# Strong quantum state transfer on graphs via loop edges

Gabor Lippner, Yujia Shi

###### Abstract

We quantify the effect of weighted loops at the source and target nodes of a
graph on the strength of quantum state transfer between these vertices. We
give lower bounds on loop weights that guarantee strong transfer fidelity that
works for any graph where this protocol is feasible. By considering local
spectral symmetry, we show that the required weight size depends only on the
maximum degree of the graph and, in some less favorable cases, the distance
between vertices. Additionally, we explore the duration for which transfer
strength remains above a specified threshold.

##  1 Introduction

We study quantum walk on a simple connected graph. Our goal is to understand
the effect of adding weighted loops on the source and target nodes on the
transfer fidelity. Let $A$ denote the adjacency matrix of the network, and let
$H=A+Q\cdot D_{u,v}$ where $D_{u,v}$ is the diagonal matrix with 1 at the $u$
and $v$ entries and 0 elsewhere.

The continuous-time quantum walk is given by

$U(t)=e^{itH},$ (1.1)

and, in particular, the $u\to v$ transfer strength at time $t$ is defined as

$p_{u,v}(t)=p(t)=|U(t)_{u,v}|^{2}.$

We are interested in $F(Q)=F(Q,u,v)=\sup_{t}p(t)$, the $u\to v$ _transfer
fidelity_ of the network, as a function of $Q$.

The relevance of transfer strength as defined above comes from its connection
to spin systems. Locally coupled networks of spin particles have been used
extensively as a protocol to transfer quantum information over physical
distances. See [1] and [4]. It is well-known [5] that the time evolution (in
the so-called 1-excitation subspace, which is most relevant for information
transfer) of such a spin network is exactly described by the continuous-time
quantum walk (1.1).

Weighted loops on the graph, in this context, correspond to applying
transverse magnetic fields to certain particles in the network [10]. Transfer
strength in this protocol has also been numerically investigated in [2, 4, 8,
3]

In terms of rigorous results, it has been shown in [7] that
$\lim_{Q\to\infty}F(Q)=1$ if and only if the network has a certain spectral
symmetry around $u$ and $v$ (for more details see Section 1.1), but there were
no quantitative bounds were given. Then, in [6] Kirkland and van Bommel gave
very precise behavior of $F(Q)$ in the case of path graphs, $u,v$ being the
endpoints of the path. It turned out that in order to achieve
$F(Q)>1-\varepsilon$, one only needs $Q>\varepsilon^{-1/2}$. What was
striking, in particular, is that the threshold did not depend on the length of
the path. Their method relies on a somewhat explicit computation of the
eigenvalues and eigenvectors of the Hamiltonian $H$, and thus it doesn’t
immediately lend itself to analyze more complicated networks.

In [9], we devised a method to generalize the bounds for the path to arbitrary
graphs that have an involution that maps $u$ to $v$. We showed that in order
to achieve $F(Q)>1-\varepsilon$ on such graphs, it is sufficient to choose
$Q>c(m,\varepsilon)$ where $m$ is the maximum degree of the network. This
method relied on the introduction of approximate eigenvectors instead of
explicit formulas. While it resulted in a worse $\varepsilon^{-2}$ \-
dependence on $\varepsilon$, it still had the important feature of not
depending on the size of the network, only its maximum degree.

In the current paper, we present a generalization of the quantitative bound to
all graphs where $\lim_{Q\to\infty}F(Q)=1$. For other graphs, the question in
this form is meaningless as the fidelity converges to a number strictly less
than 1. These new bounds depend on the maximum degree, and in certain cases
also on the distance of $u$ and $v$. However, the improved methods yield a
dependence of $1/\sqrt{\varepsilon}$ in almost all cases. In particular, we
recover the asymptotics of the Kirkland - van Bommel paper whenever $u,v$ are
cospectral in the classical sense, including any graph with an involution.

######  Theorem 1.1.

Let $G(V,E)$ be a finite graph with maximum degree $m$, and $u,v\in V$ fixed.
Assume $c\geq d$ where $c=co(u,v)$ denotes their _cospectrality_ (see
Definition 2.6) and $d=d(u,v)$ denotes the distance of the two nodes. For any
$\varepsilon>0$ we have

$Q>16\frac{1}{\varepsilon^{1/\min(2,c-d+1)}}m^{1+\max(\frac{1}{2},\frac{d}{c-d+1})}\Longrightarrow
F(Q)>1-\varepsilon.$

######  Remark 1.2.

In the above theorem, the dependence on the prescribed error is almost always
$1/\sqrt{\varepsilon}$, except for the $c=d$ case when it is $1/\varepsilon$.
The exponent of $m$ can be as large as $d+1$ when $c=d$ but goes down
gradually as $c$ increases relative to $d$, reaching the optimal $3/2$ when
$c\geq 3d-1$. In particular, when $u,v$ are cospectral in the usual sense
(i.e., $co(u,v)=\infty$), then we get $F(Q)>1-\varepsilon$ as long as
$Q=\Omega(\varepsilon^{-1/2}m^{3/2})$.

We also generalize our previous bounds [9] on the readout time $t_{0}$ at
which large transfer strength is guaranteed. While $t_{0}$ is exponential in
the distance of the source and target nodes (as was the case in [6, 9] as
well), we prove that within a large interval around $t_{0}$ the transfer
strength remains close to 1.

######  Theorem 1.3.

If $Q$ satisfies the lower bound of Theorem 1.1 then there is a time
$t_{0}<2\pi(Q+m)^{d-1}$ such that $p_{u,v}(t_{0})\geq 1-\varepsilon$.
Furthermore, for any $\delta>0$ if $t_{0}(1-\delta/\pi)\leq t\leq
t_{0}(1+\delta/\pi)$ we have $p_{u,v}(t)\geq 1-\varepsilon-2\delta$.

The outline of this paper is as follows. In Section 1.1 we outline the main
strategy of the proof. In Section 2, we establish the key estimates on the
components of the eigenvectors. Section 3 concludes the proof of the main
result using the estimates from previous sections. Finally, we address the
bounds on the readout time and readout interval in Section 4.

###  1.1 Proof strategy

The Hamiltonian $H$ is a real symmetric matrix so it has real eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}$. Let us denote the
corresponding orthonormal eigenvectors by
$\varphi_{1},\varphi_{2},\dots,\varphi_{n}$. Then we can compute the
probability by

$\displaystyle
p(t)=|U(t)_{u,v}|^{2}=\left|\sum_{j=1}^{n}\varphi_{j}(u)\varphi_{j}(v)e^{it\lambda_{j}}\right|^{2}.$
(1.2)

If $Q$ is large enough, the Hamiltonian $H$ has two diagonal entries that are
significantly larger than the rest. This results in the two largest
eigenvalues $\lambda_{1}$ and $\lambda_{2}$ approximately equal to $Q$,
whereas, in comparison, the rest eigenvalues are small in absolute value.
Furthermore, the corresponding eigenvectors $\varphi_{1},\varphi_{2}$ will be
concentrated on the $\\{u,v\\}$ subspace, and by orthogonality of the
eigenvectors we should expect
$\operatorname{sgn}(\varphi_{1}(u)\varphi_{1}(v))=-\operatorname{sgn}(\varphi_{2}(u)\varphi_{2}(v))$.
Thus, it is natural to consider the process at time
$t_{0}=\frac{\pi}{\lambda_{1}-\lambda_{2}}$. With this choice, we can estimate

$|U(t_{0})_{u,v}|\geq|\varphi_{1}(u)\varphi_{1}(v)-\varphi_{2}(u)\varphi_{2}(v)|-\sum_{j=3}^{n}|\varphi_{j}(u)\varphi_{j}(v)|\\\
\geq|\varphi_{1}(u)\varphi_{1}(v)-\varphi_{2}(u)\varphi_{2}(v)|-\sqrt{(1-\varphi_{1}(u)^{2}-\varphi_{2}(u)^{2})(1-\varphi_{1}(v)^{2}-\varphi_{2}(v)^{2})}$
(1.3)

Showing that this last expression can be arbitrarily close to 1 by selecting
an appropriate potential $Q$ is going to be our main goal.

##  2 Estimates

In this section, we establish the main estimates and introduce the key notions
to bound the values appearing in (1.3).

###  2.1 Eigenvector extension

Our main tool for studying $\varphi_{1}$ and $\varphi_{2}$ for large values of
$Q$ is adapted from [7].

######  Definition 2.1.

Given a graph $G(V,E)$ and $u,v\in V$, let $\mathcal{P}_{xy}$ denote the set
of walks in $G$ that start at $x$ and end at $y$ and avoid $u$ and $v$, except
for possibly the first and last vertex. In the case of $x=y$, we do not
include the 0-length walk. Note that this set is typically infinite. For a
path $P\in\mathcal{P}_{xy}$, let $|P|$ denote its length, i.e., its number of
edges. Finally, define a function $Z_{xy}(\lambda)$ via the formula

$Z_{xy}(\lambda)=\sum_{P\in\mathcal{P}_{xy}}\lambda^{-|P|}.$

We will often use $n_{k}(xy)=|\\{P\in\mathcal{P}_{xy}:|P|=k\\}|$ to denote the
number of allowed walks of length exactly $k$ from $x$ to $y$. Then we can
rewrite $Z$ as a power series in $\lambda^{-1}$ as

$Z_{xy}(\lambda)=\sum_{k=0}^{\infty}\frac{n_{k}(xy)}{\lambda^{k}}=\sum_{k=d(x,y)}^{\infty}\frac{n_{k}(xy)}{\lambda^{k}}.$

We omit the dependence on $u$ and $v$ from the notation in both $\mathcal{P}$
and $Z$, but this will not lead to confusion as $u,v$ will be fixed throughout
the paper.

######  Remark 2.2.

If the maximum degree of $G$ is $m$, then clearly $n_{k}(xy)\leq m^{k}$. Hence
$Z_{xy}(\lambda)$ is an absolutely convergent power series for $\lambda>m$. In
particular,

$Z_{xy}(\lambda)\leq\sum_{k=d(x,y)}^{\infty}\frac{m^{k}}{\lambda^{k}}\leq\left(\frac{m}{\lambda}\right)^{d(x,y)}\frac{1}{1-\frac{m}{\lambda}}.$

Now we have the following theorem, whose proof is based on [7].

######  Theorem 2.3 ([9]).

Let $G$ be a connected graph $u,v\in V(G)$, and $H$ as above. Let $m$ denote
the maximum degree of $G$ and let $\lambda>m$. Let further $\mu,\nu$ be given
real numbers. Then $H$ has an eigenvector $\varphi$ with eigenvalue $\lambda$
and $\varphi(u)=\mu,\varphi(v)=\nu$ if and only if

$\displaystyle\begin{bmatrix}Z_{uu}(\lambda)&Z_{uv}(\lambda)\\\
Z_{uv}(\lambda)&Z_{vv}(\lambda)\end{bmatrix}\begin{bmatrix}\mu\\\
\nu\end{bmatrix}=\left(1-\frac{Q}{\lambda}\right)\begin{bmatrix}\mu\\\
\nu\end{bmatrix}$ (2.1)

When $\varphi$ exists it is unique and satisfies

$\forall x\neq u,v:\varphi(x)=\mu Z_{xu}(\lambda)+\nu Z_{xv}(\lambda).$ (2.2)

###### Proof.

First consider a function $f:V(G)\rightarrow\mathbb{R}$ such that $f(u)=\mu$,
$f(v)=\nu$, and $Hf(x)=\lambda f(x)$ for all $x\in V(G)\backslash\\{u,v\\}$.
For any $x\neq u,v$ we then have

$f(x)=\frac{1}{\lambda}Hf(x)=\sum_{y\sim x}\frac{f(y)}{\lambda}.$

We iterate this for all $y\neq u,v$ appearing in the above sum for up to $k$
steps. Whenever $u$ or $v$ shows up during the iteration, we keep that term as
it is and only iterate on the other terms. Then we get

$f(x)=\sum_{y\neq
u,v}\sum_{P\in\mathcal{P}_{xy}:|P|=k}\frac{f(y)}{\lambda^{k}}+f(u)\sum_{P\in\mathcal{P}_{xu}:|P|\leq
k}\frac{1}{\lambda^{|P|}}+f(v)\sum_{P\in\mathcal{P}_{xv}:|P|\leq
k}\frac{1}{\lambda^{|P|}}.$

When $k\to\infty$, we see from Remark 2.2 that the first sum on the right hand
side goes to 0 because it is bounded by
$\left(\frac{m}{\lambda}\right)^{k}\sum_{y\neq u,v}|f(y)|$ since $\lambda>m$.
The other two terms converge to $f(u)Z_{xu}(\lambda)$ and
$f(v)Z_{xv}(\lambda)$ respectively. So we get that

$\displaystyle\forall x\neq u,v:f(x)=\mu Z_{xu}(\lambda)+\nu
Z_{xv}(\lambda);f(u)=\mu;f(v)=\nu,$ (2.3)

in particular we see that (2.2) holds.

Conversely, it is easy to see that (2.3) indeed defines a function
$f:V(G)\rightarrow\mathbb{R}$ such that $f(u)=\mu$, $f(v)=\nu$, and
$Hf(x)=\lambda f(x)$ for all $x\in V(G)\backslash\\{u,v\\}$. Hence there is
always a unique function $f:V(G)\to\mathbb{R}$ that satisfies these
conditions.

Thus, to finish the proof, it is sufficient to determine, when is $f$ actually
an eigenvector of $H$. It happens exactly when $\lambda f(x)=Hf(x)$ is also
satisfied at $x=u,v$,

$\displaystyle\lambda f(u)=Qf(u)+\sum_{x\sim u}f(x)\text{ ; }\lambda
f(v)=Qf(v)+\sum_{x\sim v}f(x)$ (2.4)

Dividing both sides of equations (2.4) by $\lambda$, rearranging, and plugging
in (2.3) for each $f(x)$ in the sums, we get the following equality for both
$z=u$ and $z=v$:

$\displaystyle\left(1-\frac{Q}{\lambda}\right)f(z)=\mu\cdot\sum_{x\sim
z}\frac{1}{\lambda}Z_{xu}(\lambda)+\nu\cdot\sum_{x\sim
z}\frac{1}{\lambda}Z_{xv}(\lambda)=\mu Z_{zu}(\lambda)+\nu Z_{zv}(\lambda)$
(2.5)

which, for $z=u,v$ gives exactly the $2\times 2$ linear system (2.1). ∎

######  Remark 2.4.

As we see from the preceding theorem, for two real numbers $\mu$ and $\nu$ to
be the entries $\varphi(u)$, $\varphi(v)$ of the eigenvector $\varphi$ of the
matrix $H=A+Q\cdot D_{u,v}$ corresponding to eigenvalue $\lambda$, they should
satisfy the condition of forming an eigenvector of the $2\times 2$ matrix
described in (2.1).

A particularly nice feature of (2.1) is that the only dependence on $Q$ is in
the factor on the right-hand side. So, for any given $\lambda>m$ one can
consider the $2\times 2$ matrix on the left, compute an eigenvector
$(\mu,\nu)$ and corresponding eigenvalue $\rho<1$, and then choose
$Q=\lambda(1-\rho)$. For this choice of $Q$, then $\lambda$ is guaranteed to
be an eigenvalue of $H$ and the corresponding eigenvector $\varphi$, scaled
appropriately, will satisfy $\varphi(u)=\mu$ and $\varphi(v)=\nu$.

######  Corollary 2.5.

Using the previous notation, if $\lambda\geq C\cdot m^{3/2}$ is an eigenvalue
of $H$ for some $C>2$ then the corresponding eigenvector satisfies

$\varphi(u)^{2}+\varphi(v)^{2}\geq||\varphi||_{2}^{2}\left(1-\frac{22}{C^{2}}\right)$
(2.6)

###### Proof.

First we bound $f(x)$ pointwise using Remark 2.2:

$|f(x)|\leq(|\mu|+|\nu|)\max\\{Z_{xu}(\lambda),Z_{xv}(\lambda)\\}\leq\frac{|\mu|+|\nu|}{1-\frac{m}{\lambda}}\left(\frac{m}{\lambda}\right)^{r(x)}$

where $r(x)=\min\\{d(x,u),d(x,v)\\}$. This implies

$\sum_{x\neq
u,v}f(x)^{2}\leq\left(\frac{|\mu|+|\nu|}{1-\frac{m}{\lambda}}\right)^{2}\sum_{x\neq
u,v}\left(\frac{m}{\lambda}\right)^{2r(x)}\leq
2(\mu^{2}+\nu^{2})\left(1-\frac{m}{\lambda}\right)^{-2}\sum_{k=1}^{\infty}\left(\frac{m}{\lambda}\right)^{2k}|\\{x:r(x)=k\\}|.$

In a graph of maximum degree $m$ there can be no more than $m^{k}$ nodes at
distance $k$ from any given node, hence $|\\{x:r(x)=k\\}|\leq 2m^{k}$. This
yields

$\frac{\sum_{x\neq
u,v}f(x)^{2}}{\mu^{2}+\nu^{2}}\leq\frac{4}{\left(1-\frac{m}{\lambda}\right)^{2}}\sum_{k=1}^{\infty}\left(\frac{m^{3}}{\lambda^{2}}\right)^{k}\leq\frac{4}{(1-\frac{1}{C\sqrt{m}})^{2}C^{2}(1-\frac{1}{C^{2}})}\leq\frac{64}{3C^{2}}$

and hence

$||\varphi||_{2}^{2}\leq(\mu^{2}+\nu^{2})\left(1+\frac{64}{3C^{2}}\right)\leq(\varphi(u)^{2}+\varphi(v)^{2})\left(1+\frac{22}{C^{2}}\right)$

whence the claim follows. ∎

###  2.2 Cospectrality

Let us recall the following definitions and theorem from [7].

######  Definition 2.6.

Two vertices, $u,v\in V(G)$ are $k$-cospectral if, for any positive integer
$t\leq k$ the number of closed walks of length $t$ at $u$ and $v$ are equal to
each other, or equivalently, if $\forall t\leq k:n_{t}(uu)=n_{t}(vv)$. The
cospectrality of $u$ and $v$, denoted by $co(u,v)$ is the maximum number $m$
such that $u$ and $v$ are $m$-cospectral.

######  Definition 2.7.

Define the tunneling coefficient from state $u$ to $v$ to be

$\displaystyle TC(u,v)=\liminf_{Q\rightarrow\infty}{\sup_{t\geq 0}p(t)}.$
(2.7)

We say there is

  1. 1.

asymptotic tunneling (strong state transfer) if $TC(u,v)=1$

  2. 2.

partial tunneling if $0<TC(u,v)<1$

  3. 3.

no tunneling if $TC(u,v)=0$

The relationship between cospectrality and tunneling behavior was studied in
[7] where the following was shown for the random walk transition matrix in
place of the adjacency matrix (however the proofs carry over without much
modification to the current case):

######  Theorem 2.8.

[7] Let there be two vertices $u$ and $v$ and let $d=\text{dist}(u,v)$, then
the following is true:

  1. 1.

if $co(u,v)\geq d$ then there is asymptotic tunneling

  2. 2.

if $co(u,v)=d-1$ then there is partial tunneling

  3. 3.

if $co(u,v)<d-1$ then there is no tunneling

This theorem suggests that $co(u,v)\geq d$ is a reasonable assumption in order
to achieve strong state transfer using large values of $Q$. We have seen that,
for large enough $Q$, strong state transfer depends on
$|\varphi_{j}(u)|\approx|\varphi_{j}(v)|$ for $j=1,2$. We will now show that
$co(u,v)\geq d$ indeed implies this in a quantitative way.

######  Lemma 2.9.

Using the notation from Section 2.1, if $\lambda>m$ is an eigenvalue of $H$
and $c\geq d$ then the corresponding eigenvector $\varphi$ satisfies

$\left|\frac{\varphi(u)}{\varphi(v)}-\frac{\varphi(v)}{\varphi(u)}\right|\leq\frac{m^{c+1}}{\lambda^{c-d}(\lambda-m)}$

where $c=co(u,v)$ and $d=\text{dist}(u,v)$.

###### Proof.

Let $\mu=\varphi(u)$ and $\nu=\varphi(v)$. By Theorem 2.3,
$\begin{bmatrix}\mu\\\ \nu\end{bmatrix}$ is an eigenvector of the $Z$ matrix
in (2.1), and dividing the two resulting equations to cancel the common
eigenvalue yields

$\frac{\mu}{\nu}=\frac{\mu Z_{uu}(\lambda)+\nu Z_{uv}(\lambda)}{\mu
Z_{uv}(\lambda)+\nu Z_{vv}(\lambda)}.$

After rearranging we get

$(\mu^{2}-\nu^{2})Z_{uv}(\lambda)=\mu\nu(Z_{uu}(\lambda)-Z_{vv}(\lambda)),$

and finally

$\displaystyle\left|\frac{\mu}{\nu}-\frac{\nu}{\mu}\right|=\left|\frac{Z_{uu}(\lambda)-Z_{vv}(\lambda)}{Z_{uv}(\lambda)}\right|=\frac{\left|\sum_{k=2}^{\infty}\frac{n_{k}(u,u)}{\lambda^{k}}-\sum_{k=2}^{\infty}\frac{n_{k}(v,v)}{\lambda^{k}}\right|}{\sum_{k=d}^{\infty}\frac{n_{k}(u,v)}{\lambda^{k}}}$

Here the denominator is at least $1/\lambda^{d}$ while the $k\leq c$ terms
cancel in the numerator, hence

$\left|\frac{\mu}{\nu}-\frac{\nu}{\mu}\right|\leq\lambda^{d}\sum_{k>c}\frac{|n_{k}(uu)-n_{k}(vv)|}{\lambda^{k}}\leq\lambda^{d}\sum_{k>c}\left(\frac{m}{\lambda}\right)^{k}=\frac{m^{c+1}}{\lambda^{c-d}(\lambda-m)}$

as claimed. ∎

##  3 Bounding Fidelity in the Presence of Quantum Tunneling

In this section, we prove our main estimate: a lower bound of the transfer
strength of the quantum walk between two vertices that exhibit asymptotic
state transfer as defined in Definition 2.7. This bound is a function of the
potential $Q$ that is applied to the nodes $u$ and $v$, along with the maximum
degree of the graph, and in certain cases, the distance and cospectrality of
the nodes $u,v$.

######  Theorem 3.1.

Consider a simple connected graph $G=(V,E)$ with maximum degree $m$. Let $u$
and $v$ be two vertices exhibiting asymptotic state transfer, meaning their
cospectrality $c=co(u,v)$ is at least their distance $d=d(u,v)$. After adding
a potential $Q$ to both vertices, the transfer strength from $u$ to $v$ at
time $t_{0}=\pi/(\lambda_{1}-\lambda_{2})$ satisfies

$p(t_{0})\geq
1-8\max\left\\{\frac{22m^{3}}{(Q-m)^{2}},\frac{m^{c+1}}{(Q-2m)^{c-d+1}}\right\\}$

Theorem 1.1 follows directly from Theorem 3.1 by a simple calculation.

###### Proof.

Since $Q>2m$, Gershgorin’s Circle Theorem implies that there are exactly two
eigenvalues in the interval $[Q-m,Q+m]$ and all other eigenvalues are in the
$[-m,m]$ interval. Let $\lambda_{1},\lambda_{2}$ denote the two eigenvalues in
$[Q-m,Q+m]$ and let $\varphi_{1},\varphi_{2}$ denote the corresponding unit
length eigenvectors.

For simplicity, we denote the $u,v$ entries in the normalized eigenvectors in
the following manner from now on:

$\varphi_{1}(u,v)=\begin{bmatrix}\mu_{1}\\\
\nu_{1}\end{bmatrix},\varphi_{2}(u,v)=\begin{bmatrix}\mu_{2}\\\
\nu_{2}\end{bmatrix}.$

Without loss of generality we may assume $\mu_{1}\geq\nu_{1}$. From Corollary
2.5 and Lemma 2.9, we know that

$\displaystyle 1\geq\mu_{1}^{2}+\nu_{1}^{2}$ $\displaystyle\geq
1-\frac{22m^{3}}{\lambda_{1}^{2}}\geq 1-\frac{22m^{3}}{(Q-m)^{2}}$
$\displaystyle 1\leq\frac{|\mu_{1}|}{|\nu_{1}|}$ $\displaystyle\leq
1+\frac{m^{c+1}}{(\lambda_{1}-m)^{c-d+1}}\leq
1+\frac{m^{c+1}}{(Q-2m)^{c-d+1}}.$

Thus, with
$\varepsilon^{\prime}=\max\\{\frac{22m^{3}}{(Q-m)^{2}},\frac{m^{c+1}}{(Q-2m)^{c-d+1}}\\}$
we get

$1-\varepsilon^{\prime}\leq\mu_{1}^{2}+\nu_{1}^{2}\leq\nu_{1}^{2}(1+(1+\varepsilon^{\prime})^{2})$

so

$\nu_{1}^{2}\geq\frac{1-\varepsilon^{\prime}}{2+2\varepsilon^{\prime}+\varepsilon^{\prime
2}}\geq\frac{1}{2}-\varepsilon^{\prime}.$

Hence

$\frac{1}{2}-\varepsilon^{\prime}\leq\nu_{1}^{2}\leq\mu_{1}^{2}\leq\frac{1}{2}+\varepsilon^{\prime}$

The same is clearly true for $\mu_{2}$ and $\nu_{2}$. Due to Perron-Frobenius
Theorem, $\mu_{1}$ and $\nu_{1}$ have the same sign. However,
$\mu_{2},\nu_{2}$ must have opposite signs. To see this, note that
$\varphi_{1}$ and $\varphi_{2}$ are orthogonal to each other, hence
$0=\mu_{1}\mu_{2}+\nu_{1}\nu_{2}+\sum_{x\neq
u,v}\varphi_{1}(x)\varphi_{2}(x)$. This implies

$|\mu_{1}\mu_{2}+\nu_{1}\nu_{2}|=\left|\sum_{x\neq
u,v}\varphi_{1}(x)\varphi_{2}(x)\right|\leq\sqrt{\sum_{x\neq
u,v}\varphi_{1}(x)^{2}}\sqrt{\sum_{x\neq u,v}\varphi_{2}(x)^{2}}\\\
=\sqrt{1-\mu_{1}^{2}-\nu_{1}^{2}}\sqrt{1-\mu_{2}^{2}-\nu_{2}^{2}}\leq
2\varepsilon^{\prime}$

But if $\mu_{2}$ and $\nu_{2}$ had the same sign then the left-hand side would
be at least $1-2\varepsilon^{\prime}$ which implies $\varepsilon^{\prime}\geq
1/4$ and there is nothing to prove. Therefore, employing (1.3) with
$t_{0}=\frac{\pi}{\lambda_{1}-\lambda_{2}}$, the transfer strength is at least

$\displaystyle
p(t_{0})\geq(1-2\varepsilon^{\prime}-\sqrt{(2\varepsilon^{\prime})(2\varepsilon^{\prime})})^{2}\geq
1-8\varepsilon^{\prime}$

∎

##  4 Readout Time

We have seen in Theorem 3.1 that at
$t_{0}=\frac{\pi}{\lambda_{1}-\lambda_{2}}$ the transfer strength $p(t_{0})$
is guaranteed to be at least $1-\varepsilon$ as long as $Q$ satisfies the
lower bound set forth in Theorem 1.1. Unfortunately, the scaling of $t_{0}$ is
always exponential in the distance $d=d(u,v)$ between the nodes $u$ and $v$.
Nonetheless, our main goal in this section is to give an upper bound on
$t_{0}$, or equivalently, a lower bound on $\lambda_{1}-\lambda_{2}$, the gap
between the two largest eigenvalues of the Hamiltonian. This will be done by
adapting our methods from [9]. The first part of Theorem 1.3 follows from the
subsequent statement.

######  Theorem 4.1.

Under the assumptions of Theorem 1.1 the following bound holds for
$t_{0}=\frac{\pi}{\lambda_{1}-\lambda_{2}}$:

$t_{0}\leq 2\pi(Q+m)^{d-1}$

where $d=d(u,v)$, the distance of the source and target nodes in the graph.

###### Proof.

Our starting point is the $Z$ matrix (2.1), and we apply it to both
$\varphi_{1}$ and $\varphi_{2}$. Using an indexed version of the notation
there we find

$\displaystyle(\lambda_{1}-Q)\mu_{1}=$
$\displaystyle\lambda_{1}Z_{uu}(\lambda_{1})\mu_{1}+\lambda_{1}Z_{uv}(\lambda_{1})\nu_{1}$
$\displaystyle(\lambda_{2}-Q)\mu_{2}=$
$\displaystyle\lambda_{2}Z_{uu}(\lambda_{2})\mu_{2}+\lambda_{2}Z_{uv}(\lambda_{2})\nu_{2}$

∎

which gives, after further rearrangement and taking difference of the two
equations:

$(\lambda_{1}-\lambda_{2})-(\lambda_{1}Z_{uu}(\lambda_{1})-\lambda_{2}Z_{uu}(\lambda_{2}))=\lambda_{1}\frac{\nu_{1}}{\mu_{1}}Z_{uv}(\lambda_{1})-\lambda_{2}\frac{\nu_{2}}{\mu_{2}}Z_{uv}(\lambda_{2}).$
(4.1)

Note that, according to the proof of Theorem 1.1,
$\nu_{1}/\mu_{1}>0>\nu_{2}/\mu_{2}$ and hence all terms on the right-hand side
are positive. Hence, we can lower bound it by the first respective terms in
the series of $Z_{uv}$. Also, expanding the left-hand side using the
definition of $Z_{uu}$ we get the following inequality

$(\lambda_{1}-\lambda_{2})+\sum_{k=1}^{\infty}n_{k+1}(uu)\left(\frac{1}{\lambda_{2}^{k}}-\frac{1}{\lambda_{1}^{k}}\right)\geq\left(\left|\frac{\nu_{1}}{\mu_{1}}\right|\frac{1}{\lambda_{1}^{d-1}}+\left|\frac{\nu_{2}}{\mu_{2}}\right|\frac{1}{\lambda_{2}^{d-1}}\right)\geq\frac{1}{\lambda_{1}^{d-1}}.$

Here $d$ denotes $d(u,v)$, and in the last step, we used that both
$|\nu_{1}/\mu_{1}|\geq 1-\varepsilon$ and $|\nu_{2}/\mu_{2}|\geq
1-\varepsilon$, so their sum is clearly at least 1. We can further rewrite the
left-hand side to get

$(\lambda_{1}-\lambda_{2})\left(1+\sum_{k=1}^{\infty}n_{k+1}(uu)\sum_{j=1}^{k}\frac{1}{\lambda_{1}^{j}\lambda_{2}^{k+1-j}}\right)\geq\frac{1}{\lambda_{1}^{d-1}}.$

Now using the usual $n_{k}(uu)\leq m^{k}$ bound we find

$\frac{1}{\lambda_{1}^{d-1}}\leq(\lambda_{1}-\lambda_{2})\left(1+\sum_{k=1}^{\infty}\frac{k\cdot
m^{k+1}}{\lambda_{2}^{k+1}}\right)=(\lambda_{1}-\lambda_{2})\left(1+\frac{1}{1-m/\lambda_{2}}\sum_{k=2}^{\infty}\frac{m^{k}}{\lambda_{2}^{k}}\right)=\\\
(\lambda_{1}-\lambda_{2})\left(1+\frac{m^{2}}{(\lambda_{2}-m)^{2}}\right)\leq(\lambda_{1}-\lambda_{2})\left(1+\frac{m^{2}}{(Q-2m)^{2}}\right)\leq(\lambda_{1}-\lambda_{2})\left(1+\frac{1}{m^{2}}\right),$

hence, finally,

$t_{0}=\frac{\pi}{\lambda_{1}-\lambda_{2}}\leq\lambda_{1}^{d-1}\frac{m^{2}+1}{m^{2}}\leq
2(Q+m)^{d-1}$

###  4.1 Readout interval

In this section, we establish the second part of Theorem 1.3 regarding the
length of the interval centered around $t_{0}$.

######  Theorem 4.2.

Let $\varepsilon>0$ be fixed and assume $Q$ satisfies the bounds set forth in
Theorem 1.1. Then for all
$t\in[\frac{\pi-\delta}{\lambda_{1}-\lambda_{2}},\frac{\pi+\delta}{\lambda_{1}-\lambda_{2}}]$,
the following bound is true:

$p(t)\geq 1-\varepsilon-2\delta$

###### Proof.

A simple computation shows that the lower bound on $Q$ implies that
$\varepsilon^{\prime}=\max\\{\frac{22m^{3}}{(Q-m)^{2}},\frac{m^{c+1}}{(Q-2m)^{c-d+1}}\\}\leq\varepsilon/8$.
(This is the same computation that shows Theorem 3.1 implies Theorem 1.1.) A
straightforward generalization of (1.3) shows that

$|U(t)_{u,v}|\geq|e^{it(\lambda_{1}-\lambda_{2})}\varphi_{1}(u)\varphi_{1}(v)-\varphi_{2}(u)\varphi_{2}(v)|-\sqrt{(1-\varphi_{1}(u)^{2}-\varphi_{2}(u)^{2})(1-\varphi_{1}(v)^{2}-\varphi_{2}(v)^{2})}.$

Notice that for real numbers $a>0>b$ and angle $\pi-\delta<\rho<\pi+\delta$ we
have

$|a+e^{i\rho}b|^{2}=a^{2}+b^{2}+ab(e^{i\rho}+e^{-i\rho})=(a-b)^{2}+ab(2+e^{i\rho}+e^{-i\rho})\\\
\geq(a-b)^{2}-2ab(1-\cos\delta)\geq(a-b)^{2}-\delta^{2}ab$

and thus

$|a+e^{i\rho}b|\geq|a-b|-\delta\sqrt{ab}.$

Combining this with estimates from the proof of Theorem 3.1, and noticing that
$\pi-\delta\leq t(\lambda_{1}-\lambda_{2})\leq\pi+\delta$, we get

$|U(t)_{u,v}|\geq|\varphi_{1}(u)\varphi_{1}(v)+e^{it(\lambda_{1}-\lambda_{2})}\varphi_{2}(u)\varphi_{2}(v)|-2\varepsilon^{\prime}\\\
\geq|\varphi_{1}(u)\varphi_{1}(v)-\varphi_{2}(u)\varphi_{2}(v)|-\delta\sqrt{|\varphi_{1}(u)\varphi_{1}(v)\varphi_{2}(u)\varphi_{2}(v)|}-2\varepsilon^{\prime}\geq
1-4\varepsilon^{\prime}-\delta$

and thus

$p(t)=\left|U(t)_{u,v}\right|^{2}\geq(1-4\varepsilon^{\prime}-\delta)^{2}\geq
1-8\varepsilon^{\prime}-2\delta=1-\varepsilon-2\delta.$

∎

## References

  * [1] Sougato Bose.  Quantum communication through an unmodulated spin chain.  Physical Review Letters, 91(20), November 2003.
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  * [7] Yong Lin, Gábor Lippner, and Shing-Tung Yau.  Quantum tunneling on graphs.  Communications in Mathematical Physics, 311(1):113–132, 02 2012.
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# Super-resolution of two unbalanced point sources assisted by the entangled
partner

Abdelali Sajia  Xiao-Feng Qian<EMAIL_ADDRESS>Center for Quantum Science
and Engineering, and Department of Physics, Stevens Institute of Technology,
Hoboken, New Jersey 07030, USA

###### Abstract

Sub-diffraction-limit resolution, or super-resolution, had been successfully
demonstrated by recent theoretical and experimental studies for two-point
sources with ideal equal-brightness and strict incoherenceness. Unfortunately,
practical situations of either non-equal brightness (i.e., unbalancenss) or
partial coherence are shown to have fatal effects on resolution precision. As
a step toward resolving such issues, we consider both effects together by
including an entangled partner of the two-point sources. Unexpectedly, it is
found that the two negative effects can counter affect each other, thus
permitting credible super-resolution, when the measurement is analyzed in the
entangled partner’s rotated basis. A least resolvable finite two-source
separation is also identified analytically. Our result represents useful
guidance towards the realization of super-resolution for practical point
sources. The vector-structure analog of quantum and classical light sources
also suggests that our analysis applies to both contexts.

##  I Introduction

Resolution of two point sources is one of the most crucial elements in the
science of imaging and sensing. The quality of fine resolution relies on two
major factors: resolution capability (how small a separation can be
discriminated) and measurement estimation credibility (how much the
measurement can be trusted). For over a century, the empirical Abbe-Rayleigh
diffraction criterion Abbe1873 ; Rayleigh1896 ; BornWolf ; Rayleigh1979 ,
relating to the ratio of light wavelength and aperture diameter, has been
regarded as a roadblock that limits the resolution capability with practically
feasible parameters Goodman2005 ; Kolobov2000PRL . This is due to the fact
that when the two sources are getting closer, the blurred overlapping signals
are harder to discriminate through direct intensity measurements. Moreover,
the second factor, measurement credibility (or measurement precision), will
also reduce as noise effects become relatively more prominent when the
separation of the two sources decreases. Studies have shown that while
statistical methods were able to improve the resolution capability by
determining the source locations, the precision of measurement vanishes as the
separation of the two sources goes beyond the diffraction limit, approaching
zero Bettens1999UM ; Aert2002JSB ; Ram2006PNAS ; Chao2016JOSA . This
phenomenon consolidates the common wisdom of the Abbe-Rayleigh diffraction
criterion from a more rigorous foundation, and is termed as the Rayleigh’s
curse by many authors, see for example in Tsang2015O ; Tsang2016PRL ;
Tsang2016PRX ; Paur2016O ; Rehacek2017PRA ; Larson2018O ; Hradil2019O ;
Liang2021O ; Wadood2021OE ; Faber2000OL .

Recently, the pioneering works of Tsang and coworkers Tsang2015O ;
Tsang2016PRX ; Tsang2019arxiv ; Tsang2019O  showed that it is possible in
principle to improve both factors by analyzing the signal in a different
(e.g., the Hermite-Gaussian mode) spatial basis instead of direct intensity
measurement. The new technique leaves the Abbe-Rayleigh diffraction criterion
irrelevant and at the same time guarantees a finite desired estimation
accuracy via the parameter Fisher information (FI) Helstrom1976 ; Holevo2001 .
Experimental confirmations have also been demonstrated, see for examples in
Paur2016O ; Wadood2021OE . While working perfectly in ideal situations, this
technique has two constraints, requiring (1) incoherence and (2) balance
(equal brightness) of the two point sources. It has been shown that releasing
either one of the two may lead to the resurgence of Rayleigh’s curse by
Rehacek et al., with unbalanced incoherent sources Rehacek2017PRA  and by
Larson-Saleh Larson2018O ; Larson2019O  and De, et al., DePRR2021  with
balanced but partially coherent sources .

Figure 1: The point spread functions, $h_{+}(x)$ and $h_{-}(x)$, of two
unbalanced point sources separated by $s$ via a shift-invariant imaging
system. The coefficients $a$ and $b$ characterize a continuous unbalanceness.

To address this issue, we attack both restrictions at the same time by
investigating the resolution of two unbalanced and partially coherent point
sources with the assistance of an entangled partner, see a schematic
illustration in Fig. 1. The effect of partial coherence is analyzed by
continuous basis rotation of the entangled partner Plenio2017RMP . It is found
that the effect of unbalanceness on the two-source separation estimation
parameter, i.e., Fisher Information, is equivalent to that of the basis
rotation-induced partial coherence. Unexpectedly, the joint effect of the two
restrictions by entanglement permits the realization of super-resolution with
a finite Fisher Information even when the separation of the two sources is
approaching zero. It is also determined analytically that there exists a
“least resolvable” (non-zero) distance of which the Fisher information
experiences a non-zero minimum that is determined by Lambert W functions
Corless1996ACM . This provides a guidance to employ optimum practical
parameters to achieve super-resolution with a given accuracy requirement.

##  II Model and Methods

We consider two unbalanced sources located at $x\pm s/2$ respectively with a
separation $s$. Through a shift-invariant imaging system, their normalized
point spread functions (PSF) are assumed to take the Gaussian form, i.e.,
$h_{\pm}(x)=h(x\pm s/2)$ with
$h(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp[{-\frac{x^{2}}{2\sigma^{2}}}]$ and
$\sigma$ as the width, see an illustration in Fig. 1. We include the partial
coherence property of the two sources by introducing an entangled partner of
the spatial degree of freedom. The optical field in the image plane can be
described as

$\displaystyle\ket{\Psi_{ent}}$
$\displaystyle=a\ket{h_{+}}\ket{\phi_{1}}+be^{i\varphi}\ket{h_{-}}\ket{\phi_{2}},$
(1)

where $\ket{h_{\pm}}$ represent the two non-orthogonal point spread functions
(vectors) of the spatial degree of freedom with $\langle
x\ket{h_{\pm}}=h_{\pm}(x)$, $\ket{\phi_{1}},\ket{\phi_{2}}$ are two generic
states describing the remaining degrees of freedom (including polarization,
temporal domain, etc.), $a$ and $b$ are real normalized coefficients with
$a^{2}+b^{2}=1$, and $\varphi$ is an arbitrary phase. The degree of
unbalanceness of the two sources can be simply quantified by the ratio
$r=b/a$, where $r=0$ means completely unbalanced and $r=1$ indicating
balanceness ($r>1$ is equivalent to $r<1$ since $a$ and $b$ are symmetric.
Here the spatial state space $\\{\ket{h_{+}},\ket{h_{-}}\\}$ is entangled with
the remaining state space $\\{\ket{\phi_{1}},\ket{\phi_{2}}\\}$. The entangled
state (1) can either be a quantum state of single photons or a macroscopic
classical optical field Spreeuw1998FP ; Luis2009OC ; Simon2010PRL ;
Borges2010PRA ; Qian2011OL ; Kagalwala2013NP ; Toppel2014NJP ; Zela2014PRA ;
Forbes2017Nat ; Goldberg2021AOP .

In general, $\ket{\phi_{1}},\ket{\phi_{2}}$ can be non-orthogonal and their
overlap, $\langle\phi_{1}|\phi_{2}\rangle$, determines the degree of
coherence. For the convenience of following discussions, we take the two to be
orthogonal but analyze the operations (or detections) in an arbitrarily
rotated basis, i.e.,
$|\phi^{\alpha}_{1}\rangle=\cos\alpha|\phi_{1}\rangle-\sin\alpha|\phi_{2}\rangle$
and
$|\phi^{\alpha}_{2}\rangle=\sin\alpha|\phi_{1}\rangle+\cos\alpha|\phi_{2}\rangle$.
This is an equivalent way of introducing partial coherence due to its basis-
dependent nature Plenio2017RMP ; Qian2011OL ; Kagalwala2013NP ;
Goldberg2021AOP , see an illustrative analysis in supplemental material
section 1.

The optical field can be rewritten in the new basis as

$\displaystyle\ket{\Psi_{ent}}=\ket{h_{1}}|\phi^{\alpha}_{1}\rangle+\ket{h_{2}}|\phi^{\alpha}_{2}\rangle,$
(2)

where $\ket{h_{1}}=a\cos\alpha\ket{h_{+}}-b\sin\alpha
e^{i\varphi}\ket{h_{-}}$, $\ket{h_{2}}=a\sin\alpha\ket{h_{+}}+b\cos\alpha
e^{i\varphi}\ket{h_{-}}$ are two new spatial functions that are in general
non-normalized and non-orthogonal.

To quantify the likely-hood of being an accurate estimation (or the degree to
which measurements can be trusted) of the two-source separation $s$, the
conventional Fisher information (FI) Helstrom1976 ; Pang2017NC ; Shukur2020Nat
is employed. It’s definition is based on the Cramér-Rao bound
Zmuidzinas2003OSA ; Kay1993 ; Pang2014PRL  ${\rm Var}(s)\geq 1/F$. The optimal
estimation of the unknown parameter $s$ corresponds to the maximization of the
Fisher information $F$ which corresponds to a minimum of the estimator
variance ${\rm Var}(s)$.

A recent debate about the nonphysical divergence of Fisher information
Larson2018O ; Tsang2019O ; Larson2019O ; Hradil2019O  suggests that the Fisher
information for analyzing two-source super-resolution needs to be
appropriately normalized. Here we adopt the approach proposed by Hradil et al.
Hradil2019O , to account the total FI as a sum of weighted components for all
probabilistic events. For the general entangled state (2), the FI is defined
as

$F_{tot}=\langle h_{1}|h_{1}\rangle F_{\rho_{1}}+\langle h_{2}|h_{2}\rangle
F_{\rho_{2}},$ (3)

where $\rho_{1}=|h_{1}\rangle\langle h_{1}|/N_{1}$,
$\rho_{2}=|h_{2}\rangle\langle h_{2}|/N_{2}$ are two corresponding normalized
states, by factors $N_{1}$, $N_{2}$, of the spatial domain with corresponding
weights $\langle h_{1}|h_{1}\rangle$, $\langle h_{2}|h_{2}\rangle$. Here
Fisher information takes the form $F_{\rho}=2{\rm
Tr}[(\partial_{s}\rho(s))^{2}]$ for an arbitrary normalized state $\rho(s)$.
This measure is based on the conditional outcome of a measurement in the
rotated basis $\\{|\phi^{\alpha}_{1}\rangle,|\phi^{\alpha}_{2}\rangle\\}$ of
the entangled partner.

It is important to note that here we treat the measurement of signal as a
single repetition (e.g., an independent single photon detection event,
detecting a bunch of identical photons within the coherence time, or a single
measurement of light intensity). For given number (e.g., $N$) of multiple
repetitions of measurement, our results will remain the same up to the
constant factor $N$. Our analysis doesn’t cover environment-induced loss cases
where the photon numbers are unknown.

##  III Results and Messages

In our consideration, the unbalanceness of the two sources $r=b/a$ is also an
unknown parameter. However, it will be shown later that measurement in the
rotated basis $\\{|\phi^{\alpha}_{1}\rangle,|\phi^{\alpha}_{2}\rangle\\}$ is
always able to cancel the unbalanceness effect. Therefore, one needs only to
consider the single unknown parameter $s$ for the calculation of the Fisher
Information. Then the Fisher information for the entangled field (2) can be
explicitly obtained as

$\displaystyle F_{tot}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{4\sigma^{2}}$ (4) $\displaystyle-$
$\displaystyle\frac{(r\sin
2\alpha\cos\varphi)^{2}s^{2}}{16[(\cos^{2}\alpha+r^{2}\sin^{2}\alpha)e^{s^{2}/8\sigma^{2}}-r\sin
2\alpha\cos\varphi]}$ $\displaystyle\times$
$\displaystyle\frac{1}{(r^{2}\cos^{2}\alpha+\sin^{2}\alpha)e^{s^{2}/8\sigma^{2}}+r\sin
2\alpha\cos\varphi},$

which depends on the two-source separation $s$, the entangled partner’s
measurement basis characterized by rotation angle $\alpha$, the two-source
unbalanceness through the amplitude ratio $r$, and the relative phase
$\varphi$. Fig. 2 (a) illustrates the specific behaviors of the Fisher
Information on $s$ for a fixed partial coherence (rotation angle
$\alpha=\pi/6$) and relative phase $\varphi=0$, but for different
unbalanceness $r=0,1/4,1/2,1$. Qualitative behaviors for other degrees of
partial coherence (i.e., other rotation angles) are similar to Fig. 2 (a)
except for some special cases which will be discussed later. The detailed
derivation of (4) is provided in supplemental material section 2.

Figure 2: Dependence of the FI on the displacement s for $\varphi=0$ with
$\sigma=1$. Different color represents different value of $r$. The red line is
for the balanced case ($r=1$), and the blue and green lines corresponds to
$r=\frac{1}{2}$, and $r=\frac{1}{4}$ respectively. For $r=0$ are represented
by the black line. (a) FI for the state $\ket{\Psi_{1}}$ in the case
$\alpha=\frac{\pi}{6}$. (b) FI is for the state $\ket{\Psi_{2}}$.

There are three important messages delivered by the obtained Fisher
information (4) for the entangled source.

The first major message of result (4) is that super-resolution is achievable
for various practical unbalanceness settings as the value of Fisher
information remains finite even when the separation $s$ goes to zero, as shown
in Fig. 2 (a). This unexpected behavior is a result of the existence of the
entangled partner. To have a clearer picture of effects of the entangled
partner, we also analyze the Fisher information of two unentangled point
sources with the same unbalanceness characterized by $r=b/a$ and relative
phase $\varphi$, i.e.,

$\displaystyle\ket{\Psi^{\prime}}$
$\displaystyle=(a\ket{h_{+}}+be^{i\varphi}\ket{h_{-}})|\phi\rangle,$ (5)

where $|\phi\rangle$ is a generic state of the remaining degrees of freedom.
In this non-entangled case, the measurement basis of $|\phi\rangle$ is
irrelevant to the spatial domain. Therefore, the Fisher information can be
directly computed as $F_{\rho^{\prime}}=2{\rm
Tr}[(\partial_{s}\rho^{\prime})^{2}]$ where
$\rho^{\prime}=|\Psi^{\prime}\rangle\langle\Psi^{\prime}|/N^{\prime}$ with
$N^{\prime}$ being the normalization factor. It can be obtained
straightforwardly as

$\displaystyle F_{\rho^{\prime}}(s)$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{4\sigma^{2}}-ab\cos\varphi
e^{-\frac{s^{2}}{8\sigma^{2}}}\frac{(-s^{2}+4\sigma^{2})}{8\sigma^{4}})}{1+2ab\cos\varphi
e^{-s^{2}/8\sigma^{2}}}$ $\displaystyle-\frac{1}{4}\frac{(ab\cos\varphi
e^{-\frac{s^{2}}{8\sigma^{2}}}s)^{2}}{(1+2ab\cos\varphi
e^{-s^{2}/8\sigma^{2}})^{2}}.$ (6)

Fig. 2 (b) illustrates the specific behaviors of $F_{\rho^{\prime}}$ for the
nonentangled field with the same unbalanceness values and fixed phase
$\varphi$. The detailed derivation of the above result is also provided in
supplemental material section 3.

By comparing the Fisher information of the two cases as illustrated in Fig. 2
(a) and (b), one notes apparently that the entangled field has a significant
enhancement for all unbalanceness values in the small separation $s$ regime.
Particularly, for the frequently studied balanced source case ($r=1$), the
nonentangled field Fisher information vanishes at zero separation while the
entangled field one achieves its maximum finite value. It is also interesting
to note that for the entangled field case, the Fisher information of various
different unbalanceness simply converge to the same finite value when the
source separation decreases to zero.

The second major message of result (4) lies in the underlying competing
mechanism between coherence and unbalanceness in affecting the Fisher
information. As is mentioned, neither partial coherence nor unbalancess is
able to achieve super-resolution Rehacek2017PRA ; Larson2018O . Surprisingly,
as is shown here above, the combination of the two works! This is due to the
fact that coherence and unbalanceness have counter effects against each other
on the Fisher information.

To understand this point better, we perform a detailed analysis of effects
from both properties to explore the analogous behavior of the two. We first
analyze the coherence effect by fixing at the two-source balanced case, i.e.,
setting the unbalanceness parameter to be $r=1$. Then the Fisher information
simply reduces to

$\displaystyle F_{tot}(s,r=1)$
$\displaystyle=\frac{1}{4\sigma^{2}}-\frac{1}{16}\frac{\sin^{2}2\alpha\cos^{2}\varphi
s^{2}}{e^{s^{2}/4\sigma^{2}}-\sin^{2}2\alpha\cos^{2}\varphi},$ (7)

which depends on the coherence rotation angle $\alpha$ and two-source
separation $s$. Fig 3 (a) illustrates its behavior for different coherence
parameter values of $\alpha$.

Next we analyze the unbalanceness effect by fixing the coherence rotation
angle $\alpha=\pi/4$. To be comparable we define the unbalanceness parameter
as $r=b/a=\tan\eta$, where $a=\cos\eta$ and $b=\sin\eta$ to satisfy the
normalization condition $a^{2}+b^{2}=1$. Then the Fisher information reduces
to

$\displaystyle F_{tot}(s,\alpha=\frac{\pi}{4})$
$\displaystyle=\frac{1}{4\sigma^{2}}-\frac{1}{16}\frac{\sin^{2}2\eta\cos^{2}\varphi
s^{2}}{e^{s^{2}/4\sigma^{2}}-\sin^{2}2\eta\cos^{2}\varphi},$ (8)

which depends on the unbalanceness angle $\eta$ and two-source separation $s$.
Fig 3 (b) illustrates the behavior for different unbalanceness values of $r$
or equivalently $\eta$.

By comparing the two expressions (7) and (8), one immediately notes that they
are equivalent except for the change of parameter $\alpha$ into $\eta$. This
shows that unbalanceness and the coherence are affecting the Fisher
information in the same mechanism. Therefore, the opposite variations of the
two parameters are able to cancel each other’s negative effects on the Fisher
information, thus permitting a super-resolution. Fig 3 (a), (b) illustrates
specifically the equivalence of the two Fisher information behaviors. When the
parameters are chosen appropriately, the two effects behave exactly the same
as shown by same color lines in the two plots. In addition, the fact that
angle $\alpha$ is controllable by the analyzer, one can always achieve finite
FI for arbitrary unknown parameter $r$ as shown in Fig 3. This justifies that
in the calculation of the Fisher Information (4), it is not necessary to
estimate the unknown parameter $r$.

Also, we observe that in the limit $s\rightarrow 0$, Fisher Information
$F_{tot}(s,r)$ will never vanish for any unbalanced (i.e., $r\neq 1$) sources,
see detailed proof in Supplement material section 4. From the equivalence of
coherence and unbalanceness effect, it can therefore be concluded that by
adjusting the coherence rotation angle $\alpha$ one can always avoid the
balanced situation and thus avoid the vanishing Fisher Information at zero
separation.

Figure 3: Total FI versus the displacement $s$ for $\varphi=0$ when
$\sigma=1$. (a) Rotating basis angle $\alpha$ effect on FI in the balanced
case. The minimum of the FI corresponds to $s_{\rm least}$ in the expression
(10) (b) Unbalancenees intensities effect in the case of
$\alpha=\frac{\pi}{4}$.

The third major message of the Fisher information (4) is its counter intuitive
decreasing behavior within the small separation regime, see illustrations in
Fig. 2 (a) and Fig. 3. Normally one would expect that as the separation $s$ of
the two sources increases the Fisher information would also increase because
it is natural to assume that larger separation means less relative error in
measurements. However, as shown in Fig. 2 (a) and Fig. 3, the Fisher
information experiences a decrease and then an increase as the separation $s$
increases from zero. This behavior is due to the competing nature of coherence
($\alpha$) and unbalanceness ($r$) in affecting the Fisher information for any
fixed relative phase $\varphi$ .

This interesting behavior suggests the existence of a least resolvable
separation $s_{\rm least}$ that leads to a minimum Fisher information
$F_{tot}^{\rm min}$. All practical situations should avoid analyzing distances
at the vicinity of this critical separation $s_{\rm least}$. To achieve this
separation quantitatively, we analyze the derivative of the Fisher information
(2). The vanishing derivative of $F_{tot}$ leads to the solutions of the
following equation

$\displaystyle\Lambda(s)e^{s^{2}/4\sigma^{2}}+\Pi(s)e^{s^{2}/8\sigma^{2}}+\Omega=0,$
(9)

where,
$\Lambda(s)=(a^{2}\cos^{2}\alpha+b^{2}\sin^{2}\alpha)(b^{2}\cos^{2}\alpha+a^{2}\sin^{2}\alpha)(1-\frac{2s^{2}}{8\sigma^{2}}),\Pi(s)=[2(a^{2}\cos^{2}\alpha+b^{2}\sin^{2}\alpha)-1](1-\frac{s^{2}}{8\sigma^{2}})ab\sin
2\alpha\cos\varphi$ and $\Omega=-(ab\sin 2\alpha\cos\varphi)^{2}$. There are
two trivial solutions $s=0$ and $s\rightarrow\infty$ as it is shown in Fig.
2(a) and Fig. 3. The nontrivial solution can in general be always achieved
numerically. For the commonly studied balanced case when $r=1$, the nontrivial
solution of (9) can be obtained analytically as (see detailed analysis in
supplemental material section 5)

$\displaystyle s_{\rm
least}=\sigma\sqrt{4+4\mathcal{W}[-\frac{\sin^{2}2\alpha\cos^{2}\varphi}{e}]}$
(10)

where $\mathcal{W}[.]$ is a special function known as the Lambert
$\mathcal{W}$-function, which is an increasing function with a minimum at
$\mathcal{W}[\frac{-1}{e}]=-1$. When $\sin^{2}2\alpha\cos^{2}\varphi=1$,
$s_{\rm least}=0$ which is exactly the case analyzed in Hradil2019O  for
$\varphi=0,\pi$ and $\alpha=\pi/4$.

Due to the equivalence of the coherence effect and unbalanceness effect as
analyzed earlier, the least resolvable distance with respect to different
values of the parameter $r$ can be obtained in exactly the same way as (10),
see also illustrations of $s_{\rm least}$ for different curves in Fig. 3 (a)
and (b). Since the detection basis (in terms of $\alpha$) can be controlled by
the observer, all unbalanced cases can be treated equivalently as balanced
cases but with a corresponding coherence angle $\alpha$.

The least resolvable distance analysis provides an important guidance to avoid
resolution of two-source separation at the vicinity of $s_{\rm least}$ in
various practical situations.

##  IV Conclusion

To summarize, we have investigated sub-diffraction-limit resolution of two
point-sources under two practical situations: arbitrary two-source
unbalanceness and partial coherence. By including an entangled partner of the
spatial property of the two sources to account the partial coherence, it is
found that super-resolution can be achieved with high measurement estimation
credibility (quantified by maximum Fisher information) even when the two-
source separation reduces to zero. It is revealed that such achievement is due
to the fact that the effect on Fisher information from partial coherence is
equivalent to that of the two-source unbalanceness. Appropriate control of the
rotated basis (i.e., adjustment of partial coherence) by the analyzer is able
to counter effect arbitrary unbalanceness. Such a capability indicates that
the realization super-resolution is independent of whether the unbalanceness
and partial coherence are known or not. This justifies the exclusion of
unknown parameters unbalancenss and partial coherence in analyzing Fisher
information.

We have also carried out a detailed analysis of the counter intuitive
decreasing behavior of Fisher information as the two-source separation
increases. This allows the discovery a characteristic equation to determine
the “least resolvable” distance. Analytical solutions in terms of Lambert W
function are also achieved. Our results provide an important guidance for
practical optical designs and engineering in the realization of optimum fine
resolution.

Funding: U.S. Army under contact No. W15QKN-18-D-0040.

Disclosures: The authors declare no conflicts of interest

Acknowledgments: We acknowledge partial financial support by Stevens Institute
of Technology.

## References

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