# Everyday the Same Picture: Popularity and Content Diversity

Alessandro Bessi
IUSS Pavia, Italy &Fabiana Zollo
IMT Lucca, Italy &Michela Del Vicario
IMT Lucca, Italy Antonio Scala
ISC CNR,Rome, Italy &Fabio Petroni
Sapienza University of Rome, Italy Bruno Gonçalves
Aix Marseille, France &Walter Quattrociocchi
IMT Lucca, Italy
<EMAIL_ADDRESS>
###### Abstract

Facebook is flooded by diverse and heterogeneous content, from kittens up to
music and news, passing through satirical and funny stories. Each piece of
that corpus reflects the heterogeneity of the underlying social background. In
the Italian Facebook we have found an interesting case: a page having more
than $40K$ followers that every day posts the same picture of a popular
Italian singer. In this work, we use such a page as a control to study and
model the relationship between content heterogeneity on popularity. In
particular, we use that page for a comparative analysis of information
consumption patterns with respect to pages posting science and conspiracy
news. In total, we analyze about $2M$ likes and $190K$ comments, made by
approximately $340K$ and $65K$ users, respectively. We conclude the paper by
introducing a model mimicking users selection preferences accounting for the
heterogeneity of contents.

## Introduction

Online social networks such as Facebook foster the aggregation of people
around common interests, narratives, and worldviews. Indeed, the World Wide
Web caused a shift of paradigm in the production and consumption of contents
that increased volume and heterogeneity of available contents. Users can
express their attitudes by producing and consuming heterogeneous information —
e.g. conspiracists avoid mainstream news and follow their own information
sources, whereas debunkers try to inhibit the diffusion of false claims.
Images of kittens and pets, political memes, gossip, scandals spread on
Facebook. By liking, commenting, and sharing their preferred contents, users
can express their passions and emotions — and, among these latter, sarcasm in
not an exception. Indeed, not rarely we can find pages promoting parodistic
and sarcastic imitations of online social dynamics — e.g., Ebola and Kittens
kittens  or In favor of chem-trails chemtrails . An interesting case in the
Italian Facebook is a page TotoCutugno  with more than $40K$ followers that
posts everyday the exactly alike picture of Toto Cutugno, a famous Italian
pop-singer.

In this work, we use the intriguing case of that page as a baseline to study
and model the effect of content diversity on popularity. Specifically, we
analyze user activity and post consumption patterns on the baseline page for a
timespan of about $4$ months. Through a comparative analysis between two sets
of pages producing heterogeneous contents, we show that there are no
remarkable differences in user activity patterns, whereas significant
dissimilarities between post consumption patterns emerge. Such a comparative
analysis allows to derive a model of information consumption accounting for
the heterogeneity of contents. Hence, we show that the proposed model is able
to reproduce the phenomenon observed from empirical data. In particular, we
show the effects of different levels of contents’ heterogeneity on posts
consumption patterns.

## Background and Related Works

A large body of literature addresses the study of social dynamics on socio-
technical systems from social contagion up to social reinforcement Onnela2010
; Ugander2012 ; Lewis2012 ; Mocanu2012 ; Adamic2005 ; Kleinberg2013 ; eRep ;
Quattrociocchi2011 ; Quattrociocchi2009 ; Bond2012 ; Moreno2011 ; Centola2010
; Castellano2007 ; Quattrociocchi2014 ; Bennaim2003 ; Adamic2013 ; Hannak2012
; Cheng2014 ; Goncalves2012 . Among these, one of the most defining topic of
computational social science is the understanding of driving forces behind the
popularity of contents tatar2014survey . Such a challenge has been addressed
looking at the sentiment of comments, contents, or users’ attention
tatar2011predicting ; figueiredo2014does ; zadeh2014modeling ;
ratkiewicz2011detecting ; bandari2012pulse ; gomez2008statistical ;
szabo2010predicting ; lerman2010using ; lerman2010information . However, the
mechanisms behind popularity remain largely unexplored Goldhaber1997 ;
Watts2002 ; Leskovec2009 . In salganik2006experimental  the authors address
such a challenge experimentally by measuring the impact of content quality and
social influence on the eventual popularity or success of cultural artifacts.
The effects of specific contents on the formation of communities of interest,
their permeability to false information, and the resistance to changes have
been recently characterized in JTM2014 ; Rojecki2014 ; SOCINFO2014 ; PLOS2014
. In particular, in WWW2015  the authors point out that connectivity patterns
of the Facebook social network are prominently driven by homophily of users —
i.e., the tendency of individuals to associate with similar others — towards
specific kinds of contents. Microblogging platforms such as Facebook and
Twitter Tapscott2006  have lowered the cost of information production and
broadcasting, boosting the potential reach of each idea or meme Dawkins1989 ;
Bauckhage2011  — i.e., content or concepts that spread rapidly on the Web.
Still, the abundance of information to which we are exposed through online
social networks and other socio-technical systems is exceeding our capacity to
consume it Weng2012 . As a result, the dynamics of information is driven more
than ever before by the economy of attention Simon1971 ; dukas2001limited ;
Lehmann2012 . We address this challenge by studying the interlink between
contents diversity and popularity. More specifically, we investigate the
effects of sources producing always the same information on users’ activity,
consumption, and attention patterns.

## Data Description

In this work, we aim at investigating the role of content diversity on the
dynamics of information consumption in online social networks. To this end, we
use a set of Facebook pages promoting heterogeneous contents and a Facebook
page promoting always the same picture. The set of pages promoting
heterogeneous contents is composed by $73$ public Facebook pages, whereof $34$
are about scientific news and $39$ about conspiratorial news; we refer to the
former as _science pages_ and to the latter as _conspiracy pages_. The page
promoting homogeneous contents is called ”La stessa foto di Toto Cutugno ogni
giorno” (”Everyday the same photo of Toto Cutugno”, a well-known Italian pop
singer-songwriter); such a page, by publishing everyday the same picture of
the Italian singer — and nothing else —, represents the perfect control for
studying content diversity; we refer to this page as the _baseline page_.
Starting from these pages, we downloaded all the posts, we collected all the
_likes_ and _comments_ to the posts, and we counted the number of _shares_.
Data related to science and conspiracy pages have been collected from August
22, 2013 to December 31, 2013, whereas data related to the baseline page have
been collected from August 22, 2014 (birthdate of the page) to December 31,
2014. In total, we collected around $2M$ likes and $190K$ comments, made by
about $340K$ and $65K$ users, respectively. In Table 1 we summarize the
details of our data collection. Likes, shares, and comments have a different
meaning from the user viewpoint. Most of the times, a like stands for a
positive feedback to the post; a share expresses the will to increase the
visibility of a given information; and a comment is the way in which online
collective debates take form. Comments may contain negative or positive
feedbacks with respect to the post.

| Total | Science | Conspiracy | Baseline
---|---|---|---|---
Pages | $74$ | $34$ | $39$ | $1$
Posts | $49,354$ | $13,028$ | $36,169$ | $157$
Likes | $2,095,677$ | $614,078$ | $1,184,084$ | $297,515$
Comments | $192,967$ | $40,608$ | $138,138$ | $14,221$
Shares | $3,782,480$ | $477,457$ | $3,297,687$ | $7,336$
Likers | $344,367$ | $162,146$ | $159,524$ | $22,697$
Commenters | $64,903$ | $18,358$ | $41,666$ | $4,875$

Table 1:  Dataset breakdown. The number of pages, posts, likes, comments,
shares, likers, and commenters for science pages, conspiracy pages, and the
baseline page.

## Results and Discussion

In this section, we first present the statistical signatures characterizing
users activity on pages with diversified content on specific topics (science
and conspiracy news) against the case of the page posting every day the same
picture (baseline). Then we derive a model of information consumption
mimicking user preferences with respect to contents.

### Content and Users Activity

Let us focus some regularities concerning users’ activity on science pages and
conspiracy pages compared with the baseline page. Figure 1 shows the
probability density function (PDF) for the normalized111We performed the
unity–based normalization to bring all values in the range $[0,1]$. number of
likes for each user. We find that the activity of users presents an
heavy–tailed distribution.

Figure 1: Users’ activity patterns. Probability density function (PDF) for the
normalized number of likes by each user.

In Figure 2 we show the PDF of the users’ lifetime in terms of their liking
activity — i.e. the temporal interval between the first and the last like of
the user on a given page. We find a slight difference in the lifetime of the
baseline users with respect to science and conspiracy users.

Figure 2: Users’ lifetime. Probability density function (PDF) of the users’
lifetime in terms of their liking activity. The PDF shows a slight difference
in the lifetime of the baseline users with respect to science and conspiracy
users.

These figures show that users activity patterns are similar and present
heavy–tailed distributions despite the different nature of the contents, and
we can not find any significant difference between the users interaction
patterns induced by heterogeneous or homogeneous contents.

Conversely, by analyzing consumption patterns related to posts, we find a
significant difference in the information consumption dynamics. Figure 3 shows
the PDF for the number of likes received by posts belonging to science pages,
conspiracy pages, and the baseline page.

Figure 3: Posts’ consumption patterns. Probability distribution function (PDF)
for the normalized number of likes received by posts belonging to science
pages, conspiracy pages, and the baseline page. The PDFs show remarkable
differences between consumption patterns’ distributions related to pages
promoting heterogeneous contents and those related to the page promoting
homogeneous contents.

The number of likes received by posts are heavy–tailed distributed if the
posts belong to pages promoting heterogeneous contents (science and conspiracy
pages); whereas they are approximately distributed according to a Gaussian if
the posts belong to a page promoting homogeneous content (baseline page).

Summarizing, users’ activities always present heavy–tailed distributions
resolving in heavy–tailed distributed consumption patterns on posts in the
heterogeneous contents case. Still, when the content promoted by a page is
homogeneous – i.e., always the same – we find that the heavy–tailed
distributed users’ activities resolve in posts’ consumption patterns that are
approximately Gaussian.

### Modeling Contents Consumption

Here we introduce a model of pattern consumption that exploits the Beta
distribution properties to generate different levels of posts’ attractiveness,
thus varying content–heterogeneity in the simulated collection of posts.

The Beta distribution is a family of continuous probability distributions
defined in the interval $[0,1]$ and characterized by two real parameters,
$\alpha>0$ and $\beta>0$, which control the shape of the distribution. In
particular, for $\alpha=1$ and $\beta=1$ the Beta distribution
$\mathcal{B}e(\alpha,\beta)$ is equivalent to the Uniform distribution
$\mathcal{U}(0,1)$. Conversely, if $\alpha=1$ and $\beta\gtrsim 20$, the Beta
distribution $\mathcal{B}e(\alpha,\beta)$ is a right heavy–tailed
distribution. Figure 4 shows the Beta probability density function with
respect to the two shape parameters $\alpha$ and $\beta$.

Figure 4: Beta distribution $\mathcal{B}e(\alpha,\beta)$. Two parameters,
$\alpha$ and $\beta$, control the shape of the distribution. In particular,
for $\alpha=1$ and $\beta=1$ the Beta distribution
$\mathcal{B}e(\alpha,\beta)$ is equivalent to the Uniform distribution
$\mathcal{U}(0,1)$. Conversely, if $\alpha=1$ and $\beta\gtrsim 20$, the Beta
distribution $\mathcal{B}e(\alpha,\beta)$ is a right heavy–tailed
distribution.

In our model, each post has a value drawn from a Beta distribution
$v\sim\mathcal{B}e(1,\beta)$, with $\beta$ ranging between $1$ and
$1,000,000$, indicating its attractiveness. We let the parameter $\beta$
assume those extreme values in order to obtain different distributions for
posts’ attractiveness. Indeed, notice that when $\beta=1$ the Beta
distribution $\mathcal{B}e(1,\beta)$ is equivalent to a uniform distribution
$\mathcal{U}(0,1)$, so that we have a collection of homogeneous–content posts
— i.e., each post has the same degree of attractiveness; whereas when
$\beta\to\infty$ the Beta distribution $\mathcal{B}e(1,\beta)$ is equivalent
to a right heavy–tailed distribution, so that we have a collection of
heterogeneous–content posts — i.e., there are few posts with a high level of
attractiveness, while the vast majority of the posts is characterized by a low
level of attractiveness. Moreover, each user is characterized by two
parameters randomly drawn from power law distributions: her volume of
activity, $a\sim p(x)$; and her fixed–preference about the posts, $b\sim
p(x)$, where $p(x)=x^{-\gamma}$ with $\gamma=1.5$. Each user can not exceed
her assigned volume of activity, $a$, and she likes a given post if and only
if her normalized222Note that we performed a unity–based normalization in
order to bring all values of $b\sim p(x)=x^{-1.5}$ in the range $[0,1]$, so
that the fixed–preference of the user is comparable with the attractiveness of
the posts. fixed–preference, $b$, is smaller than the attractiveness, $v$, of
that post. Note that in our model we do not take into account the users’
network: since Facebook network is very dense — indeed, the diameter of
Facebook social network is $3.74$ Backstrom2012 ; WWW2015  — the connections
between users are not likely to influence posts’ consumption dynamics.

We run simulations for $\beta$ ranging between $1$ and $1,000,000$, with
$P=10,000$ (posts) and $U=20,000$ (users). Results are averaged over $100$
iterations.

Figure 5 shows the probability density function (PDF) of the users activity
and the posts consumption patterns generated by a simulation of the model with
$\beta=1,000,000$ — i.e., in the case of extremely heterogeneous–content
posts. Observe that users’ activity is heavy–tailed, and the distribution of
posts’ consumption is skewed. Such a result is consistent with empirical data
shown in the previous section: if the content promoted by a page is
heterogeneous, the heavy–tailed users’ activity resolves in skewed posts
consumption’s patterns.

Figure 5: Users activity and post consumption patterns with extremely
heterogenous–content posts. Probability density function (PDF) of the users
activity and the posts consumption patterns generated by a simulation of the
model with $\beta=1,000,000$. If the content promoted by a page is
heterogeneous, the heavy–tailed users’ activity resolves in skewed posts
consumption’s patterns.

Figure 6 shows the probability density function (PDF) of the users activity
and the posts consumption patterns generated by a simulation of the model with
$\beta=1$ — i.e., in the case of homogeneous–content posts. Notice that users’
activity is heavy–tailed, whereas posts’ consumption is approximately
Gaussian. Such a result is consistent with empirical data shown in the
previous section: if the content promoted by a page is always the same, the
heavy–tailed users’ activity resolves in approximately Gaussian posts
consumption’s patterns.

Figure 6: Users activity and post consumption patterns with
homogeneous–content posts. Probability density function (PDF) of the users
activity and the posts consumption patterns generated by a simulation of the
model with $\beta=1$. If the content promoted by a page is always the same,
the heavy–tailed users’ activity resolves in approximately Gaussian posts
consumption’s patterns.

## Concluding Remarks

Facebook is overflowed by different and heterogeneous contents, from the
latest news up to satirical and funny stories. Each piece of that corpus
reflects the heterogeneity of the underlying social background. Indeed, the
World Wide Web caused a shift of paradigm in the production and consumption of
information that increased the amount and heterogeneity of contents available
to users. On online social networks such as Twitter and Facebook, people can
express their attitudes, passions, and emotions by producing and consuming
heterogeneous information.

In the Italian Facebook, we have found a fascinating case of contents’
homogeneity: a page with more than $40K$ followers that every day posts the
same picture of Toto Cutugno, a popular Italian singer. In this work, we use
such a page as a benchmark to investigate and model the effect of contents’
heterogeneity on popularity. In particular, we use that page for a comparative
analysis of information consumption patterns with respect to pages posting
heterogeneous contents related to science and conspiracy.

We show that there are not remarkable differences in user activity patterns,
whereas we find significant dissimilarities between post consumption patterns
of the page promoting homogeneous contents and those of the pages producing
heterogeneous contents. Finally, we derive a model of information consumption
that accounts for the heterogeneity of contents. Hence, we show that the
proposed model is able to reproduce the phenomenon observed from empirical
data.

## Acknowledgments

Funding for this work was provided by EU FET project MULTIPLEX nr. 317532 and
SIMPOL nr. 610704. The funders had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript. We want
to thank Prof. Guido Caldarelli for precious insights and contribution on the
data analysis. Special thanks to Josif Stalin, Stefano Alpi, Michele Degani
for giving access to the Facebook page of La stessa foto di Toto Cutugno ogni
giorno.

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# Process-optimized phase covariant quantum cloning

Chloe Kim and Eric Chitambar

(August 27, 2024)

###### Abstract

After the appearance of the no-cloning theorem, approximate quantum cloning
machines (QCMs) have become one of the most well-studied subject in quantum
information theory. Among several measures to quantify the performance of a
QCM, single-qudit fidelity and global fidelity have been most widely used. In
this paper we compute the optimal global fidelity for phase-covariant cloning
machines via semi-definite programming optimization, thereby completing a
remaining gap in the previous results on QCMs. We also consider optimal
simulations of the cloning and transpose cloning map, both by a direct
optimization and by a composition of component-wise optimal QCMs. For the
cloning map the composition method is sub-optimal whereas for the transpose
cloning map the method is asymptotically optimal.

##  I Introduction

The famous no-cloning theorem tells us that there exists no quantum operation
that copies an arbitrary quantum state [29, 15, 34]. Despite this
impossibility, the notion of approximate quantum cloning machines was
introduced in 1996 [6], and the topic has received extensive development over
the past twenty-five years. The inability to universally copy states is a
fundamental signature of “quantumness” that has significant applications. In
particular, the no-cloning theorem is essential for ensuring security in
cryptographic tasks like quantum key distribution (QKD) [3]. A general
overview of quantum cloning and its relevance to quantum information
processing can be found in review papers such as [30, 21].

An ideal $N\to M$ quantum cloner is a non-linear map

$\rho^{\otimes N}\mapsto\rho^{\otimes M}$ (1)

for any density matrix $\rho$ of some $d$-dimensional system $\mathsf{S}$, and
a quantum cloning machine (QCM) approximates this operation via a physically
realizable quantum channel. That is, an $N\to M$ QCM is a completely positive
trace-preserving (CPTP) map $\mathcal{E}$ satisfying

$\displaystyle\mathcal{E}\left(\rho^{\otimes
N}\right)=\sigma^{\mathsf{S}_{1}\cdots\mathsf{S}_{M}}\approx\rho^{\otimes M}.$
(2)

Different criteria are used to categorize QCMs. For instance, the QCM is
called symmetric if
$\sigma^{\mathsf{S}_{i}}:=\operatorname{Tr}_{\overline{\mathsf{S}}_{i}}\sigma$
is the same reduced state for all $\mathsf{S}_{i}$, where
$\operatorname{Tr}_{\overline{\mathsf{S}}_{i}}$ indicates a partial trace over
all subsystems but $\mathsf{S}_{i}$. An economic QCM is one that restricts to
maps using no ancillary system other than the $\mathsf{S}_{i}$ output systems
[8]. Finally, universality of a QCM means that it clones all the input states
equally well, and a QCM is optimal if the cloning process is simulated
optimally with respect to a given figure of merit [30].

The most commonly used figure of merit for cloning is the average single-qudit
fidelity, which measures the distance between the input state $\rho$ and the
reduced state of each output copy $\sigma^{\mathsf{S}_{i}}$. Another possible
measure evaluates the average global fidelity between the QCM output state
$\sigma^{\mathsf{S}_{1}\cdots\mathsf{S}_{M}}$ and the ideal output state
$\rho^{\otimes M}$. In [28], this global fidelity was called _process
fidelity_ , and the authors used the semidefinite programming (SDP) techniques
introduced in [2] to optimize the process fidelity for cloning one qubit
($d=2$). The goal of this paper is to extend this result by computing the
optimal process fidelity for cloning qudits ($d>2$). In particular, we
consider the cloning of maximally coherent pure states having the form

$\left|\text{\boldmath{\hbox{\theta}}}\right\rangle=\frac{1}{\sqrt{d}}\sum_{k=1}^{d}e^{i\theta_{k}}\left|k\right\rangle,$
(3)

where $\theta$ denotes $d$-tuple of angles $(\theta_{1},\cdots,\theta_{d})$,
$\theta_{i}\in\left[0,2\pi\right)$. For $d=2$, this represents the family of
equatorial states on the Bloch sphere. Up to a unitary transformation, the
four BB84 states
$\\{\left|0\right\rangle,\left|1\right\rangle,\left|\pm\right\rangle=\frac{1}{\sqrt{2}}(\left|0\right\rangle\pm\left|1\right\rangle)\\}$
belong to this family [3].

A (non-universal) QCM whose domain is restricted to the phase-encoded input
states $\left|\text{\boldmath$\theta$}\right\rangle$ is called a phase-
covariant quantum cloner. Bruß et al. first studied the cloning of BB84 states
[4], and they showed that the optimal map that clones BB84 states also
optimally clones any arbitrary phase-covariant states on the Bloch sphere.
They obtained an optimal single-qubit fidelity of
$\frac{1}{2}+\frac{1}{\sqrt{8}}$, which is greater than that of the universal
cloner ($\frac{5}{6}$). Subsequent work has been conducted on the phase-
covariant cloning of $1\to M$ qubits [20], $1\to 2$ qutrits [14, 13, 10],
$1\to M$ qudits [19], optimizing over the single qudit fidelity.

In this paper, we present an explicit formula for an optimal $1\to 2$ phase
covariant QCM, using the process fidelity as the figure of measure. This
completes the picture of $1\to 2$ optimal QCMs, as summarized in Table 1. The
values indicated in the table are in agreement with those that can be achieved
by an alternative SDP method proposed for optimal channels in [22, 11],
although we do not make the assumption that the domain of an optimal channel
is the set of pure states. The organization of this paper as follows. In
Section II we specify notation and definitions related to the process
fidelity. In Section III we present our main results on optimal process
fidelity of phase-covariant cloning. We introduce transpose cloning in Section
IV and provide the optimal single-qudit and process fidelities. This allows us
to consider in Section IV.5 the modular construction of larger QCMs by
composing smaller ones. Finally, some concluding remarks are provided in
Section V.

dimension |  type |  optimal single-qubit fidelity |  optimal process fidelity
---|---|---|---
$1\to 2$ qubits | universal |  $\frac{5}{6}$ [6, 5, 23, 24] |  $\frac{2}{3}$[6]
$1\to 2$ qudits | universal |  $\frac{d+3}{(2d+2)}$ [9] |  $2/(d+1)$[33]
$1\to 2$ qubits | phase covariant |  $\frac{1}{2}+\frac{1}{\sqrt{8}}$ [1, 4, 16, 25] | 0.75 [28]
$1\to 2$ qutrits | phase covariant |  $\frac{5+\sqrt{17}}{12}$[14, 13, 10] | $5/9$
$1\to 2$ qudits | phase covariant |  $\frac{1}{d}+\frac{d-2+\sqrt{d^{2}+4d-4}}{4d}$ [19] | $(2d-1)/d^{2}$
Table 1: Optimal single qudit fidelity and process fidelity of symmetric UQCM
and phase covariant QCM

##  II Notations and definitions

Let $B(\mathcal{H})$ denote the set of bounded operators on a Hilbert space
$\mathcal{H}$. Let $\mathcal{D}\subset B(\mathcal{H})$ be the set of density
operators, i.e. positive trace-one elements,
$\mathcal{D}_{\text{pure}}\subset\mathcal{D}$ the set of pure states, i.e.
rank-one density operators, and
$\mathcal{D}_{\text{\boldmath$\theta$}}\subset\mathcal{D}_{\text{pure}}$ the
set of states
$|\text{\boldmath$\theta$}\rangle\langle\text{\boldmath$\theta$}|$ having the
form of (3). The fidelity [27] of any two $\rho,\sigma\in\mathcal{D}$ is given
by

$F(\rho,\sigma):=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2}.$
(4)

Suppose $\mathcal{E}_{\text{ideal}}:B(\mathcal{H})\to B(\mathcal{K})$ is an
“ideal” map that is not necessarily a quantum channel (not even necessarily
linear), i.e, completely-positive and trace-preserving (CPTP). We are
interested in approximating $\mathcal{E}_{\text{ideal}}$ by a physically
realizable quantum channel $\mathcal{E}$, or possibly approximating just the
action of $\mathcal{E}_{\text{ideal}}$ on some restricted set of inputs $S$.
There are various approaches to quantifying how well $\mathcal{E}$
approximates $\mathcal{E}_{\text{ideal}}$, and here we consider the process
fidelity.

######  Definition 1.

Let $S\subset\mathcal{D}$ be the set of states that are generated by the
action of a compact group $G$, and let $\mu$ be the Haar measure induced by
$G$. The process fidelity between an arbitrary ideal map
$\mathcal{E}_{\text{ideal}}$ and a CPTP map $\mathcal{E}$ is defined as

$F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|S):=\int_{S}F(\mathcal{E}_{\text{ideal}}(\sigma),\mathcal{E}(\sigma))d\mu(\sigma).$
(5)

If $\mathcal{E}_{\text{ideal}}$ maps pure state to pure states and
$S\subset\mathcal{D}_{\text{pure}}$, then the process fidelity is equivalent
to

$F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|S)=\int_{S}\operatorname{Tr}\Big{[}\mathcal{E}_{\text{ideal}}(\sigma)\mathcal{E}(\sigma)\Big{]}\;d\mu(\sigma).$
(6)

A key feature of this expression is that the integrand is linear in
$\mathcal{E}(\sigma)$. This will allow us to exploit symmetry properties of
Haar measure below. On the other hand, note that $\mathcal{E}_{\text{ideal}}$
need not be linear, such as with the cloning map $\rho\mapsto\rho\otimes\rho$.

######  Definition 2.

Let $\text{CPTP}(\mathcal{H}\to\mathcal{K})$ be the set of all quantum
channels mapping $B(\mathcal{H})$ to $B(\mathcal{K})$. The optimal process
fidelity of a map $\mathcal{E}_{\text{ideal}}:B(\mathcal{H})\to
B(\mathcal{K})$ is the maximum process fidelity over all possible quantum
channels, i.e.

$F^{*}_{\text{proc}}(\mathcal{E}_{\text{ideal}}|S):=\max_{\mathcal{E}\in\text{CPTP}(\mathcal{H}\to\mathcal{K})}\;F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|S).$
(7)

Given an orthonormal basis $\\{\left|i\right\rangle\\}_{i=1}^{d}$ for
$\mathcal{H}$, the Choi-Jamiołkowski isomorphism [26, 12] establishes an
equivalence between every channel
$\mathcal{E}\in\text{CPTP}(\mathcal{H}\to\mathcal{K})$ and an operator
$J(\mathcal{E})\in B(\mathcal{H})\otimes B(\mathcal{K})$,

$\mathcal{E}\leftrightarrow J(\mathcal{E}):=\sum_{i,j}|i\rangle\langle
j|\otimes\mathcal{E}(|i\rangle\langle j|).$ (8)

The operator $J(\mathcal{E})$ is called the Choi matrix of $\mathcal{E}$ and
the action of $\mathcal{E}$ can be directly expressed in terms of
$J(\mathcal{E})$ as

$\mathcal{E}(\rho)=\operatorname{Tr}_{\mathcal{H}}(J(\mathcal{E})(\rho^{T}\otimes\mathds{1})),$
(9)

where $\mathds{1}$ is the identity on $\mathcal{K}$. When
$S\subset\mathcal{D}_{\text{pure}}$ and
$\mathcal{E}_{\text{ideal}}:\mathcal{D}_{\text{pure}}\to\mathcal{D}_{\text{pure}}$,
the process fidelity can be expressed in terms of the Choi matrix as

$\displaystyle
F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|S)=\int_{S}\operatorname{Tr}\Big{[}J(\mathcal{E})\rho^{T}\otimes\mathcal{E}_{\text{ideal}}(\rho)\Big{]}\;d\mu(\rho).\qquad$
(10)

To compute $F_{\text{proc}}^{*}(\mathcal{E}_{\text{ideal}},\mathcal{E}|S)$,
the problem then reduces to finding the Choi matrix $J(\mathcal{E})$ that
maximizes Eq. (10).

##  III Process-optimized phase-covariant quantum cloning

####  III.0.1 Characterization of an average Choi map

The $1\to 2$ ideal phase covariant cloner $\mathcal{E}_{\text{ideal}}$ is
given by (11)

$\mathcal{E}_{\text{ideal}}(\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|)=\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|^{\otimes
2}$ (11)

for an arbitrary
$\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|\in\mathcal{D}_{\text{\boldmath$\theta$}}$.
The goal is to find an optimal quantum channel $\mathcal{E}$ satisfying

$\displaystyle
F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{\boldmath$\theta$}})=F^{*}_{\text{proc}}(\mathcal{E}_{\text{ideal}}|\mathcal{D}_{\text{\boldmath$\theta$}}).$
(12)

It is useful to write $\left|\text{\boldmath$\theta$}\right\rangle$ as
$U(\text{\boldmath$\theta$})\left|\varphi^{+}_{d}\right\rangle$, where
$U(\text{\boldmath$\theta$})=\sum_{k=1}^{d}e^{i\theta_{k}}|k\rangle\langle k|$
and
$\left|\varphi^{+}_{d}\right\rangle=\frac{1}{\sqrt{d}}\sum_{k=1}^{d}\left|k\right\rangle$.
By the definition expressed in (10), the process fidelity for $\mathcal{E}$
can be written as

$F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}})=\operatorname{Tr}\left[\left|\varphi_{d}^{+}\right\rangle\left\langle\varphi_{d}^{+}\right|^{\otimes
3}\mathcal{T}(J(\mathcal{E}))\right],$ (13)

where $\mathcal{T}(\cdot)$ denotes the “twirling” operation

$\mathcal{T}(X):=\int U(\text{\boldmath{\hbox{\theta}}})\otimes
U(-\text{\boldmath{\hbox{\theta}}})^{\otimes
2}\left(X\right)U(-\text{\boldmath{\hbox{\theta}}})\otimes
U(\text{\boldmath{\hbox{\theta}}})^{\otimes
2}\;d\mu(\text{\boldmath{\hbox{\theta}}}).$ (14)

We exploit the symmetric properties of $\mathcal{T}(J(\mathcal{E}))$ to
formulate the constraints of a semidefinite program to obtain maximum process
fidelity. This is a standard trick [31], but we include it as lemmas for the
sake of self-containment.

Let $S_{d}$ be the permutation group of $d$ elements. Given an orthonormal
basis $\\{\left|k\right\rangle\\}_{k}$, define

$U_{\pi}=\sum_{k=1}^{d}|\pi(k)\rangle\langle k|,\qquad\pi\in S_{d}.$ (15)

In other words, $U_{\pi}$ permutes the basis vectors given a permutation
$\pi\in S_{d}$. Next, let $V_{\sigma}$ be the operator permuting three sub-
systems, where $\sigma\in S_{3}$. For example,

$V_{(23)}\left|\varphi_{1}\right\rangle\otimes\left|\varphi_{2}\right\rangle\otimes\left|\varphi_{3}\right\rangle=\left|\varphi_{1}\right\rangle\otimes\left|\varphi_{3}\right\rangle\otimes\left|\varphi_{2}\right\rangle.$
(16)

######  Lemma 1.

Suppose $\mathcal{E}$ is a quantum channel with Choi matrix $J(\mathcal{E})$.
If we define the average Choi map $\widetilde{J}$ for the channel
$\mathcal{E}$ as

$\widetilde{J}(\mathcal{E}):=\frac{1}{2\left\lvert
S_{d}\right\rvert}\sum_{\begin{subarray}{c}\pi\in S_{d}\\\
\sigma\in\\{\textrm{id},(23)\\}\end{subarray}}V_{\sigma}U_{\pi}^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U_{\pi}^{\dagger\otimes 3}V_{\sigma}^{\dagger},$
(17)

then we have

  1. (i)

$\widetilde{J}(\mathcal{E})$ is invariant under conjugation by
$U(-\text{\boldmath$\theta$})\otimes U(\text{\boldmath$\theta$})^{\otimes 2}$
for any $\theta$;

  2. (ii)

$\widetilde{J}(\mathcal{E})$ is invariant under conjugation by
$U_{\pi}^{\otimes 3}$ for all $\pi\in S_{d}$;

  3. (iii)

$\widetilde{J}(\mathcal{E})$ is invariant under conjugation by $V_{(23)}$;

  4. (iv)

$\widetilde{J}(\mathcal{E})$ is positive and
$\operatorname{Tr}_{23}(\widetilde{J}(\mathcal{E}))=\mathds{1}$.

Note that property (iv) of Lemma 1 assures that $\widetilde{J}(\mathcal{E})$
is a valid Choi matrix of a quantum channel.

######  Lemma 2.

Let $\mathcal{E}_{\text{ideal}}$ be the ideal phase-covariant cloning map and
$\mathcal{E}$ an arbitrary quantum channel. Then

$F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}})=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\widetilde{J}(\mathcal{E})\right].$ (18)

###### Proof.

Let $\pi^{\prime}\in S_{d}$. We can observe that

$\mathcal{T}(X)=U_{\pi^{\prime}}^{\dagger\otimes
3}\mathcal{T}(U_{\pi^{\prime}}^{\otimes 3}XU_{\pi^{\prime}}^{\dagger\otimes
3})U_{\pi^{\prime}}^{\otimes 3}.$ (19)

Using (ii) of the Lemma 1, it follows that

$\displaystyle
F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{\boldmath$\theta$}})$
$\displaystyle=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\mathcal{T}(J(\mathcal{E}))\right]$ (20)
$\displaystyle=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}U_{\pi^{\prime}}^{\dagger\otimes 3}\mathcal{T}(U_{\pi^{\prime}}^{\otimes
3}J(\mathcal{E})U_{\pi^{\prime}}^{\dagger\otimes 3})U_{\pi^{\prime}}^{\otimes
3}\right]$ $\displaystyle=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\frac{1}{\left\lvert S_{d}\right\rvert}\sum_{\pi\in S_{d}}U_{\pi}^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U_{\pi}^{\dagger\otimes 3}\right]$
$\displaystyle=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\frac{1}{2\left\lvert S_{d}\right\rvert}\sum_{\begin{subarray}{c}\pi\in
S_{d}\\\
\sigma\in\\{\textrm{id},(23)\\}\end{subarray}}V_{\sigma}U_{\pi}^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U_{\pi}^{\dagger\otimes
3}V_{\sigma}^{\dagger}\right]=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\widetilde{J}(\mathcal{E})\right],$ (21)

where we have used the facts that $\varphi_{d}^{+}$ is invariant under
$\pi^{\prime}$ and $\varphi_{d}^{+\otimes 3}$ is invariant under $V_{(23)}$. ∎

In summary, we have

$\displaystyle
F^{*}_{\text{proc}}(\mathcal{E}_{\text{ideal}}|\mathcal{D}_{\text{\boldmath$\theta$}})$
$\displaystyle=\max\operatorname{Tr}\left[\varphi_{d}^{+\otimes 3}X\right]$
(22)

in which the maximization is taken over all operators $X\in
B(\mathbb{C}^{d})^{\otimes 3}$ satisfying properties i, ii, iii, iv in Lemma
1.

####  III.0.2 Optimization via semidefinite programming

Using the invariant properties of the average Choi matrix for quantum channel
$\mathcal{E}$ in Lemma 2, we will construct a semi-definite program (SDP) to
obtain an optimal $\mathcal{E}$ that maximizes the process fidelity. A
particularly nice reference for applying semidefinite programming to quantum
information problems is [32], and we apply the basic results here.

First, we characterize a $d^{3}\times d^{3}$ hermitian operator $X\neq 0$ that
satisfies i, ii, iii, iv in Lemma 1. If we write

$X=\sum_{i,j,k,l,m,n}x_{ijklmn}|ijk\rangle\langle lmn|,$ (23)

then the invariance under $U(-\text{\boldmath$\theta$})\otimes
U(\text{\boldmath$\theta$})^{\otimes 2}$ gives

$x_{ijklmn}(1-e^{i(-\theta_{i}+\theta_{j}+\theta_{k}+\theta_{l}-\theta_{m}-\theta_{n})})=0.$
(24)

This can be satisfied if and only if $x_{ijklmn}=0$ or
$-\theta_{i}+\theta_{j}+\theta_{k}+\theta_{l}-\theta_{m}-\theta_{n}$ is
identically zero. If we next add the invariance under under $V_{(23)}$ and
$U_{\pi}^{\otimes 3}$, then $X$ has the form $\sum_{i=1}^{9}x_{i}X_{i}$, where
$x_{i}\in\mathbb{C}$ and

$\displaystyle X_{1}$ $\displaystyle=\sum_{i}|iii\rangle\langle iii|,\qquad
X_{2}=\sum_{i\neq k}|iik\rangle\langle iik|+|iki\rangle\langle iki|,\qquad$
$\displaystyle X_{3}$ $\displaystyle=\sum_{i\neq k}|kii\rangle\langle kii|,$
$\displaystyle X_{4}$ $\displaystyle=\sum_{i\neq k}|kik\rangle\langle
iii|+|iii\rangle\langle kik|+|kki\rangle\langle iii|+|iii\rangle\langle kki|,$
$\displaystyle X_{5}$ $\displaystyle=\sum_{i\neq k}|iik\rangle\langle
iki|+|iki\rangle\langle iik|,$ $\displaystyle X_{6}$
$\displaystyle=\sum_{i\neq k\neq\ell}|ik\ell\rangle\langle ik\ell|,$
$\displaystyle X_{7}$ $\displaystyle=\sum_{i\neq
k\neq\ell}|kk\ell\rangle\langle i\ell i|+|\ell k\ell\rangle\langle iik|,$
$\displaystyle X_{8}$ $\displaystyle=\sum_{i\neq
k\neq\ell}|kk\ell\rangle\langle ii\ell|+|\ell k\ell\rangle\langle iki|,$
$\displaystyle X_{9}$ $\displaystyle=\sum_{i\neq
k\neq\ell}|ik\ell\rangle\langle i\ell k|.$ (25)

If we denote $\mathbf{x}=(x_{i})\in\mathbb{C}^{9}$, then

$\displaystyle\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}X\right]=\frac{1}{d^{2}}x_{1}+\frac{(d-1)}{d^{2}}(2x_{2}+x_{3}+4x_{4}+2x_{5})+\frac{(d-1)(d-2)}{d^{2}}(x_{6}+2x_{7}+2x_{8}+x_{9}).$
(26)

The condition $\operatorname{Tr}_{23}X=\mathds{1}$ corresponds to the equality

$A(\mathbf{x}):=x_{1}+(d-1)(2x_{2}+x_{3})+(d-1)(d-2)x_{6}=1.$ (27)

Finally, the positivity constraint $X\geq 0$ allows us to express (22) in a
simplified SDP. Let $\mathbf{a}=(a_{i})$ where $a_{i}$ is the coefficient of
$x_{i}$’s in $A(\mathbf{x})$, and $\mathbf{c}=(c_{i})$ be the, the coefficient
of $x_{i}$ in $F_{\mathcal{E}}(\mathbf{x})$. Define $F_{0}=0_{d^{3}\times
d^{3}}\oplus[1]$, $F_{i}=X_{i}\oplus[-a_{i}].$ Then we have the primal form of
the SDP

$\displaystyle\text{minimize}\quad-\mathbf{c}^{T}\mathbf{x}$ (28)
$\displaystyle\text{subject to}\quad F_{0}+\sum_{i}x_{i}F_{i}\geq
0,\quad\mathbf{a}^{T}\mathbf{x}=1.$ (29)

By Eq. (22), this SDP yields the value of
$F^{*}_{\text{proc}}(\mathcal{E}_{\text{ideal}}|\mathcal{D}_{\text{\boldmath$\theta$}})$.

Now we present our main results.

######  Theorem 1.

Let $\mathcal{E}_{\text{ideal}}$ be the ideal $1\to 2$ phase covariant cloner.
Then, the optimal process fidelity of $\mathcal{E}_{\text{ideal}}$ is

$F^{*}_{\text{proc}}(\mathcal{E}_{\text{ideal}}|\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}})=\frac{2d-1}{d^{2}},$
(30)

where $d$ is the dimension of the input system.

###### Proof.

Define

$\displaystyle k_{d}=\frac{1}{2d-1},$ (31)
$\displaystyle\mathbf{x}=(k_{d},k_{d},0,k_{d},k_{d},0,k_{d},k_{d},0),$ (32)

where $d$ is the dimension. It is straightforward to show that
$X=\sum_{i}x_{i}X_{i}$ satisfies the positivity and the trace condition. So
this $\mathbf{x}$ is a primal feasible solution and it yields
$F_{\mathcal{E}}(\mathbf{x})=\frac{2d-1}{d^{2}}$. Consider the dual form

$\displaystyle\text{maximize}\quad-\operatorname{Tr}F_{0}Z$ (33)
$\displaystyle\text{subject to}\quad\operatorname{Tr}F_{i}Z=-c_{i},Z\geq 0.$
(34)

We will construct $Z$ such that $\operatorname{Tr}F_{0}Z=(2d-1)/d^{2}$ and
show that this $Z$ is dual feasible. By strong duality, this implies that
$(2d-1)/d^{2}$ is indeed the optimal solution. Let $Z=\hat{Z}\oplus z$, where
$\hat{Z}=\sum_{i}b_{i}X_{i}$, $b_{i}\in\mathbb{R}$ and $z=\frac{2d-1}{d^{2}}$.
Then

$\operatorname{Tr}F_{0}Z=\frac{2d-1}{d^{2}},$ (35)

and the constraints of the dual form become

$\displaystyle\begin{aligned}
&\operatorname{Tr}[X_{1}\hat{Z}]=z-\frac{1}{d^{2}},&\operatorname{Tr}[X_{2}\hat{Z}]&=2\operatorname{Tr}[X_{3}\hat{Z}]=2(d-1)z-\frac{2(d-1)}{d^{2}}\\\
&\operatorname{Tr}[X_{4}\hat{Z}]=2\operatorname{Tr}[X_{5}\hat{Z}]=-\frac{4(d-1)}{d^{2}},&\operatorname{Tr}[X_{6}\hat{Z}]&=(d-1)(d-2)z-\frac{(d-1)(d-2)}{d^{2}}\\\
&\operatorname{Tr}[X_{7}\hat{Z}]=\operatorname{Tr}[X_{8}\hat{Z}]=2\operatorname{Tr}[X_{9}\hat{Z}]=-\frac{2(d-1)(d-2)}{d^{2}}.\end{aligned}$
(36)

Solving these gives $b_{1}=b_{2}=b_{3}=b_{6}=\frac{2(d-1)}{d^{3}},$
$b_{4}=b_{5}=b_{7}=b_{8}=b_{9}=-\frac{1}{d^{3}}.$ We can rewrite $\hat{Z}$ as
a linear combination of projections

$\displaystyle\hat{Z}=\frac{1}{d^{3}}\Big{[}(2d-2-\sqrt{2})X_{1}+\left(2d-\frac{9}{2}-\frac{3}{2\sqrt{2}}X_{2}\right)$
$\displaystyle+(2d-2)X_{3}+(2d-3)X_{6}+2\sqrt{2}P_{A}$ (37)
$\displaystyle+2\left(\frac{2}{\sqrt{2}}+1\right)P_{B}+2P_{C}+2P_{D}+2P_{E}\Big{]},$
(38)

where we define

$\displaystyle P_{A}$
$\displaystyle=-\frac{1}{2\sqrt{2}}X_{4}+\frac{1}{4}(X_{1}+X_{2})+\frac{1}{4}(X_{1}+X_{5})$
$\displaystyle P_{B}$ $\displaystyle=\frac{1}{2}(X_{2}-X_{5}),\quad
P_{C}=\frac{1}{2}(X_{2}-X_{7}),\quad P_{D}=\frac{1}{2}(X_{2}-X_{8}),\quad
P_{E}=\frac{1}{2}(X_{6}-X_{9}).$ (39)

When $d\geq 3$ all the coefficients are positive, and when $d=2$ it can be
easily checked that $\hat{Z}\geq 0$. This shows that $Z$ is also positive, and
therefore $\frac{2d-1}{d^{2}}$ is indeed the optimal solution. ∎

Note that one can calculate the single-qudit fidelity using a map
$\mathcal{E}$ that is optimal for the process fidelity. Doing so leads to a
single-qudit fidelity of $(d+1)/(2d-1)$, which is strictly smaller than the
optimal value [19]. This means that the map yielding optimal process fidelity
does not give optimal single-qudit fidelity unlike the universal cloners [33].

##  IV Transposition cloning

###  IV.1 Optimal phase-covariant transposition

The transposition map is another famous non-CP map. The ideal operation of
transposition map is given by

$\mathcal{E}_{\text{ideal}}:|\psi\rangle\langle\psi|\mapsto|\psi\rangle\langle\psi|^{T},\qquad\forall\rho\in\mathcal{D}_{\text{pure}},$
(40)

It has been shown that the optimal fidelity for approximate transposition
channel of arbitrary pure states is $\frac{2}{d+1}$ [7]. What happens if we
restrict the input states to phase covariant states? In this case, Eq. (40)
takes the form

$\mathcal{E}_{\text{ideal}}:|\text{\boldmath{\hbox{\theta}}}\rangle\langle\text{\boldmath{\hbox{\theta}}}|\mapsto|\text{\boldmath{\hbox{\theta}}}\rangle\langle\text{\boldmath{\hbox{\theta}}}|^{T}=|-\text{\boldmath{\hbox{\theta}}}\rangle\langle-\text{\boldmath{\hbox{\theta}}}|,\qquad\forall\rho\in\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}},$
(41)

where $-\text{\boldmath$\theta$}=(-\theta_{1},\cdots,$ $-\theta_{d})$. To
compute the optimal process fidelity in this restricted case, we can use the
same SDP analysis. The calculation yields to $F_{\text{proc}}=\frac{2}{d}$,
and the Choi operator of an optimal map is given as

$\widetilde{J}(\mathcal{E})=\frac{1}{d-1}\sum_{i\neq j}(|ij\rangle\langle
ij|+|ij\rangle\langle ji|).$ (42)

Hence we see that restricting to maximally coherent states
$\left|\text{\boldmath$\theta$}\right\rangle$ only yields an improved process
fidelity of $\frac{2}{d}$ versus $\frac{2}{d+1}$ in the universal case. These
have the same asymptotic scaling, yet in the qubit case perfect transposition
can be achieved only for equatorial states since the map reduces to a
$\pi$-rotation on the Bloch sphere.

###  IV.2 Process-optimized universal transposition cloning map

We can also define the ideal transposition cloning operation as the map that
outputs two copies of the transpose of the input state, i.e.

$\mathcal{E}_{\text{ideal}}:\rho\mapsto\rho^{T}\otimes\rho^{T},\qquad\forall\rho\in\mathcal{D}_{\text{pure}}.$
(43)

Here we compute the process fidelity of the $1\to 2$ transposition cloning map
for an arbitraty pure state as its input. Let $\mu$ be the induced Haar
measure for the pure states $\mathcal{D}_{\text{pure}}$ and $\sigma$ be the
Haar measure on the set of unitary operators $\mathcal{U}(\mathcal{H})$.
Define the average Choi operator

$\widetilde{J}(\mathcal{E}):=\int_{\mathcal{U}(\mathcal{H})}U^{\dagger\otimes
3}J(\mathcal{E})U^{\otimes 3}\;d\sigma(U).$ (44)

Notice that by the Haar invariance we get

$\displaystyle
F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{pure}})$
$\displaystyle=\int_{\mathcal{D}_{\text{pure}}}\operatorname{Tr}\left[J(\mathcal{E})(\rho^{T})^{\otimes
3}\right]\;d\mu(\rho)$
$\displaystyle=\operatorname{Tr}\left[\int_{\mathcal{U}(\mathcal{H})}J(\mathcal{E})(U|0\rangle\langle
0|U^{\dagger})^{\otimes 3}\;d\sigma(U)\right]$ (45)
$\displaystyle=\operatorname{Tr}\left[\int_{\mathcal{U}(\mathcal{H})}U^{\dagger\otimes
3}J(\mathcal{E})U^{\otimes 3}\;d\sigma(U)\left|0\right\rangle\left\langle
0\right|^{\otimes
3}\right]=\operatorname{Tr}\left[\widetilde{J}(\mathcal{E})\left|0\right\rangle\left\langle
0\right|^{\otimes 3}\right].$ (46)

To find the optimal process fidelity, we exploit the fact that
$\widetilde{J}(\mathcal{E})$ is $U^{\otimes 3}$-invariant. We then obtain the
following proposition as a direct consequence of the results from work by
Eggeling and Werner [18].

######  Proposition 1.

The optimal process fidelity of the $1\to 2$ universal transpose cloning map
is $6/(d^{2}+3d+2)$.

###### Proof.

Eggeling and Werner’s have shown that any quantum state $\rho$ satisfying
$U^{\otimes 3}\rho U^{\dagger\otimes 3}=\rho$ can be uniquely expressed as

$\widetilde{J}(\mathcal{E})=\sum_{i\in\\{+,-,0,1,2,3\\}}c_{i}R_{i},\quad
c_{i}\in\mathbb{R},$ (47)

where

$\displaystyle
R_{+}=\frac{1}{6}(\mathds{1}+V_{(12)}+V_{(23)}+V_{(31)}+V_{(123)}+V_{(132)})$
(48) $\displaystyle
R_{-}=\frac{1}{6}(\mathds{1}-V_{(12)}-V_{(23)}-V_{(31)}+V_{(123)}+V_{(132)})$
(49) $\displaystyle
R_{0}=\frac{1}{3}(2\cdot\mathds{1}-V_{(123)}-V_{(321)}),\quad
R_{1}=\frac{1}{3}(2\cdot\mathds{1}-V_{(31)}-V_{(12)}),$ $\displaystyle
R_{2}=\frac{1}{\sqrt{3}}(V_{(12)}-V_{(31)}),\quad
R_{3}=\frac{i}{\sqrt{3}}(V_{(123)}-V_{(321)}).$ (50)

Furthermore, it satisfies the conditions (i) $c_{+},c_{-},c_{0}\geq 0$,
(ii)$c_{1}^{2}+c_{2}^{2}+c_{3}^{2}\leq c_{0}^{2}$, and (iii)
$\operatorname{Tr}[\rho R_{+}]+\operatorname{Tr}[\rho
R_{-}]+\operatorname{Tr}[\rho R_{-}]=1$. Let $\widetilde{J}(\mathcal{E})$ be
the average Choi operator defined as (44). Then $\widetilde{J}(\mathcal{E})$
is invariant under $U^{\otimes 3}$ action, and the preceding statement holds
except replacing (iii) with

$\operatorname{Tr}_{23}[R_{+}\widetilde{J}(\mathcal{E})+R_{-}\widetilde{J}(\mathcal{E})+R_{0}\widetilde{J}(\mathcal{E})]=\mathds{1}.$
(51)

Suppose $\widetilde{J}(\mathcal{E})=\sum c_{k}R_{k}$. Then the objective
function to maximize becomes

$\displaystyle\operatorname{Tr}\left[\widetilde{J}(\mathcal{E})\left|0\right\rangle\left\langle
0\right|^{\otimes 3}\right]=c_{+},$ (52)

while the constraints $\widetilde{J}(\mathcal{E})\geq 0$ is equivalent to
$c_{+},c_{-},c_{0}\geq 0$ and $c_{1}^{2}+c_{2}^{2}+c_{3}^{2}\leq c_{0}^{2}$
and the trace condition $\operatorname{Tr}\widetilde{J}(\mathcal{E})=d$ is
equivalent to

$\frac{d^{2}+3d+2}{6}c_{+}+\frac{d^{2}-3d+2}{6}c_{-}+\frac{2(d^{2}-1)}{3}c_{0}=1.$
(53)

Solving this linear programming gives

$c_{-}=c_{0}=c_{1}=c_{2}=c_{3}=0,\qquad c_{+}=\frac{6}{d^{2}+3d+2}.$ (54)

∎

Eggeling and Werner use results from representation theory to characterize
$U^{\otimes 3}$-invariant positive matrices, and details of the proof are in
[17] and [18].

###  IV.3 Process-optimized phase-covariant transposition cloning map

We next compare the previous result with the same type of cloner except with
the domain further restricted to $\mathcal{D}_{\text{\boldmath$\theta$}}$. The
ideal map for phase covariant transposition cloner is given by

$\mathcal{E}_{\text{ideal}}:\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|\mapsto\left|-\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle-\text{\boldmath{\hbox{\theta}}}\right|^{\otimes
2}\qquad\forall\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|\in\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}}.$
(55)

Different from the cloning case, the average Choi operator is defined as

$\widetilde{J}(\mathcal{E}):=\frac{1}{\left\lvert S_{d}\right\rvert\left\lvert
S_{3}\right\rvert}\sum_{\pi\in S_{d}}\sum_{\sigma\in
S_{3}}V_{\sigma}U_{\pi}^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U_{\pi}^{\dagger\otimes 3}V_{\sigma}^{\dagger},$
(56)

where the twirling operation is given by

$\mathcal{T}(X):=\int U(-\text{\boldmath{\hbox{\theta}}})^{\otimes
3}\left(X\right)U(\text{\boldmath{\hbox{\theta}}})^{\otimes
3}\;d\mu(\text{\boldmath{\hbox{\theta}}}).$ (57)

The positivity and trace condition of $\widetilde{J}$ remains the same and its
invariance under the permutations of the basis vector also remains the same.
The main difference is that $\widetilde{J}$ is invariant under
$U(\text{\boldmath$\theta$})^{\otimes 3}$ instead of
$U(-\text{\boldmath$\theta$})\otimes U(\text{\boldmath$\theta$})^{\otimes 2}$,
and it is also invariant under permuting any subsystems instead of just 2 and
3. Despite the difference, how we solve the optimal process fidelity is
analogous to that of the cloning case. Here we just state the result as a
theorem and include the proof in the appendix for interested readers.

######  Theorem 2.

Let $\mathcal{E}_{\text{ideal}}$ be given by Eq. (55). Then, the optimal
process fidelity of $\mathcal{E}_{\text{ideal}}$ is

$F_{\text{proc}}^{*}(\mathcal{E}_{\text{ideal}}|\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}})=\begin{cases}3/4&d=2\\\
6/d^{2}&d\geq 3.\end{cases}$ (58)

###  IV.4 Phase-covariant hybrid transposition cloning

Now consider a hybrid transposition cloning map

$\mathcal{E}_{\text{ideal}}:\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|\mapsto\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|\otimes\left|-\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle-\text{\boldmath{\hbox{\theta}}}\right|\qquad\forall\left|\text{\boldmath{\hbox{\theta}}}\right\rangle\left\langle\text{\boldmath{\hbox{\theta}}}\right|\in\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}}.$
(59)

It is easy to see that the symmetries of the process fidelity between the
optimal map and the above ideal map are identical to that of the cloning case,
except we permute the system 1 and 3. In other words, we can reuse the
constraints (i)-(iii) in Lemma 1 to construct the average Choi map for the
hybrid cloning and make a modification for (iv) to be
$\operatorname{Tr}_{12}(\widetilde{J}(\mathcal{E}_{\text{ideal}}))=\mathds{1}$,
instead of taking the partial trace over system 2 and 3. In fact,
$U(-\text{\boldmath$\theta$})\otimes U(\text{\boldmath$\theta$})^{\otimes 2}$
invariance guarantees the partial trace over any two system of
$\widetilde{J}(\mathcal{E})$ to be a scalar multiple of $\mathds{1}$. Hence,
we can conclude that the process fidelity of the hybrid map is $(2d-1)/d^{2}$.
Furthermore, if we let $\mathcal{E}_{\text{hybrid}}$ denote an optimal
approximation of the hybrid transposition cloner (phase-covariant) and
$\mathcal{E}_{\text{cloner}}$ an optimal approximation of the $1\to 2$ cloner
(phase-covariant), then

$\widetilde{J}(\mathcal{E}_{\text{hybrid}})=V_{(13)}\widetilde{J}(\mathcal{E}_{\text{clone}})V_{(13)}.$
(60)

It is interesting that the two maps

$\displaystyle\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|$
$\displaystyle\mapsto\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|\otimes\left|-\text{\boldmath$\theta$}\right\rangle\left\langle-\text{\boldmath$\theta$}\right|$
$\displaystyle\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|$
$\displaystyle\mapsto\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|\otimes\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|$
(61)

can be approximated with the same process fidelity. One might not expect this
since the second can be obtained from the first by applying a transpose on the
first system, and as computed above, the transposition itself has a process
fidelity of $2/d$. Hence a composition of maps yields a highly non-optimal
approximation of the $1\to 2$ phase-covariant cloner. We make this comparison
more explicitly in the next section.

###  IV.5 Modular building of QCMs

In this section we consider a modular approach of simulating some non-physical
process by breaking the latter down into smaller parts and then simulating
each of those. We have just observed that a $1\to 2$ phase-covariant cloner
can be obtained by combining the hybrid cloner (Eq. (59)) with the transpose
map (Eq. (41)). The construction is depicted in Fig. 1, and the question we
consider here is how well the combined optimal maps for the individual parts,
$\mathcal{E}_{\text{hybrid}}$ and $\mathcal{E}_{T}$ respectively, compare to
the optimal map computed in Theorem 1. The composition of
$\mathcal{E}_{\text{hybrid}}:A\to B\widetilde{C}$ and
$\mathcal{E}_{T_{C}}:\widetilde{C}\to C$ can be expressed in terms of their
Choi matrices as

$(\mathcal{E}_{T_{C}}\circ\mathcal{E}_{\text{hybrid}})(\rho)=\operatorname{Tr}_{A\widetilde{C}}\left[\left(J(\mathcal{E}_{\text{hybrid}})((\rho^{T})^{A}\otimes\mathds{1}^{B\widetilde{C}})\right)^{T_{\widetilde{C}}}\otimes\mathds{1}^{C}(J(\mathcal{E}_{T})^{C\widetilde{C}}\otimes\mathds{1}^{AB})\right],$
(62)

where $A,B,C$ denote the quantum systems indicated in Figure 1 and the
subscript $T_{\widetilde{C}}$ denotes the partial transpose of system
$\widetilde{C}$.

Similarly, the transposition cloning of Eq. (55) can be obtained by combining
the hybrid cloner $\mathcal{E}_{\text{hybrid}}:A\to\widetilde{B}C$ (Eq. (59))
with the transpose map $\mathcal{E}_{T_{B}}:\widetilde{B}\to B$ (Eq. (41)).
The construction is depicted in Fig. 2, and we compare to the optimal map
computed in Theorem 2. The composition of maps has the form

$(\mathcal{E}_{T_{B}}\circ\mathcal{E}_{\text{hybrid}})(\rho)=\operatorname{Tr}_{A\widetilde{B}}\left[\left(J(\mathcal{E}_{\text{hybrid}})((\rho^{T})^{A}\otimes\mathds{1}^{\widetilde{B}C})\right)^{T_{\widetilde{B}}}\otimes\mathds{1}^{B}(J(\mathcal{E}_{T})^{B\widetilde{B}}\otimes\mathds{1}^{AC})\right],$
(63)

where $A,B,C$ denote the quantum systems indicated in Figure 1 and the
subscript $T_{\widetilde{B}}$ denotes the partial transpose of system
$\widetilde{B}$.

The process fidelity of both maps are $\frac{3d-4}{d(d-1)(2d-1)}$, which is
factor of $d$ smaller than the optimal cloner in Theorem 1. However, it is
comparable to that of optimal transposition cloning channel with process
fidelity of $6/d^{2}$. Hence in this case, the modular approach is
asymptotically optimal in the process fidelity.

Figure 1: Modular cloning channel. Figure 2: Modular transpose cloning
channel.

##  V Conclusion

We have shown that the optimal process fidelity of the $1\to 2$ phase
covariant-cloner is $\frac{2d-1}{d^{2}}$. This should be compared to the
optimal single-qudit fidelity, which is $(d+1)/(2d-1)$. Phase covariant
cloning map with optimal process fidelity does not necessarily yield optimal
single qudit fidelity, which is the case for cloning arbitrary pure states. In
fact, the previously known optimal phase covariant cloner [19] also does not
yield the optimal process fidelity. We also defined a transpose cloning map,
and showed that the optimal process fidelity for transpose cloning of
arbitrary pure states is $6/(d^{2}+3d+2)$ using the results from [18]. In
comparison, the optimal fidelity for transpose cloning of phase-covariant
states is $6/d^{2}$.

Future work in this direction would include finding an explicit formula for
$1\to M$ phase-covariant cloners. This will require a representation-theoretic
approach involving representations of $S_{M}$ with additional positivity and
phase covariant structure. In general, the $1\to M$ phase covariant cloner is
interesting for applications in phase estimation. We expect that the modular
approach presented in Fig. 1 may also be helpful in simulating such a map and
other multi-system maps like it.

##  Appendix A Solving the process fidelity for $1\to 2$ phase covariant
transpose cloner

Let
$\mathcal{E}_{\text{ideal}}:\left|-\text{\boldmath$\theta$}\right\rangle\left\langle-\text{\boldmath$\theta$}\right|\mapsto\left|\text{\boldmath$\theta$}\right\rangle\left\langle\text{\boldmath$\theta$}\right|^{\otimes
2}$ be the $1\to 2$ ideal phase covariant transpose cloning map.

######  Lemma 3.

Let $\mathcal{E}$ be a quantum channel. Define the average Choi operator
$\widetilde{J}$ for the channel $\mathcal{E}$ as

$\widetilde{J}(\mathcal{E}):=\frac{1}{\left\lvert S_{d}\right\rvert\left\lvert
S_{3}\right\rvert}\sum_{\begin{subarray}{c}\pi\in S_{d}\\\ \sigma\in
S_{3}\end{subarray}}V_{\sigma}U_{\pi}^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U_{\pi}^{\dagger\otimes 3}V_{\sigma}^{\dagger},$
(64)

where the twirling operation $\mathcal{T}$ is defined as

$\mathcal{T}(X):=\int U(\text{\boldmath{\hbox{\theta}}})^{\otimes
3}\left(X\right)U(-\text{\boldmath{\hbox{\theta}}})^{\otimes
3}\;d\mu(\text{\boldmath{\hbox{\theta}}}).$ (65)

then we have

  1. (i)

$\widetilde{J}(\mathcal{E})$ is invariant under conjugation by
$U(\text{\boldmath$\theta$})^{\otimes 3}$ for any $\theta$;

  2. (ii)

$\widetilde{J}(\mathcal{E})$ is invariant under conjugation by
$U_{\pi}^{\otimes 3}$ for all $\pi\in S_{d}$;

  3. (iii)

$\widetilde{J}(\mathcal{E})$ is invariant under permuting subsystems;

  4. (iv)

$\widetilde{J}(\mathcal{E})$ is positive and
$\operatorname{Tr}_{2}(\widetilde{J}(\mathcal{E}))=\mathds{1}$.

###### Proof.

(i) directly follows from

$U(\text{\boldmath{\hbox{\theta}}})^{\otimes
3}\mathcal{T}(J(\mathcal{E}))U(-\text{\boldmath{\hbox{\theta}}})^{\otimes
3}=\mathcal{T}(J(\mathcal{E})).$ (66)

Same as the cloning case, (ii), (iii) also directly follow from the definition
of $\widetilde{J}$ and (iv) follows from the fact that twirling operation,
permutation of basis or the system do not change the positivity of
$J(\mathcal{E})$. Trace is also invariant under $\mathcal{T}$ and under
conjugation by $U_{\pi}^{\otimes 3}$ or $V_{\sigma}$. ∎

With the above characterization of $\widetilde{J}$, we can establish the
following in an analogous manner to the cloning case.

######  Lemma 4.

Suppose $\widetilde{J}$ is defined as (17) for a quantum channel
$\mathcal{E}$. Then we have

$F_{\text{proc}}(\mathcal{E}_{\text{ideal}},\mathcal{E}|\mathcal{D}_{\text{\boldmath{\hbox{\theta}}}})=\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}\widetilde{J}(\mathcal{E})\right].$ (67)

We omit the proof since it is more or less identical to Lemma 2 for the
cloning case.

Similar to the cloning case, we characterize a $d^{3}\times d^{3}$ hermitian
operator $X\neq 0$ that satisfies i, ii, iii, iv in Lemma 3. As a result, we
can express $X$ as $\sum_{i=1}^{6}x_{i}X_{i}$, where $x_{i}\in\mathbb{C}$ and

$\displaystyle X_{1}=\sum_{i}|iii\rangle\langle iii|,\;X_{2}=\sum_{i\neq
k}|iik\rangle\langle iik|+|iki\rangle\langle iki|+|kii\rangle\langle kii|,$
(68) $\displaystyle X_{3}=\sum_{i\neq k}|iik\rangle\langle
kii|+|kii\rangle\langle iik|+|kii\rangle\langle iki|+|iik\rangle\langle
iki|+|iki\rangle\langle iik|,$ (69) $\displaystyle X_{4}=\sum_{i\neq
k\neq\ell}|ik\ell\rangle\langle ik\ell|,\;X_{5}=\sum_{i\neq
k\neq\ell}|ik\ell\rangle\langle k\ell i|+|ik\ell\rangle\langle\ell ik|,$ (70)
$\displaystyle X_{6}=\sum_{i\neq k\neq\ell}|ik\ell\rangle\langle
ki\ell|+|ik\ell\rangle\langle\ell ki|+|ik\ell\rangle\langle i\ell k|.$ (71)

If we denote $\mathbf{x}=(x_{i})\in\mathbb{C}^{6}$, then

$\displaystyle\operatorname{Tr}\left[\varphi_{d}^{+\otimes
3}X\right]=\frac{1}{d^{2}}x_{1}+\frac{(d-1)}{d^{2}}(3x_{2}+6x_{3})+\frac{(d-1)(d-2)}{d^{2}}(x_{4}+2x_{5}+3x_{6}),$
(72)

and the trace condition

$A(\mathbf{x})=x_{1}+3(d-1)x_{2}+(d-1)(d-2)x_{4}=1.$ (73)

We construct the SDP the identical way as (33) to prove the optimality of the
process fidelity. Now we present the proof of the Theorem 2.

###### Proof.

When the dimension $d=2$, $\mathbf{x}=(0,1/3,1/3,0,0,0)$ yields a primal
feasible solution of $3/4$, and $\hat{Z}\oplus[3/4]$ is dual feasible, where

$\hat{Z}=\frac{1}{4}(X_{1}+X_{2}+X_{4})-\frac{1}{8}(X_{3}+X_{5}+X_{6}).$

Let $d\geq 3$ and define

$\displaystyle
k_{d}=\frac{1}{(d-1)(d-2)},\;\mathbf{x}=(0,0,0,k_{d},k_{d},k_{d}).$ (74)

It can be easily be checked that the resulting operator satisfies the
positivity and trace condition. Analogous to the cloning case, we prove the
optimality of $F_{\mathcal{E}}$ by finding a dual feasible
$Z=\hat{Z}\oplus[z]$, where $z=6/d^{2}$ and $\hat{Z}=\sum_{i}b_{i}X_{i}$,
$b_{i}\in\mathbb{R}$. The constraints $\operatorname{Tr}F_{i}Z=c_{i}$ yields

$\displaystyle\operatorname{Tr}[X_{1}\hat{Z}]-z$
$\displaystyle=-\frac{1}{d^{2}},$
$\displaystyle\operatorname{Tr}[X_{2}\hat{Z}]-3(d-1)z$
$\displaystyle=-\frac{3(d-1)}{d^{2}}$
$\displaystyle\operatorname{Tr}[X_{3}\hat{Z}]$
$\displaystyle=-\frac{6(d-1)}{d^{2}},$
$\displaystyle\operatorname{Tr}[X_{4}\hat{Z}]-(d-1)(d-2)z$
$\displaystyle=-\frac{(d-1)(d-2)}{d^{2}}$
$\displaystyle\operatorname{Tr}[X_{5}\hat{Z}]$
$\displaystyle=-\frac{2(d-1)(d-2)}{d^{2}}$
$\displaystyle\operatorname{Tr}[X_{6}\hat{Z}]$
$\displaystyle=-\frac{3(d-1)(d-2)}{d^{2}},$ (75)

and we get

$b_{1}=b_{2}=b_{4}=\frac{5}{d^{3}},\quad\quad\quad
b_{3}=b_{5}=b_{6}=-\frac{1}{d^{3}}.$ (76)

Next, we rewrite $\hat{Z}$ as the positive linear combination of projections
to show positivity. Define the projections

$\displaystyle P_{A}$ $\displaystyle=\frac{2}{3}X_{2}-\frac{1}{3}X_{3}$
$\displaystyle P_{B}$
$\displaystyle=\frac{1}{2}\Big{(}1+\frac{1}{\sqrt{2}}\Big{)}X_{4}-\frac{1}{4\sqrt{2}}X_{5}-\frac{1}{4\sqrt{2}}X_{6}.$
(77)

Then we can write $\hat{Z}$ as

$\displaystyle\hat{Z}$
$\displaystyle=\frac{5}{d^{3}}\Bigg{(}X_{1}+X_{2}+X_{4}\Bigg{)}-\frac{1}{d^{3}}\Bigg{(}X_{3}+X_{5}+X_{6}\Bigg{)}$
$\displaystyle=\frac{1}{d^{3}}\left(3P_{1}+4\sqrt{2}P_{B}+5X_{1}+3X_{2}+(3-2\sqrt{2})X_{4}\right),$
(78)

and hence $\hat{Z}\geq 0$ and also $Z\geq 0$. Therefore, $\frac{6}{d^{2}}$ is
indeed the optimal solution.

∎

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