{Y}}_{\mathit{N}}\mathbf{)}^{\mathit{t}},\mathit{\hat{X}}_{\mathit{k}}=(\hat{a}_{{}_{k}}+\hat{a}_{{}_{k}}^{\dagger})$
and
$\mathit{\hat{Y}}_{\mathit{k}}=i(\hat{a}_{{}_{k}}^{\dagger}-\hat{a}_{{}_{k}})$
are the amplitude and phase quadratures of an optical mode, respectively. The
element of covariance matrix $\mathbf{V}$ is defined as
$\mathbf{V}_{ij}=\frac{1}{2}\langle\hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i}\rangle-\langle\hat{x}_{i}\rangle\langle\hat{x}_{j}\rangle$.
The diagonal Gaussian states (incoherent states) are thermal states Xu2016 ,
so the incoherent state $\hat{\rho}_{diag}$ in Eq. (1) is replaced by an
$N$-mode thermal state $\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})$
whose mean number of particle is the same to
$\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ for each mode. Thus, the Gaussian
quantum coherence of an $N$-mode Gaussian state can be represented as Xu2016

$\mathcal{C}_{rel.~{}ent.}\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right]=S\left[\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})\right]-S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right],$
(2)

where
$S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right]\\!=-\\!\underset{i=1}{\overset{N}{\sum}}\\!\left[\\!\left(\frac{\nu_{i}-1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\nu_{i}-1}{2}\right)\\!-\\!\left(\frac{\nu_{i}+1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\nu_{i}+1}{2}\right)\\!\right]$
and
$S\left[\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})\right]\\!\\!=\\!-\\!\underset{i=1}{\overset{N}{\sum}}\\!\left[\\!\left(\frac{\mu_{i}-1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\mu_{i}-1}{2}\right)\\!-\\!\left(\frac{\mu_{i}+1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\mu_{i}+1}{2}\right)\\!\right]$
are the von-Neumann entropy of $\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ and
$\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})$, respectively. $\nu_{i}$
and $\mu_{i}$ are the symplectic eigenvalues of $\mathbf{V}$ and
$\mathbf{V}_{th}$, respectively. Here the displacements
$\mathbf{\bar{x}}_{th}=0$ and the elements of the diagonal covariance matrix
$\mathbf{V}_{th}$ are given by $\mathbf{V}$ with $V_{th}\
{}_{2i-1,2i-1}=V_{th}\
{}_{2i,2i}=\frac{1}{2}\left(V_{2i-1,2i-1}+V_{2i,2i}+\left[\mathbf{\bar{x}}_{2i-1}\right]^{2}+\left[\mathbf{\bar{x}}_{2i}\right]^{2}\right)$.

Since the displacements $\bar{\mathbf{x}}$ of the Gaussian states we used in
our experiment are zero, the Gaussian state can be completely represented by
its covariance matrix $\mathbf{V}$. The covariance matrix of the amplitude
squeezed state is given by

$\displaystyle\mathbf{V}_{squ}=\left(\begin{array}[]{cc}V_{s}&0\\\
0&V_{as}\end{array}\right),$ (5)

where $V_{s}$ and $V_{as}$ are the variances of squeezed and antisqueezed
noise of the squeezed state, respectively. The squeezed and antisqueezed noise
levels of the squeezed state are quantified by $10\log_{10}V_{s}$ dB and
$10\log_{10}V_{as}$ dB, respectively. The symplectic eigenvalue of the
squeezed state can be determined by $\sqrt{\textup{Det}\mathbf{V}_{squ}}$.

The covariance matrix of the EPR entangled state is given by

$\displaystyle\mathbf{V}_{ent}$
$\displaystyle=\left(\begin{array}[]{cccc}\mathbf{A}&\mathbf{C}\\\
\mathbf{C}^{t}&\mathbf{B}\end{array}\right),$ (8)

where $\mathbf{A}=\mathbf{B}=\frac{1}{2}(V_{s}+V_{as})~{}\mathbf{I}$,
$\mathbf{C}=\frac{1}{2}(V_{s}-V_{as})~{}\mathbf{Z}$, $t$ denotes transpose,
$\mathbf{I}=\begin{pmatrix}\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\end{pmatrix}$ and
$\mathbf{Z}=\begin{pmatrix}\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\end{pmatrix}$. The symplectic eigenvalues can be
determined by
$\sqrt{\frac{\Delta\pm\sqrt{\Delta^{2}-4\textup{Det}\mathbf{V}_{ent}}}{2}}$,
where
$\Delta=\textup{Det}\mathbf{A}+\textup{Det}\mathbf{B}+2\textup{Det}\mathbf{C}$.
The positive partial transposition (PPT) criterion Simon2000  is applied to
describe the entanglement of the EPR entangled state, which is a sufficient
and necessary condition for a two-mode entanglement state with continuous
variables. The PPT value can be determined by
$\sqrt{\frac{\Gamma-\sqrt{\Gamma^{2}-4\textup{Det}\mathbf{V}_{ent}}}{2}}$,
where
$\Gamma=\textup{Det}\mathbf{A}+\textup{Det}\mathbf{B}-2\textup{Det}\mathbf{C}$.
When the PPT value is less than 1, the two mode quantum states are entangled
Simon2000 .

A one-mode Gaussian state $\rho(\mathbf{\bar{x},V})$ transmitting in a
Gaussian channel can be represented by Holevo1999 ; Eisert2005

$\displaystyle\begin{aligned}
\mathbf{\bar{x}}\rightarrow&T\mathbf{\bar{x}}+\bar{d},\\\
\mathbf{V}\rightarrow&T\mathbf{V}T^{t}+\varLambda,\end{aligned}$ (9)

where $\bar{d}$ is displacement operator in phase space, $T$ is amplification
or attenuation and rotation operator in phase space, and $\varLambda$ is a
noise term that may consist of quantum as well as classical noise. The thermal
noise channel, which belongs to incoherent channel Xu2016 , of CV Gaussian
quantum system can be written as Eisert2005

$\displaystyle\begin{aligned} T&=\sqrt{\eta}~{}\mathbf{I},\\\
\varLambda&=\left(1-\eta\right)\left(\delta+\upsilon\right)~{}\mathbf{I},\\\
\bar{d}&=0,\end{aligned}$ (10)

where $\eta$ and $\delta$ are the transmission efficiency and the excess noise
of the Gaussian channel, respectively, and $\upsilon=1$ represents the vacuum
noise. When $\delta=0$, the Gaussian channel is a lossy channel. The loss of
the channel is given by $L=1-\eta$.

##  IV IV. Results

First, we quantify the quantum coherence of the amplitude squeezed state and
the EPR entangled state by the covariance matrices we reconstructed (see
Appendix A) in a lossy environment. The decoherence of the squeezing of the
squeezed state and the entanglement of the EPR entangled state in the lossy
channel is shown in Fig. 2(a) and 2(b), respectively. It is obvious that the
squeezing and entanglement are robust against loss in lossy channel. Quantum
coherence of the squeezed state and the EPR entangled state in lossy channel
is shown in Fig. 2(c) and 2(d), respectively. We can find that the quantum
coherence of the squeezed state and the EPR entangled state are both decreased
with the increase of loss, which reaches zero only when the maximal loss is
reached. When the loss equals 1, the squeezed state turns into a vacuum state
and the EPR state turns into a separable state. We can see that quantum
coherence of these Gaussian states is also robust against loss in a lossy
channel.

Figure 2: Experimental results in a lossy channel. (a) Dependence of squeezing
(red solid line) and antisqueezing (blue dotted line) of the squeezed state on
the loss. The dashed line is the shot noise limit (SNL). (b) Dependence of PPT
value of the EPR entangled state on the loss. The dashed line is the boundary
of entangled and separable state. (c) and (d), Dependence of quantum coherence
of the squeezed state and the EPR entangled state on the loss, respectively.
The initial squeezed and antisqueezed noise levels are -2.95 dB and 4.15 dB,
respectively. The error bars represent one standard deviation and are obtained
based on the statistics of the data.

Then we quantify the quantum coherence of a squeezed state and an EPR
entangled state in a noisy environment. In the case of noisy channel, we fix
the channel losses to $L=0.4$, and change the excess noise $\delta$ added on
the amplitude squeezed state and one mode of the EPR entangled state. The
decoherence of the squeezing of the squeezed state and entanglement of the the
EPR entangled state in the noisy channel is shown in Fig. 3(a) and 3(b),
respectively. Different from the results in the lossy channel, the noise level
of squeezing is beyond the shot noise limit (SNL) when the excess noise overs
$\delta=0.74$ and the PPT value is greater than 1 when the excess noise is
greater than $\delta=2.14$, which means the squeezing of the squeezed state
and the entanglement of the EPR entanglement state are destroyed.

Figure 3: Experimental results in a noisy channel. (a) Dependence of squeezing
(red solid line) and antisqueezing (blue dotted line) of the squeezed state on
the excess noise. The dashed line is SNL. (b) Dependence of PPT value of the
EPR entangled state on the excess noise. The dashed line is the boundary of
entangled and separable state. (c) and (d), Dependence of quantum coherence of
the squeezed state and the EPR entangled state on the excess noise,
respectively. (e) and (f), Quantum coherence of the squeezed state and the EPR
entangled state parameterized by loss and excess noise, respectively. The red
dots represent the experimental results in (c) and (d).

The quantum coherence of the squeezed state and the EPR entangled state in a
noisy channel is shown in Fig. 3(c) and 3(d), respectively. It is obvious that
the quantum coherence of the squeezed state and the EPR entangled state is
both decreased with the increase of the excess noises. However, the quantum
coherence of these Gaussian states still exists even if the squeezing and
entanglement disappear. It is interesting that quantum coherence of these two
Gaussian states will vanish only when infinite excess noises are involved in
the case of fixed channel loss. The dependence of quantum coherence of the
squeezed state and the EPR entangled state on loss and excess noise is shown
in Fig. 3(e) and 3(f), respectively. It is obvious that the quantum coherence
of these Gaussian states is robust against noise in a noisy channel.

We note that the squeezing and entanglement are destroyed at different excess
noise levels as shown in Fig. 3(a) and 3(b). The reason for this result is
that the excess noise is only added on one mode of the EPR entangled state. It
shows that the EPR entanglement can tolerate more noise than squeezed state in
one-mode noise channel. The squeezing and entanglement will be destroyed at
same excess noise level if we add the excess noise on both modes of the EPR
entangled state simultaneously (see Appendix B).

We also demonstrate the monotonicity of quantum coherences of the squeezed
state and the entanglement state in lossy and noisy channels as shown in Fig.
2 and 3, respectively. The quantum coherences of these two Gaussian states are
decreasing with the increase of loss and noise in quantum channels, which is
because the lossy and noisy channels are all incoherent operations and quantum
coherence will decrease under incoherent operations Baumgratz2014 ; Xu2016 .
The physical reason for the robustness of quantum coherences of these Gaussian
states in a noisy channel is that the proportion of quantum coherence is
decreased when it is mixed with thermal noise, but the quantum coherence
disappears completely only when infinite thermal noise is involved.

##  V V. Conclusion

In summary, we experimentally demonstrate the quantum coherence of Gaussian
states in lossy and noisy channel. The results confirm that quantum coherences
of the squeezed state and the EPR entangled state are robust against loss and
noise in a Gaussian thermal noise channel, although the squeezing and
entanglement of a Gaussian states disappear at a certain noise level in a
noisy channel. Thus, the quantum coherence of Gaussian state can resist the
decoherence when it is used as quantum resource. Our investigation makes a
step toward the application of quantum coherence as a quantum resource in
quantum communication.

It is interesting to accomplish quantum information tasks that only require
quantum coherence of a Gaussian state due to its unique property of it in the
presence of loss and noise. However, a suitable application for only applying
the quantum coherence of a Gaussian state remains an open question. Recently,
it has been shown that the Gaussian entanglement can be transferred in a
single-mode cavity Bougouffa2020 , which is an application of robustness of
quantum coherence of Gaussian states. Based on the presented results of
robustness for quantum coherences of Gaussian states, the potential
application of quantum coherence is worthy of further investigation.

## APPENDIX A: Reconstruction of the covariance matrix

To reconstruct the covariance matrix of an EPR entangled state, the variances
and the cross correlations of the amplitude or phase quadratures are obtained
by simultaneously measuring the amplitude or phase quadratures of two modes of
the EPR entangled state in the time domain. The diagonal elements of the
covariance matrix are the variances of the amplitude and phase quadratures
$\langle\Delta^{2}(\hat{x}_{i})\rangle$, and the nondiagonal elements are the
covariances of the amplitude or phase quadratures, which are calculated via
the measured variances Steinlechner2013

$V_{ij}=[\langle\Delta^{2}(\hat{x}_{i}+\hat{x}_{j})\rangle-\langle\Delta^{2}(\hat{x}_{i})\rangle-\langle\Delta^{2}(\hat{x}_{j})\rangle]/2,$
(11)
$V_{ij}=-[\langle\Delta^{2}(\hat{x}_{i}-\hat{x}_{j})\rangle-\langle\Delta^{2}(\hat{x}_{i})\rangle-\langle\Delta^{2}(\hat{x}_{j})\rangle]/2,$
(12)

where $\langle\Delta^{2}(\hat{x}_{i}-\hat{x}_{j})\rangle$ and
$\langle\Delta^{2}(\hat{x}_{i}+\hat{x}_{j})\rangle$ are the correlation
variances of amplitude and phase quadratures, which can be obtained from the
measured variances in the time domain. Based on the reconstructed covariance
matrix, the quantum coherence of the EPR entangled state can be quantified
according to Eq. (2).

## APPENDIX B: Two Gaussian thermal noise channels

Here, we consider the situation in which two modes of the Gaussian EPR
entangled state are distributed through two Gaussian thermal noise channels,
where the loss and excess noise added on both modes of EPR entangled state, as
shown in Fig. B1. The two output states are measured by Alice’s and Bob’s
homodyne detectors.

Figure B1: Schematic of transmitting an EPR entangled state in two Gaussian
thermal noise channels.

The decoherence of the entanglement of the EPR entangled state in the two
lossy and noisy channels is shown in Fig. B2(a) and B2(b), respectively. The
entanglement of the EPR entangled state is robust against loss in two lossy
channels, where we assume the losses in the two channels are the same. In the
case of the two noisy channels, we fix the losses of two Gaussian thermal
noise channels to $L=0.4$, and add the same excess noise $\delta$ on the two
modes of the EPR entangled state, respectively. The entanglement of the EPR
entangled state is destroyed when the excess noise $\delta>0.74$, which is
same as the case in which the squeezing of the squeezed state is destroyed as
shown in Fig. 3(a) in the main text. Comparing Fig. B2(b) and Fig. 3(b) in the
main text, it is obvious that critical point where the disappearance of
entanglement happens is different when the excess noise is added on one or two
modes of the EPR entangled state.

The quantum coherences of the EPR entangled state in two lossy and noisy
channels are shown in Fig. B2(c) and B2(d), respectively. The dependence of
the quantum coherence of the EPR entangled state on loss and excess noise is
shown in Fig. B2(e). It is obvious that quantum coherence is robust against
loss and noise when two modes of the EPR entangled state are transmitted
through two Gaussian thermal channels respectively, which is the same as the
case in which one mode of the EPR entangled state is transmitted through a
Gaussian thermal channel as shown in Fig. 3 in the main text.

Figure B2: (a) and (b), Dependence of PPT values of the EPR entangled state on
the loss and the excess noise, respectively. The dashed line is the boundary
of the entangled and separable state. (c) and (d), Dependence of quantum
coherence of the EPR entangled state on loss and the excess noise,
respectively. (e) Quantum coherence of the EPR entangled state parameterized
by loss and excess noise.

Funding. National Natural Science Foundation of China (11834010, 11804001);
Natural Science Foundation of Anhui Province (Grant Nos 1808085QA11); National
Key R&D Program of China (Grant No. 2016YFA0301402); Fund for Shanxi “1331
Project” Key Subjects Construction.

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