# Park visitation and walkshed demographics in the United States

Kelsey Linnell  Vermont Complex Systems Center, University of Vermont  Mikaela
Fudolig  Vermont Complex Systems Center, University of Vermont  Laura
Bloomfield  Gund Institute for Environment, University of Vermont  Thomas
McAndrew  College of Health, Lehigh University, Bethlehem, PA, United States
Taylor H. Ricketts  Gund Institute for Environment, University of Vermont
Jarlath P. M. O’Neil-Dunne  Gund Institute for Environment, University of
Vermont  Peter Sheridan Dodds  Vermont Complex Systems Center, University of
Vermont  Christopher M. Danforth  Vermont Complex Systems Center, University
of Vermont

###### Abstract

A large and growing body of research demonstrates the value of local parks to
mental and physical well-being. Recently, researchers have begun using passive
digital data sources to investigate equity in usage; exactly who is benefiting
from parks? Early studies suggest that park visitation differs according to
demographic features, and that the demographic composition of a park’s
surrounding neighborhood may be related to the utilization a park receives.
Employing a data set of park visitations generated by observations of roughly
50 million mobile devices in the US in 2019, we assess the ability of the
demographic composition of a park’s walkshed to predict its yearly visitation.
Predictive models are constructed using Support Vector Regression, LASSO,
Elastic Net, and Random Forests. Surprisingly, our results suggest that the
demographic composition of a park’s walkshed demonstrates little to no utility
for predicting visitation.

††preprint: APS/123-QED

##  I  Introduction

The positive impact of park access, proximity, and use on well-being has been
well established. Populations with park access have been found to be more
active, have lower rates of obesity, and overall better cardiovascular health
[1, 2, 3, 4]. As a form of nature exposure, park access also benefits
cognition and mental well-being and is associated with lower stress and
reduced rates of depression [5, 6, 7, 8]. Independent of time spent in the
park, residential proximity is positively correlated with improved mental
health [9].

Inequity in park resources can arise as a function of accessibility, or in
qualitative differences in parks [10, 11, 12, 13, 14, 15, 16]. Studies have
found that even when spatial access is equitable, income and race are linked
to park quality. Parks associated with whiter and more affluent parks have
more acreage, more tree canopy, and different amenities —including more
playgrounds [15, 12, 14, 17, 18, 19].

Differences in park quality are significant because they may drive differences
in visitation, which is the assumed mechanism by which parks offer health
benefits. For example, while studies find correlations between residential
proximity and lower rates of obesity, the underlying mechanism assumed to
create this correlation is often that proximity to a park increases the
likelihood of exercise in the park [20, 21, 22, 23]. Similarly, studies on the
impact of green space on mental health indicate that benefits are either
accessed or increased by visitation to the space [5, 24, 25]. Park-based
physical activity, for example, can mediate the relationship between park
proximity and mental health benefits [26]. Thus, realized usage, or
visitation, is an important variable when studying equity and the heath impact
of parks [27]. Measuring park usage is particularly useful in equity studies
because it can be used to quantify the effect of non-geographic barriers.

Non-geographic barriers to access—such as time constraints and safety —create
inequity in the benefits that communities receive from parks. A much studied
example of such a barrier is the perceived safety of parks [28, 29, 30, 31,
26, 32]. Scholars have found that parks in higher income areas had fewer
safety concerns than parks in low income areas and that perceptions of park
safety are strongly tied to the odds of visiting a park [32, 30]. The impact
of perceived safety as a barrier is visible in a 2020 study by Orstad, where
mental health benefits were only associated with living in proximity to a park
for residents who did not have concerns about park crime [26].

Classifying non-geographic barriers, and measuring their impact on communities
remains an active area of research. Investigations with regards to these
barriers requires measuring realized usage. Historically, park visits have
been quantified using either surveys or by in-person observation of visitors
[33, 4, 9, 34, 35, 30, 31, 20, 21]. These methods are limited both
geographically and temporally, and can be expensive to implement.

Recently, these temporal and geographic limitations have been managed by using
digital data sources such as social media and GPS data from mobile devices
[36, 37, 38, 25, 39]. Digital data sources offer the ability to measure park
usage at all times of the day, for prolonged periods of time, over a large
geographic scale. Using over 2 million observations of smartphone use,
texting, calling, and environmental exposures for 700 young adults, Minor et
al. saw calling and texting actually increase during short recreational
greenspace visits [40]. Longer exposures led to smartphone use decline
however, suggesting that nature exposure may support digital impulse
inhibition. Importantly, participants with elevated baseline screen-time
significantly reduced device use in nature.

By using Twitter and Flickr data, Hamstead et al. [37] were able to observe
differences in visitation to all of the parks in New York City; their findings
suggest visitation varies based on not only characteristics of the park, but
on the demographic composition of the neighborhood surrounding the park. If
park visitation varies with the demographic composition of its neighborhood
for the US in general, it may be possible to predict park visitation using
demographic data from its neighborhood.

Predicting which parks are being underutilized could inform decisions about
which parks need more investment, better infrastructure, new programming, or
could benefit from investigative studies into the barriers preventing full
utilization. To explore whether park visitation can be predicted from
demographic features, we use a novel dataset of daily park visitation counts
obtained through observations of approximately 50 million mobile devices for
2,506 parks throughout the contiguous United States. In particular, we focus
on the population residing within a ten minute walk of the park. This area is
referred to in the literature as a “walkshed” and represents the residential
area for which a park is considered “accessible” [37, 41, 42]

Historically, residents of the walkshed have been considered the primary users
of the park, and walksheds have been conceptualized of as a fixed radius
buffer around the park. Here we establish a park’s walkshed using the convex
hull of a pedestrian walking network within ten minutes walk of a park
boundary. Using this method, the walkshed is limited to areas that have a
walkable route to the park, and parks which cannot be reached by foot are
excluded. Employing census data, we attribute demographic characteristics to
the residents of a park’s walkshed, and evaluate the ability of these
characteristics to predict the visitations received by the park itself.
Echoing the literature, we explore dimensions of race, income, educational
attainment, gender, and age on park visitation in the United States.

##  II  Data

###  II.1 Park Visitation Data

Our data set consists of daily records of the number of unique visitors to
7,997 non-commercial parks in the contiguous United States for the year of
2019. Non-commercial parks include city, municipal, and neighborhood parks.
Because we wish to focus on parks which are frequented by those living near
them, we excluded national and state parks which often attract tourists and
infrequent visitors from a wide geographic area.

The number of daily visitors to each park was estimated using the number of
unique mobile devices that reported GPS data from within the park bounds on
that day to the company UberMedia (since acquired by Near) [43].

UberMedia—a data vendor specializing in GPS data acquired from
cellphones–collects GPS coordinates with timestamps from billions of devices.
GPS coordinates are collected when a person uses one of over 400 location
collecting apps or when they are exposed to advertising through real-time
bidding.

In 2019 the average number of unique devices that UberMedia observed each day
varied between 44 and 60 million, representing approximately 10% of the adult
population in the United States [43].

Figure 1:  A conceptual model of the walkshed. In the left most image the park
is represented by a green square, and the walkshed is represented by the blue
polygon surrounding the green square. The three black bordered shapes labeled
“Tract” demonstrate how a walkshed could intersect multiple Census Tracts. The
intersections of the walkshed with the census tracts are labeled $C_{i}$,
where $C$ indicates the region is a component of the walkshed, and $i$ refers
to the census tract that the particular component lies within. The middle
image demonstrates the assumed uniform spatial distribution of a homogenous
population within the tract, where the people associated with the component
have the same features as the tract, and the population is proportional to the
area of the census tract covered by the component. The final image is of the
walkshed, with each of its components populated with respect to the census
tract in which they lie. The total walkshed population is considered to be the
aggregation of the populations of each component.

###  II.2 Census and Geographic Data

Demographic data were retrieved from the 2019 American Community Survey 5 year
data using the Census API. Census Tracts were chosen as the geographic unit
because they are designed to be relatively homogeneous units with respect to
the demographic features of their residents. We collected the following
demographic data: race, ethnicity, age, median income, and educational
attainment of residents over 25.

Demographic data were linked to park visitation data generated by Ubermedia
based on the intersection between census tracts boundaries and areas within
walking distance of the park (which we term a walkshed). The process of
defining walksheds and linking demographic data to parks is described in
section III.1.

##  III  Methods

Exploring the predictive value of the demographic features of the surrounding
neighborhood for a park’s visitation required joining the data sets mentioned
in II, confirming the suitability of the resulting data for the study, and
evaluating the predictive ability of several models.

Each of these steps are detailed in the subsections below. Section III.1
explains how demographic data from the US Census were joined to park
visitation data using a geographic unit called a walkshed. This section also
discusses how population data were aggregated for the walkshed. The inclusion
criteria chosen for the study is delineated in section III.2. The resulting
data set is described in section III.3. Compared to the US population, the
walksheds included in our data have a large distribution of demographic
features. Finally, section III.4 explains the methodology by which models were
chosen, fit, and evaluated.

Figure 2: Heat map of the population of the contiguous United States overlaid
with the locations of the parks used in the study, with each park demarcated
by a black point, and distributions of park sizes and popularity below.
Histograms display the log normal distributions of parks in the dataset across
walkshed population, park area, and yearly visits. The median park has 2785
people in its walkshed, covers 5.68 acres, and received 2528 visits in 2019.

###  III.1  Walkshed Construction

For each of the parks in our dataset we constructed a walkshed and calculated
an estimated walkshed population. The walkshed was defined to be the convex
hull of the graph of the walking network obtained from OpenStreetMaps that
represented a ten minute walk to the boundary of the park. This walkshed was
then considered to be composed of the disjoint components lying in unique
Census Tracts (see II.2). Census Tracts are treated as homogenous populations
uniformly distributed over a geographic area. Walksheds for each park in our
study were created through a series of steps (Figure 1). The estimated number
of people in a walkshed associated with park $p$, $P^{p}_{\text{walkshed}}$
was computed as

$P^{p}_{walkshed}=\sum_{i=0}^{n}P_{tract_{i}}\frac{A_{i}}{A_{tract}}$ (1)

where $P_{tract_{i}}$ is the estimated number of residents in the $i^{th}$
census tract intersecting with the walkshed, ${A_{i}}$ is the area of that
intersection, and $n$ is the total number of census tracts intersecting with
the walkshed.

In addition to the number of people in the walkshed, we estimated the income,
gender, age, educational attainment, ethnicity, and racial composition of the
walkshed population. The income of a person within the walkshed was calculated
as the population-weighted average of the median incomes of the tracts
intersecting that walkshed:

$I_{walkshed}=\frac{1}{P_{walkshed}}\sum_{i=0}^{n}P_{i}I_{i}$ (2)

where $I_{walkshed}$ is our estimate of the median income of a person in the
walkshed and $I_{i}$ is the median income of the census tract containing
component $i$.

The average age of the residents was calculated similarly. The proportion of
the walkshed belonging to a racial group, ethnicity, age range, sex, or having
reached a given educational attainment was calculated as:

$P_{walkshed,j}=\frac{1}{P_{walkshed}}\sum_{i=0}^{n}P_{i,j}$ (3)

.

Where $i$ refers to the component, and $j$ refers to the classification of
interest, for example $j\in\\{race_{1},race_{2},\cdots race_{J}\\}$.

###  III.2  Inclusion Criteria

A park was included in analysis if (i) the walkshed had an area greater than
zero (i.e. the park could be accessed on foot) and (ii) demographic data were
available from the US census for each census tract intersecting with the
walkshed. Parks were omitted from the study if their walkshed included a
component contained in a tract for which the US Census reported no data.

Because we assumed people in a census tract were distributed uniformly, we
excluded a walkshed if it contained a census tract with less than one person
per quarter-acre.

Parks with walksheds containing fewer than 500, or more than 12,500 people
were also excluded. Parks with high population walksheds were considered large
enough to attract tourists. Parks with fewer than 500 residents were
considered too rural, with potentially inconsistent visitation. These parks
accounted for 154 of the parks in the dataset. Of the initial 7,997 parks,
2,506 met our inclusion criteria. The primary reason for exclusion was
insufficient population density.

###  III.3  Study Set Description

Our data set contained 2506 parks, which we observed to exhibit log-normal
distributions in walkshed population, park area, and yearly visits (Figure
LABEL:fig:park_info). The average park in our data set was 0.023
$\textrm{km}^{2}$ (5.68 acres), contained 2,785 people in its walkshed, and
received 2,528 visits each year.

Figure 3:  Distributions of the parks in the study set across race, ethnicity,
sex, educational attainment, age and measures of wealth. The first row
displays the distribution of parks in the study set across fraction of
walkshed population in four racial categories (Black, Asian, Multiracial, and
White), and fraction identifying as Hispanic. The second row indicates the
distribution of parks in the study set across average age -defined as the
population weighted average of the medians for each component, and the
fraction of walkshed residents falling into each of four child age ranges
(under 5, 6-10, 11-14, and 15-17).The third row displays the distribution of
parks in the study set across fraction of walkshed population over 25 having
earned at least a high school diploma (or equivalent) (blue), at least a
Bachelor’s degree (orange), and with more than a Bachelor’s degree (green), as
well as the fraction that is male (right). The final row displays the
distribution of walkshed populations across the log of the average household
income, and the fraction living below federal poverty level. Dashed vertical
lines in each of the plots indicate the corresponding value for the general
population of the United States in 2019.

The income associated with walksheds was also log-normally distributed. The
average household income associated with a park’s walkshed was $76,155 per
year, which is $7,542 more than the US national median income. Approximately
54% of communities associated with a walkshed had a lower proportion of
individuals living below the Federal Poverty Level than the United States as a
whole.

Figure 3 presents a summary of the distribution of walksheds with regards to
race, ethnicity, educational attainment, and age. 55% and 46% of the parks in
our dataset had a greater proportions of Asians and Hispanics than in the
general population, while only 22% and 37% of parks had greater proportions of
Black and White people than the general population.The average proportion of
females within a walkshed was 51.0% compared to 49.8%, the estimated
proportion of females in the US.

Comparing the educational attainment of walkshed populations to that of the
general population, we find that approximately 60% of the walkshed populations
had more highschool graduates, college graduates, and persons with advanced
degrees than the general population. Overall, walkshed populations tended to
have higher educational attainment than the general population. On average the
parks in our data set had a similar median age, and similar proportions of
children in each age group, as the general population.

###  III.4  Analytic Methodology

In our data set, the area of the park in sqkm and walkshed population are both
positively correlated with yearly visitation on a log-log scale (Pearson
correlation of 0.61 and 0.47 respectively, Spearman correlation of 0.59 and
0.47). Both factors effect the spatial accessibility of the park, but do not
necessarily reflect the users of the park, or the demographic environment of
the park.

To adjust for the area of the park and the walkshed population, we normalize
for the increased visitation that is likely only related to the increased
population to which the park is available. To normalize we fit simple linear
regressions for park area and walkshed population in the log-log space. The
resulting slopes were used as exponents in the following normalization
parameter:

$n_{p}=\frac{1}{A_{p}^{0.58}*P_{w}^{0.84}}$ (4)

where $A_{p}$ is the area of the park in $\mathrm{km}^{2}$, and $P_{w}$ is the
population of the walkshed. The exponents for population and area were
determined using the slope of the line of best fit for each variable when
plotted against yearly visits in log-log space (See Figure 4).

Normalized visitation $n_{p}$ was used as the target value for a set of
predictive models that took as training data demographic features associated
with walksheds. Both input and target values were standardized by subtracting
the mean and dividing by the sample standard deviation, prior to fitting, to
account for different scales of measurement for the variables. The models
tested were: Support Vector Machine Regression with linear, polynomial, and
radial basis function kernels; Random Forest Regression; LASSO Regression; and
Elastic Net Regression. These models were selected for the variety of
functional forms offered, and for the ability of some to consider subsets of
features. Random Forests were included to account for possible interaction
effects between demographic features. Model fitting and analysis were
performed in Python using Scikit-learn[44].

Model hyper parameters were optimized using a grid search of the parameter
space (see Table 1) over which models were evaluated using 5-fold cross
validation. The best performing parameters and the model scores are presented
in Table 1.

Model performances were compared against a null model: a constant function set
to the mean of the normalized visitation. The null model asserts that all
parks receive the same visitation The empirical distribution of normalized
visitation suggests that parks do not receive an equal number of annual
visitors (4) and so we expect the null model to perform poorly.

##  IV  Model Performance

Model | Hyperparameters | Hyperparameter Grid | Tuned Hyperparameters | Mean Absolute Error |
---|---|---|---|---|---
Constant (null) | - | - | - | 0.2810 |
SVR - linear kernel | C | [$10^{-5},10^{-4},10^{-3},10^{-2}$] | $10^{-4}$ | 0.2806 |
SVR - rbf kernel |  | C
---
$\gamma$
| [0.01,0.02,0.03…1] ,
---
[$\frac{1}{n_{features}}$, $\frac{1}{(n_{features}*X.var()}$]
| 0.99
---
$\frac{1}{n_{features}}$
0.2793 |
SVR - polynomial kernel |  | C
---
$\gamma$
degree
| [$10^{-5},10^{-4},10^{-3},10^{-2}$] ,
---
[$\frac{1}{n_{features}}$, $\frac{1}{(n_{features}*X.var()}$]
[1,2,3,4,5]
| $10^{-1}$
---
$\frac{1}{n_{features}}$
3
0.2736 |
LASSO | $\alpha$ | [$10^{-5},10^{-4},10^{-3},10^{-2}$] | $10^{-4}$ | 0.2759 |
Elastic Net |  | $\alpha$
---
l1_ratio
| [$10^{-5},10^{-4},10^{-3},10^{-2}$],
---
[0.00, 1, 0.01]
| $10^{-4}$
---
0.01
0.2758 |
Random Forest |  | $n_{estimators}$
---
$max_{features}$
$max_{depth}$
min samples to split
min samples at leaf
bootstrap
| [200, 400]
---
$n_{features},\sqrt{n_{features}}$
[10,30,50,70,90,110]
[2,5,10]
[1,2,4]
[True, False]
| 200
---
$\sqrt{n_{features}}$
10
2
1
False
0.2664 |
Table 1:  Results of training and testing predictive models for predicting
normalized yearly park visitation using the demographic features associated
with that park’s walkshed. Model types are presented along with the
hyperparameters they use. The hyperparameters were tuned using a 5 fold cross-
validation and a grid search over the parameter space. For each model the grid
used for the grid search is reported followed by the tuned hyperparameters and
the Mean Absolute Error of the tuned model. The notation $n_{features}$ refers
to the number of independent variables, and $X.var()$ refers to the variance
of the feature array. The l1_ratio is the ratio of the weights of the $L1$ and
$L2$ penalties, such that l1_ratio $=1$ is the LASSO penalty. All
hyperparameters refer to those in the scikit-learn library [44].

Modeling the data using a constant function set to the mean of the normalized
visitation yielded a Mean Absolute Error (MAE) of 0.2810. The random forest
model was most successful of the regression models, achieving a MAE of 0.2664,
or an 5.20% improvement over the null model.

Figure 4:  Log-log plots displaying the relationship between yearly visits and
park area and walkshed population, and the distribution of normalized
visitation. In log space park area is positively correlated with yearly
visits, as is population of the walkshed. The slope of the line of best fit is
0.58 for park area, and 0.84 for walkshed population. These slopes were used
to determine the normalized visitation value, given by Equation 4. Visitation
remains log-normally distributed after normalization. See Figure 1 for
distribution of visits prior to normalization.

Support Vector Regression (SVR) with a polynomial kernel was the second most
accurate predictive model. This model resulted in a MAE of 0.2746, which is
2.33% less than the null model. SVR using radial basis function kernels had
slightly worse performance, obtaining a MAE of 0.2793; only a 0.60%
improvement over the null model. When paired with a linear kernel SVR
performed comparably to the null model, with a MAE of 0.2810.

The other two linear models performed equally poorly. LASSO had a minimum MAE
of 0.2759, and Elastic Net managed a very similar minimum MAE of 0.2758,
indicating that they probably converged to very similar linear models. This is
only an 1.81%, and 1.85% improvement upon the null model.

Given that the most successful model, the random forest, only achieved an
improvement of 5.20% over the null model, we can conclude that demographic
information did not significantly contribute to more accurate predictions of
normalized yearly visits.

##  V Discussion

Our approach investigated a variety of functional forms, employed methods for
decreasing noise created by unimportant features (LASSO and Elastic Net), and
incorporated the potential for interaction effects through sequential variable
consideration in the Random Forest model. Additionally, the hyperparameters of
each model were tuned. None of the predictive models performed substantially
better than the null model. The null model provides no information about
individual park usage, and has very weak predictive power. It also assumes
that the demographic features considered in this study are irrelevant to
visitation prediction, as it does not use them at all. Since a comprehensive
set of models was tested, and none performed considerably better than the null
model, we can infer that the demographic features have little predictive value
for average yearly park visitation.

This result is perhaps unexpected in light of Hamstead et al’s 2018 findings
in New York City [37]. In contextualizing this discrepancy, it is important to
note that our inclusion criteria focuses on community parks in suburban or
urban residential areas, while previous work focused on parks in general. This
effort decreased the influence of tourism on visitation by decreasing the
number of parks that would be considered tourist destinations. In addition,
exclusion of these parks naturally excludes potential demographic disparity in
residential proximity to destination parks (i.e. homes near destination parks
may be more expensive) as a confounding effect. Thus it is possible that our
results differ because of the type of parks considered.

This study is limited by the inability to determine the residential proximity
of visitors. We observed a correlation between the number of people in a
park’s walkshed and the number of visits received by a park. This suggests
that the number of people a park “serves” is related to the number of people
who visit it. In order to control for the effect of park size, and
geographical accessibility, a normalization constant was applied to control
for this relationship. In doing so, we made an assumption that a ten minute
walk to the park encompassed the ‘service area’ of every park. It is possible
that depending on demographic features, some populations may travel farther to
parks on average than others, but maintain similar visitation rates. If this
were the case, the normalization constant for parks serving those populations
would be too small, inflating the normalized visitation observed, and
potentially obscuring important trends.

It is worth reiterating that these results are park-centered; results speak to
the usage that a park receives, but do not reflect who uses the park. Our data
does not provide the origin of the mobile device visiting the park there is no
way to determine if visits are made by the people living in the walkshed, or
by persons living farther away. Therefore, whether park usage differs for
different populations is not addressed by this work.

Future work should include more detailed data on the origin of the devices
observed in each park. This data would allow for a more accurately constructed
walkshed. It would also allow further exploration into who visits parks, and
which parks they visit. In addition, consideration should be given to park
quality; this work did not incorporate features of the park itself into the
predictive modeling. Park amenities, condition, and environment are all
important contributors to visitor attraction. Since aspects of park quality
could be confounding factors with walkshed demographic features, a logical
extension of this work is to control for this confounding.

Further future analyses would benefit from consideration of park visitation
relative to season and weather. In the current study yearly visitation is
used, which obscures the difference in “visitation-season” length between
parks; parks in Southern California may be used more days in a year relative a
park in Boston, which creates a different story of population and visitation.

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figurec

figurec

Revisiting the static spherically symmetric solutions of gravity with a
conformally coupled scalar

Sourya Ray
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile,
Valdivia, Chile

Email<EMAIL_ADDRESS>

Abstract

> We revisit the static spherically symmetric solutions of Einstein’s General
> Relativity with a conformally coupled scalar field in arbitrary dimensions.
> Using a four rank tensor introduced earlier we recast the field equations in
> a manifestly symmetric form to elucidate a somewhat less-known feature of
> dual mapping between solutions. We also show that there is a two-parameter
> subfamily of solutions which enjoy a duality symmetry and in four dimensions
> both the BBMB black hole and the Barcelo-Visser wormhole belong to this
> subfamily. Along the way, we rederive the full three-parameter family of
> solutions by direct integration of the field equations and a natural choice
> of ansatz which arguably has several advantages over other previously known
> methods.

##  1 Introduction

The energy-momentum tensor for a massless scalar field conformally coupled to
gravity has long been known to have some interesting properties. It was shown
in [1] to have finite matrix elements in every order of renormalized
perturbation theory. Soon after Bekenstein showed that the conformally coupled
scalar field system admits a black hole with a scalar charge thus providing
one of the first counterexamples of the no-hair theorem [2, 3]. The solution
was also independently obtained by Bocharova, Bronnikov and Melnikov in [4]
and is commonly known as the BBMB black hole. It has the same geometry as that
of an extreme Reissner-Nordström black hole although the scalar field becomes
unbounded on the horizon.

Growing interest in higher dimensional gravity led to the study of the
conformally coupled scalar in higher dimensions. However, it was found that
there are no higher dimensional analogues of the BBMB black hole. In fact a
systematic search for black hole solutions among the general static
spherically symmetric family resulted in the conclusion that the BBMB black
hole is unique [5] (see also [6, 7]). Subsequently a two-parameter subfamily
among the general class in four dimensions was identified to represent
traversable wormholes connecting two asymptotically flat regions [8]. The
existence of the wormhole solution is attributed to the violation of the null
energy condition.

It is often claimed that the field equations of this system are quite
involved. So, a popular method to solve the system, discovered by Bekenstein
himself, is to map the solutions of a minimally coupled scalar to that of a
conformally coupled scalar. This map involves a conformal transformation
followed by a suitable reparametrization of the scalar field. Although for
static spherically symmetric solutions, direct integration of the field
equations have been carried out [10], they are usually, in our viewpoint,
cumbersome and unintuitive.

In this short article, we reformulate the field equations in a manifestly
symmetric form in order to simplify the field equations. This is done by using
a four-rank tensor introduced previously which has the same algebraic
symmetries as the Riemann tensor. We first demonstrate a duality relation
between the solutions in arbitrary dimensions. The fact that the solutions of
this system come in pairs was first observed by Bekenstein in four dimensions.
We then solve the field equations for static spherically symmetric spacetimes
by direct integration. To this end, we use an ansatz and an auxiliary scalar
$\Psi$ which naturally exploits the symmetry of the equations and enables us
to obtain the full three-parameter family of solutions in a straightforward
manner. We then identify the duality relation between the solutions within
this family and further point out a two-parameter subfamily which are self-
dual. It is interesting to note that both the BBMB black hole and the Barcelo-
Visser traversable wormhole (and its higher dimensional counterparts) belong
to this subfamily, with the former as the limiting case of the latter. We also
analyze the singularities of the quadratic curvature invariants for the whole
family. Finally, we offer some further potential applications of our work.

##  2 Conformally coupled scalar and duality of solutions

In four spacetime dimensions, Einstein’s General Relativity supplemented by a
conformally coupled scalar field is usually given by the action

$\displaystyle
I=\int\left[\left(1-\dfrac{1}{6}\kappa\phi^{2}\right)R+\kappa\phi\Box\phi\right]\sqrt{g}\,d^{4}x$
(1)

where $\kappa$ is a coupling constant. More generally, in arbitrary dimensions
$D$, it takes the form

$\displaystyle
I=\int\left[\left(1-\dfrac{(D-2)}{4(D-1)}\kappa\phi^{2}\right)R+\kappa\phi\Box\phi\right]\sqrt{g}\,d^{D}x$
(2)

However, the action and the corresponding field equations can be recast in a
more symmetrical manner by introducing the following four-rank tensor which
has the same algebraic symmetries as the Riemann tensor [11]

$\displaystyle
S_{\alpha\beta}^{\mu\nu}=\phi^{2}R_{\alpha\beta}^{\mu\nu}+\dfrac{4}{s}\phi\delta_{[\alpha}^{[\mu}\phi_{;\beta]}^{\
\nu]}+\dfrac{4(1-s)}{s^{2}}\delta_{[\alpha}^{[\mu}\phi_{;\beta]}\phi_{;}^{\
\nu]}-\dfrac{2}{s^{2}}\delta_{[\alpha}^{[\mu}\delta_{\beta]}^{\nu]}\phi_{;\lambda}\phi_{;}^{\
\lambda}$ (3)

where $\phi_{;\alpha}=\nabla_{\alpha}\phi$ and $\phi_{;\alpha}^{\
\beta}=g^{\beta\mu}\nabla_{\alpha}\nabla_{\mu}\phi$. Let us also note that
this tensor can be obtained from the Riemann tensor by a conformal rescaling
of the metric. More precisely,

$\displaystyle
S_{\alpha\beta}^{\mu\nu}=\phi^{\frac{2(s-1)}{s}}R_{\alpha\beta}^{\mu\nu}[\phi^{-\frac{2}{s}}g_{\lambda\sigma}]$
(4)

Then, under the conformal transformation

$\displaystyle
g_{\mu\nu}\rightarrow\Omega^{2}g_{\mu\nu},\qquad\phi\rightarrow\Omega^{s}\phi$

this tensor transforms as

$\displaystyle
S_{\alpha\beta}^{\mu\nu}\rightarrow\Omega^{2(s-1)}S_{\alpha\beta}^{\mu\nu}.$

The real quantity $s$ characterizes the conformal weight of the scalar field
and can be chosen arbitrarily except $0$. We also define the contractions

$\displaystyle S_{\alpha}^{\beta}=S_{\alpha\mu}^{\beta\mu}\qquad\&\qquad
S=S_{\alpha}^{\alpha}$

which are analogously related to the Ricci tensor and scalar curvature through
a conformal rescaling. In terms of this tensor, the action is compactly
expressed as

$\displaystyle I=\int\left[R-\kappa\phi^{m}S\right]\sqrt{g}\,d^{D}x$

where $m=(2(1-s)-D)/s$. Furthermore, we can set $m=0$ by taking $s=1-D/2$
without any loss of generality. This essentially amounts to a (conformal)
rescaling of the scalar field $\phi$. This particular choice reduces the
action to the more familiar form given earlier.

Varying the resulting action with respect to the metric gives

$\displaystyle
G_{\alpha}^{\beta}=\kappa\left[S_{\alpha}^{\beta}-\dfrac{1}{2}S\delta_{\alpha}^{\beta}\right]$

whereas variation with respect to the scalar field gives the conformally
invariant equation $S=0$, which in turn, along with the trace of the equation
above, leads to $R=0$. Thus, the equations of motion can be summarized as

$\displaystyle\qquad R_{\alpha}^{\beta}=\kappa S_{\alpha}^{\beta}$ (5)
$\displaystyle S=0\quad\Leftrightarrow\quad R=0.$ (6)

Finally, before we confine ourselves to static solutions, let us point out
another of the advantages of our formulation. Suppose we write
$g_{\mu\nu}=\Omega^{2}\tilde{g}_{\mu\nu}$, where $\Omega$ is any real
function. Then, due to the property (4), the field equation (5) becomes

$\displaystyle R_{\alpha}^{\beta}[\Omega^{2}\tilde{g}_{\mu\nu}]=\kappa
S_{\alpha}^{\beta}=\kappa\phi^{\frac{2D}{D-2}}R_{\alpha}^{\beta}[\Omega^{2}\phi^{\frac{4}{D-2}}\tilde{g}_{\mu\nu}]$

We next introduce an auxiliary field $\Psi=\Omega\phi^{2/(D-2)}$ 111For
arbitrary $s$, $\Psi=\Omega\phi^{-1/s}$. The rest of the equations are then
independent of $s$.. After multiplying both sides by $\Omega^{D}$ and setting
$\kappa=1$, we get

$\displaystyle\Omega^{D}R_{\alpha}^{\beta}[\Omega^{2}\tilde{g}_{\mu\nu}]-(\Omega\rightarrow\Psi)=0.$
(7)

where the second term inside parenthesis can simply be obtained from the first
by replacing $\Omega$ with $\Psi$. The equation for the scalar field can be
expressed as

$\displaystyle R[\Psi^{2}\tilde{g}_{\mu\nu}]=0\quad\Leftrightarrow\quad
R[\Omega^{2}\tilde{g}_{\mu\nu}]=0.$ (8)

Note that the equations (7) and (8) are manifestly symmetric under
$\Omega\leftrightarrow\Psi$ interchange. Explicitly, this implies that for
every solution $(g_{\mu\nu},\phi)$ to the system, there is a dual solution
$(\phi^{4/(D-2)}g_{\mu\nu},\phi^{-1})$. This duality in four dimensions was
first observed by Bekenstein (Theorem 2 in [2]) and more recently in [12] in
higher dimensions. Here we clarify the underlying origin of the duality
explicitly without any reference to solutions to the minimally coupled scalar
field system.

There is a class of trivial solutions to the equations (7) and (8) for which
the scalar field $\phi$ takes a constant value $\phi_{0}$ everywhere. In this
case the equations reduce to

$\displaystyle(1-\phi_{0}^{2})\Omega^{D}R_{\alpha}^{\beta}[\Omega^{2}\tilde{g}_{\mu\nu}]=0\quad\text{and}\quad
R[\Omega^{2}\tilde{g}_{\mu\nu}]=0$

If $\phi_{0}\neq\pm 1$, then any Ricci-flat metric solves the field equations.
However, if $\phi_{0}=\pm 1$, then any metric with $R=0$ is a solution. We
shall exclude these trivial solutions from further discussions below.

##  3 Static spherically symmetric solution

We now analyze the static spherically symmetric solutions of the field
equations (5) and (6). These solutions were previously obtained in [9] using
transformations mapping solutions from the minimally coupled scalar field to
those coupled conformally and in [10] via direct integration of the field
equations. In either case, the solutions were obtained in isotropic
coordinates where two unknown metric functions are solved for. Here, we will
show that the reformulation of the conformal coupling in terms of the
$S$-tensor naturally motivates an alternate ansatz which not only elucidates
several aspects of the phase space of the solutions but also allows a simpler
and a more systematic direct integration of the field equations (See also
[12]).

Considering the fact that the $S$-tensor is related to the Riemann tensor by a
conformal rescaling, it is then natural to take one of the unknown functions
in the metric ansatz to be an overall conformal factor. The other unknown
function can be chosen in multiple equivalent ways, here we choose it to be
the $g_{tt}$ component of the conformally related metric i.e., our ansatz is

$\displaystyle
ds^{2}=\Omega(r)^{2}\left[-f(r)^{2}\,dt^{2}+dr^{2}+d\Sigma_{D-2}^{2}\right]$
(9)

where $d\Sigma_{D-2}^{2}=\tilde{g}_{ij}d\tilde{x}^{i}d\tilde{x}^{j}$ is the
line element of a $(D-2)-$dimensional sphere. The Ricci tensor components are
then given by

$\displaystyle
R^{t}_{t}=-\dfrac{\Omega^{-D}f^{-1}}{D-2}\partial_{r}(f^{-(D-3)}\partial_{r}(\Omega
f)^{D-2})$ (10) $\displaystyle
R^{r}_{r}=-\Omega^{-2}\left[\partial_{r}^{2}\ln|f\Omega^{D-1}|+(\partial_{r}\ln|f|)(\partial_{r}\ln|\Omega
f|)\right]$ (11) $\displaystyle
R_{j}^{i}=-\dfrac{\Omega^{-D}}{(D-2)}\left[-(D-3)(D-2)\Omega^{D-2}+f^{-1}\partial_{r}(f\partial_{r}\Omega^{D-2})\right]\delta_{j}^{i}$
(12)

while the scalar curvature is given by

$\displaystyle R$
$\displaystyle=-\Omega^{-2}\left[-(D-3)(D-2)+2\partial_{r}^{2}\ln|f\Omega^{D-1}|+(D-2)(D-1)(\partial_{r}\ln|\Omega|)^{2}\right.$
$\displaystyle\left.\>\>\>+2(D-1)(\partial_{r}\ln|\Omega|)(\partial_{r}\ln|f|)+2(\partial_{r}\ln|f|)^{2}\right]$
(13)

We first solve the following combination of the scalar equations

$\displaystyle\Omega^{2}R-(\Omega\rightarrow\Psi)=\dfrac{2(D-1)}{f(\Omega\Psi)^{\frac{D-2}{2}}}\partial_{r}(f(\Omega\Psi)^{\frac{D-2}{2}}\partial_{r}\ln|\Omega/\Psi|)=0$
(14)

The above equation suggests the following reparametrizations of $\Omega$ and
$\Psi$ in terms of $z(r)$ and $\omega(r)$

$\displaystyle\Omega^{\frac{D-2}{2}}=e^{\left(\frac{D-2}{D-3}\right)z}\cosh\omega\qquad\&\qquad\Psi^{\frac{D-2}{2}}=e^{\left(\frac{D-2}{D-3}\right)z}\sinh\omega$
(15)

which brings the solution to the form

$\displaystyle fe^{\frac{2(D-2)z}{D-3}}\partial_{r}\omega=C_{4}$ (16)

If $C_{4}=0$, then $\omega=$ constant which in turn implies that the scalar
field $\phi=(\Psi/\Omega)^{\frac{D-2}{2}}=\phi_{0}$ is constant everywhere and
the solution belongs to the trivial class discussed earlier. Henceforth, we
shall consider $C_{4}\neq 0$. We then solve the ${}^{t}_{t}$ -component of the
field equations (7) which gives

$\displaystyle
f^{-(D-3)}\partial_{r}(f^{D-2}e^{\frac{2(D-2)z}{D-3}})=2(D-2)C_{4}C_{2}$ (17)

which upon using (16) becomes

$\displaystyle\partial_{\omega}\ln|fe^{\frac{2z}{D-3}}|=2C_{2}$ (18)

and can be integrated to obtain

$\displaystyle fe^{\frac{2(D-2)z}{D-3}}=\pm
e^{2\tilde{z}}=C_{4}\partial_{\omega}r$ (19)

where $\tilde{z}=z+C_{2}\omega+C_{3}$. Next, we solve the ${}^{i}_{j}$
-component of the field equations (7) giving

$\displaystyle
f^{-1}\partial_{r}(f\partial_{r}e^{\frac{2(D-2)z}{D-3}})=(D-3)(D-2)e^{\frac{2(D-2)z}{D-3}}$

which on using (19) reduces to

$\displaystyle\partial_{\omega}^{2}\tilde{z}=\dfrac{(D-3)^{2}}{2C_{4}^{2}}e^{4\tilde{z}}$
(20)

and can be solved using elementary methods noting that the right hand side
does not depend on the variable $\omega$ explicitly. The solution is given by

$\displaystyle
e^{4\tilde{z}}=\dfrac{4C_{4}^{2}C_{1}^{2}}{(D-3)^{2}}\sinh^{2}[(D-3)(r-r_{0})]$
and $\displaystyle\tanh[(D-3)(r-r_{0})/2]=\pm\exp[2C_{1}(\omega-\omega_{0})]$

The remaining field equations are solved provided the integration constants
$C_{1}$ and $C_{2}$ satisfy the relation

$\displaystyle C_{1}^{2}-C_{2}^{2}=\dfrac{(D-1)(D-3)}{(D-2)^{2}}$ (21)

The integration constant $r_{0}$ can be set to zero by a translation of the
coordinate $r$. Similarly, the constant $C_{4}(\neq 0)$ can be chosen
arbitrarily by rescaling the time coordinate $t$. Hence, we obtain a
3-parameter family of solutions characterized by $C_{1}$ (or $C_{2}$), $C_{3}$
and $\omega_{0}$. Plugging everything back and expressing in terms of the
rescaled coordinate $\tilde{r}=(D-3)r/2$ we obtain

$\displaystyle
f(r)^{2}=\left(\operatorname{sech}\tilde{r}\right)^{\frac{4}{D-3}}\left(\tanh\tilde{r}\right)^{\frac{2((D-2)C_{2}-C_{1})}{(D-3)C_{1}}}$
(22)
$\displaystyle\Omega(r)^{D-2}-\Psi(r)^{D-2}=A\left(\cosh\tilde{r}\right)^{2\left(\frac{D-2}{D-3}\right)}\left(\tanh\tilde{r}\right)^{\frac{(D-2)(C_{1}-C_{2})}{(D-3)C_{1}}}$
(23)

while

$\displaystyle\phi(r)=\left(\dfrac{\Psi(r)}{\Omega(r)}\right)^{\frac{D-2}{2}}=\pm\dfrac{(\tanh\tilde{r})^{\frac{1}{C_{1}}}-B}{(\tanh\tilde{r})^{\frac{1}{C_{1}}}+B}$
(24)

where the constants $C_{3}$ and $\omega_{0}$ have been traded for $A(\neq 0)$
and $B(\neq 0,\infty)$ respectively.

Notice that for a fixed $\\{C_{1},C_{2}\\}$ the mapping
$\\{A,B\\}\to\\{-A,-B\\}$ corresponds to a mapping to the dual solution i.e.,
it is equivalent to interchanging $\Omega$ and $\Psi$.

We reiterate the simplicity of the direct integration of the field equations
in this formulation and the natural choice of our ansatz in contrast to the
ones existing in the literature. In the following sections we demonstrate some
further advantages.

##  4 Self-dual solutions

As was explained earlier, since the field equations are invariant under
$\Omega\leftrightarrow\Psi$ interchange, there is a dual mapping between
solutions. In general, the dual solutions correspond to distinct points in the
parameter space and are physically distinct. However, we show below that there
exist special points in the parameter space of the solutions for which the
dual solutions are simply related by a (discontinuous) coordinate
transformation and hence are self-dual.

Among the non-trivial class, the condition for self duality is
$(D-2)C_{2}=C_{1}=\pm 1$. Imposing this condition the solution reduces to

$\displaystyle f(r)^{2}=(\operatorname{sech}\tilde{r})^{\frac{4}{D-3}};$
$\displaystyle\Omega(r)^{D-2}-\Psi(r)^{D-2}=A\left(\tanh\tilde{r}\right)(\cosh\tilde{r})^{2\left(\frac{D-2}{D-3}\right)}$
$\displaystyle\phi(r)=\left(\dfrac{\Psi(r)}{\Omega(r)}\right)^{\frac{D-2}{2}}=\pm\dfrac{b-e^{-2\tilde{r}}}{1-be^{-2\tilde{r}}}=\pm\left(\dfrac{b-e^{2\tilde{r}}}{1-be^{2\tilde{r}}}\right)^{-1}$

where $B$ in the previous section has been replaced by $(1-b)/(1+b)$, where
$b$ now takes on any real values except $\pm 1$. It is evident that for the
above two-parameter family, the solutions related by
$\Omega\leftrightarrow\Psi$ interchange simply corresponds to the
transformation $r\leftrightarrow-r$.

It is interesting to note that the two well-known physically relevant
spacetimes belong to this subclass of solutions. In $D=4$, this solution was
shown in [8] to represent a traversable wormhole for negative values of $b$.
It can be easily expressed in the isotropic coordinates used in [8] by making
the transformation $\bar{r}=(\eta/2)e^{r}$ and $b=\tan{\Delta/2}$. While for
$b=0$ in $D\geq 4$ the above family of asymptotically flat self-dual solutions
were termed Bekenstein spacetimes in [10], which contains the BBMB black hole
in $D=4$.

##  5 Singularities, horizons and throats

In this section we examine some physical properties of the solutions. We begin
by performing a simple analysis of the curvature singularities. Since the
scalar curvature vanishes, there are only two independent squared curvature
invariants. We shall first check for possible singularities of these
invariants at $r=0$. Due to spherical symmetry and our particular choice of
the coordinates it is straightforward to calculate the Weyl invariant
$C_{\alpha\beta}^{\rho\sigma}C^{\alpha\beta}_{\rho\sigma}$. In fact, due to
spherical symmetry every non-zero component of the conformal (Weyl) tensor are
proportional to each other while the conformal factor only appears as a
multiplicative prefactor. Explicitly,

$\displaystyle
C_{\alpha\beta}^{\rho\sigma}C^{\alpha\beta}_{\rho\sigma}=\dfrac{4(D-3)}{(D-1)}\Omega(r)^{-4}\left(\dfrac{f^{\prime\prime}(r)}{f(r)}-1\right)^{2}$

We first evaluate the term inside the parenthesis. A little algebra shows

$\displaystyle\dfrac{f^{\prime\prime}(r)}{f(r)}-1=\dfrac{(D-2)}{4C_{1}^{2}}\left[\left((D-2)C_{2}-C_{1}\right)\left(C_{2}-C_{1}\right)\coth^{2}\tilde{r}\right.$
$\displaystyle\qquad\qquad\qquad\left.+\left((D-2)C_{2}+C_{1}\right)\left(C_{2}+C_{1}\right)\tanh^{2}\tilde{r}-2((D-2)C_{2}^{2}+C_{1}^{2})\right]$

So unless the coefficient of the hyperbolic cotangent (squared) on the right
hand side vanishes, as $r\to 0$

$\displaystyle\left(\dfrac{f^{\prime\prime}(r)}{f(r)}-1\right)^{2}\sim\coth^{4}\tilde{r}$
(25)

Next, we examine the behavior of the conformal factor $\Omega$ near $r=0$.
Using the expressions in (22) and (24) we find that as $r\to 0$,

$\displaystyle\Omega(r)^{-4}\sim\tanh^{\alpha}\tilde{r}$ (26)

where

$\displaystyle\alpha=4\left(\dfrac{1}{(D-2)|C_{1}|}-\dfrac{1-C_{2}/C_{1}}{D-3}\right)<\dfrac{4}{(D-2)|C_{1}|}\leqslant\dfrac{4}{\sqrt{(D-3)(D-1)}}$

where we have used the constraint (21). Hence, $\alpha<4$ for $D\geqslant 4$.
Combining (25) and (26) we see that the conformal invariant is singular at
$r=0$ for any dimensions $D\geqslant 4$ unless $(D-2)C_{2}=C_{1}$ which is
precisely the condition for self-duality of the solutions. Therefore, all the
non-self-dual solutions have a naked singularity at $r=0$.

We now look at the singularities of the two-parameter family of self-dual
solutions, in which case

$\displaystyle
C_{\alpha\beta}^{\rho\sigma}C^{\alpha\beta}_{\rho\sigma}=(D-3)(D-1)\left(\dfrac{4B}{A}\right)^{4/(D-2)}(\tanh\tilde{r}+B)^{\frac{-8}{(D-2)}}\left(\operatorname{sech}\tilde{r}\right)^{\frac{4(D-1)}{D-3}}$

For $|B|\geqslant 1$ the above invariant is finite for all values of $r$ and
in particular it vanishes as $r\to\pm\infty$, while for $|B|<1$ it is singular
at $\tilde{r}=\operatorname{arctanh}(-B)$ since the conformal factor
$\Omega(r)$ blows up there.

Next we look for singularities in the second independent quadratic invariant
$R_{\alpha\beta}R^{\alpha\beta}$ for the self-dual family. The non-zero
components of the Ricci tensor for this family are

$\displaystyle R^{t}_{t}$
$\displaystyle=-\dfrac{D-3}{2}\left(\dfrac{D-3}{D-2}\right)\dfrac{A}{4B}\Omega(r)^{-D}(\operatorname{sech}{\tilde{r}})^{\frac{2(D-4)}{D-3}}$
$\displaystyle R^{r}_{r}$
$\displaystyle=-\dfrac{1}{2}\left(\dfrac{D-3}{D-2}\right)\dfrac{A}{4B}\Omega(r)^{-D}(\operatorname{sech}{\tilde{r}})^{\frac{2(D-4)}{D-3}}\left[1-(D-2)^{2}(1-B^{2})\cosh^{2}\tilde{r}\right]$
$\displaystyle R^{i}_{j}$
$\displaystyle=\delta^{i}_{j}\dfrac{1}{2}\left(\dfrac{D-3}{D-2}\right)\dfrac{A}{4B}\Omega(r)^{-D}(\operatorname{sech}{\tilde{r}})^{\frac{2(D-4)}{D-3}}\left[1-(D-2)(1-B^{2})\cosh^{2}\tilde{r}\right]$

where

$\displaystyle\Omega(r)^{D-2}=\dfrac{A}{4B}(\tanh\tilde{r}+B)^{2}\left(\cosh\tilde{r}\right)^{\frac{2(D-2)}{D-3}}$
(27)

First note that for $|B|<1$, exactly as the conformal tensor, each component
of the Ricci tensor above is singular at
$\tilde{r}=\operatorname{arctanh}(-B)$ due to the singularity of the conformal
factor $\Omega(r)$. Next, for $|B|=1$ all the components are proportional to
the function

$\displaystyle\Omega(r)^{-D}(\operatorname{sech}{\tilde{r}})^{\frac{2(D-4)}{D-3}}=\exp{\left(\mp\frac{2D\tilde{r}}{D-2}\right)}(\operatorname{sech}\tilde{r})^{\frac{4(D-2)}{D-3}-\frac{2D}{D-2}}$

Since, as $\tilde{r}\to\pm\infty$, $\operatorname{sech}\tilde{r}\to
2e^{-|\tilde{r}|}$, the expression above is non-singular at $r=\pm\infty$ only
if

$\displaystyle\dfrac{4(D-2)}{D-3}-\dfrac{4D}{D-2}=-\dfrac{4(D-4)}{(D-2)(D-3)}\geqslant
0$

which holds only for $D=4$. Hence the invariant
$R_{\alpha\beta}R^{\alpha\beta}$ is singular at $r=\infty$ or $-\infty$ except
for $D=4$ in which case it reduces to $\Omega(r)^{-8}$ up to a proportionality
constant while $\Omega(r)$ is finite at either of the two locations and
becomes infinite at the other depending on the sign of $B$. The point where
$\Omega(r)$ is finite represents a horizon since $f(r)$ vanishes there and the
metric can be analytically extended beyond this point by a coordinate
transformation while the other represents an asymptotic region. This
exceptional four dimensional spacetime represents the well-known BBMB black
hole and has the same geometry as the extreme Reissner-Nordström black hole.
Finally, for $|B|>1$ each component is regular for the whole range
$-\infty<r<\infty$ and in particular vanishes at both the boundaries and hence
so is the curvature invariant $R_{\alpha\beta}R^{\alpha\beta}$. It is also not
too difficult to show from (27) that in this case $\Omega(r)\to\infty$ as
$r\to\pm\infty$ and has a minimum somewhere in between. In order to see the
latter, we equate the derivative $\Omega^{\prime}(r)$ to $0$ and obtain the
following quadratic equation in $\tanh\tilde{r}$

$\displaystyle
P(\tanh\tilde{r})\coloneqq\tanh^{2}\tilde{r}+(D-2)B\tanh\tilde{r}+(D-3)=0$

which for $|B|>1$ has two real roots $\xi_{1},\xi_{2}$. Since
$\xi_{1}\xi_{2}=(D-3)>0$ while $\xi_{1}+\xi_{2}=-(D-2)B$, the two roots have
the same sign and opposite to that of $B$. Suppose $|\xi_{1}|<|\xi_{2}|$.
Moreover, $P(\pm 1)=(D-2)(1\pm B)$. Hence,

$\displaystyle\text{If}\>B>1,\>\text{then}\>P(-1)<0\implies\xi_{2}<-1<\xi_{1}<0$
$\displaystyle\text{If}\>B<-1,\>\text{then}\>P(1)<0\implies
0<\xi_{1}<1<\xi_{2}$

i.e., $\xi_{1}$ lies between $0$ and $\pm 1$ depending on whether $B$ is
positive or negative. Therefore, $\Omega(r)$ has exactly one minima at
$\tanh\tilde{r}_{m}=\xi_{1}$ where

$\displaystyle\xi_{1}=-\dfrac{(D-2)B}{2}\left(1-\sqrt{1-\dfrac{4(D-3)}{(D-2)^{2}B^{2}}}\right)$

The spacetime represents the well-known asymptotically flat ‘traversable’
wormhole [8] obtained by Barcelo and Visser and its higher dimensional
generalization with $\tilde{r}=\tilde{r}_{m}$ as the wormhole throat. The
scalar field $\phi$ approaches mutually reciprocal values at the two
asymptotic regions.

##  6 Conclusions

Recasting the field equations for a scalar field conformally coupled to
gravity in a manifestly symmetrical form, we have shed light on a duality
feature among the solutions. Next, focusing on the static spherically
symmetric solutions we have identified a two-parameter subfamily which are
self-dual and are related by a point reflection transformation. In four
spacetime dimensions this subfamily is shown to include both the BBMB black
hole and the Barcelo-Visser ‘traversable’ wormhole. In the process, we have
provided a direct integration method to obtain the full three-parameter family
of solutions in any dimension $D\geqslant 4$ in a simpler way than existing in
the literature.

The duality among solutions is generally broken in presence of other matter
fields or even in presence of a cosmological constant [13]. It is also broken
if the gravity theory is arbitrarily modified. However, if both the gravity
part and the matter coupling part are modified in the ‘same way’ then the
duality symmetry will continue to exist. Note that the tensor
$S_{\alpha\beta}^{\mu\nu}$ used here was originally introduced in [11] in
order to construct higher curvature analogues of conformal coupling and
interpreted as a scalar field conformally coupled to Lovelock gravity which is
the higher curvature generalization of General Relativity. So, if a Lovelock
theory of a fixed order (also known as ‘pure’ Lovelock theory) is appended by
the corresponding higher curvature conformal coupling, then the corresponding
equations of motion will exhibit the duality symmetry. It would be interesting
to see if the symmetry can be similarly exploited in such a theory to
integrate the field equations and obtain exact solutions.

Although, here we have confined ourselves to static spherically symmetric
spacetimes, as explained in section 2, the duality among solutions is valid
for any solution and could be useful to obtain other exact solutions. In
particular, to the best of our knowledge, there are no known (non-trivial)
time-dependent solutions within the spherically symmetric family. On the other
hand, there is no known Birkhoff-like theorem which prohibits such time-
dependent spherically symmetric solutions. It is worth studying if such
solutions do exist and our method could be useful in such an investigation.
Work along these lines is in progress.

Acknowledgements: The author gratefully acknowledges many useful discussions
with Marcello Ortaggio.

## References

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# Condition on n-Qubit State For Getting Perfect Quantum Teleportation111This
paper is dedicated in the memory of Late Prof. R. Prakash and Late Prof. H.
Prakash

Shamiya<EMAIL_ADDRESS>Phool Singh
<EMAIL_ADDRESS>Ranjana Prakash and Hari Prakash
Physics Department, University of Allahabad, Prayagraj, India

###### Abstract

It is shown that standard quantum teleportation (SQT) with multi-qubit
resource result in fidelity $(2+C)/3$ where $C$ is concurrence of the resource
in bipartite entanglement between qubit going to receiver and rest of the
qubits. For perfect SQT, obviously, $C=1$. For a general 3-qubit resource, we
find conditions for getting perfect SQT for state expressed in any basis
states. Zha et al [Mod. Phys. Lett. B 22, 2523-2528 (2008)], who studied
perfect SQT using 3-qubit resource, reported conditions for perfect SQT for
only those resource states which are given in the 3-qubit canonical form of
Acin representation. We show that there is an alternative easily derivable
representation which gives more generalized results. To illustrate the
difference between the two schemes, we build an example of 3-qubit entangled
state, giving perfect SQT and not included in Zha et al results.
Keywords: Standard quantum teleportation, Multi-qubit resource, Bipartite
entanglement.

##  1 Introduction

Quantum entanglement stands as the strange phenomenon of quantum mechanics
which was first predicted by Einstein in his famous paper [1]. Since then it
has been studied by several authors [2, 3]. It has many applications in
quantum information theory such as quantum teleportation [4], quantum
cryptography [5], superdense coding [6], quantum computation [7], quantum
secure direct communication [8, 9, 10], quantum machine learning [11] and so
on. Quantum teleportation (QT) is one of the most important applications and
was first proposed by Bennet et al. [4]. In standard quantum teleportation
(SQT), the information of a quantum two level system is transmitted from a
sender (Alice) to a far apart receiver (Bob) using a 2-qubit maximally
entangled (ME) state and a 2-bit classical communication channel. Afterwards,
many researchers made significant advances in theory [12, 13, 14, 15, 16, 17]
and experiment [18, 19] with QT. Many recent progress in the direction of QT
has also been made such as, long distance QT [16], high dimensional QT [19],
logic qubit QT [20, 21] and bidirectional QT [22, 23, 24, 25]. Due to the
tight connection between quantum entanglement and QT, numerous studies have
been conducted on the use of multi-particle entangled quantum states other
than two-particle entangled states for QT. In particular, 3-qubit entangled
GHZ state[12] and 3-qubit entangled W state[13, 14] have been explored. The QT
involving ME states (concurrence [26], $C=1$) achieves with unit fidelity and
unit success probability, which is termed as perfect QT. The perfect QT of a
single qubit via 3-qubit entangled GHZ state [12] and 3-qubit entangled GHZ
like state [15] is well studied. However, using a 3-qubit W state [13, 14]
perfect teleportation with non-unit success probability is obtained, which is
called probabilistic QT. Since in real scenario the ME state interacts with
the environment resulting in the decoherence[27, 28] of the ME state and
changes it to a non-maximally entangled (NME) state. Therefore, several
authors [29, 30, 31, 32] have also investigated the probabilistic QT of a
single qubit using various NME states.

Agrawal and Pati [33] proposed that a class of 3-qubit W-state resource leads
to perfect SQT. Further, Li et al. [34] generalized the results of Agrawal and
Pati [33] to higher dimensional quantum systems. Recently, much attentions
have been taken to this topic [35, 36, 37, 38, 39, 40] and various schemes
[41, 42] have been proposed for generating such states giving perfect QT. Zuo
et al. [35] gave a criterion on W-states being suitable for perfect
teleportation and superdense coding. A multi-qubit W-type state has also been
suggested by Singh et al. [36]to accomplish this task. Zha et al. [37] gives
two forms of 3-qubit entangled states and reported that perfect SQT can be
obtained using only those general 3-qubit states which can be expressed in one
of their two forms. For an n-qudit entangled resource, Li et al. [38]
suggested that in order to provide perfect SQT the state must satisfy the
total $\frac{1}{2}(d+2)(d-1)$ constraints. In all these schemes various
entangled resource states have been suggested for perfectly teleporting an
arbitrary single qubit state. However, no one gives a general form of n-qubit
entangled resource that will leads to perfect SQT. It should be interesting to
find conditions on n-qubit entangled state, so that one can use it as a
resource to teleport a single qubit state perfectly.

We study SQT of a single qubit using a general n-qubit resource state
expressed in any arbitrary basis. We provide a simple scheme to find maximal
average fidelity (MAF) in this general case and conditions for perfect SQT.
This result has been used to obtain conditions for perfect SQT using a general
3-qubit resource given in any basis. Zha et al [37] have also given results
for perfect SQT with 3-qubit states. However, their results are usable only if
the 3-qubit resource state is expressed in one particular Acin representation
[43, 44].

Figure 1: Schematic diagram of the proposed scheme of perfect SQT via n-qubit
entangled resource.

##  2 Perfect SQT scheme using a multi-qubit entangled resource

The sketch diagram of the SQT scheme using a general n-qubit resource is shown
in Fig. 1. We consider an arbitrary n-qubit non-maximally entangled resource,
$|E\rangle_{2,3,.....,n,n+1}$, with n-1 qubits (2,3,4,…..n) go to Alice and
one qubit (n+1) goes to Bob. First Alice makes an n-qubit projective
measurement or von Neumann measurement (VNM) in the orthogonal basis
$\\{|\Psi^{(r)}\rangle\\}$ on information qubit and her shared entangled
qubits. Then she sends her measurement result $r=0,1,2,3$ via 2 bit classical
channel to Bob, who makes a suitable unitary transformation (UT), $u^{(r)}$ on
his qubit to recover the information state. Let Alice has an unknown
information state,

$|{I}\rangle_{1}=a|{0}\rangle_{1}+b|{1}\rangle_{1}$ (1)

with the normalization $|a|^{2}+|b|^{2}=1$, required to be teleported to a far
distant receiver Bob. Also, consider n-qubit entangled resource of which n-1
qubits (numbered as 2,3,4….n) go to Alice and one qubit (numbered as n+1) goes
to Bob. We separate Bob’s qubit and write resource state as

$|{E}\rangle_{2,3,....n+1}=A|{\psi_{0}}\rangle_{2,3,....n}|{0}\rangle_{n+1}+B|{\psi_{1}}\rangle_{2,3,....n}|{1}\rangle_{n+1}$
(2)

where $|{\psi_{0}}\rangle$ and $|{\psi_{1}}\rangle$ are two states of n-1
qubits going to Alice, which may or may not be orthogonal or entangled. For
qubit n+1, let us use new basis states
$|{\bar{0}}\rangle=U|{0}\rangle\equiv(1+|z|^{2})^{-1/2}(|{0}\rangle+z|{1}\rangle)$
and
$|{\bar{1}}\rangle=U|{1}\rangle\equiv(1+|z|^{2})^{-1/2}(|{1}\rangle-z^{*}|{0}\rangle)$
respectively, where $z$ is a complex number, and write the Eq.(2) as,

$\displaystyle|{E}\rangle_{2,3,....n+1}$ $\displaystyle=$
$\displaystyle\frac{1}{1+|z|^{2}}[A|{\psi_{0}}\rangle_{2,3,....n}\\{|{\bar{0}}\rangle_{n+1}-z|{\bar{1}}\rangle_{n+1}\\}+B|{\psi_{1}}\rangle_{2,3,....n}\\{z^{*}|{\bar{0}}\rangle_{n+1}+|{\bar{1}}\rangle_{n+1}\\}],$
(3) $\displaystyle=$
$\displaystyle\frac{1}{1+|z|^{2}}[A|{\psi_{0}}\rangle_{2,3,....n}+Bz^{*}|{\psi_{1}}\rangle_{2,3,....n}]|{\bar{0}}\rangle$
$\displaystyle+\frac{1}{1+|z|^{2}}[-Az|{\psi_{0}}\rangle_{2,3,....n}+B|{\psi_{1}}\rangle_{2,3,....n}]|{\bar{1}}\rangle.$

This enables us to write the entangled resource (2) in the form

$|{E}\rangle_{2,3,....n+1}=\bar{A}|{\bar{\psi_{0}}}\rangle_{2,3,....n}|{\bar{0}}\rangle_{n+1}+\bar{B}|{\bar{\psi_{1}}}\rangle_{2,3,....n}|{\bar{1}}\rangle_{n+1}$
(4)

where $|{\bar{\psi_{0}}}\rangle$ and $|{\bar{\psi_{1}}}\rangle$ are orthogonal
and $\bar{A}$ and $\bar{B}$ are real and positive, $z$ is then a solution of

$Kz^{2}+(A^{2}-B^{2})z-K^{*}=0,$ (5)

with

$K=\langle{\psi_{1}}|{\psi_{0}}\rangle,$ (6)
$\bar{A}=(1+|z|^{2})^{-1/2}[A^{2}+B^{2}|z|^{2}+AB(Kz+K^{*}z^{*})]^{-1/2},$ (7)
$\bar{B}=(1+|z|^{2})^{-1/2}[B^{2}+A^{2}|z|^{2}-AB(Kz+K^{*}z^{*})]^{-1/2}.$ (8)

This state given by Eq.(4) has bipartite entanglement between (n-1) qubits of
Alice and 1 qubit of Bob with concurrence $C=2\bar{A}\bar{B}$. It should be
noted that in the calculation of bipartite concurrence between sender and
receiver we considered all the (n-1) qubits of Alice as a single qubit.
If Alice makes projective measurements on qubits 1,2,….n on four mutually
orthogonal states

$\displaystyle|{\Psi^{(0,1)}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[|{0}\rangle_{1}|{\bar{\psi_{0}}}\rangle_{2,....n}\pm|{1}\rangle_{1}|{\bar{\psi_{1}}}\rangle_{2,....n}],$
$\displaystyle|{\Psi^{(2,3)}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[|{0}\rangle_{1}|{\bar{\psi_{1}}}\rangle_{2,....n}\pm|{1}\rangle_{1}|{\bar{\psi_{0}}}\rangle_{2,....n}]$
(9)

the probabilities $P^{(r)}$ for detection of states $|{\Psi^{(r)}}\rangle$,
where $r=0,1,2,3$ are given by

$\displaystyle P^{(0)}$ $\displaystyle=$ $\displaystyle
P^{(1)}=\frac{1}{2}[|a\bar{A}|^{2}+|b\bar{B}|^{2}],$ $\displaystyle P^{(2)}$
$\displaystyle=$ $\displaystyle
P^{(3)}=\frac{1}{2}[|b\bar{A}|^{2}+|a\bar{B}|^{2}]$ (10)

respectively. The corresponding Bob’s states are

$\displaystyle|Bob^{(0)}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2P^{(0)}}}[a\bar{A}|{\bar{0}}\rangle+b\bar{B}|{\bar{1}}\rangle],$
$\displaystyle|Bob^{(1)}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2P^{(1)}}}[a\bar{A}|{\bar{0}}\rangle-b\bar{B}|{\bar{1}}\rangle],$
$\displaystyle|Bob^{(2)}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2P^{(2)}}}[b\bar{A}|{\bar{0}}\rangle+a\bar{B}|{\bar{1}}\rangle],$
$\displaystyle|Bob^{(3)}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2P^{(3)}}}[b\bar{A}|{\bar{0}}\rangle-a\bar{B}|{\bar{1}}\rangle]$
(11)

respectively. If Bob then makes unitary transformations $U^{\dagger}$,
$\sigma_{z}U^{\dagger}$, $\sigma_{x}U^{\dagger}$,
$\sigma_{x}\sigma_{z}U^{\dagger}$ respectively, SQT with fidelities

$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle
F^{(1)}=\frac{1}{2P^{(0)}}||a|^{2}\bar{A}+|b|^{2}\bar{B}|^{2},$ $\displaystyle
F^{(2)}$ $\displaystyle=$ $\displaystyle
F^{(3)}=\frac{1}{2P^{(2)}}||b|^{2}\bar{A}+|a|^{2}\bar{B}|^{2}$ (12)

respectively are obtained. On using average values
$\langle|\bar{a}|^{4}\rangle=\langle|\bar{b}|^{4}\rangle=1/3$ and
$\langle|ab|^{2}\rangle=1/6$ for randomly changing values of $a$ and $b$, we
obtain maximal average fidelity (MAF) as,

$MAF=<\sum_{r}P^{(r)}F^{(r)}>=\frac{(2+C)}{3}$ (13)

where $C=2\bar{A}\bar{B}$, concurrence of the resource in bipartite
entanglement between qubit going to receiver and rest of the qubits. For
perfect SQT, the right hand side of Eq.(13) should be $1$, which means $C$
should be $1$. To fulfill this, we demand $\bar{A}=\bar{B}=1/\sqrt{2}$, which
gives $A=B=1/\sqrt{2}$ and $\langle{\psi_{1}}|{\psi_{0}}\rangle=0$ also. This
simply implies that to achieve perfect SQT, the resource state should have
perfect bipartite entanglement between qubit going to Bob and rest of the
qubits.

##  3 Perfect SQT scheme using a 3-qubit entangled resource

The most general 3-qubit state in any basis can be written as

$|{\Psi}\rangle_{123}=A|{000}\rangle+B|{010}\rangle+C|{100}\rangle+D|{110}\rangle+E|{001}\rangle+F|{011}\rangle+G|{101}\rangle+H|{111}\rangle$
(14)

with the normalization condition
$|A|^{2}+|B|^{2}+|C|^{2}+|D|^{2}+|E|^{2}+|F|^{2}+|G|^{2}+|H|^{2}=1$.
This is written in the form of Eq.(2), equality of two constants and the
condition $\langle{\psi_{1}}|{\psi_{0}}\rangle=0$ gives

$|A|^{2}+|B|^{2}+|C|^{2}+|D|^{2}-|E|^{2}-|F|^{2}-|G|^{2}-|H|^{2}=0,\;A^{*}E+B^{*}F+C^{*}G+D^{*}H=0$
(15)

as the conditions for getting perfect SQT with qubit 3 going to Bob, in any
measurement basis.
In order to write a 3-qubit state satisfying both the conditions of Eq.(15),
we separate Bob’s qubit and write a general 3-qubit resource state as,

$|{\Psi}\rangle_{123}=A_{0}|{\psi_{0}}\rangle_{12}|{0}\rangle_{3}+A_{1}|{\psi_{1}}\rangle_{12}|{1}\rangle_{3}.$
(16)

Here $|{\psi_{1}}\rangle$ may be separable or entangled. If it is separable,
one can cast $|{\psi_{1}}\rangle$ in the form $|11\rangle_{12}$ with proper
choice of states $|1\rangle_{1}$ and $|1\rangle_{2}$. Orthogonal state
$|{\psi_{0}}\rangle_{12}$ then involves states $|00\rangle_{12}$,
$|01\rangle_{12}$, $|10\rangle_{12}$ and $|\Psi\rangle_{123}$ can be written
in the general form by choosing suitable phases of states as,

$\displaystyle|\Psi\rangle_{123}=[a|{00}\rangle+b|{01}\rangle+\sqrt{\frac{1}{2}-a^{2}-b^{2}}|{10}\rangle]_{12}|0\rangle_{3}+\frac{1}{\sqrt{2}}|{111}\rangle_{123}.$
(17)

If $|{\psi_{1}}\rangle$ is not separable it can be expressed in 2-qubit
Schmidt representation [45] in the form

$\displaystyle|{\psi_{1}}\rangle_{12}=a|00\rangle+\sqrt{1-a^{2}}|11\rangle$
(18)

with properly chosen states $|0\rangle_{i}$ and $|1\rangle_{i}$, ($i=1,2$) and
real coefficient $a$. The most general form of orthogonal
$|{\psi_{0}}\rangle_{12}$ can then be written as

$\displaystyle|{\psi_{0}}\rangle_{12}=\kappa(\sqrt{1-a^{2}}|00\rangle-a|11\rangle)+be^{i\beta}|01\rangle+\sqrt{1-\kappa^{2}-b^{2}}|10\rangle.$
(19)

Here $a,b,\beta,\kappa$ are real. Then $|\Psi\rangle_{123}$ can be written in
the form

$\displaystyle|{\Psi}\rangle_{123}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[\\{\kappa(\sqrt{1-a^{2}}|00\rangle_{12}-a|11\rangle_{12})+be^{i\beta}|01\rangle_{12}$
(20)
$\displaystyle+\sqrt{1-\kappa^{2}-b^{2}}|10\rangle_{12}\\}|0\rangle_{3}+\\{a|00\rangle_{12}+\sqrt{1-a^{2}}|11\rangle_{12}\\}|1\rangle_{3}].$

This is more generalized form of the result suggested by Zha et al [37], for 3
qubit states available in the Acin representation [43],

$|{\Psi}\rangle_{123}=\kappa_{0}e^{i\theta}|{000}\rangle+\kappa_{1}|{001}\rangle+\kappa_{2}|{010}\rangle+\kappa_{3}|{100}\rangle+\kappa_{4}|{111}\rangle$
(21)

like the simplifying Schmidt representation [45] for 2 qubit states given in
terms of only one real and positive parameter and two terms, the Acin et al
representation expresses 3-qubit state in terms of only five terms which
involves only four independent real and positive parameters (four of real and
positive parameters $\kappa_{0}$, $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$,
$\kappa_{4}$) and an angle $\theta$. Zha et al reported that perfect SQT will
result if $|{\psi}\rangle_{123}$ is in one of the following forms:

$|{\psi}\rangle_{123}=\kappa_{0}e^{i\theta}|{000}\rangle+\kappa_{2}|{010}\rangle+\sqrt{\frac{1}{2}-\kappa^{2}_{0}-\kappa^{2}_{2}}|{100}\rangle+\frac{1}{\sqrt{2}}|{111}\rangle,$
(22)
$|{\psi}\rangle_{123}=\kappa_{1}|{001}\rangle+\kappa_{2}|{010}\rangle+\sqrt{\frac{1}{2}-\kappa^{2}_{2}}|{100}\rangle+\sqrt{\frac{1}{2}-\kappa^{2}_{1}}|{111}\rangle.$
(23)

This result is obvious, if one writes Eq.(21) in the form of Eq.(2) and then
applies orthogonality condition $\langle{\psi_{1}}|{\psi_{0}}\rangle=0$, which
gives $\kappa_{0}\kappa_{1}=0$, resulting in $|{\psi}\rangle_{123}$ in the
form of Eq.(22) or Eq.(23), according as $\kappa_{0}=0$ is taken or
$\kappa_{1}=0$ is taken.

Here, this must also be noted that one could use the alternative Acin
representation [44] of 3 qubit states, which also involves five real and
positive parameters (four of $a,b,c,d,f$ and an angle $\theta$) and only five
terms, viz,

$|{\psi}\rangle_{123}=a|{000}\rangle+b|{100}\rangle+c|{101}\rangle+d|{110}\rangle+fe^{i\theta}|{111}\rangle$
(24)

For this the orthogonality condition $\langle{\psi_{1}}|{\psi_{0}}\rangle=0$
gives $df=0$ as the condition for resource $|{\psi}\rangle_{123}$ to result in
perfect SQT. Hence the two permitted forms of $|{\psi}\rangle_{123}$ parallel
to Eqs. (22) and (23) should contain the first three terms of right hand side
of Eq. (24) and any one of the last two. Using our scheme it can easily be
shown that there exist 3-qubit entangled states which can lead to perfect SQT
and not included in earlier results of Zha et al.
For example, let us consider a 3-qubit entangled state,

$|{E}\rangle_{123}=\frac{1}{\sqrt{2}}e^{i\theta}|000\rangle_{123}+a|011\rangle_{123}+be^{i\delta}|101\rangle_{123}+\sqrt{\frac{1}{2}-a^{2}-b^{2}}e^{i\gamma}|111\rangle_{123}$
(25)

where particles 1, 2 belongs to Alice and particle 3 goes to Bob. It can also
be written in the form of Eq.(2) as,

$|{E}\rangle_{123}=\frac{1}{\sqrt{2}}e^{i\theta}|00\rangle_{12}|0\rangle_{3}+[a|01\rangle+be^{i\delta}|10\rangle+\sqrt{\frac{1}{2}-a^{2}-b^{2}}e^{i\gamma}|11\rangle]_{12}|1\rangle_{3}.$
(26)

Here $|{\psi_{0}}\rangle=|00\rangle$ and
$|{\psi_{1}}\rangle=\sqrt{2}[a|01\rangle+be^{i\delta}|10\rangle+\sqrt{\frac{1}{2}-a^{2}-b^{2}}e^{i\gamma}|11\rangle]$
implying $\langle{\psi_{0}}|{\psi_{1}}\rangle=0$. Also, this state is
satisfying the second condition i.e. unit bipartite entanglement between
sender’s and receiver’s qubits. Hence, it can be used to teleport a single
qubit perfectly.

Our results also shows directly that the only W-state which can give perfect
SQT is [33]

$|{\psi}\rangle_{123}=B|{100}\rangle+C|{010}\rangle+E|{001}\rangle$ (27)

with $|B|^{2}+|C|^{2}=|E|^{2}=\frac{1}{2}$, which also includes the
normalization i.e. $|B|^{2}+|C|^{2}+|E|^{2}=1$.

##  4 Conclusions

We study perfect SQT of a single qubit and find conditions on a most general
form of n-qubit resource to give perfect SQT. To achieve this, we consider an
n-qubit general entangled resource, where n-1 qubits go to Alice and 1 qubit
goes to Bob. Alice makes an n-qubit projective measurement on her particles
along with the information particle and sends 2-c bit message to Bob. Then Bob
applies a suitable unitary transformation on his particle to replicate the
information. We obtained maximal average fidelity for a general n-qubit
resource. For perfect SQT, there should be unit bipartite entanglement between
qubit belongs to the receiver and rest of the qubits. Using this result we
discussed a 3-qubit entangled state, written in a derivable computational
basis and giving perfet SQT. This reslut is different from earlier, where
entangled resource is written in Acin representation. We have also given an
example of 3-qubit entangled state which can give perfect SQT and not included
in previous studies. Additionally, we demonstrate how our findings clearly
show the existence of a class of 3-qubit W-states that can result in perfect
SQT.

## Acknowledgement

The authors are gratefully acknowledge Prof. Naresh Chandra, Dr. Onkar Nath
Verma, Dr. Vikram Verma and Mr. Ravi Kamal Pandey for their valuable and
helpful discussions. One of the authors SJ is also thankful to University
Grant Commission scheme of Maulana Azad National Fellowship for providing
financial support.

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