# Strong lensing of the nonsingular black hole in nonlinear electrodynamics
framework

M. F. Ishlah  F. P. Pratama  I. Alfredo  H. S. Ramadhan  Departemen Fisika,
FMIPA, Universitas Indonesia, Depok, 16424, Indonesia.

###### Abstract

We model supermassive black hole as the nonsingular black hole viewed from the
point of view of Ayon-Beato-Garcia nonlinear electrodynamics (NLED) and
present a complete study of their corresponding strong gravitational lensing.
The NLED modifies the the photon’s geodesic, and our calculations show that
such effect increases the corresponding photon sphere radius and image
separation, but decreases the magnification. We also present the intensity map
of the black hole’s shadow image with thin accretion disk.

##  I Introduction

Black hole (BH) is one of the most straightforward yet profound prediction of
General Relativity (GR). Its extreme gravity distorts its surrounding
spacetime and bends light, creating (among many things) the gravitational
lensing phenomenon. The recent observation by Event Horizon Telescope (EHT)
that successfully captured the visual images of the superheavy BHs M87*
EventHorizonTelescope:2019dse  and Sgr A* akiyama_first_2022  has established
a triumph for the gravitational lensing as a means to empirically prove black
hole’s existence. By “image” here is the corresponding shadow Falcke:1999pj
surrounded by accreting materials that emits and lenses light from the nearby
background source.

Theoretically, the study of gravitational lensing is as old as GR itself (see,
for example, schneider:1992 ; Perlick:2004tq  and the references therein), but
it was Darwin who first applied it for Schwarzschild BH Darwin_gravity_1959 .
His exact calculation on the deflection angle shows that at small impact
parameters there exists a critical value (close to the corresponding photon
sphere) where the deflection angle suffers from logarithmic divergence
Bozza:2010xqn , beyond which photons fall into the horizon. His results were
later rediscovered and developed by other authors, for example in
Luminet:1979nyg ; Chandrasekhar:1985kt . In the last two decades the study of
gravitational lensing in the strong deflection limit received revival and
extensive elaboration Frittelli:1999yf ; Virbhadra:1999nm ;
bozzaStrongFieldLimit2001 ; bozzaGravitationalLensingStrong2002 . In
particular, Bozza shows in bozzaGravitationalLensingStrong2002  that the
analytical expansion of the strong deflection angle in the limit of
$r\rightarrow r_{ps}$ ($r_{ps}$ being the photon sphere radius) is given by

$\alpha(b)=-\tilde{a}_{1}\log\left(\frac{b}{b_{c}}-1\right)+\tilde{a}_{2}+O(b-b_{c}).$
(1)

with $\tilde{a}_{1}$ and $\tilde{a}_{2}$ some constants. Upon closer
inspection, Tsukamoto gave correction to the higher order expansion
Tsukamoto:2016qro ; Tsukamoto:2016jzh . This result on Schwarszchild was
extended to the case of Reissner-Nordstrom by Eiroa et al Eiroa:2002mk , while
the strong lensing in Kerr BH was studied in Bozza:2002af ; Vazquez:2003zm ;
Bozza:2006nm .

Probably the most intriguing property of black holes is the existence of
singularity due to the gravitational collapse Oppenheimer:1939ue . It was
believed that such singularity is an inherent solution of general relativity,
but the stable ones (like all observable black holes) are disconnected from
the observers by event horizon Penrose:1964wq . Nevertheless, Bardeen in 1968
constructed a metric function that produced nonsingular spacetime
bardeenNonsingularGeneralrelativisticGravitational1968 . The metric and all
invariants are devoid of singularity everywhere, including at $r=0$. Instead,
we have regular de Sitter space at the core. (For an excellent review on
regular BH see, for example, Ansoldi:2008jw .) The strong lensing phenomenon
around Bardeen BH has been studied in Eiroa:2010wm ; Wei:2015qca .

At first nobody knows what kind of matter that sources the Bardeen geometry,
but later Ayon-Beato and Garcia (ABG) realized that this nonsingular metric
can be obtained as solutions of Einstein’s equations coupled to some nonlinear
electrodynamics (NLED) source Ayon-Beato:1998hmi ; ayon-
beatoBardeenModelNonlinear2000 . Invoking NLED turns out to have profound
impact on the geodesic of test photon. Novello et al showed that in nonlinear
electrodynamics background, photon moves in an effective modified geometry
Novello:1999pg , and this radically modifies the corresponding optical
observables. In this work, we discuss the effect of effective geometry to the
lensing phenomenon in the Bardeen BH using one of the ABG’s NLED model. In
particular, we calculate the image separation and magnification. We also
investigate its shadow with thin accretion disks. This paper is organized as
follows. In Sec. II we briefly review the regular Bardeen solution and its
corresponding ABG models. In Sec. III we present the effective metric of ABG
and the corresponding photon sphere. Sec. IV is devoted to applying the
Bozza’s and Tsukamoto’s strong lensing formalism to our model. Sec. V is
devoted to calculating the strong lensing observables. In Sec. VI we compare
the shadow images of the NLED model to the RN as well as the ordinary Bardeen
in the thin accretion disk approximation. Finally, we summarize our findings
in Sec. VIII.

##  II Bardeen Spacetime

The Bardeen metric is given by
bardeenNonsingularGeneralrelativisticGravitational1968 :

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (2)

with

$f(r)\equiv 1-\frac{2mr^{2}}{(r^{2}+q^{2})^{\frac{3}{2}}},$ (3)

and $q$ is the charge. This spacetime is regular at $r=0$, as can easily be
seen from the Kretschmann scalar:

$\lim_{r\rightarrow
0}R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}=\frac{96m^{2}}{q^{8/3}},$
(4)

while the metric behaves de Sitter-like

$f(r)\approx 1-\frac{2m}{q^{3}}r^{2}.$ (5)

The horizons $r_{h}$ are given by the roots of

$\left(r_{h}^{2}+q^{2}\right)^{3}-4m^{2}r_{h}^{4}=0.$ (6)

Bardeen black hole can, in general, possess up to two horizons. The extremal
condition is achieved when ayon-beatoBardeenModelNonlinear2000

$q^{2}=\frac{16}{27}m^{2}\ \rightarrow r_{extr}=\sqrt{\frac{32}{27}}m.$ (7)

Ayon-Beato and Garcia proposed the NLED matter to source the Bardeen
spacetime, given in Ayon-Beato:1998hmi . This model, however, produces a
slightly different metric function than the original Bardeen,

$f(r)=1-\frac{2mr^{2}}{(r^{2}+q^{2})^{\frac{3}{2}}}+\frac{q^{2}r^{2}}{\left(r+2+q^{2}\right)^{2}}.$
(8)

Strong lensing of this particular model has been discussed in
ghaffarnejadGravitationalLensingCharged2018 . Later ABG considered a simpler
NLED sourced by magnetic monopole as follows ayon-
beatoBardeenModelNonlinear2000 ,

$\mathcal{L}=\frac{3}{2sq^{2}}\left(\frac{\sqrt{2q^{2}F}}{1+\sqrt{2q^{2}F}}\right)^{5/2},$
(9)

where $F\equiv\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ and $s\equiv q/2m$. By
inserting the monopole ansatz
$A_{\mu}=\delta^{\varphi}_{\mu}q\left(1-\cos\theta\right)$, the field strength
becomes $F=q^{2}/2r^{4}$ and the Lagrangian produces the metric solution Eq.
(3).

##  III Effective Geometry

The NLED Lagrangian above induces the effective metric tensor Novello:1999pg

$g^{\mu\nu}_{eff}=g^{\mu\nu}-\frac{4\mathcal{L}_{FF}}{\mathcal{L}_{F}}F^{\mu}_{\alpha}F^{\alpha\nu},$
(10)

where $\mathcal{L}_{A}\equiv\partial\mathcal{L}/\partial A$. This, in turn,
yields the effective length element

$ds^{2}_{eff}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+h_{m}(r)r^{2}d^{2}\Omega,$ (11)

where

$h_{m}(r)=\left(1+\frac{4\mathcal{L}_{FF}}{\mathcal{L}_{F}}\frac{q^{2}}{r^{4}}\right)^{-1}.$
(12)

Inserting the Lagrangian (9) we obtain

$h_{ABG}(r)=\left(1-\frac{2(6q^{2}-r^{2})}{(q^{2}+r^{2})}\right)^{-1}.$ (13)

From the corresponding geodesic equation it is not difficult to see that the
radial equation satisfies

$\frac{1}{2}\dot{r}^{2}+V_{eff}=0,$ (14)

where we define the effective potential $V_{eff}$ as

$V_{eff}\equiv\frac{f(r)}{h(r)}\frac{\mathbb{L}^{2}}{r^{2}}.$ (15)

The corresponding photon sphere radius is given by the largest positive root
of the following condition, bozzaGravitationalLensingStrong2002

$\frac{f^{\prime}(r_{ps})}{f(r_{ps})}-\frac{2}{r_{ps}}-\frac{h^{\prime}(r_{ps})}{h(r_{ps})}=0.$
(16)

This yields,

$\displaystyle\frac{2}{r_{ps}}+\frac{28q^{2}r_{ps}}{11q^{4}+8q^{2}r_{ps}^{2}-3r_{ps}^{4}}+\frac{m\left(4q^{2}r_{ps}-2r_{ps}^{3}\right)}{\left(q^{2}+r_{ps}^{2}\right)\left[\left(q^{2}+r_{ps}^{2}\right)^{3/2}-2mr_{ps}^{2}\right]}=0.$
(17)

Figure 1: Photon Sphere of Bardeen with ABG source as function of charge ($q$)
for $m=1$.

By solving the roots numerically, the behavior of $r_{ps}$ as a function of q,
$r_{ps}=r_{ps}(m=1,q)$, is shown in Fig.1. It is shown that the photon sphere
decreases as the charge increases until some critical value $q_{crit}$ where
$r_{ps}(m=1,q=q_{crit})$ is minimum, beyond which $r_{ps}$ starts increasing
without bound. Interestingly, the critical value $q_{crit}$ is not given by
the extremal charge $q=\sqrt{16/27}m$, as in the Bardeen case. Rather,
$q_{crit}=16/27\ m=0.592$.

Figure 2: $r_{ps}$ as function of $m$ for a variation of charge $q$. Figure 3:
$r_{ps}$ as function of $q$ for different models.

In Fig. 2 we show how $r_{ps}$ varies with $m$ for several values of $q$. They
differ only at small $m$. When the mass is large, $r_{ps}$ for different $q$
asymptote to a single gradient. In Fig. 3 we show the deviation of $r_{ps}$ as
a function of $q$ from the Schwarzschild (charge-less condition). The ABG
model we consider here falls between the Schwarzschild and the RN at large
$q$, unlike the original Bardeen which falls the fastest.

##  IV Deflection Angle in the Strong Field Limit

From the spherical symmetry and staticity conditions, Noether’s theorem
dictates that this spacetime has constants of motion, the total test
particle’s energy $E$ and angular momentum $\mathbb{L}$, related to the
$\partial_{t}$ and $\partial_{\varphi}$ Killing vectors, respectively. We
define the impact parameter for photon as

$b(r_{0})\equiv\frac{\mathbb{L}}{E}=\sqrt{\frac{h_{m}(r_{0})r_{0}^{2}}{f(r_{0})}}.$
(18)

Solving the null geodesic equation, the general expression for bending angle
of light rays can be expressed as (see, for example, in Weinberg:1972kfs )

$\alpha(r_{0})=2\int_{r_{0}}^{\infty}\sqrt{\frac{1}{r^{2}f(r)h(r)R(r)}}dr-\pi,$
(19)

where

$R(r)=\frac{r^{2}h(r)}{b^{2}f(r)}-1.$ (20)

The integral is divergent at $r_{0}\rightarrow r_{ps}$. To circumvent this
problem we define $z\equiv 1-r_{0}/r$ Tsukamoto:2016jzh  and write Eq. (19) as

$\alpha(r_{0})=\int^{1}_{0}\mathcal{H}(z,r_{0})dz-\pi,$ (21)

with

$\mathcal{H}(z,r_{0})\equiv\frac{2\left(1-z\right)^{2}}{\sqrt{f\left(\frac{r_{0}}{1-z}\right)h\left(\frac{r_{0}}{1-z}\right)R\left(\frac{r_{0}}{1-z}\right)}}.$
(22)

The singular part can be isolated by defining

$\mathcal{H}(z,r_{0})\equiv\mathcal{H}_{R}(z,r_{0})+\mathcal{H}_{D}(z,r_{0}),$
(23)

where the subscript R(D) refers to the regular (divergent) part, respectively.

To handle the divergent part:

$\mathcal{I}_{D}(r_{0})\equiv\int^{1}_{0}\mathcal{H}_{D}(z,r_{0})dz,$ (24)

we define $\mathcal{H}(z,r_{0})\equiv 2r_{0}/\sqrt{\mathcal{G}(z,r_{)}}$ and,
by expanding it around $z\rightarrow 0$, obtain the expression for
$\mathcal{G}(z,r_{0})$ up to second-order:

$\mathcal{G}_{0}(z,r_{0})=c_{1}(r_{0})z+c_{2}(r_{0})z^{2},$ (25)

where

$\displaystyle c_{1}(r_{0})$ $\displaystyle=$ $\displaystyle
C_{0}\mathcal{D}_{0}r_{0}f(r_{0}),$ $\displaystyle c_{2}(r_{0})$
$\displaystyle=$ $\displaystyle
C_{0}r_{0}f_{0}\bigg{\\{}\mathcal{D}_{0}\left[\mathcal{D}_{0}\left(\mathcal{D}_{0}+\frac{f^{\prime}_{0}}{f_{0}^{3}}\right)r_{0}-3\right]$
(26) $\displaystyle+\frac{\mathcal{D}_{0}r_{0}}{2}\bigg{\\}},$

with $X_{0}\equiv X(r=r_{0})$, $C(r)\equiv h(r)r^{2}$, and

$\mathcal{D}(r)\equiv\frac{C^{\prime\prime}(r)}{C(r)}-\frac{f^{\prime\prime}(r)}{f(r)}.$
(27)

For the ABG model, the values of $c_{1}$ and $c_{2}$ are

$\displaystyle c_{1}(r_{0})$ $\displaystyle=$
$\displaystyle-\frac{2(r_{0}^{2}(11q^{4}+22q^{2}r_{0}^{2}-3r_{0}^{4})\tilde{j}_{1}^{3/2}+m(9r_{0}^{8}-61q^{2}r_{0}^{6})))}{(11q^{2}-3r^{2})^{2}\tilde{j}_{1}^{3/2}},$
$\displaystyle c_{2}(r_{0})$ $\displaystyle=$
$\displaystyle\frac{mr_{0}^{6}\tilde{j}_{2}+r_{0}^{2}(\tilde{j}_{3}\tilde{j}_{1}^{5/2}-\tilde{j}_{3})}{(11q^{2}-3r_{0}^{2})^{3}\tilde{j}_{1}^{5/2}},$

with

$\displaystyle\tilde{j}_{1}$ $\displaystyle\equiv$ $\displaystyle
q^{2}+r_{0}^{2},$ $\displaystyle\tilde{j}_{2}$ $\displaystyle\equiv$
$\displaystyle 2013q^{6}+3104q^{4}r_{0}^{2}+57q^{2}r_{0}^{4}-54r_{0}^{6},$
$\displaystyle\tilde{j}_{3}$ $\displaystyle\equiv$ $\displaystyle
121q^{6}+825q^{4}r_{0}^{2}-99Q^{2}r_{0}^{4}+9r_{0}^{6}.$ (29)

Following Bozza bozzaGravitationalLensingStrong2002  and Tsukamoto
Tsukamoto:2016jzh  it can be shown that the divergent integral in the strong
field limit $r\rightarrow r_{ps}$ (or equivalently $b\rightarrow b_{ps}$) is
expressed as

$\displaystyle\mathcal{I}_{D}(b)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{f_{ps}C_{ps}^{\prime\prime}-f^{\prime\prime}_{ps}C_{ps}}}$
$\displaystyle\times\bigg{\\{}\log\left(\frac{b}{b_{c}}-1\right)+\log\left[r_{ps}^{2}\left(\frac{C_{ps}^{\prime\prime}}{C_{ps}}-\frac{f_{ps}^{\prime\prime}}{f_{ps}}\right)\right]\bigg{\\}}.$

Figure 4: The values of $\tilde{a}_{1}$ and $\tilde{a}_{2}$ for Bardeen with
ABG source

The regular part is

$\mathcal{I}_{R}\equiv\int^{1}_{0}\mathcal{H}_{R}(z,r_{0})dz.$ (31)

Likewise, by expanding around $r\rightarrow r_{ps}$ the integral can be
expressed as

$\displaystyle\frac{\mathcal{I}_{R}(r_{ps})}{2r_{ps}}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\frac{dz}{F(r_{ps},z)}-\int^{1}_{0}\sqrt{\frac{2}{C_{ps}r_{ps}f_{r}{ps}\mathcal{D}_{ps}z^{2}}}dz,$

with

$F(r_{ps},z)\equiv\sqrt{C\left(\frac{r_{ps}}{1-z}\right)R\left(\frac{r_{ps}}{1-z}\right)f\left(\frac{r_{ps}}{1-z}\right)\left(1-z\right)^{4}}.$
(33)

For the ABG model, the integral can be evaluated numerically. Putting (IV) and
(IV) into (21), we can express it as Eq. (1) by identifying

$\displaystyle\tilde{a}_{1}$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2A_{ps}B_{ps}}{A_{ps}C^{\prime\prime}_{ps}-A^{\prime\prime}_{ps}C_{ps}}},$
$\displaystyle\tilde{a}_{2}$ $\displaystyle\equiv$
$\displaystyle\tilde{a_{1}}\log\tilde{b}+I_{R}(r_{ps})-\pi,$ (34)

with

$\tilde{b}\equiv
r_{ps}^{2}\left(\frac{C^{\prime\prime}_{ps}}{C_{ps}}-\frac{A^{\prime\prime}_{ps}}{A_{ps}}\right).$
(35)

In terms of the ABG model we consider, their expressions are

$\displaystyle\tilde{a}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(11q^{2}-3r_{ps}^{2})^{3}\tilde{k}_{1}^{5/2}}{\tilde{k}_{2}-\tilde{k}_{3}\tilde{k}_{1}^{5/2}}},$
$\displaystyle\tilde{a}_{2}$ $\displaystyle=$
$\displaystyle-\pi+\mathcal{I}_{R}(r_{ps})+\sqrt{\frac{(11q^{2}-3r_{ps}^{2})^{3}\tilde{k}_{1}^{5/2}}{\tilde{k}_{2}-\tilde{k}_{3}\tilde{k}_{1}^{5/2}}}$
$\displaystyle\times\log\left[\frac{2\tilde{k}_{1}^{5/2}\tilde{k}_{3}-2\tilde{k}_{2}}{(11q^{2}-3r_{ps}^{2})^{2}(\tilde{k}_{1}^{3/2}-2mr_{ps}^{2})\tilde{k}_{1}^{2}}\right],$

with

$\displaystyle\tilde{k}_{1}$ $\displaystyle=$ $\displaystyle
q^{2}+r_{ps}^{2},$ $\displaystyle\tilde{k}_{2}$ $\displaystyle=$
$\displaystyle m(3355q^{6}r_{ps}^{4}+466q^{4}r_{ps}^{6}+51q^{2}r_{ps}^{8}),$
$\displaystyle\tilde{k}_{3}$ $\displaystyle=$
$\displaystyle(121q^{6}+825q^{4}r_{ps}^{2}-99q^{2}r_{ps}^{4}+9r_{ps}^{6}).$
(37)

Figure 5: Comparison of light deflection for Regular Bardeen and Bardeen with
ABG source

The calculated $\tilde{a}_{1}$ and $\tilde{a}_{2}$ are shown in Fig. 4. The
$\tilde{a}_{1}$ slowly increases until $q=q_{crit}$ and then decreases
linearly. The $\tilde{a}_{2}$, on the other hand, decreases as $q$ goes up
until $q=q_{crit}$, then it starts increasing. The deflection angles are
depicted in Fig. 5. It is shown that the critical impact parameters for the
NLED models are smaller than for the pure Bardeen. This critical value
increases with increasing charge.

##  V Lensing

The most straightforward effect of light deflection due to gravitational field
is the notion of “gravitational lensing”. The lensing mechanism can be
inferred from Fig. 6. The straight segment $\overline{SO}$ is the path the
light would have taken had it not been deflected due to the lens (BH) at $L$.
The angle $\beta$ denotes the angular position of the source $S$ from the
observer $O$ if there were no lensing. What the $O$ observes is the “image” of
$S$ located at $I$ whose angular position is given by $\theta$. The deflection
angle is given by $\alpha$. From simple geometry the reation between $\beta$
and $\theta$ can be expressed as Virbhadra:1999nm ; Bozza:2008ev

$\tan\beta=\tan\theta-\frac{D_{LS}}{D_{OL}}\left[\tan\theta+\tan(\alpha-\theta)\right],$
(38)

known as the lens equation.

In the strong field limit ($r_{0}\rightarrow r_{ps}$) $\beta$ and $\theta$ are
small, $\alpha$ can exceed $2\pi$ and light can loop around the black hole
several ($n$) times before escaping out to the observer.

Figure 6: Gravitational lensing diagram. S, L, and O are the source, the lens,
and the observer, respectively.

In this sense, $\alpha=2n\pi+\Delta\alpha_{n}$. We can then expand
$\tan(\alpha-\theta)\sim\Delta\alpha_{n}-\theta$ bozzaStrongFieldLimit2001 .
The lens equation thus becomes

$\beta=\theta-\frac{D_{LS}}{D_{OL}}\Delta\alpha_{n}.$ (39)

We also have the relation

$b=D_{OL}\theta.$ (40)

Substituting it into (1) and inverting it results in
bozzaGravitationalLensingStrong2002

$\theta(\alpha)\simeq\frac{b_{c}}{D_{OL}}\left(1+e^{\frac{\tilde{a}_{2}-\alpha}{\tilde{a}_{1}}}\right).$
(41)

From Fig. 6, the innermost image is given by

$\theta_{\infty}=\frac{b_{c}}{D_{OL}}.$ (42)

Expanding around $\alpha$ yields

$\theta_{n}=\theta_{n}^{0}-\gamma_{n}\Delta\alpha_{n},$ (43)

with

$\displaystyle\theta_{n}^{0}$ $\displaystyle=$
$\displaystyle\frac{b_{c}}{D_{OL}}\left(1+e_{n}\right),$ (44)
$\displaystyle\gamma_{n}$ $\displaystyle=$
$\displaystyle\frac{b_{c}}{\tilde{a}_{2}D_{OL}}e_{n},$ (45) $\displaystyle
e_{n}$ $\displaystyle=$ $\displaystyle
e^{\frac{\tilde{a}_{2}-2n\pi}{\tilde{a}_{1}}}.$ (46)

We can eliminate $\Delta\alpha_{n}$ using the Eq. (39). This results in the
equation for the n-th shadow position bozzaStrongFieldLimit2001 ;
bozzaGravitationalLensingStrong2002 ; Tsukamoto:2016qro

$\theta_{n}=\theta_{n}^{0}+\frac{D_{LS}}{D_{OS}}\gamma_{n}(\beta-\theta_{n}^{0}),$
(47)

where the second term on the right-hand side is small compared to the first
one. For the Einstein ring case, $\beta=0$ and

$\theta_{nE}=\left(1-\frac{D_{LS}}{D_{OS}}\gamma_{n}\right)\theta_{n}^{0}.$
(48)

The observables we wish to calculate are the image separation and the
magnification. The separation $s$ is the difference between the outermost and
innermost images,

$s=\theta_{1}-\theta_{\infty}=\theta_{\infty}e^{\frac{\tilde{a}_{2}-2\pi}{\tilde{a}_{1}}}.$
(49)

The magnification $\mu$ is the inverse of the corresponding Jacobian
determinant for the critical curve schneider:1992 ; bozzaStrongFieldLimit2001
. The $n^{th}$ image magnification is defined to be

$\mu_{n}=\frac{1}{|\det
J_{\theta_{n}^{0}}|}=\frac{1}{\frac{\beta}{\theta_{n}^{0}}\left.\frac{\partial\beta}{\partial\theta}\right|_{\theta_{n}^{0}}},$
(50)

from which the (relativistic) flux ratio is expressed as

$r=\frac{\mu_{1}}{\sum\limits_{n=2}^{\infty}\mu_{n}}=e^{\frac{2\pi}{\tilde{a}_{1}}}.$
(51)

In this paper we calculate the strong lensing from the Sgr A* black hole
modeled as the Bardeen with ABG source. We use data from GRAVITY collaboration
where the black hole mass and its distance from the Earth (observer) are
$m=4.154\times 10^{6}M_{\odot}$ and $D_{OL}=8.178\ kpc$, respectively
abuterGeometricDistanceMeasurement2019 . These values are consistent with the
EHT results akiyama_first_2022 . In Table 1 we show the observables. Here the
value the magnification is converted to magnitudes $r_{m}=2.5\log_{10}r$
bozzaGravitationalLensingStrong2002 . From the Table it can be seen that the
observable values for the Bardeen does not differ much from that of RN.
However, the observables for the ABG are significantly different from both
Bardeen and RN; i.e., the ABG’s are smaller. This shows that the NLED effect
is quite significant here.

Model | $Q/m$ |  $\theta_{\infty}$ ($\mu$ as) |  $s$($\mu$ as) | $r_{m}$
---|---|---|---|---
Schwarzschild | - | 26.0592 | 0.0327 | 6.8184
Bardeen | 0 | 26.0592 | 0.0327 | 6.8184
| 0.1 | 26.0156 | 0.0332351 | 6.79937
| 0.2 | 25.8833 | 0.0349143 | 6.74083
| 0.3 | 25.6569 | 0.0381989 | 6.63828
| 0.4 | 25.3266 | 0.044135 | 6.48266
| 0.5 | 24.8751 | 0.0552564 | 6.25695
| 0.6 | 24.2718 | 0.0788401 | 5.92697
| 0.7 | 23.4566 | 0.144124 | 5.41046
RN | 0 | 26.0592 | 0.0327 | 6.8184
| 0.1 | 26.0157 | 0.0330446 | 6.81081
| 0.2 | 25.8841 | 0.0340718 | 6.7875
| 0.3 | 25.6612 | 0.0359467 | 6.74699
| 0.4 | 25.3412 | 0.0389696 | 6.68646
| 0.5 | 24.9146 | 0.043696 | 6.60104
| 0.6 | 24.3668 | 0.0511167 | 6.48246
| 0.7 | 23.6747 | 0.0628463 | 6.31574
ABG | 0 | 15.0453 | 1.01295 | 3.93662
| 0.1 | 15.0593 | 1.015 | 3.93252
| 0.2 | 15.1029 | 1.02079 | 3.92061
| 0.3 | 15.1806 | 1.02911 | 3.90248
| 0.4 | 15.3007 | 1.03751 | 3.88228
| 0.5 | 15.4768 | 1.04185 | 3.87016
| 0.6 | 15.7286 | 1.03655 | 3.8886
| 0.7 | 16.0813 | 1.01629 | 3.9788
Table 1: Observables for the Sgr A* Schwarzschild, Bardeen, RN, and ABG. The
$\theta$ and its corresponding separation are expressed in $\mu$ arc-second
(as) unit.

From Table 1 it can be inferred that the ABG has $1.5\times$ smaller values of
$\theta_{\infty}$ compared to Bardeen. This means that photon can orbit ABG
with smaller radius. While the separation $s$ for the ABG is surprisingly
$30\times$ larger, its magnification $r_{m}$ is smaller than for Bardeen. The
NLED thus strengthens the gravitational field by decreasing the innermost
distance while at the same time increasing its corresponding separation with
the outermost image. Interestingly, the observables in the ABG behave in such
opposite ways with the the ones in the Bardeen. In Figs. 7 it is shown that
while the $\theta_{\infty}$ in Bardeen decreases monotonically, in the ABG
case it increases.

Figure 7: $\theta_{\infty}$ as function of $q$ for: [top] The ABG model and,
[down] the original Bardeen.

From Figs. 8 the separation in the Bardeen model increases monotonically,
while in the ABG there exists some maximum value $s=s_{max}$ before which it
initially increases and after which it starts decreasing.

Figure 8: $s$ as function of $q$ for: [top] the ABG model, [down] the Bardeen
model.

Similarly, in Figs. 9 we see that the $r_{m}$ decreases unboundedly for the
Bardeen as $q$ increases, whereas it decreases to its minimum value before
increasing monotonically for the ABG.

Figure 9: $r_{m}$ as function of $q$ for: [top] the ABG model, [down] the
Bardeen model.

##  VI Shadow with Accretion Disk

The radius of the black hole shadow is defined as the dark region due to the
inability of the photon from the background light source to escape the
gravitational potential around the black hole. The size of the shadow
corresponds to the critical impact parameter of the photon orbit
stuchlik_shadow_2019 ; bisnovatyi-kogan_gravitational_2017 ; lu_size_2020 .
The idea that the Sgr A* shadow can be observed was suggested in Falcke:1999pj
; Melia:2001dy ; kruglovShadowM87Black2020 .

The shadow radius $R_{sh}$ is represented by the critical impact parameter,
i.e., the impact parameter evaluated at the photon sphere radius $r_{ps}$

$R_{sh}=b_{c}=r_{ps}\sqrt{\frac{h(r_{ps})}{f(r_{ps})}}.$ (52)

For Bardeen with ABG source we will have

$R_{shABG}=r_{c}\sqrt{\frac{\left(1-\frac{2(6q^{2}-r_{c}^{2})}{(q^{2}+r_{c}^{2})}\right)^{-1}}{\left(1-\frac{2mr_{c}^{2}}{(r_{c}^{2}+q^{2})^{3/2}}\right)}}.$
(53)

Due to the effect of NLED source, the emitted photon from the accretion disk
will be affected by the effective geometry of the spacetime. The intensity of
the observed emission from the accretion disk can be obtained by integrating
the specific emissivity $j(\nu_{e})$ along photon path $\gamma$ bambi_can_2013
,

$I_{obs}=\int_{\gamma}g^{3}j(\nu_{e})dl_{prop},$ (54)

where $dl_{prop}$ is the infinitesimal proper length, and $g$, also known as
the redshift factor $g\equiv\nu_{obs}/\nu_{e}$, can be obtained from

$g=\frac{k_{\alpha}u^{\alpha}_{obs}}{k_{\beta}u^{\beta}_{e}}.$ (55)

We use the model from Bambi bambi_can_2013  defined as

$j(\nu_{e})\propto\frac{\delta(\nu_{e}-\nu_{*})}{r^{2}},$ (56)

with $k^{\mu}=(1,0,0,0)$ the 4-velocity of the photon, $u^{\mu}_{obs}$ the
4-velocity of the observer, and $u^{\mu}_{e}$ the 4-velocity of the accreting
gas emitting the radiation. For the gas in free fall in a static and
spherically symmetric spacetime,

$\displaystyle u^{\mu}_{obs}=(1,0,0,0),$ $\displaystyle u^{t}_{e}$
$\displaystyle=$ $\displaystyle\frac{1}{f(r)},$ $\displaystyle u^{r}_{e}$
$\displaystyle=$ $\displaystyle-\sqrt{\frac{g(r)}{f(r)}[1-f(r)]},$
$\displaystyle u^{\theta}_{e}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle u^{\phi}_{e}$ $\displaystyle=$ $\displaystyle 0.$ (57)

The redshift factor $g$, after taking into account the effective metric, is

$\displaystyle g$ $\displaystyle=$
$\displaystyle\frac{k_{\alpha}u^{\alpha}_{o}}{k_{\beta}u^{\beta}_{e}}$ (58)
$\displaystyle=$
$\displaystyle\frac{1}{\frac{1}{f(r)}-\frac{k_{r}}{k_{t}}\sqrt{\frac{g(r)}{f(r)}-[1-f(r)]}}.$

From $k_{\mu}k^{\mu}=0$ we obtain the relation between $k^{\mu}$ as

$\frac{k_{r}}{k_{t}}=\pm
f(r)\sqrt{g(r)\left(\frac{1}{f(r)}-\frac{b^{2}}{h(r)r^{2}}\right)}.$ (59)

We put all the ingredients together into the Eq. (54) to obtain the intensity
of the accretion disk emission. For this purpose we modify the EinsteinPy
einsteinpy  python library to accomodate the metric functions and the null
effective geometry. Then, we can compare the result to the known charged
solutions: the Reissner-Nordstrom and the original Bardeen; they are shown in
Fig. 10. For the ABG black hole, the shadow radius seems to be smaller
compared to the other two.

Figure 10: Comparison of shadow of black holes with accretion disk. The lower
panels describe the corresponding intensity variation along the $x$-axis.

##  VII Conclusion

##  VIII Conclusion

In this work, by assuming that our galaxy’s supermassive Sgr A* is a
nonsingular magnetically-charged black hole, we study both its strong lensing
as well as its shadow image in the thin accretion disk approach. Our approach
is different either from Eiroa:2010wm ; Wei:2015qca  in that we regard the
nonsingularity coming from the NLED charge, or from
ghaffarnejadGravitationalLensingCharged2018  where we used simpler Ayon-Beato-
Garcia NLED model.

The NLED reduces photon sphere radius with increasing $q$ until it reaches
$q=q_{crit}$, after which it starts to increase monotonically. The NLED
reduces the radius of the photon orbit, and also increases the gravitational
field by pulling the innermost distance closer but at the same time stretching
its separation distance from the outermost image. Our observable quantities
behave differently from the ordinary Bardeen Eiroa:2010wm ; Wei:2015qca  in
terms of the innermost image (Figs. 7), the separation between innermost and
outermost images (Figs. 8), and the magnification (Figs. 9). Interestingly,
the NLED type that we particularly choose to model Bardeen in this paper gives
distinct observable results compared to other types. While the model
ghaffarnejadGravitationalLensingCharged2018  predicts that the angular
separation $s$ decreases while the magnification $r_{m}$ increases with
increasing $q$, our results show the opposite. From Table 1 we can observe
that as the charge increases, the angular separation does increase while the
magnification does decrease. Lastly, our analysis with thin accretion disk
shows that the nonsingular black hole with ABG metric gives the smallest
shadow radius when compared to the RN and ordinary Bardeen.

###### Acknowledgements.

We thank Reyhan Lambaga and Imam Huda for the fruitful discussions on the
preliminary stage of this work.

## Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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# Two quantum algorithms for communication between spacelike separated
locations

Amitava Datta
Department of Computer Science and Software Engineering,
University of Western Australia,
35 Stirling Highway, Perth, WA6009, Australia

###### Abstract

The ‘no communication’ theorem prohibits superluminal communication by showing
that any measurement by Alice on an entangled system cannot change the reduced
density matrix of Bob’s state, and hence the expectation value of any
measurement operator that Bob uses remains the same. We argue that the proof
of the ‘no communication’ theorem is incomplete and superluminal communication
is possible through state discrimination in a higher-dimensional Hilbert space
using ancilla qubits. We propose two quantum algorithms through state
discrimantion for communication between two observers Alice and Bob, situated
at spacelike separated locations. Alice and Bob share one qubit each of a Bell
state $\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. While sending classical
information, Alice measures her qubit and collapses the state of Bob’s qubit
in two different ways depending on whether she wants to send $0$ or $1$.
Alice’s first measurement is in the computational basis, and the second
measurement is again in the computational basis after applying Hadamard
transform to her qubit. Bob’s first algorithm detects the classical bit with
probability of error $<\frac{1}{2^{k}}$, but Alice and Bob need to share $k$
Bell states for communicating a single classical bit. Bob’s second algorithm
is more complex, but Bob can detect the classical bit deterministically using
four ancilla qubits. We also discuss possible applications of our algorithms.

##  1 Introduction

Nonlocality inherent in quantum mechanics was first pointed out by Einstein,
Podolsky and Rosen [1] in their seminal 1935 paper. Their argument was that
the apparent instantaneous effect of measuring one qubit of an entangled state
on the other qubit in spacelike separated locations was an indication that
quantum mechanics was an incomplete theory. However Bell [2] in his 1965 paper
showed that no deterministic hidden variable theory can make predictions that
are consistent with quantum mechanics. This firmly established the inherent
nonlocal nature of quantum mechanics. One natural question that arose after
Bell’s work is, whether this nonlocal nature of quantum mechanics can be
utilized for sending information between spacelike seprated parties
instantaneously. Several proposals were made to exploit the nonlocal
correlations for superluminal signalling, or transfer of information [3, 4, 5,
6, 7]. However, all of these proposals either assumed operations that are not
permitted by quantum mechanics, e.g., cloning of a quantum state, non-unitary
transformations, or complex measurement processes that are impractical or not
achievable with known technologies. The problems with these proposals were
pointed out in several papers [8, 9, 10, 11, 15, 12]. In particular Ghirardi
[11] summarized and refuted many of these proposals in a review article. The
proposals and their refutations have resulted in two important no go theorems
in quantum mechanics, the no cloning and no communication theorems. In
particular, the no communication theorem was proven to show that the nonlocal
correlations in EPR like scenarios cannot be used for sending information. The
correletaions get nullified due to the probabilistic nature of the quantum
measurement process.

Several proofs of the no communication theorem have been proposed over the
years [14, 10, 13], and all of them have an implicit assumption that Bob can
only try to get classical information from the expectation value of a
measurement operator. The first party Alice measures her qubits and the second
party Bob then measures his collapsed qubits to detect the changes in his
collapsed states in a statistical sense, by collecting measurement statistics.
The proofs of the no communication theorem show that the reduced density
matrix of Bob remains the same irrespective of Alice’s measurement. As a
result, the expectation value of any measurement operator Bob uses is
identical irrespective of Alice’s measurement. We argue that these proofs of
the ‘no communication’ theorem do not rule out communication through state
discrimination, when Bob identifies the single collapsed states due to Alice’s
measurements.

The rest of the paper is organized as follows. We discuss communication
through state discrimination in Section 2, we discuss the two algorithms in
Section 3, and we conclude in Section 4.

##  2 Communication through state discrimination

Quantum mechanics is a statistical theory as the measurement results can only
give a probability distribution of possible outcomes. However, there are
specific instances when a single measurement result on a state can give
important information when a priori information is available about the
measured state [17]. We argue that though the proof of the ‘no communication’
theorem is correct in a statistical sense, it does not prevent communication
through discrimination of quantum states.

We reduce the communication problem to a state discrimination problem in the
following way. Alice and Bob share one qubit each of a Bell state
$\frac{1}{2}(\ket{00}+\ket{11})$. When Alice makes a measurement on her qubit,
she can collapse the state of Bob’s qubit in two different ways, in the first
case by measuring in the computational basis and in the second case measuring
in the computational basis after applying Hadamard transform to her qubit.
Alice’s first measurement in the computational basis collapses Bob’s qubit to
the set of states $S_{1}=\\{\ket{0},\ket{1}\\}$, and Alice’s second
measurement in the computational basis after applying Hadamard transform to
her qubit collapses Bob’s qubit to the set of states
$S_{2}=\\{\frac{1}{2}(\ket{0}+\ket{1}),\frac{1}{2}(\ket{0}-\ket{1}\\}$. Hence
the result of Alice’s measurement is the collapse of Bob’s state to either a
state from $S_{1}$ or a state from $S_{2}$. Note that Bob’s reduced density
matrix for these two sets are identical, and Bob cannot differentiate between
these two sets statistically by using a local measurement operator. However,
Bob knows these two possibilities a priori and Bob’s job is to detect which
set his state belongs to after each of Alice’s measurements. If Bob detects
his state to be in $S_{1}$, Alice is communicating the classical bit $0$, and
if Bob’s state is in $S_{2}$, Alice is communicating the classical bit $1$. We
show that Bob can detect this either probabilistically or deterministically in
two different algorithms, proving that it is possible for Alice and Bob to
communicate even when they are at spacelike separated locations. We discuss
these two algorithms in the next section and discuss the relevant work on
state discrimination in this section.

The state discrimination problem is trivial if the two states $\ket{\psi_{1}}$
and $\ket{\psi_{2}}$ Alice prepares are orthogonal, as Bob can determine the
state in a single measurement. Hence distinguishing between non-orthogonal
states is the challenge, i.e.,when $\braket{\psi_{1}}{\psi_{2}}\neq 0$. It is
impossible to discriminate between non-orthogonal states in general [16].
Chefles [18] showed that it is also impossible to distinguish between states
that are linearly dependent. Ivanovic [19] was the first to consider this
problem, and showed that it is possible to discriminate between non-orthogonal
states if inconclusive results are allowed, in other words two non-orthogonal
states cannot be discriminated perfectly. Ivanovic’s procedure can be
implemented using von Neumann measurements. If the measurement directions are
$\ket{\psi_{1}}$ and $\ket{\psi_{1}^{\perp}}$, where $\ket{\psi_{1}^{\perp}}$
is orthogonal to $\ket{\psi_{1}}$, a success in measuring
$\ket{\psi_{1}^{\perp}}$ in the $\\{\ket{\psi_{1}},\ket{\psi_{1}^{\perp}}\\}$
basis will indicate with certainty that the state is $\ket{\psi_{2}}$.
However, a measurement result of ${\ket{\psi_{1}}}$ is inconclusive (the same
argument can be repeated with a $\\{\ket{\psi_{2}},\ket{\psi_{2}^{\perp}}\\}$
basis. Ivanovic [19] showed that a sequence of measurements (potentially
infinite in number) can improve the detection probability. Later Dieks [20]
and Peres [21] showed that it is possible to implement Ivanovic’s scheme with
a single POVM (positive operator valued measurement), though the measurement
can be inconclusive with some probability. Peres [21] showed that the
probability of inconclusive measurement is $|\braket{\psi_{1}}{\psi_{2}}|$,
and this is optimal [24].

Neumark’s theorem [22, 23] states that every POVM can be realised as von
Neumann projective measurements in a larger Hilbert space compared to the
Hilbert space of the original system. These larger Hilbert spaces are usually
constructed by using additional qubits called ancilla [23]. Our problem is to
distinguish between a state in $S_{1}$, and a state in $S_{2}$. The states in
these two sets are linearly independent, but not orthogonal. Our approach is
to use a two level ancilla system, we first increase the dimenionality of the
system by entangling the original states in $S_{1}$ and $S_{2}$ with an
ancilla qubit, using the CNOT gate. The aim is to make the systems in $S_{2}$
orthogonal to the systems in $S_{1}$. The systems in $S_{2}$ are now the two
Bell states $\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ and
$\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$. Note that these two states are still
not orthogonal to the states in $S_{1}$, $\ket{00}$ and $\ket{11}$ (after
entangling with the first ancilla). However, applying the Hadamard transform
to the two qubits of the state $\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$, gives
us the state $\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$ in the Hadamard basis
[17], when $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ and
$\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$. This can be shown either by
direct calculation, by replacing $\ket{0}=\frac{1}{\sqrt{2}}(\ket{+}+\ket{-})$
and $\ket{1}=\frac{1}{\sqrt{2}}(\ket{+}-\ket{-})$, or by applying the
$4$-dimensional Hadamard transform $H\otimes
H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\otimes\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&-1&1&-1\\\
1&1&-1&-1\\\ 1&-1&-1&1\end{pmatrix}$. Hence, $H^{\otimes
2}\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\
1&-1&1&-1\\\ 1&1&-1&-1\\\
1&-1&-1&1\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ 0\\\ 0\\\
-1\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 1\\\ 1\\\
0\end{pmatrix}$. This state is $\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$ in the
Hadamard basis. It can be seen easily that $H^{\otimes 2}\ket{00}=\ket{++}$
and $H^{\otimes 2}\ket{11}=\ket{--}$. Hence, the Hadamard transform of
$\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$ is orthogonal to the Hadamard
transforms of the states in $S_{1}$. This can be detected by using another
ancilla qubit. Though $\frac{1}{\sqrt{2}}(\ket{++}+\ket{--})$ is not
orthogonal to the Hadamard transforms of the states in $S_{1}$, we can flip
the relative phase of $\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ for converting it
to the state $\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$ and subsequently to the
state $\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$. We give the details in the next
section.

##  3 The algorithms

Our algorithms use entanglement for collapsing Bob’s state in two different
ways by Alice depending on the classical bit $0$ or $1$ she wants to send. Bob
then identifies these different states by using ancilla qubits. Our algorithms
have different qubit-circuit tradeoffs. Bob’s first algorithm can detect the
two classsical bits with probability of error $<\frac{1}{2^{k}}$ if Alice and
Bob share $k$ Bell states for communicating a single classical bit, and Bob
uses $k$ ancilla qubits. Bob can detect the classical bit deterministically
with a more complex circuit and $4$ ancilla qubits in the second algorithm. As
discussed before, our algorithms are based on discrimination of individual
collapsed states at Bob’s end. The current lower bound of ‘spooky action at a
distance’, or speed of collapse of entanglement is four orders of magnitude of
the speed of light [25], hence our algorithms can be used for communicating
between spacelike separated locations.

###  3.1 Alice’s encoding algorithm

We consider communication as a mechanism to send an ordered sequence of
classical bits by the sender to the receiver. Alice wants to send a sequence
of classical bits of her choice to the receiver Bob. Alice prepares maximally
entangled Bell states $\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$
for sending classical bits. Alice keeps the first qubit of $\ket{\Phi^{+}}$
and Bob takes the second qubit to a spacelike separated location. Alice
measures her qubit in two different ways when she wants to send the classical
bit $0$ or $1$.

If Alice wants to send $0$, she measures her qubit in the computational basis.
She gets the state $\ket{0}$ with probability $\frac{1}{2}$ and Bob’s qubit is
in the state $\psi_{1}=\ket{0}$. Alice gets the state $\ket{1}$ with
probabilty $\frac{1}{2}$ and Bob’s qubit is in the state $\psi_{2}=\ket{1}$.

If Alice wants to send $1$, she first applies Hadamard transform to her qubit
and then measures it in the computational basis. The result of applying
Hadamard transform is:
$\frac{1}{2}(\ket{0}+\ket{1})\ket{0}+\frac{1}{2}(\ket{0}-\ket{1})\ket{1}=\frac{1}{2}\ket{0}(\ket{0}+\ket{1})+\frac{1}{2}\ket{1}(\ket{0}-\ket{1})$.
Hence Bob’s state collapses to either
$\ket{\psi_{3}}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$, or
$\ket{\psi_{4}}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$ after Alice measures her
qubit in the computational basis.

###  3.2 Bob’s decoding algorithms

We use the single qubit Hadamard transform
$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$, and two qubit
Hadamard transform $H^{\otimes 2}=H\otimes
H=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&-1&1&-1\\\ 1&1&-1&-1\\\
1&-1&-1&1\end{pmatrix}$. We also use the phase-flip or $Z$ transform (Pauli
$\sigma_{z}$ operator) $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$. We need the
two Bell states $\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$,
$\ket{\Phi^{-}}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$. The single qubit basis
states in the Hadamard basis are $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$
and $\ket{-}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$. Conversely, the
computational basis states in the Hadamard basis are
$\ket{0}=\frac{1}{\sqrt{2}}(\ket{+}+\ket{-})$ and
$\ket{1}=\frac{1}{\sqrt{2}}(\ket{+}-\ket{-})$. It can be verified easily by
substituting for $\ket{0}$ and $\ket{1}$ that the Hadamard transforms of the
two Bell states are: $H^{\otimes
2}\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{++}+\ket{--})$ and $H^{\otimes
2}\ket{\Phi^{-}}=\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$.

We will use the CNOT gate both in the computational and Hadamard bases. The
first qubit of a two-qubit CNOT gate is called the control qubit, and the
second qubit is the target qubit. The target qubit is flipped when the control
qubit is $\ket{1}$, the target qubit remains the same when the control qubit
is $\ket{0}$. The CNOT gate in the Hadamard basis has a reverse effect on the
control and the target qubits. The control is flipped when the target qubit is
$\ket{-}$, and the control remains the same when the target is $\ket{+}$. For
example, (assuming the first qubit as control and the second as target)
$\ket{+-}=\frac{1}{2}(\ket{0}+\ket{1})(\ket{0}-\ket{1})=\frac{1}{2}(\ket{00}-\ket{01}+\ket{10}-\ket{11})\xRightarrow{CNOT}\frac{1}{2}(\ket{00}-\ket{01}+\ket{11}-\ket{10})=\ket{--}$.
Similarly,
$\ket{--}=\frac{1}{2}(\ket{0}-\ket{1})(\ket{0}-\ket{1})=\frac{1}{2}(\ket{00}-\ket{01}-\ket{10}+\ket{11})\xRightarrow{CNOT}\frac{1}{2}(\ket{00}-\ket{01}-\ket{11}+\ket{10})=\ket{+-}$.
It is easy to show that $\ket{++}\xRightarrow{CNOT}\ket{++}$, and
$\ket{-+}\xRightarrow{CNOT}\ket{-+}$.

(a) Bob needs $Nk$ qubits for receiving $N$ classical bits.

$\Qcircuit{\lstick{\ket{\psi}}&\ctrl{+1}\gate{H}\meter\cw\\\
\lstick{\ket{0}}\targ\gate{H}\meter\cw}$ (b) Bob’s first decoding circuit.

Figure 1: Illustration for Bob’s first decoding algorithm.

####  3.2.1 Bob’s first algorithm

Alice and Bob share $k$ Bell states $\ket{\Phi^{+}}$ for each classical bit
that Alice wants to send, as shown in Fig. 1(a). Bob recovers the classical
bit with high probability depending on the value of $k$. The circuit is shown
in Fig. 1(b).

For $\ket{\psi_{1}}$, the result of applying the first CNOT gate is
$\ket{00}$. The two Hadamard transforms give the state $\ket{++}$; for
$\ket{\psi_{2}}$, the state after the Hadamard transforms is $\ket{--}$; for
$\ket{\psi_{3}}$, the CNOT operation creates the state $\ket{\Phi^{+}}$ and
the Hadamard gates give the state $\frac{1}{\sqrt{2}}(\ket{++}+\ket{--})$. The
two measurements are done in the Hadamard basis. Hence, there is no way of
distinguishing between these three states through these two measurements, as
the measurement results will be either both $\ket{+}$, or both $\ket{-}$.
However, if Alice is trying to send the classical bit $1$, she has a
probability of $\frac{1}{2}$ to collapse Bob’s state to $\ket{\psi_{3}}$ and
probability of $\frac{1}{2}$ to collapse to $\ket{\psi_{4}}$. If the state
collapses to $\ket{\psi_{4}}$, the CNOT gate produces the state
$\ket{\Phi^{-}}$, and the two Hadamard gates produce the state
$\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$. Hence the two measurement results
will be either $\ket{+},\ket{-}$, or $\ket{-},\ket{+}$. This algorithm
requires multipe qubits for sending a single classical bit. If Alice wants to
send the classical bit $1$, the probability that Bob’s state always collapses
to $\ket{\psi_{3}}$ is $\frac{1}{2^{k}}$ for $k$ qubits. Hence the probability
that Bob will always measure $\ket{+},\ket{+}$, or $\ket{-},\ket{-}$ is
$\frac{1}{2^{k}}$ for $k$ qubits. Hence, if Alice uses $k$ qubits to send a
single classical bit, the probability that Bob will get the wrong classical
bit is $<\frac{1}{2^{k}}$, which is $<0.001$ for $k=10$. Bob needs $k$ ancilla
qubits for decoding each classical bit.

####  3.2.2 Bob’s second algorithm

However, Bob can deterministically get the correct classical bit sent by
Alice, if he uses a more complex circuit and some ancilla qubits. The circuit
is shown in Fig. 2. Both the measurements are done in the Hadamard basis. The
circuit detects the state $\ket{-}$ in the first measurement if the input
state is $\ket{\psi_{4}}$, hence Bob can detect the classical bit $1$ if
Alice’s measurement has collapsed Bob’s state to $\ket{\psi_{4}}$. However,
the first measurement is inconclusive if the measurement result is $\ket{+}$,
all three states $\ket{\psi_{1}}$, $\ket{\psi_{2}}$ and $\ket{\psi_{3}}$ give
this result. The rest of the circuit is basically a repeat of the first part
after restoring the original state and applying the $Z$ gate to flip the
relative phase. This phase flip has no effect on $\ket{\psi_{1}}$, introduces
a global phase for $\ket{\psi_{2}}$, which again has no effect. But transforms
$\ket{\psi_{3}}$ to $\ket{\psi_{4}}$. Hence the result of the second
measurement is $\ket{-}$ if the input state is $\ket{\psi_{3}}$. The
measurement results are both $\ket{+}$ if the input state is either
$\ket{\psi_{1}}$ or $\ket{\psi_{2}}$. These measurement results are summarized
in Table 1. We now discuss the details of the circuit in Figure 2.

$\ket{\psi_{1}}$: The first CNOT gate gives the state $\ket{00}$, and the two
Hadamard transforms give the state $\ket{++}$. The second ancilla qubit is in
the state $\ket{+}$ after the Hadamard transform, and the two CNOT gates with
the second ancilla qubit as control do not change the second ancilla qubit
from the discussion above. Hence the first measurement result is $\ket{+}$.
The two Hadamard gates change the state to $\ket{00}$. The state is $\ket{00}$
after the CNOT gate. The input qubit is restored to $\ket{\psi_{1}}=\ket{0}$,
and the $Z$ gate has no effect. The rest of the circuit is identical to the
first part of the circuit, hence the second measurement result is $\ket{+}$.

$\Qcircuit<EMAIL_ADDRESS>\lstick{\ket{0}}\targ\gate{H}\qw\targ\qw\gate{H}\targ\qw\\\ \lstick{\ket{0}}\qw\gate{H}\ctrl{-2}\ctrl{-1}\meter\cw\\\ \lstick{\ket{0}}\qw\qw\qw\qw\qw\qw\qw\qw\targ\gate{H}\qw\targ\qw\\\ \lstick{\ket{0}}\qw\qw\qw\qw\qw\qw\qw\qw\qw\gate{H}\ctrl{-4}\ctrl{-1}\meter\cw}$ Figure 2: Bob’s second decoding circuit. state | first measurement | second measurement
---|---|---
$\ket{\psi_{1}}$ | $\ket{+}$ | $\ket{+}$
$\ket{\psi_{2}}$ | $\ket{+}$ | $\ket{+}$
$\ket{\psi_{3}}$ | $\ket{+}$ | $\ket{-}$
$\ket{\psi_{4}}$ | $\ket{-}$ | not required
Table 1: The measurement results for the second circuit.

$\ket{\psi_{2}}$: The first CNOT gate gives the state $\ket{11}$, and the two
Hadamard gates give the state $\ket{--}$. The second ancilla qubit is
$\ket{+}$ after the Hadamard transform. It is flipped by the first qubit to
$\ket{-}$ by the first CNOT gate and flipped back to $\ket{+}$ by the second
CNOT gate. Hence the first measurement result is $\ket{+}$. The two Hadamard
gates transform the state to $\ket{11}$. The next CNOT gate gives the state
$\ket{10}$, and hence the input qubit is restored to $\ket{\psi_{2}}=\ket{1}$.
The $Z$ gate introduces a global phase and the state is $-\ket{1}$. The global
phase has no effect on the second part of the circuit and the second
measurement result is also $\ket{+}$.

$\ket{\psi_{3}}$: The first CNOT gate gives the entangled state
$\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}$. The two Hadamard gates give the state
$\frac{1}{\sqrt{2}}(\ket{++}+\ket{--})$. The first part of the entangled state
has no effect on the second ancilla qubit and the second ancilla qubit is
flipped twice by the second part and is in the state $\ket{+}$ after the two
CNOT gates. Hence the first measurement result is $\ket{+}$. The two Hadamard
transforms transform the state to $\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. The
CNOT gate breaks the entanglement and restores the input qubit to
$\ket{\psi_{3}}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$. The $Z$ gate transforms
the state to $\ket{\psi_{4}}=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$, and the
effect of the second part of the circuit is identical to the case of
$\ket{\psi_{4}}$ below.

$\ket{\psi_{4}}$: The first CNOT gate gives the state
$\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$. The Hadamard transforms of the two
qubits gives the state $\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$. Each term of
this state flips the second ancilla qubit $\ket{+}$ exactly once due to the
two CNOT gates. Hence the state of the ancilla qubit after the two CNOT gates
is $\ket{-}$, and the first measurement in the Hadamard basis gives $\ket{-}$.
There is no need for the second measurement in this case.

##  4 Conclusion

We have presented two algorithms for communication between spacelike separated
locations. The algorithms have different qubit-circuit tradeoffs. Bob’s first
decoding algorithm requires a simple circuit, but $k$ entangled pairs and $k$
ancilla qubits for transmitting a single classical bit. The second algorithm
requires a more complex circuit and four ancilla qubits. Actually the circuit
in Figure 2 can be modified to reuse the second ancilla qubit if the first
measurement result is inconclusive and hence three ancilla qubits are
sufficient. According to Chefles’ theorem [18], two states can be
unambiguously discriminated if and only if they are linearly independent. But
the states have to be transformed to an orthogonal set before von Neumann
measurement is performed. In the second algorithm, only the state
$\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})$ transformed to
$\frac{1}{\sqrt{2}}(\ket{+-}+\ket{-+})$ is orthogonal to both $\ket{++}$ and
$\ket{--}$, and hence can be discriminated in the first measurement. The state
$\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ transformed to
$\frac{1}{\sqrt{2}}(\ket{++}+\ket{--})$ is not orthogonal to $\ket{++}$ and
$\ket{--}$, hence the relative phase of $\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$
needs to be flipped and the second measurement is necessary.

One important consequence of our algorithm is that there is no need for
classical communication in the quantum teleportation algorithm [26], if Alice
and Bob share extra entangled pairs. Alice can send Bob her two measurement
results and Bob can get the two classical bits using Bob’s second decoding
algorithm in this paper.

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