# Coherence, superposition, and Löwdin symmetric orthogonalization

Gökhan Torun<EMAIL_ADDRESS>TUBITAK Research Institute for Fundamental
Sciences, 41470 Gebze, Kocaeli, Türkiye

###### Abstract

The notions of coherence and superposition are conceptually the same; however,
an important distinction exists between their resource-theoretic formulations.
Namely, while basis states are orthogonal in the resource theory of coherence,
they are not necessarily orthogonal in the resource theory of superposition.
Owing to the nonorthogonality, the manipulation and characterization of
superposition states require significant efforts. Here, we show that Löwdin
symmetric orthogonalization (LSO) provides a convenient way of characterizing
and manipulating the superposition states. We analyze the superposition states
transformation by taking advantage of LSO in cooperation with coherence
transformations and show that LSO can improve the efficiency of superposition
states transformation. The principal property of LSO is that the structure and
symmetry of the original nonorthogonal basis states are preserved to the
greatest extent possible, which prompts us to study the role of LSO in
identifying the hierarchical relations of states. Notably, we reveal that the
maximally coherent states turn into the states with maximal superposition with
the help of LSO. We also extend our discussion to the distillation of pure
superposition states, one of the leading state conversion tasks. Our results
facilitate further connections between coherence and superposition, where LSO
is the main tool.

##  I Introduction

Characterizing the intrinsic physical features of quantum mechanics, which is
what provides the way to understand their pivotal role as a resource in
information processing tasks, is among the essential topics in quantum
information science. In this regard, the study of quantum resource theories
[1, 2, 3, 4, 5] has gained prominence as an active area of research over time
as well. A fundamental concept with particular relevance is that of quantum
coherence [6, 7] which lies at the heart of many quantum information
processing applications, including quantum metrology [8, 9, 10, 11], quantum
computation [12], and quantum thermodynamics [13, 14, 15, 16, 17]. Basically,
the resource theory of coherence (RTC) [7] is formulated with respect to a
reference basis that forms an orthonormal set. In addition, a resource-
theoretic formulation based on nonorthogonal basis states generalizes the
concept of coherence, known as the resource theory of superposition (RTS)
[18], where a wide range of tasks such as communication complexity [19, 20]
and channel discrimination [18] need the presence of superposition as a
resource. Thus, from the perspective of resource theory, coherence is accepted
as a special case of superposition, i.e., in the limit of overlapping between
basis states going to zero, RTS reduces to RTC. Comparing the amount of
studies based on RTC and RTS, the former has made significant progress [21],
whereas the latter needs further research. Nonorthogonality is a major factor
that makes it difficult to contribute to this need. As a consequence, it is
natural and necessary to delve deeper into the links between RTC and RTS,
which enhances our capabilities in exploring and exploiting the latter.

It is our purpose in this paper to present a systematic approach to the
manipulation and characterization of superposition states with the assistance
of coherence by using Löwdin symmetric orthogonalization (LSO) [22, 23, 24,
25] as an instrumental tool. To the best of our knowledge, our study is the
first to shed light on the aforementioned problem in this way, which offers a
great potential to extend the results obtained for the coherence to the
superposition. A number of studies have included LSO, such as [26] which
considered the Schmidt decomposition of the two-mode Schrödinger cat states —
a superposition of two distinct coherent states — and [27] whose objective was
to obtain a set of orthogonal (Löwdin) pulse. Among the different types of
orthogonalization schemes [28], our main object of interest is the LSO where
the importance of this method is due to the fact that it preserves as much as
possible of the structure and symmetry of the original nonorthogonal basis
[22, 25]. For instance, this attractive feature is the main reason why LSO is
preferred when one aims to obtain a set of orthonormal orbitals generated from
the Hilbert space spanned by the nonorthogonal orbitals [29, 30]. These
insights allow us to probe more deeply into the connections between the
coherence and the superposition — providing a picture of the utility of LSO
for characterizing and manipulating the superposition states.

The task of state transformation, as a central subject of study within
resource theories, is concerned with converting a given state into a target
state, probabilistic or deterministic, via quantum operations that do not
create resource. In the case of RTS, a partial order structure was uncovered
for the probabilistic transformations of pure superposition states [18]. Later
on, deterministic transformations of superposition states was investigated in
Ref. [31]. Of course, it is always essential to consider a well-established
method that offers an alternative and practical view for the state transitions
between pairs of quantum states, which also enables a broad characterization
of states. In this work, we demonstrate an integrated approach in which LSO
and a suitable coherence transformation protocol cooperate to address
superposition state transformations, see Figure 1 below. For a given pair of
superposition states
$\\{\mathinner{|{\psi}\rangle},\mathinner{|{\varphi}\rangle}\\}$, a pair of
coherent states
$\\{\mathinner{|{\bar{\psi}}\rangle},\mathinner{|{\bar{\varphi}}\rangle}\\}$
can be obtained with the use of LSO. The scenario where we aim to convert
$\mathinner{|{\psi}\rangle}$ into $\mathinner{|{\varphi}\rangle}$ is replaced
by transforming $\mathinner{|{\bar{\psi}}\rangle}$ into
$\mathinner{|{\bar{\varphi}}\rangle}$. In order to reach the target
superposition state $\mathinner{|{\varphi}\rangle}$, after a coherence
transformation from $\mathinner{|{\bar{\psi}}\rangle}$ to
$\mathinner{|{\bar{\varphi}}\rangle}$ achieved with a probability $p\in(0,1]$,
we apply LSO to $\mathinner{|{\bar{\varphi}}\rangle}$ in the opposite
direction. Thus, the initial superposition state $\mathinner{|{\psi}\rangle}$
is transformed into the target state $\mathinner{|{\varphi}\rangle}$, where
$p$ is the probability of success. Moreover, one of our main results shows
that LSO can improve the success probability of superposition states
transformation [31]. In Ref. [32] it was shown that catalytic effect, which is
a crucial ingredient for quantum resource theories [33], can increase the
efficiency of entanglement concentration procedures for finite states. In this
sense, it should be reasonable to infer that LSO shows a catalyst-like effect
for superposition states transformation. We present an explicit example to
illustrate this point.

More to the point, our approach has a significant implication for the
characterization of pure superposition states. The recent work of Ref. [34]
has established a structural way to investigate maximal states of RTS, i.e.,
golden states, utilizing the eigenvalues and eigenvectors of the Gram matrix
constructed from the given set of arbitrary (nonorthogonal) basis states. Here
we show that it is possible to effectively accomplish the same goal, where LSO
allows us to do so. We present an explicit construction of maximal
superposition states, starting with the maximally coherent states followed by
LSO in the opposite direction. As part of our study, we also consider the task
of superposition state distillation. Our way of handling the mentioned problem
needs coherence distillation protocols which were well discussed in the
literature, such as [35, 36, 37, 38, 39, 40, 41, 42]. We believe that LSO,
which is a well-known scheme in quantum chemistry, has the potential to bring
advantages that are state of the art for our understanding of the notions of
coherence and superposition within the scope of quantum resource theories.

The structure of the paper is as follows. In Sec. II, we give the background
that contains the necessary ingredients and tools needed to construct the
paper: the basics of RTS (Sec. II.1) and RTC (Sec. II.3), and the details of
LSO (Sec. II.2). We proceed in Sec. III by discussing the role of LSO in the
characterization and manipulation of superposition states. We start with
superposition states transformation (Sec. III.1); then, we employ LSO to
reveal the form of the states with maximal superposition (Sec. III.2); after
that, we study superposition distillation (Sec. III.3); and finally, we
briefly discuss resource values of the states connected via LSO (Sec. III.4).
We conclude our work in Sec. IV.

##  II Background

###  II.1 Resource theory of superposition

The essentials of RTS was introduced and discussed in Ref. [18]. Here we give
a succinct overview of RTS, including the free states ($\mathcal{F}$),
resource states ($\mathcal{R}$), and the restricted set of free operations
($\mathcal{O}$), which are the three major elements of a general quantum
resource theory [4]. As in [18], let $\mathcal{H}_{d}$ be a $d$-dimensional
Hilbert space and $\\{\mathinner{|{c_{i}}\rangle}:i=0,1,\dots,d-1\\}$ be the
set of nonorthogonal, normalized, and linearly independent basis of
$\mathcal{H}_{d}$. Then, states defined as
$\rho=\sum_{i}\rho_{i}|{c_{i}}\rangle\langle{c_{i}}|$ are called
superposition-free, where $\rho_{i}\geq 0$ form a probability distribution.
The set of superposition-free density operators is denoted by $\mathcal{F}$
and forms the set of free states. All density operators which are not an
element of $\mathcal{F}$ are called superposition states and form the set of
resource states $\mathcal{R}$. A linear combination of
$\\{\mathinner{|{c_{i}}\rangle}\\}$ gives us the pure superposition states

$\displaystyle\mathinner{|{\psi}\rangle}=\sum_{i=0}^{d-1}\psi_{i}\mathinner{|{c_{i}}\rangle},$
(1)

where the coefficients $\\{\psi_{i}\\}$ are complex number. Let $S$ be overlap
matrix of basis states$\\{\mathinner{|{c_{i}}\rangle}\\}$, i.e., Gram matrix,
with the elements $S_{ij}=\langle{c_{i}}|c_{j}\rangle$. Then the normalization
condition reads
$\langle{\psi}|\psi\rangle=\sum_{i,j=0}^{d-1}\psi_{i}^{\ast}S_{ij}\psi_{j}=1$,
where $S_{ij}$ is complex in general.

In the context of RTS, a Kraus operator $K_{n}$ is called superposition-free
if $K_{n}\rho K_{n}^{\dagger}\in\mathcal{F}$ for all $\rho\in\mathcal{F}$, and
is of the form

$\displaystyle
K_{n}=\sum_{k}c_{k,n}{\mathinner{|{c_{f_{n}(k)}}\rangle}\mathinner{\langle{c_{k}^{\perp}}|}},$
(2)

where $c_{k,n}$ are complex number, ${f_{n}(k)}$ are arbitrary index functions
[18], and $\langle{c_{i}^{\perp}}|c_{j}\rangle=\delta_{ij}$ where the vectors
$\mathinner{|{c_{k}^{\perp}}\rangle}$ are not normalized. Also, a quantum
operation $\Phi(\cdot)$ is called superposition-free if it is trace preserving
and can be written such that $\Phi(\rho)=\sum_{n}K_{n}\rho K_{n}^{\dagger}$,
where all $K_{n}$ are free.

###  II.2 Löwdin symmetric orthogonalization

Recall that $\\{\mathinner{|{c_{i}}\rangle}:i=0,1,\dots,d-1\\}$ be a set of
nonorthogonal, normalized, and linearly independent basis vectors as in the
previous section. We aim to construct an orthonormal basis set
$\\{\mathinner{|{l_{i}}\rangle}:i=0,1,\dots,d-1\\}$ by a suitable linear
transformation. To this end, we utilize the Löwdin’s symmetric
orthogonalization [22, 25] abbreviated by LSO, so that the orthogonal basis
vectors $\\{\mathinner{|{l_{i}}\rangle}\\}$ are called as “Löwdin basis”. We
start by defining column vectors

$\bm{C}:={\left(c_{0},c_{1},\dots,c_{d-1}\right)}^{\intercal},\quad\bm{L}:={\left(l_{0},l_{1},\dots,l_{d-1}\right)}^{\intercal}.$
(3)

Here the components $\\{c_{i}\\}$ and $\\{l_{i}\\}$ represent the basis sets
$\\{\mathinner{|{c_{i}}\rangle}\\}$ and $\\{\mathinner{|{l_{i}}\rangle}\\}$,
respectively. We can define a general linear transformation $T$ such that

$\displaystyle\bm{L}=T\bm{C},\quad\Big{(}TT^{\dagger}=S^{-1}\Big{)}.$ (4)

In order to get the linear transformation $T$, we first construct the overlap
matrix $S$, which is positive semidefinite in general, such that
$S_{ij}=\langle{c_{i}}|c_{j}\rangle$ for $i,j=0,1,\dots,d-1$. By a unitary
matrix $U$, we can diagonalize the overlap matrix $S$, that is,
$S_{\text{diag}}=U^{\dagger}SU$. Let $S$ has eigenvalues $\\{\lambda_{i}\\}$.
Since the eigenvalues of $S$ are positive, the elements of $S_{\text{diag}}$
can be replaced by $\\{\lambda_{i}\\}$, from which we can get
${S^{{1}/{2}}_{\text{diag}}}=\text{diag}(\sqrt{\lambda_{0}},\sqrt{\lambda_{1}},\dots,\sqrt{\lambda_{d-1}})$.
By using this matrix, we define the matrix
${S^{{1}/{2}}}=US^{{1}/{2}}_{\text{diag}}U^{\dagger}$ and finally get
${S^{-{1}/{2}}}={(S^{{1}/{2}})}^{-1}$. Then the linear transformation given by
Eq. (4) can be rewritten as

$\displaystyle\big{(}\bm{L}\big{)}_{i}=\sum_{j=0}^{d-1}\big{(}{S^{-{1}/{2}}}\big{)}_{ij}\big{(}\bm{C}\big{)}_{j},$
(5)

for $i,j=0,1,\dots,d-1$, which is obviously invertible. Thus, we can obtain
the orthogonal basis vectors $\\{\mathinner{|{l_{i}}\rangle}\\}$ as a linear
combination of the nonorthogonal basis vectors
$\\{\mathinner{|{c_{i}}\rangle}\\}$, that is, from Eq. (5) one has
$\mathinner{|{l_{i}}\rangle}=\sum_{j}({S^{-{1}/{2}}})_{ij}\mathinner{|{c_{j}}\rangle}$.

In fact, the ways of taking a nonorthogonal set into an orthonormal set are
multifarious. The LSO has a meaningful feature among all possible
orthogonalization methods: namely, the symmetric orthogonalization ensures

$\displaystyle\sum_{i}\|c_{i}-l_{i}\|^{2}=\min,$ (6)

where $\|c_{i}-l_{i}\|^{2}\equiv\langle{c_{i}-l_{i}}|c_{i}-l_{i}\rangle$. From
Eq. (6), one can infer that the symmetrically orthogonalized vectors
$\\{\mathinner{|{l_{i}}\rangle}\\}$ are the least distant in the Hilbert space
from the original nonorthogonal vectors $\\{\mathinner{|{c_{i}}\rangle}\\}$,
i.e., LSO implies the gentlest pushing of the directions of the nonorthogonal
vectors in order to get them orthogonal [22, 25].

###  II.3 Resource theory of coherence

The resource theory of coherence (RTC), which is a special case of RTS, is a
basis dependent concept. Importantly, one may choose the reference basis in
accordance with the physics of the problem under consideration. In that
connection, we here designate the Löwdin basis as our preferred (complete and
orthonormal) basis. Namely, let
$\\{\mathinner{|{l_{k}}\rangle}:k=0,1,\dots,d-1\\}$ be a particular
orthonormal basis of the $d$-dimensional Hilbert space $\mathcal{H}_{d}$; then
the quantum states that are diagonal with respect to this specific basis
$\\{\mathinner{|{l_{k}}\rangle}\\}$ are called as incoherent, constituting a
set labeled by $\mathcal{I}$ [6, 7]. Hence, all incoherent states
$\rho\in\mathcal{I}$ are of the form
$\rho=\sum_{k=0}^{d-1}p_{k}\mathinner{|{l_{k}}\rangle}\mathinner{\langle{l_{k}}|}$,
where $p_{k}\in[0,1]$ and $\sum_{k}p_{k}=1$. On the other hand, a finite
$d$-dimensional pure coherent state is given by

$\displaystyle\mathinner{|{\phi}\rangle}=\sum_{k=0}^{d-1}\phi_{k}e^{i\theta_{k}}\mathinner{|{l_{k}}\rangle},$
(7)

where $\\{\phi_{k}:k=0,1,\dots,d-1\\}$ are real numbers such that
$\phi_{k}\geq\phi_{k+1}>0$, satisfying $\sum_{k}\phi_{k}^{2}=1$. Note that all
complex phases $\theta_{k}\in[0,2\pi)$ in Eq. (7) can be eliminated by
incoherent operations. We denote by $\mathinner{|{\Phi_{d}}\rangle}$ the
maximally coherent state in the reference basis with entries
$\phi_{k}=\frac{1}{\sqrt{d}}$, that is,

$\displaystyle\mathinner{|{\Phi_{d}}\rangle}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{i\theta_{k}}\mathinner{|{l_{k}}\rangle}.$
(8)

Moreover, since all other $d$-dimensional coherent states can be generated
from $\mathinner{|{\Phi_{d}}\rangle}$ by means of the free operations, this
definition regards as a unit of RTC — coherence bit.

Following standard definitions of RTC, Kraus operators $\\{K_{n}\\}$ are
called incoherent operators, which are characterized as the set of trace
preserving completely positive maps, such that

$\displaystyle\rho\rightarrow\rho_{n}=\frac{K_{n}\rho
K_{n}^{{\dagger}}}{\text{Tr}[K_{n}\rho K_{n}^{{\dagger}}]}\in\mathcal{I}$ (9)

for all $n$, where $\rho\in\mathcal{I}$ and
$\sum_{n}K_{n}^{{\dagger}}K_{n}=I$. Also, the quantum operations
$\Lambda(\cdot)$ consisting of the Kraus operators $\\{K_{i}\\}$ satisfying
Eq. (9) are the free operations in the context of RTC.

###  II.4 Quantifying superposition and coherence

At this point, we give two of superposition (and coherence) measures, where
the conditions when a function $M$ is called a superposition measure were
detailed in Refs. [6, 18]. In fact, the following two measures of
superposition were introduced [18] by extending the method used for
quantifying coherence [7]. Within certain types of resource measures, the
$l_{1}$ measure of superposition is defined as

$\displaystyle M_{l_{1}}(\rho)=\sum_{i\neq j}|\rho_{ij}|,$ (10)

for
$\rho=\sum_{i,j}\rho_{ij}\mathinner{|{c_{i}}\rangle}\mathinner{\langle{c_{j}}|}$
[18]. Also, Eq. (10) corresponds to the $l_{1}$ norm of coherence such that
$\rho=\sum_{i,j}\rho_{ij}\mathinner{|{l_{i}}\rangle}\mathinner{\langle{l_{j}}|}$
[7]. The relative entropy of superposition is given by

$\displaystyle
M_{\text{rel.ent}}(\rho)=\min_{\sigma\in\mathcal{F}}S(\rho\|\sigma),$ (11)

where $S(\rho\|\sigma)=\text{tr}[\rho\log\rho]-\text{tr}[\rho\log\sigma]$
denotes the quantum relative entropy. From the definition of (11), the
relative entropy of coherence reads $S(\rho_{\text{diag}})-S(\rho)$, where
incoherent state $\rho_{\text{diag}}$ denotes the state obtained from $\rho$
by deleting all off-diagonal elements [7].

##  III Setting the Approach

This section comprises four parts: in Sec. III.1, we discuss superposition
states transformation and show how to implement LSO for this particular
problem with the aid of a coherence transformation task; in Sec. III.2, we
show that LSO helps to uncover the connections between maximally resourceful
states of coherence and superposition; in Sec. III.3, we study the
superposition state distillation; and in Sec. III.4, we briefly discuss
$l_{1}$ measures of the states connected via LSO.

###  III.1 Superposition states transformation

FIG. 1: The superposition state transformations procedure assisted by the
Löwdin symmetric orthogonalization (LSO). The goal is to transform the
superposition state $\mathinner{|{\psi}\rangle}$ into the superposition state
$\mathinner{|{\varphi}\rangle}$, in which the transformation succeed with some
probability $p\in(0,1]$. Here, while the linear map $\text{LSO}^{\mapsto}$
corresponds the LSO itself, the linear map $\text{LSO}^{\mapsfrom}$
corresponds to the LSO, but in the opposite direction. The procedure has three
stages. First, by using $\text{LSO}^{\mapsto}$, we achieve the transformations
$\mathinner{|{\psi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle}$
and
$\mathinner{|{\varphi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\varphi}}\rangle}$,
and therefore, we have an initial state $\mathinner{|{\bar{\psi}}\rangle}$ and
know the target state $\mathinner{|{\bar{\varphi}}\rangle}$. Second, we
transform the coherent state $\mathinner{|{\bar{\psi}}\rangle}$ into the
coherent state $\mathinner{|{\bar{\varphi}}\rangle}$ via some suitable
coherence transformations (CTs) tasks, which can be achieved either
deterministically or probabilistically. In the last stage, by using
$\text{LSO}^{\mapsfrom}$, we achieve the transformation
$\mathinner{|{\bar{\varphi}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\varphi}\rangle}$.
The detailed analysis is given in the paper.

To formulate our approach based on LSO, we consider superposition states
transformation involving single copies of states. Figure 1 illustrates the
strategy that transforms an initial superposition state
$\mathinner{|{\psi}\rangle}$ into a final superposition state
$\mathinner{|{\varphi}\rangle}$ with some probability $p\in(0,1]$. The general
structure of our strategy has three stages (see Fig. 1). In the following,
after delineating the first and third steps respectively, we finally discuss
the second step that involves a coherence transformation task. In pondering
the proposed approach, it is natural to start with exploring the two-
dimensional systems. Let us then consider the nonorthogonal basis states
$\mathinner{|{c_{0}}\rangle}$ and $\mathinner{|{c_{1}}\rangle}$ with
$\langle{c_{0}}|c_{1}\rangle=s\in(-1,1)$. After concluding the symmetric
orthogonalization given by Eq. (5), we obtain

$\displaystyle\mathinner{|{l_{0}}\rangle}=\alpha\mathinner{|{c_{0}}\rangle}+\beta\mathinner{|{c_{1}}\rangle},\quad\mathinner{|{l_{1}}\rangle}=\beta\mathinner{|{c_{0}}\rangle}+\alpha\mathinner{|{c_{1}}\rangle},$
(12)

where $\alpha=({1}/{\sqrt{\lambda_{1}}}+{1}/{\sqrt{\lambda_{0}}})/2$ and
$\beta=({1}/{\sqrt{\lambda_{1}}}-{1}/{\sqrt{\lambda_{0}}})/2$. Here,
$\lambda_{0}={1-s}$ and $\lambda_{1}={1+s}$ are the eigenvalues of the overlap
matrix $S$. Thus, from Eq. (12), we can complete the linear map
$\\{\mathinner{|{c_{0}}\rangle},\mathinner{|{c_{1}}\rangle}\\}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\\{\mathinner{|{l_{1}}\rangle},\mathinner{|{l_{2}}\rangle}\\}$.
Now consider the (normalized) superposition state
$\mathinner{|{\psi}\rangle}=\psi_{0}\mathinner{|{c_{0}}\rangle}+\psi_{1}\mathinner{|{c_{1}}\rangle}$
with $\langle{c_{0}}|c_{1}\rangle=s\in(-1,1)$. The transformation
$\mathinner{|{\psi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle}$
can be attained with the help of Eq. (12), from which we have that
$\mathinner{|{c_{0}}\rangle}=\sqrt{1-s^{2}}(\alpha\mathinner{|{l_{0}}\rangle}-\beta\mathinner{|{l_{1}}\rangle})$
and
$\mathinner{|{c_{1}}\rangle}=\sqrt{1-s^{2}}(\alpha\mathinner{|{l_{1}}\rangle}-\beta\mathinner{|{l_{0}}\rangle})$.
From these, we find that

$\displaystyle\mathinner{|{\psi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle}$
$\displaystyle=$
$\displaystyle\sqrt{1-s^{2}}\Big{[}\big{(}\alpha\psi_{0}-\beta\psi_{1}\big{)}\mathinner{|{l_{0}}\rangle}+\big{(}\alpha\psi_{1}-\beta\psi_{0}\big{)}\mathinner{|{l_{1}}\rangle}\Big{]}$
(13) $\displaystyle=$
$\displaystyle{\bar{\psi}}_{0}\mathinner{|{l_{0}}\rangle}+{\bar{\psi}}_{1}\mathinner{|{l_{1}}\rangle},$

Here in Eq. (13) one can see that ${\bar{\psi}}_{0}$ (${\bar{\psi}}_{1}$) is
equal to the conjugate of ${\bar{\psi}}_{1}$ (${\bar{\psi}}_{0}$) satisfying
$|{\bar{\psi}}_{0}|^{2}+|{\bar{\psi}}_{1}|^{2}=1$. Thus, we have an initial
superposition state $\mathinner{|{\psi}\rangle}$ and after
$\text{LSO}^{\mapsto}$ we have a coherent state
$\mathinner{|{\bar{\psi}}\rangle}$ in our hand. Moreover, our (normalized)
target superposition state is
$\mathinner{|{\varphi}\rangle}=\varphi_{0}\mathinner{|{c_{0}}\rangle}+\varphi_{1}\mathinner{|{c_{1}}\rangle}$
and if we follow the same path for $\mathinner{|{\varphi}\rangle}$, we can
readily obtain

$\displaystyle\mathinner{|{\varphi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\varphi}}\rangle}$
$\displaystyle=$
$\displaystyle\sqrt{1-s^{2}}\Big{[}\big{(}\alpha\varphi_{0}-\beta\varphi_{1}\big{)}\mathinner{|{l_{0}}\rangle}+\big{(}\alpha\varphi_{1}-\beta\varphi_{0}\big{)}\mathinner{|{l_{1}}\rangle}\Big{]}$
(14) $\displaystyle=$
$\displaystyle{\bar{\varphi}}_{0}\mathinner{|{l_{0}}\rangle}+{\bar{\varphi}}_{1}\mathinner{|{l_{1}}\rangle},$

where $|{\bar{\varphi}}_{0}|^{2}+|{\bar{\varphi}}_{1}|^{2}=1$ as usual. This
completes the first part of the procedure depicted in Fig. 1; namely, we now
know how the initial and target superposition states are written in terms of
the Löwdin basis. As we mentioned above, we proceed with the third part of the
strategy before discussing the second part. In fact, the third stage is
nothing other than the opposite of the first stage. More precisely, if we
obtain the state $\mathinner{|{\bar{\varphi}}\rangle}$ from the state
$\mathinner{|{\bar{\psi}}\rangle}$ under free operations by using some
coherence transformation tasks, then the linear transformation
$\mathinner{|{\bar{\varphi}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\varphi}\rangle}$
is all that is needed to finalize our strategy. Substituting Eq. (12) into Eq.
(14), it is straightforward to show that

$\displaystyle\mathinner{|{\bar{\varphi}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\varphi}\rangle}$
$\displaystyle=$
$\displaystyle\big{(}\alpha{\bar{\varphi}}_{0}+\beta{\bar{\varphi}}_{1}\big{)}\mathinner{|{c_{0}}\rangle}+\big{(}\alpha{\bar{\varphi}}_{1}+\beta{\bar{\varphi}}_{0}\big{)}\mathinner{|{c_{1}}\rangle}$
(15) $\displaystyle=$
$\displaystyle\varphi_{0}\mathinner{|{c_{0}}\rangle}+\varphi_{1}\mathinner{|{c_{1}}\rangle}.$

So far, we outlined how to carry out the first and third stages of our
approach shown in Fig. 1. Now, we are in a position to deal with the second
part of the strategy, which consists of a coherence transformation (CT) task.
This CT task can be chosen among the numerous kinds of coherence manipulation
studies already exist in the literature [43, 44, 45, 46, 47, 8, 48, 49, 50,
36, 51, 52, 37, 35, 39, 53, 38, 54, 55, 42, 40, 41, 56, 57] based on the
intended usage. However, we here particularly discuss the transformation
$\mathinner{|{\bar{\psi}}\rangle}{\rightarrow}\mathinner{|{\bar{\varphi}}\rangle}$
between single copies of pure states under free operations. The maximal
probability of transforming the coherent state
$\mathinner{|{\bar{\psi}}\rangle}$ given in Eq. (13) to the coherent state
$\mathinner{|{\bar{\varphi}}\rangle}$ given in Eq. (14) via incoherent
operations (IOs) is

$\displaystyle
p_{\max}(\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle})={\min}\left\\{1,\frac{\bar{\psi}_{1}^{2}}{\bar{\varphi}_{1}^{2}}\right\\}.$
(16)

Note that it is assumed that $\bar{\psi}_{0}^{2}\geq\bar{\psi}_{1}^{2}>0$ and
$\bar{\varphi}_{0}^{2}\geq\bar{\varphi}_{1}^{2}>0$. We then have two
possibilities: (1) the probability of the transformation
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
is $p=1$ for $\bar{\psi}_{1}^{2}\geq\bar{\varphi}_{1}^{2}$ and (2) $p<1$ for
$\bar{\psi}_{1}^{2}<\bar{\varphi}_{1}^{2}$. In order to achieve
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$,
one can apply a transformation task presented for RTC that serves the purpose
sufficiently, for instance Refs. [51, 56]. As a result, the maximal
probability of transforming the superposition state
$\mathinner{|{\psi}\rangle}$ into the superposition state
$\mathinner{|{\varphi}\rangle}$, that is,
$p_{\max}(\mathinner{|{\psi}\rangle}\rightarrow\mathinner{|{\varphi}\rangle})$,
equals to
$p_{\max}(\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle})$
given in Eq. (16).

Having introduced our strategy for two-dimensional systems, it is now time to
look at $d$-dimensional systems. The starting point for the strategy that we
introduce is to obtain the forms of the initial and the target states after
performing LSO. In this general case, Eq. (13) can be rewritten as

$\displaystyle\mathinner{|{\psi}\rangle}=\sum_{i=0}^{d-1}\psi_{i}\mathinner{|{c_{i}}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle}=\sum_{i=0}^{d-1}g_{i}(\psi,S)\mathinner{|{l_{i}}\rangle}.$
(17)

Here the coefficients $g_{i}(\psi,S)$ (or ${\bar{\psi}}_{i}$) are functions of
$\psi_{k}$ and $\lambda_{k}$, that is,
$g_{i}(\psi,S)=g_{i}(\psi_{0},\psi_{1},\dots,\psi_{d-1},\lambda_{0},\lambda_{d-1})$,
satisfying $\sum_{i=0}^{d-1}|g_{i}(\psi,S)|^{2}=1$, where $\\{\lambda_{k}\\}$
are the eigenvalues of the overlap matrix $S$. As a result, LSO given in Eq.
(17) leads the initial superposition state $\mathinner{|{\psi}\rangle}$ to the
initial coherent state $\mathinner{|{\bar{\psi}}\rangle}$. By following the
same way as for the state $\mathinner{|{\psi}\rangle}$, we have

$\displaystyle\mathinner{|{\varphi}\rangle}=\sum_{i=0}^{d-1}\varphi_{i}\mathinner{|{c_{i}}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\varphi}}\rangle}=\sum_{i=0}^{d-1}g_{i}(\varphi,S)\mathinner{|{l_{i}}\rangle}$
(18)

for the target superposition state $\mathinner{|{\varphi}\rangle}$, where
$\sum_{i=0}^{d-1}|g_{i}(\varphi,S)|^{2}=1$. Given the overlap matrix $S$ and
the superposition states $\mathinner{|{\psi}\rangle}$ and
$\mathinner{|{\varphi}\rangle}$ for dimension $d$, the explicit forms of the
coefficients $g_{i}(\psi,S)$ and $g_{i}(\varphi,S)$ follow by Eq. (5). For the
simplicity of the discussion, we limit ourselves with the inner product
setting $\langle{c_{i}}|c_{j}\rangle=s\in(\frac{1}{1-d},1)$, where the
eigenvalues of the constructed overlap matrix $S$ are found to be
$\lambda_{0}=\lambda_{1}=\dots=\lambda_{d-2}={1-s}$ and
$\lambda_{d-1}={1+(d-1)s}$. Setting

$\displaystyle\mu:=\frac{1}{d}\left[\frac{1}{\sqrt{1+(d-1)s}}+\frac{d-1}{\sqrt{1-s}}\right]$
(19)

and

$\displaystyle\kappa:=\frac{1}{d}\left[\frac{1}{\sqrt{1+(d-1)s}}-\frac{1}{\sqrt{1-s}}\right],$
(20)

we obtain the following relations between the orthogonal basis states
$\\{\mathinner{|{l_{0}}\rangle},\mathinner{|{l_{1}}\rangle},\dots,\mathinner{|{l_{d-1}}\rangle}\\}$
and the nonorthogonal basis states
$\\{\mathinner{|{c_{0}}\rangle},\mathinner{|{c_{1}}\rangle},\dots,\mathinner{|{c_{d-1}}\rangle}\\}$:

$\displaystyle\mathinner{|{l_{i}}\rangle}=\mu\mathinner{|{c_{i}}\rangle}+\kappa\sum_{\begin{subarray}{c}j=0\\\
(j\neq i)\end{subarray}}^{d-1}\mathinner{|{c_{j}}\rangle}\ \
\big{(}i=0,1,\dots,d-1\big{)}.$ (21)

It is easy to check that the basis states
$\\{\mathinner{|{l_{0}}\rangle},\mathinner{|{l_{1}}\rangle},\dots,\mathinner{|{l_{d-1}}\rangle}\\}$
are orthonormal. From Eq. (21), nonorthogonal basis states can also be
obtained in terms of orthogonal basis states. Then, with some calculation we
can find the coefficients $g_{i}(x,S)$ [$x\equiv\psi,\varphi$] such that

$g_{i}(x,S)={\sqrt{\lambda_{d-1}\lambda_{0}}}\bigg{[}{\Big{(}\mu+[d-2]\kappa\Big{)}x_{i}-\kappa\sum_{\begin{subarray}{c}j=0\\\
(j\neq i)\end{subarray}}^{d-1}x_{j}}\bigg{]},$ (22)

for $i=0,1,\dots,d-1$. Note that, when the overlap matrix $S$ is equal to the
identity matrix, we have $\lambda_{i}=1$, $\mu=1$, and $\kappa=0$, and
therefore $g_{i}(\psi,I)=\psi_{i}$ and $g_{i}(\varphi,I)=\varphi_{i}$, i.e.,
two notions, superposition and coherence, coincide when $s=0$.

Next, we move on to the second step of our strategy. We refer to Refs. [44,
51] for the task of coherence manipulation
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
to be considered here. Assuming that $g_{0}^{2}(x,S)\geq
g_{1}^{2}(x,S)\geq\dots\geq g_{d-1}^{2}(x,S)>0$, the maximal probability of
transforming the coherent state $\mathinner{|{\bar{\psi}}\rangle}$ to the
coherent state $\mathinner{|{\bar{\varphi}}\rangle}$ via IOs is given by

$\displaystyle
p_{\max}(\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle})=\underset{j\in[1,d-1]}{\min}\left\\{1,\frac{\sum_{i=j}^{d-1}|g_{i}(\psi,S)|^{2}}{\sum_{i=j}^{d-1}|g_{i}(\varphi,S)|^{2}}\right\\}.$
(23)

Of course, Eq. (23) has two possibilities as usual: namely, for the
probability of the transformation
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
we either have $p=1$ or $0<p<1$. If the majorization condition is satisfied
[44], that is,
$\sum_{i=j}^{d-1}|g_{i}(\psi,S)|^{2}\geq\sum_{i=j}^{d-1}|g_{i}(\varphi,S)|^{2}$
for all $j=0,1,\dots,d-1$, then the transformation
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
can be achieved deterministically; otherwise, it is probabilistic.

Let us assume that the transformation
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
succeeds with the probability $\bar{p}\in(0,1]$. In the last step, we employ
${\text{LSO}^{\mapsfrom}}$ in order to obtain the target superposition state
$\mathinner{|{\varphi}\rangle}$. To this end, we must return to Eq. (21), from
which Eq. (15) can be rewritten as

$\displaystyle\mathinner{|{\bar{\varphi}}\rangle}=\sum_{i=0}^{d-1}g_{i}(\varphi,S)\mathinner{|{l_{i}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\varphi}\rangle}=\sum_{i=0}^{d-1}\varphi_{i}\mathinner{|{c_{i}}\rangle}$
(24)

for $d$-dimensional systems. Here, the relation between the coefficients
$\varphi_{i}$ and $g_{i}(\varphi,S)$ reads as

$\displaystyle\varphi_{i}\equiv\mu
g_{i}(\varphi,S)+\kappa\sum_{\begin{subarray}{c}j=0\\\ (j\neq
i)\end{subarray}}^{d-1}g_{j}(\varphi,S)\ \ \big{(}i=0,1,\dots,d-1\big{)}.$
(25)

Thus, using the three-step strategy that we suggest, we achieve superposition
state transformation from $\mathinner{|{\psi}\rangle}$ to
$\mathinner{|{\varphi}\rangle}$, where the probability of the success is
$p=\bar{p}$, that is,
$\mathinner{|{\psi}\rangle}\rightarrow\mathinner{|{\varphi}\rangle}$ and
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
have the same probability.

We emphasize that this way of dealing with superposition states transformation
can improve the efficiency. To see this, consider the following superposition
states transformation:

$\displaystyle\mathinner{|{\psi}\rangle}=\frac{1}{\sqrt{13}}\big{(}{3\mathinner{|{c_{1}}\rangle}+\mathinner{|{c_{2}}\rangle}}\big{)}\rightarrow\mathinner{|{\varphi}\rangle}=\frac{1}{\sqrt{21}}\big{(}{4\mathinner{|{c_{1}}\rangle}+\mathinner{|{c_{2}}\rangle}}\big{)}$
(26)

with $\langle{c_{1}}|c_{2}\rangle={1}/{2}$. The conditions for a class of
superposition state transformations was given in Ref. [31]. It was shown that,
for a deterministic transformation, the initial and the target states must
satisfy two conditions: majorization for superposition and condition on
completeness abbreviated by CoC [31]. In other words, although majorization
alone is the necessary and sufficient condition for deterministic coherence
transformations [44], one additional condition is required to ensure that
superposition-free transformation can be achieved deterministically. For the
transformation given in Eq. (26) only the majorization (for superposition) is
satisfied, and therefore (26) cannot be achieved with $p=1$ under
superposition-free operations. However, from Eqs. (13) and (14) we have

$\displaystyle\mathinner{|{\psi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle}=\sqrt{0.766469}\mathinner{|{l_{1}}\rangle}+\sqrt{0.233531}\mathinner{|{l_{2}}\rangle},$
(27)
$\displaystyle\mathinner{|{\varphi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\varphi}}\rangle}=\sqrt{0.809295}\mathinner{|{l_{1}}\rangle}+\sqrt{0.190705}\mathinner{|{l_{2}}\rangle}.$
(28)

Obviously, the coherent state $\mathinner{|{\bar{\psi}}\rangle}$ given in Eq.
(27) is majorized by the coherent state $\mathinner{|{\bar{\varphi}}\rangle}$
given in Eq. (28), written
$\mathinner{|{\bar{\psi}}\rangle}\prec\mathinner{|{\bar{\varphi}}\rangle}$.
One can then achieve
$\mathinner{|{\bar{\psi}}\rangle}\rightarrow\mathinner{|{\bar{\varphi}}\rangle}$
with unit probability [44]. Therefore, we can achieve
$\mathinner{|{\psi}\rangle}\rightarrow\mathinner{|{\varphi}\rangle}$ with the
probability of success $p=1$ by using our proposed approach that combines LSO
and a coherence transformation task [51]. Also, for a given pair of
superposition states
$\\{\mathinner{|{\psi}\rangle},\mathinner{|{\varphi}\rangle}\\}$, if the
majorization for superposition [31] is satisfied, that is,
$\mathinner{|{\psi}\rangle}\prec\mathinner{|{\varphi}\rangle}$, then the
majoraiziton for coherence [44] is satisfied between the pair of coherent
states
$\\{\mathinner{|{\bar{\psi}}\rangle},\mathinner{|{\bar{\varphi}}\rangle}\\}$.
Thus, for such pairs
$\\{\mathinner{|{\psi}\rangle},\mathinner{|{\varphi}\rangle}\\}$, the above
discussion exists.

###  III.2 States with maximal superposition

In this section we turn our attention to the states with maximal
superposition. This essential subject was particularly discussed in Ref. [34],
where the eigensystems of the constructed overlap matrices involve information
on the existence of maximal superposition states. Here, we show that the same
results [34] can be obtained forthwith by LSO. Figure 2 illustrates the way
that we apply to realize the form of the states with maximal superposition,
i.e., golden states in RTS [34].

Let us consider the two-dimensional case first, assuming that we initially
have the coherent state
$\mathinner{|{\eta}\rangle}=\eta_{0}\mathinner{|{l_{0}}\rangle}+\eta_{1}\mathinner{|{l_{1}}\rangle}$.
With the help of Eq. (12), we can achieve the transformation
$\mathinner{|{\eta}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\zeta}\rangle}$
such that

$\displaystyle\mathinner{|{\eta}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\zeta}\rangle}$
$\displaystyle=$
$\displaystyle\big{(}\alpha\eta_{0}+\beta\eta_{1}\big{)}\mathinner{|{c_{0}}\rangle}+\big{(}\alpha\eta_{1}+\beta\eta_{0}\big{)}\mathinner{|{c_{1}}\rangle}$
(29) $\displaystyle=$
$\displaystyle\zeta_{0}\mathinner{|{c_{0}}\rangle}+\zeta_{1}\mathinner{|{c_{1}}\rangle}.$

Thus, we have a linear map $\text{LSO}^{\mapsfrom}$ that connects two states
$\mathinner{|{\eta}\rangle}$ and $\mathinner{|{\zeta}\rangle}$, where the
former is a coherent state and the latter is a superposition state.
Undoubtedly, one of the first questions worth addressing is what form the
state $\mathinner{|{\zeta}\rangle}$ has if $\mathinner{|{\eta}\rangle}$ is
taken as the maximally coherent state. To see that, let us consider the
following coherent state

$\eta_{0}\mathinner{|{l_{0}}\rangle}+\eta_{1}\mathinner{|{l_{1}}\rangle}=\frac{1}{\sqrt{2}}\left(\mathinner{|{l_{0}}\rangle}+\mathinner{|{l_{1}}\rangle}\right),$
(30)

which is the maximally coherent state $\mathinner{|{\Phi_{2}}\rangle}$ for
qubit systems. From Eqs. (29) and (30), it is easy to get

$\displaystyle\mathinner{|{\Phi_{2}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\Psi^{+}_{2}}\rangle}=\frac{1}{\sqrt{2(1+s)}}(\mathinner{|{c_{0}}\rangle}+\mathinner{|{c_{1}}\rangle}).$
(31)

The state given in Eq. (31) is the maximal superposition state for $d=2$ with
$s\in(-1,0]$ [18, 31, 34]. Note that the state
$(\mathinner{|{l_{0}}\rangle}-\mathinner{|{l_{1}}\rangle})/\sqrt{2}$ is also
maximally coherent state for the qubit systems. In this case one has

$\displaystyle\frac{1}{\sqrt{2}}\left({\mathinner{|{l_{0}}\rangle}-\mathinner{|{l_{1}}\rangle}}\right)\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\Psi^{-}_{2}}\rangle}=\frac{1}{\sqrt{2(1-s)}}(\mathinner{|{c_{0}}\rangle}-\mathinner{|{c_{1}}\rangle}).$
(32)

The state given in Eq. (32) is the maximal superposition state for $d=2$ with
$s\in[0,1)$ [31, 34]. Indeed, the LSO process is quite simple and
straightforward, establishing a direct connection between the maximally
coherent states and the states with maximal superposition.

FIG. 2: By using the Löwdin symmetric orthogonalization it is possible to
embody the profound connection between the maximally coherent states and
states with maximal superposition. Here, $\Psi$ and $\Phi$ denote maximal
superposition states (i.e., golden states) and maximally coherent states for
$d$-dimensional systems, respectively.

As the above discussion strongly indicates, by applying LSO in the opposite
direction, that is, $\text{LSO}^{\mapsfrom}$, it is possible to bring to light
information regarding the maximal superposition states in dimension $d$ (see
Fig. 2). To this end, consider the following as a final example. The state
$\mathinner{|{\Phi_{d}}\rangle}=({1}/{\sqrt{d}})\sum_{i=0}^{d-1}\mathinner{|{l_{i}}\rangle}$
is the maximally coherent state in dimension $d$ [7, 58]. Following the
approach presented, i.e., combining Eqs. (19), (20), (24), and (25), we can
obtain

$\displaystyle\mathinner{|{\Phi_{d}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\Psi^{+}_{d}}\rangle}=\frac{\mathinner{|{c_{0}}\rangle}+\mathinner{|{c_{1}}\rangle}+\dots+\mathinner{|{c_{d-1}}\rangle}}{\sqrt{d(1+[d-1]s)}},$
(33)

where $S_{ij}=s$. The state given in Eq. (33) is the maximal superposition
state with $s\in(\frac{1}{1-d},0]$ [34]. Extensions of the examples similar to
the case we explicitly discussed for $d=2$ in Eq. (32) are possible for
$d$-dimensional systems. We refer the reader who wants to know more about the
states with maximal superposition to Ref. [34].

In essence, one of the main advantages of LSO is now manifest: namely, if one
knows the form(s) of the maximally coherent states, then one can surely find
out what form the states with maximal superposition are in by using LSO, and
vice versa. In light of the above results, we can conclude that LSO not only
preserves the structure and symmetry of the basis vectors as stated by Eq.
(6), but also serves as a mirror facing the states with maximal coherence
(superposition) and reflecting the states with maximal superposition
(coherence).

###  III.3 Distillation of superposition states

In the previous section, we concisely discussed golden states in RTS [34].
Once the hierarchy of resource states — a structure that characterizes the
potential usefulness of resource states in information processing tasks — has
been constructed, a natural question arises: how does one extract the states
having maximal resource from a given state? Here we deal with this problem,
which is called _resource distillation_ , in the case of the theory of
superposition.

Suppose the superposition state
$\mathinner{|{\psi}\rangle}=\psi_{0}\mathinner{|{c_{0}}\rangle}+\psi_{1}\mathinner{|{c_{1}}\rangle}+\dots+\psi_{d-1}\mathinner{|{c_{d-1}}\rangle}$
with $\langle{c_{i}}|c_{j}\rangle=s\in(\frac{1}{1-d},0]$ is given. Our aim is
to achieve the transformation from $\mathinner{|{\psi}\rangle}$ to
$\mathinner{|{\Psi_{d}^{+}}\rangle}$ — superposition distillation. In order to
succeed this, we perform the procedure outlined in Fig. 1, that is,

$\displaystyle\left\\{\mathinner{|{\psi}\rangle}\overset{\text{LSO}^{\mapsto}}{\longrightarrow}\mathinner{|{\bar{\psi}}\rangle};\
\
\mathinner{|{\bar{\psi}}\rangle}{\rightarrow}\mathinner{|{\Phi_{d}}\rangle};\
\
\mathinner{|{\Phi_{d}}\rangle}\overset{\text{LSO}^{\mapsfrom}}{\longrightarrow}\mathinner{|{\Psi_{d}^{+}}\rangle}\right\\}.$
(34)

We have three transformations in Eq. (34) in a sequential manner. The first
and third ones were already given in Eqs. (17) and (33), respectively. For the
second one, we need a coherence distillation protocol [35, 36, 37, 38, 39, 40,
41, 42]. As mentioned previously, one needs to decide which of these to use
that depends on the nature of the problem being studied. Here, we limit
ourselves to a single copy of an initial state $\mathinner{|{\psi}\rangle}$ to
distill, since our priority is to show how the main approach we proposed is
ideally applied. Then, the (maximal) probability of transforming the coherent
state $\mathinner{|{\bar{\psi}}\rangle}$ to the maximally coherent state
$\mathinner{|{\Phi_{d}}\rangle}$ is given by

$\displaystyle
p(\bar{\psi}{\rightarrow}\mathinner{|{\Phi_{d}}\rangle})=d|g_{d-1}(\psi,S)|^{2}.$
(35)

Essentially, the transformation probability
$p(\mathinner{|{\psi}\rangle}{\rightarrow}\mathinner{|{\Psi_{d}^{+}}\rangle})$
equals that of $p(\bar{\psi}{\rightarrow}\mathinner{|{\Phi_{d}}\rangle})$
given in Eq. (35). We leave open the application of our results to the any of
Refs. [35, 36, 37, 38, 39, 40, 41, 42] (and more), which would be an
interesting direction to consider in the future works on the theory of
superposition.

###  III.4 $l_{1}$ measures of connected states via LSO

It would be interesting to compare the $l_{1}$ measure of the superposition
state and the coherent state that are linked via LSO. For this purpose, we
consider qubit systems, where we have the following relations:

$\displaystyle M_{l_{1}}(\rho)=2|\psi_{0}\psi_{1}|,$ (36) $\displaystyle
M_{l_{1}}(\bar{\rho})=|2\psi_{0}\psi_{1}+s(\psi_{0}^{2}+\psi_{1}^{2})|,$ (37)

for $\rho=\mathinner{|{\psi}\rangle}\mathinner{\langle{\psi}|}$ and
$\bar{\rho}=\mathinner{|{\bar{\psi}}\rangle}\mathinner{\langle{\bar{\psi}}|}$
such that
$\mathinner{|{\psi}\rangle}=\psi_{0}\mathinner{|{c_{0}}\rangle}+\psi_{1}\mathinner{|{c_{1}}\rangle}$
and the state $\mathinner{|{\bar{\psi}}\rangle}$ is given by Eq. (13). First,
from Eqs. (36) and (37) it is obvious that
$M_{l_{1}}(\rho)=M_{l_{1}}(\bar{\rho})$ for $s=0$. Second, setting
$\psi_{1}\equiv x$ and $\psi_{0}\equiv\eta x$ [for qubit systems we always
have $\psi_{0}/\psi_{1}=\eta\in\mathbb{R}\backslash\\{0\\}$], one can easily
obtain $x^{2}=1/{(\eta^{2}+2s\eta+1)}$ from $\langle{\psi}|\psi\rangle=1$.
Then we can compare Eqs. (36) and (37) for some particular value of $\eta$ and
$s\in(-1,1)$. For instance, Fig. 3 illustrate this consideration for $\eta=3$.
We have repeated this for various choices of $\eta$ and observed that there is
not a single relation between the $l_{1}$ measures of the superposition states
and the coherent states that are connected with LSO.

FIG. 3: Plots of the $l_{1}$ measure of the superposition state
$\rho=\mathinner{|{\psi}\rangle}\mathinner{\langle{\psi}|}$ (the dashed blue
line) and the coherent state
$\bar{\rho}=\mathinner{|{\bar{\psi}}\rangle}\mathinner{\langle{\bar{\psi}}|}$
(the solid orange line), where
$\mathinner{|{\psi}\rangle}=x(\eta\mathinner{|{c_{0}}\rangle}+\mathinner{|{c_{1}}\rangle})$
with $\langle{c_{1}}|c_{2}\rangle=s\in(-1,1)$ such that $x\neq 0$ and
$\eta=3$.

Moreover, for a given superposition state
$\mathinner{|{\psi}\rangle}=x(\eta\mathinner{|{c_{0}}\rangle}+\mathinner{|{c_{1}}\rangle})$
with $\langle{c_{1}}|c_{2}\rangle=-2\eta/(\eta^{2}+1)\in(-1,1)$, we have
$M_{l_{1}}(\bar{\rho})=0$ while $M_{l_{1}}(\rho)\neq 0$. In other words, for
qubit systems, there are superposition states from which no coherent state can
be obtained with the use of LSO. As an explicit example, one can consider the
superposition state
$\mathinner{|{\psi}\rangle}=\frac{\sqrt{10}}{8}(3\mathinner{|{c_{0}}\rangle}+\mathinner{|{c_{1}}\rangle})$,
which is the state discussed in Fig. 3 with
$\langle{c_{1}}|c_{2}\rangle=-0.6$. Therefore, the superposition states
transformation constructed above cannot be applied for such kind of states.
Although we here only consider qubit systems, same situation probably exists
for some particular cases in dimension $d$. Note that the above discussion has
been done using $l_{1}$ measure since it is easy to calculate — other measures
such as relative entropy of superposition requires optimization, which is
particularly difficult to calculate in the nonorthogonal setting. We hope that
the results related with LSO that connects the theory of coherence and
superposition may be useful in investigating effective and practical
superposition resource measures.

##  IV Conclusions

To summarize, we have shown that LSO offers compelling advantages in revealing
the strong connections between the notions of coherence and superposition
within their resource-theoretic formulations. The starting point of this stems
from attempts to figure out a way to utilize existing studies on the
manipulation and characterization of coherence, by which we can get more
details on the nature of superposition states. We have presented an approach
that combines LSO and a coherence transformation task to achieve superposition
state manipulation. Our three-step method has the advantage of being simple in
structure, which can improve the transition efficiency of pure superposition
states.

At the heart of our consideration is LSO [22, 25], which has a unique feature
among all existing orthogonalization techniques. Namely, we can obtain the
states with maximal superposition (i.e., golden states) from the maximally
coherent state by means of LSO. We have discussed qubit systems, and
afterwards $d$-dimensional case. We have obtained consistent results with the
comprehensive study recently presented in Ref. [34]. We have considered
superposition distillation, which is an another example that reflects the
effectiveness of our approach. We have also briefly discussed the $l_{1}$
measures of the superposition states and the coherent states, where LSO
connects them.

Looking forward, there could be various implications of the present results.
For instance, practical and efficient strategies were presented to obtain one
of a family of maximally coherent states of dimension $n=2,3,\dots,d$ from any
pure state [38] and mixed state [40] of dimension $d$. One direction for
future work would be to explore whether we can obtain a set of states with
maximal superposition of dimension $n\leq d$ from a given (pure or mixed)
superposition state, with the aid of LSO and tasks developed in Refs. [38,
40]. It would be interesting to see the consequences of these for mixed
superposition states. Moreover, LSO may provide further motivation to extend
the role of coherence as a resource in metrology [10, 11] to superposition. We
believe that LSO positively impacts our understanding of the links between
orthogonal and nonorthogonal settings, and could provide more for general
quantum resource theories.

###### Acknowledgements.

I would like to thank Onur Pusuluk for valuable remarks and fruitful
discussions about Löwdin symmetric orthogonalization.

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