# Static and spherically symmetric black holes in gravity with a background
Kalb-Ramond field

Ke<EMAIL_ADDRESS>corresponding author, Yue-Zhe Chen, Zheng-Qiao
Duan, Ju-Ying Zhao  School of Physical Science and Technology, Southwest
University, Chongqing 400715, China

###### Abstract

The Lorentz symmetry of gravity is spontaneously broken when the non-minimally
coupled Kalb-Ramond field acquires a nonzero vacuum expectation value. In this
work, we present exact solutions for static and spherically symmetric black
holes in the framework of this Lorentz-violating gravity theory. In order to
explore the physical implications of Lorentz violation, we analyze the
thermodynamic properties of the obtained solutions and evaluate its impact on
some classical gravitational experiments within the Solar System. Furthermore,
the Lorentz-violating parameter is constrained by using the measured results
of these experiments.

###### pacs:

04.70.-s, 04.50.Kd

##  I Introduction

As a fundamental concept in modern physics, Lorentz symmetry states that the
laws of physics are the same in all inertial reference frames. Although
Lorentz symmetry is widely believed to be a fundamental symmetry of the nature
and has been confirmed by numerous experiments and observations, it has been
revealed that Lorentz symmetry may be broken at some energy scale in plenty of
theories, such as string theory Kostelecky1989a , loop quantum gravity
Alfaro2002 , Horava-Lifshitz gravity Horava2009a , noncommutative field theory
Carroll2001 , Einstein-æther theory Jacobson2001 , massive gravity
Dubovsky2005 , $f(T)$ gravity Bengochea2009 , very special relativity
Cohen2006 , and so on. The study of Lorentz symmetry breaking (LSB) is
important for understanding fundamental physical processes in high-energy
physics and gravity.

LSB can occur either explicitly or spontaneously. The explicit LSB occurs when
the Lagrange density is not Lorentz invariant, i.e., the physical laws have
different forms in certain reference frames. However, the spontaneous LSB
occurs when Lagrange density is Lorentz invariant, but the ground state of a
physical system does not exhibit Lorentz symmetry. A general framework for
studying the spontaneous LSB is the Standard-Model Extension Kostelecky2004a .
The simplest field theories proposed within this framework are known as
bumblebee models Kostelecky1989a ; Kostelecky1989 ; Kostelecky1989b ;
Bailey2006 ; Bluhm2008a . In bumblebee models, a vector field known as the
bumblebee field acquires a nonzero vacuum expectation value (VEV). The nonzero
VEV selects a specific direction, leading to the violation of particle local
Lorentz invariance.

An exact solution for a static and spherically symmetric spacetime in
bumblebee gravity model was reported by Casana et al. in Ref. Casana2018 .
This Schwarzschild-like solution has been extensively studied in various
aspects, such as the gravitational lensing Ovgun2018 , Hawking radiation
Kanzi2019 , accretion process Yang2019b ; Cai2022a , quasinormal modes
Oliveira2021 . Subsequently, Maluf et al. obtained an (A)dS-Schwarzschild-like
solution by relaxing the vacuum conditions Maluf2021 . Xu et al. found new
classes of static spherical bumblebee black holes by incorporating a
background bumblebee field with a non-vanishing temporal component Xu2023 .
The associated thermodynamic properties and observational implications were
studies in Refs. Mai2023 ; Xu2023a ; Liang2023 . Rotating bumblebee black
holes were investigated by Ding et al. in Refs. Ding2020a ; Ding2021a . The
shadow Wang2022 , accretion process Liu2019 , quasinormal modes Liu2023 , and
quasi-periodic oscillations Wang2022a  were discussed in these rotating
bumblebee black holes. An exact rotating BTZ-like black hole solution was
obtained in Ref. Ding2023  and its quasinormal modes were studied analytically
in Ref. Chen2023 . A Schwarzschild-like black hole with a global monopole was
presented in Ref. Gullu2022 , and its quasinormal modes were analyzed in Refs.
Zhang2023 ; Lin2023 . Moreover, other black hole solutions in the framework of
Bumblebee gravity model were investigated in Refs. Ding2021 ; Jha2021 ;
Ding2023a . A traversable bumblebee wormhole solution was found in Ref.
Ovgun2019 . In addition, the gravitational waves in bumblebee gravity were
analyzed in Refs. Liang2022 ; Amarilo2023 .

Instead of a vector field, a rank-two antisymmetric tensor field, called Kalb-
Ramond (KR) field, has also been considered as a source for LSB Altschul2010 .
The KR field emerges in the spectrum of bosonic string theory Kalb1974 . When
the KR field non-minimally couples to gravity and acquires a nonzero VEV, the
Lorentz symmetry is spontaneously broken. An exact static and spherically
symmetric solution was reported by Lessa et al. under the VEV background of KR
field Lessa2020 . Subsequently, the motion of massive and massless particles
in the vicinity of this static spherical KR black hole was studied in Ref.
Atamurotov2022 . The gravitational deflection of light and shadow cast by the
rotating KR black holes were further investigated in Ref. Kumar2020c .
Moreover, traversable wormhole solutions were constructed in this theory
Lessa2021 ; Maluf2022 , and the implications of the VEV background on Bianchi
type I cosmology were explored Maluf2022a .

In this work, we focus on the construction of new exact solutions for the
static and spherically symmetric spacetime in the presence and absence of the
cosmological constant under the nonzero VEV background of KR field. The layout
of the paper is as follows: In Sect. II, we solve the theory to obtain
analytic solutions for the static and spherically symmetric spacetime,
considering both the cases with and without the cosmological constant. In
Sect. III, some basic thermodynamic properties of the obtained KR black holes
are analyzed. In Sect. IV, we impose constraints on the theoretical parameter
associated with Lorentz-breaking effects through classical Solar System tests.
Finally, brief conclusions are present.

##  II KR Black Hole solutions

We start from the Einstein-Hilbert action non-minimally coupled to a self-
interacting KR field in the form Altschul2010 ; Lessa2020

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int{}d^{4}x\sqrt{-g}\Bigg{[}R-2\Lambda-\frac{1}{6}H^{\mu\nu\rho}H_{\mu\nu\rho}-V(B^{\mu\nu}B_{\mu\nu})+\xi_{2}B^{\rho\mu}B^{\nu}{}_{\mu}R_{\rho\nu}+\xi_{3}B^{\mu\nu}B_{\mu\nu}R\Bigg{]}$
(1) $\displaystyle+\int{d^{4}x\sqrt{-g}\mathcal{L}_{\text{M}}},$

where $\kappa=8\pi G$ with $G$ the Newtonian constant of gravitation,
$\xi_{2}$ and $\xi_{3}$ are the coupling constants between the gravity and the
KR field, $H_{\mu\nu\rho}\equiv\partial_{[\mu}B_{\nu\rho]}$ is the strength of
KR field, and $\Lambda$ is the cosmological constant. The self-interaction
potential $V(B^{\mu\nu}B_{\mu\nu})$ depends on $B^{\mu\nu}B_{\mu\nu}$ in order
to maintain the theory invariant upon observer local Lorentz transformations.
As the cosmological constant $\Lambda$ is accounted for separately in the
action, the potential is chosen to be zero at its minimum.

By varying the action (1) with respect to the metric $g^{\mu\nu}$, one obtains
the gravitational filed equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda
g_{\mu\nu}=T^{\text{KR}}_{\mu\nu}+T^{\text{M}}_{\mu\nu},$ (2)

where $T^{\text{M}}_{\mu\nu}$ is the energy-momentum tensor of matter fields,
and

$\displaystyle T^{\text{KR}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}H_{\mu\alpha\beta}H_{\nu}^{\alpha\beta}-\frac{1}{12}g_{\mu\nu}H^{\alpha\beta\rho}H_{\alpha\beta\rho}+2V^{\prime}(X)B_{\alpha\mu}B^{\alpha}{}_{\nu}-g_{\mu\nu}V(X)$
(3)
$\displaystyle+\xi_{2}\bigg{[}\frac{1}{2}g_{\mu\nu}B^{\alpha\gamma}B^{\beta}{}_{\gamma}R_{\alpha\beta}-B^{\alpha}{}_{\mu}B^{\beta}{}_{\nu}R_{\alpha\beta}-B^{\alpha\beta}B_{\nu\beta}R_{\mu\alpha}-B^{\alpha\beta}B_{\mu\beta}R_{\nu\alpha}$
$\displaystyle+\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}\left(B^{\alpha\beta}B_{\nu\beta}\right)+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}\left(B^{\alpha\beta}B_{\mu\beta}\right)-\frac{1}{2}\nabla^{\alpha}\nabla_{\alpha}\left(B_{\mu}{}^{\gamma}B_{\nu\gamma}\right)$
$\displaystyle-\frac{1}{2}g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}\left(B^{\alpha\gamma}B^{\beta}{}_{\gamma}\right)\bigg{]},$

where the prime represents the derivative with respect to the argument of the
corresponding functions. With the Bianchi identities, it is clear that the
total energy-momentum tensor $T^{\text{KR}}_{\mu\nu}+T^{\text{M}}_{\mu\nu}$ is
conserved.

In order to generate a non-vanishing VEV for the KR field, $\langle
B_{\mu\nu}\rangle=b_{\mu\nu}$, we assume a potential with a general form given
by $V=V\left(B^{\mu\nu}B_{\mu\nu}\pm b^{2}\right)$, where the sign $\pm$ is
chosen such that $b^{2}$ is a positive constant Bluhm2008 ; Altschul2010 ;
Lessa2020 . Consequently, the VEV is determined by the constant norm condition
$b^{\mu\nu}b_{\mu\nu}=\mp b^{2}$. Upon the vacuum condensation, the gauge
invariance $B_{\mu\nu}\to B_{\mu\nu}+\partial_{[\mu}\Gamma_{\nu]}$ of the KR
field is spontaneously broken. Due to the non-minimal coupling of the KR field
to gravity, the symmetry-breaking VEV background leads to the violation of
particle local Lorentz invariance. Furthermore, the term
$\xi_{3}B^{\mu\nu}B_{\mu\nu}R$ in the action (1) transforms to
$\mp\xi_{3}b^{2}R$ in the vacuum, which can be absorbed into Einstein-Hilbert
terms through a redefinition of variables.

It is convenient to decompose the antisymmetric tensor $B_{\mu\nu}$ to be
$B_{\mu\nu}=\tilde{E}_{[\mu}v_{\nu]}+\epsilon_{\mu\nu\alpha\beta}v^{\alpha}\tilde{B}^{\beta}$
with $v^{\alpha}$ a timelike 4-vector Altschul2010 ; Lessa2020 . The pseudo-
fields $\tilde{E}_{\mu}$ and $\tilde{B}_{\mu}$ are spacelike, satisfying
$\tilde{E}_{\mu}v^{\mu}=\tilde{B}_{\mu}v^{\mu}=0$. Analogous to Maxwell
electrodynamics, these fields can be interpreted as the pseudo-electric and
pseudo-magnetic fields, respectively. Assuming that the only non-vanishing
terms are $b_{10}=-b_{01}=\tilde{E}(r)$ in the VEV, or equivalently,
$\mathbb{b}_{2}=-\tilde{E}(r)dt\wedge dr$ in terms of differential forms, it
follows that the vacuum field exhibits a pseudo-electric configuration.
Consequently, this configuration automatically vanishes the KR field strength,
i.e., $H_{\lambda\mu\nu}=0$ or $\mathbb{H}_{3}=d\mathbb{b}_{2}=0$.

Here, our focus is on investigating a static and spherically symmetric
spacetime under the nonzero VEV background of KR field. The metric ansatz is
given by

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (4)

Correspondingly, the pseudo-electric field $\tilde{E}(r)$ can be rewrite as
$\tilde{E}(r)=|b|\sqrt{A(r)B(r)/2}$, such that the constant norm condition
$b^{\mu\nu}b_{\mu\nu}=-b^{2}$ is satisfied.

Under the VEV configuration, it is advantageous to reformulate the field
equation as

$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle\Lambda
g_{\mu\nu}+V^{\prime}\left(2b_{\mu\alpha}b_{\nu}{}^{\alpha}+b^{2}g_{\mu\nu}\right)+\xi_{2}\bigg{[}g_{\mu\nu}b^{\alpha\gamma}b^{\beta}{}_{\gamma}R_{\alpha\beta}-b^{\alpha}{}_{\mu}b^{\beta}{}_{\nu}R_{\alpha\beta}-b^{\alpha\beta}b_{\mu\beta}R_{\nu\alpha}$
(5)
$\displaystyle-b^{\alpha\beta}b_{\nu\beta}R_{\mu\alpha}+\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}\left(b^{\alpha\beta}b_{\nu\beta}\right)+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}\left(b^{\alpha\beta}b_{\mu\beta}\right)-\frac{1}{2}\nabla^{\alpha}\nabla_{\alpha}\left(b_{\mu}{}^{\gamma}b_{\nu\gamma}\right)\bigg{]}.$

With the metric ansatz, the field equations (5) can be written explicitly as

$\displaystyle\frac{2A^{\prime\prime}}{A}-\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}-\frac{A^{\prime
2}}{A^{2}}+\frac{4}{r}\frac{A^{\prime}}{A}+\frac{4\Lambda}{1-\ell}B$
$\displaystyle=$ $\displaystyle 0,$ (6a)
$\displaystyle\frac{2A^{\prime\prime}}{A}-\frac{A^{\prime}}{A}\frac{B^{\prime}}{B}-\frac{A^{\prime
2}}{A^{2}}-\frac{4}{r}\frac{B^{\prime}}{B}+\frac{4\Lambda}{1-\ell}B$
$\displaystyle=$ $\displaystyle 0,$ (6b)
$\displaystyle\frac{2A^{\prime\prime}}{A}-\frac{A^{\prime}B^{\prime}}{AB}-\frac{A^{\prime
2}}{A^{2}}+\frac{1+\ell}{\ell
r}\left(\frac{A^{\prime}}{A}-\frac{B^{\prime}}{B}\right)-\left(1-\Lambda
r^{2}-b^{2}r^{2}V^{\prime}\right)\frac{2B}{\ell r^{2}}+\frac{2(1-\ell)}{\ell
r^{2}}$ $\displaystyle=$ $\displaystyle 0,$ (6c)

where $\ell\equiv\xi_{2}b^{2}/2$.

###  II.1 Absence of cosmological constant $\Lambda=0$

First, by considering the spacetime without the cosmological constant, we
would like to construct a Schwarzschild-like black hole in the theory. In this
case, we take the assumption that $V^{\prime}=0$, which corresponds to the
case where the VEV is located at the local minimum of the potential. For
instance, it can be simply realized by a potential of quadratic form,
$V=\frac{1}{2}\lambda X^{2}$, with $X\equiv B^{\mu\nu}B_{\mu\nu}+b^{2}$ and
$\lambda$ a coupling constant Bluhm2008 .

By subtracting Eq. (6b) from Eq. (6a), the following relation is obtained

$\displaystyle\frac{A^{\prime}(r)}{A(r)}+\frac{B^{\prime}(r)}{B(r)}=0.$ (7)

It simply yields

$A(r)=\frac{1}{B(r)}.$ (8)

After subtracting Eq. (6c) from Eq. (6a) and substituting the relation (8)
into it, one obtains

$A(r)=\frac{1}{1-\ell}-\frac{c_{1}}{r},$ (9)

where the integration constant $c_{1}$ can be determined using the Komar
integral. Utilizing the Killing vector $K^{\mu}=(1,0,0,0)$ and the metric on
the two-sphere at infinity
$\gamma^{(2)}_{ij}=r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)$,
the Komar mass $M$ is given by Carroll2019

$M=-\frac{1}{4\pi}\int_{\partial\Sigma}d^{2}x\sqrt{\gamma^{(2)}}\nabla^{t}K^{r}=\frac{c_{1}}{2}.$
(10)

Thus, one arrives at

$A(r)=\frac{1}{1-\ell}-\frac{2M}{r}.$ (11)

It is straightforward to verify that this set of solutions (8) and (11)
satisfies all the equations of motion (6).

Consequently, a Schwarzschild-like metric is obtained222Note that the solution
we obtained differs from that presented in the Ref. Lessa2020 .

$ds^{2}=-\left(\frac{1}{1-\ell}-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\frac{1}{1-\ell}-\frac{2M}{r}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (12)

The dimensionless parameter $\ell$ characterizes the effect of Lorentz
violation caused by the nonzero VEV of KR filed on spacetime. Due to strong
constraints from experimental observations on Lorentz-violating effects in
gravitational fields, which will be discussed later, the Lorentz-violating
parameter $\ell$ is supposed to be very small. Note that, due to the non-
vanishing of the Riemann tensor in the limit $r\to\infty$, the spacetime is
not asymptotically Minkowski in this case. Furthermore, the Kretschmann scalar
for the current spacetime is given by

$R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}=\frac{48M^{2}}{r^{6}}-\frac{16\ell
M}{(1-\ell)r^{5}}+\frac{4\ell^{2}}{(\ell-1)^{2}r^{4}}.$ (13)

This implies that the Lorentz-violating effects cannot be eliminated by mere
coordinate transformations.

It is interesting that the black hole described by the metric (12) possesses a
horizon radius $r_{\text{h}}=2(1-\ell)M$, which is shifted by the Lorentz-
violating parameter $\ell$. This is different from the case of bumblebee black
hole, where the horizon radius is identical to that of a Schwarzschild black
hole Casana2018 . At the horizon, the Kretschmann scalar is finite, given by
$R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}=\frac{3-(2-\ell)\ell}{4(1-\ell)^{6}M^{4}}$,
indicating that the singularity at the horizon can be removed through a
coordinate transformation. However, the Kretschmann scalar diverges at the
center $r=0$, signifying an intrinsic and non-removable singularity at that
point.

###  II.2 Presence of cosmological constant $\Lambda\neq 0$

When the cosmological constant is present, it is found that the assumption
$V^{\prime}=0$ does not support a self-consistent solution that satisfies all
the equations of motion. Therefore, following the approach in Ref. Maluf2021 ,
the vacuum conditions is relaxed to be $V=0$ but $V^{\prime}\neq 0$. The most
commonly considered potential form that satisfies the condition is a linear
form, given by $V=\lambda X$, where $\lambda$ is a Lagrange multiplier field
Bluhm2008 . Consequently, the derivative of the potential with respect to $X$
is $V^{\prime}(X)=\lambda$. The equation of motion of the Lagrange-multiplier
$\lambda$ restricts the theory to the extrema of the potential, satisfying
$X=0$, such that $b_{\mu\nu}$ is the VEV of KR field for the on-shell
$\lambda$. Note that the off-shell value of the Lagrange multiplier should
have the same sign as $X$ to keep the positivity of the potential $V$
Bluhm2008 . In principle, the Lagrange-multiplier field $\lambda$ can also be
expanded around its vacuum value, i.e.,
$\lambda=\langle\lambda\rangle+\tilde{\lambda}$, and $\langle\lambda\rangle$
could vary with spacetime position. However, it is convenient to fix
$\tilde{\lambda}=0$ by choosing the initial conditions and to assume that
$\langle\lambda\rangle$ is a real constant, then the on-shell value
$\lambda\equiv\langle\lambda\rangle$ is determined by the field equations
completely Bluhm2008 ; Maluf2021 .

Now, by subtracting Eq. (6b) from Eq. (6a), one obtains the same relation as
in the previous case,

$A(r)=\frac{1}{B(r)}.$ (14)

Furthermore, by subtracting Eq. (6c) from Eq. (6a) and substituting the
relation (14) into it, one arrives at

$A(r)=\frac{1}{1-\ell}-\frac{2M}{r}-\frac{(1-3\ell)\Lambda+(1-\ell)b^{2}\lambda}{3(1-\ell)^{2}}r^{2}.$
(15)

Finally, by substituting Eqs. (14) and (15) into (6a), the on-shell value of
$\lambda$ is determined by

$\lambda=\frac{2\ell\Lambda}{(1-\ell)b^{2}}.$ (16)

It is evident that the theory supports a solution with a non-vanishing
cosmological constant if only $V^{\prime}(X)=\lambda\neq 0$.

After inserting the on-shell value of $\lambda$ into (15), an (A)dS-
Schwarzschild-like metic is obtained

$ds^{2}=-\left(\frac{1}{1-\ell}-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3\left(1-\ell\right)}\right)dt^{2}+\frac{dr^{2}}{\frac{1}{1-\ell}-\frac{2M}{r}-\frac{\Lambda
r^{2}}{3\left(1-\ell\right)}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}.$
(17)

The solution reduces to the Schwarzschild-like metric (12) when the
cosmological constant vanishes. Likewise, it degenerates to the (A)dS-
Schwarzschild metric when the Lorentz-violating parameter $\ell$ is set to
zero. Furthermore, it is easy to demonstrate that the relation
$R_{\mu\nu\rho\sigma}=\frac{\Lambda_{\text{eff}}}{3}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})$
holds in the limit $r\to\infty$, where the effective cosmological constant
reads $\Lambda_{\text{eff}}\equiv\frac{\Lambda}{1-\ell}$. It indicates that
the spacetime approaches (A)dS at infinity. This behavior is also evident from
the metric (17), where the metric functions approximate
$A(r)=1/B(r)\to-{r^{2}\Lambda_{\text{eff}}}/{3}$ as $r\to\infty$ tends to
infinity, exhibiting the same asymptotic behavior as the (A)dS metric. Note
that the truly contributing cosmological constant is the effective one
$\Lambda_{\text{eff}}\equiv\Lambda/(1-\ell)$ due to the additional
contributions from the modified Einstein equations when $V^{\prime}(X)$ is
nonzero.

Figure 1: The parameter space $(l,\Lambda M^{2})$ for black hole solutions and
non-black hole solutions, where the light blue region represents the parameter
space of black hole solutions.

When the cosmological constant is negative, $\Lambda<0$, the event horizon
exists for any parameter, and thus the metric (17) always supports black hole
solutions in this case. However, as shown in Fig. 1, when the cosmological
constant is positive, $\Lambda>0$, the black hole solutions exist only for the
parameters $\ell$ and $\Lambda M^{2}$ satisfying $9(1-\ell)^{2}\Lambda
M^{2}\leq 1$, where the equality sign represents the case when the event
horizon and the cosmological horizon coincide with each other.

##  III Thermodynamics

Research on black holes has revealed that they possess thermodynamic
properties analogous to the laws of thermodynamics, which provides insights
into the nature of black holes and gravity. Asymptotically AdS black holes are
well-known for their close relationship with AdS/CFT duality and diverse
thermodynamic behaviors, including various phase transitions. However, the
presence of two distinct temperatures in asymptotically dS black holes renders
the system in a non-equilibrium state. Although some attempts have been made
to investigate the thermodynamics of these black holes Mbarek2019 , our
understanding in this area remains limited. For the sake of simplicity, we
primarily focus on analyzing some fundamental thermodynamic properties of the
obtained AdS-Schwarzschild-like black hole in this section.

By solving $A(r_{\text{h}})=0$ in Eq. (17), the black hole mass can be
expressed with the radius of the event horizon $r_{\text{h}}$,

$M=\frac{\left(3-\Lambda r_{\text{h}}^{2}\right)r_{\text{h}}}{6(1-\ell)}.$
(18)

Since we consider a static spherical spacetime with an effective cosmological
constant $\Lambda_{\text{eff}}$, the black hole mass $M$ is associated with
the black hole’s enthalpy. In this case, the effective cosmological constant
plays the role of a thermodynamic pressure, given by
$P=-\frac{\Lambda_{\text{eff}}}{8\pi}=-\frac{\Lambda}{8\pi(1-l)}$. Considering
that the Lorentz-violating parameter $\ell$ is a dimensionless constant, the
enthalpy can be expressed as a function of entropy $S$ and pressure $P$, i.e.,
$M=M\left(S,P\right)$. Consequently, the first law of the AdS-Schwarzschild-
like black hole reads

$dM=\mathcal{T}dS+\mathcal{V}dP,$ (19)

where $\mathcal{T}$ is the Hawking temperature and $\mathcal{V}$ the
thermodynamic volume.

By employing the metric (17) and Eq. (18), the Hawking temperature is given by

$\mathcal{T}=-\left.\frac{1}{4\pi}\frac{\partial g_{tt}}{\partial
r}\right|_{r_{\text{h}}}=\frac{1-\Lambda
r_{\text{h}}^{2}}{\left(1-\ell\right)4\pi r_{\text{h}}}.$ (20)

It is observed that the Lorentz-violating parameter $\ell$ has an impact on
the temperature of the black hole. Specifically, the temperature is higher
compared to that of an AdS-Schwarzschild black hole for a positive $\ell$,
while it is lower for a negative $\ell$.

Further, by utilizing the first law (19) and the Hawking temperature (20), the
entropy can be obtained as

$S=\int{}\left(\frac{dM}{\mathcal{T}}\right)_{P}=\int{}\frac{1}{\mathcal{T}}\left(\frac{\partial
M}{\partial r_{\text{h}}}\right)_{P}dr_{\text{h}}=\pi
r_{\text{h}}^{2}=\frac{A_{\text{h}}}{4},$ (21)

where $A_{\text{h}}=4\pi r_{\text{h}}^{2}$ is the horizon area. It is evident
that the entropy of the black hole still satisfies the Bekenstein-Hawking
area-entropy relation.

Correspondingly, the thermodynamic volume is calculated as

$\mathcal{V}=\left(\frac{\partial M}{\partial
P}\right)_{S}=\left(\frac{\partial
M}{\partial\Lambda}\right)_{S}\left(\frac{\partial\Lambda}{\partial
P}\right)_{S}=\frac{4\pi r_{\text{h}}^{3}}{3},$ (22)

which is exactly the volume of the event horizon of the black hole.

Further, utilizing the aforementioned results, the Smarr formula is obtained
as

$M=2\left(\mathcal{T}S-\mathcal{V}P\right).$ (23)

Therefore, the fist law and Smarr formula are identical to the results
obtained for the AdS-Schwarzschild black hole.

The specific heat is useful for analyzing the local stability of a black hole
solution. Specifically, a black hole is stable when its specific heat is
positive, while it is unstable when its specific heat is negative. The
specific heat is computed by

$C_{P}=\left(\frac{\partial
M}{\partial\mathcal{T}}\right)_{P}=\left(\frac{\partial M}{\partial
r_{\text{h}}}\right)_{P}\left(\frac{\partial
r_{\text{h}}}{\partial\mathcal{T}}\right)_{P}=2\pi
r_{\text{h}}^{2}\left(\frac{\Lambda r_{\text{h}}^{2}-1}{\Lambda
r_{\text{h}}^{2}+1}\right).$ (24)

Interestingly, although both the mass and temperature of the black hole
contain the Lorentz-violating parameter $\ell$, the specific heat is
independent of it. Consequently, the AdS-Schwarzschild-like black hole is
locally stable only if its radius is sufficiently large, satisfying
$r_{\text{h}}>\sqrt{-1/\Lambda}$. However, for the Schwarzschild-like black
hole with $\Lambda=0$, the negative specific heat indicates its locally
instability.

Furthermore, a black hole is globally stable if its Gibbs free energy is
negative, but it is globally unstable if its Gibbs free energy is positive.
Hence, in order to investigate the global stability of the black hole, we
further evaluate its Gibbs free energy in the canonical ensemble, which is
expressed as

$F=M-\mathcal{T}S=\frac{\left(3+\Lambda
r_{\text{h}}^{2}\right)r_{\text{h}}}{12(1-\ell)}.$ (25)

It is evident that the free energy is negative only when
$r_{\text{h}}>\sqrt{-3/\Lambda}$, indicating that only large AdS-
Schwarzschild-like black holes are globally stable. For the Schwarzschild-like
black hole, its free energy is always positive and hence it is globally
unstable. It is worth noting that despite the presence of the Lorentz-
violating parameter $\ell$ in the free energy, it does not affect the critical
size that determines the stability of the black holes.

##  IV Solar system tests

General relativity (GR) has undergone extensive testing, especially in the
weak field regime, such as within the Solar System. So far, no evidence
contradicting the theory has been found. However, it would be helpful to
evaluate the impact of the present Lorentz-violating theory by testing the
derived solutions within the Solar System. The theoretical parameter $\ell$
can be constrained by comparing the experimental results with the predictions
of the theory. Since the influence of the cosmological constant can be
neglected at the scale of the Solar System, we focus solely on considering the
Schwarzschild-like metric (12).

The motion of a test particle along its geodesics can be described by the
Lagrangian

$\mathcal{L}=-\frac{1}{2}g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta},$ (26)

where the dot represents the derivative with respect to the affine parameter
$\lambda$. From the normalization conditions for the four-velocity of timelike
and null particles, one can express the Lagrangian density as
$\mathcal{L}=\eta/2$. Accordingly, for massless photons, $\eta=0$, while for
massive particles, $\eta=1$ when the affine parameter $\lambda$ is chosen to
be the proper time $\tau$.

By substituting the metric (12) into the Lagrangian (26), one obtains

$\displaystyle\mathcal{L}=\frac{1}{2}\bigg{(}A(r)\dot{t}^{2}-A(r)^{-1}\dot{r}^{2}-r^{2}\dot{\theta}^{2}-r^{2}\sin^{2}\theta\dot{\phi}^{2}\bigg{)}.$
(27)

The Euler-Lagrange equation corresponding to $\theta$ is given by

$r^{2}\ddot{\theta}+2r\dot{r}\dot{\theta}-r^{2}\dot{\phi}^{2}\sin\theta\cos\theta=0.$
(28)

With the initial condition $\theta=\frac{\pi}{2}$ and $\dot{\theta}=0$, one
has $\ddot{\theta}=0$. It indicates that the particle will be restricted to
motion within the equatorial plane. In this case, the Lagrangian density (26)
is rewritten as

$A(r)\left(\frac{dt}{d\lambda}\right)^{2}-A(r)^{-1}\left(\frac{dr}{d\lambda}\right)^{2}-r^{2}\left(\frac{d\phi}{d\lambda}\right)^{2}=\eta.$
(29)

Since the spacetime is static and spherically symmetric, the Lagrangian is
independent of $t$ and $\phi$. As a result, there are two conserved
quantities: the energy $E$ and the angular momentum $L$, given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{t}}=A(r)\frac{dt}{d\lambda},$
(30) $\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\frac{d\phi}{d\lambda}.$
(31)

Now that we have all the required equations, we can proceed to examine three
classical tests within Solar System: the perihelion precession of Mercury, the
deflection of light by the Sun, and the Shapiro time delay.

###  IV.1 Perihelion precession of Mercury

Mercury can be viewed as a test particle, allowing us to choose the affine
parameter $\lambda$ as the proper time $\tau$. By utilizing Eqs. (29), (30)
and (31), the orbital equation is derived as

$\displaystyle\left[\frac{d}{d\phi}\left(\frac{1}{r}\right)\right]^{2}=\frac{E^{2}}{L^{2}}-\frac{A(r)}{L^{2}}\left(1+\frac{L^{2}}{r^{2}}\right).$
(32)

After redefining $u=\frac{L^{2}}{Mr}$ and differentiating the equation with
respect to $\phi$, we have

$\frac{d^{2}u}{d\phi^{2}}+\frac{u}{1-\ell}-1=\frac{3M^{2}}{L^{2}}u^{2}.$ (33)

It is observed that the coefficient of the second term has modified by the
Lorentz-violating parameter $\ell$, which is supposed to be very small,
$\ell\ll 1$. The equation can be solved using a perturbative method. Noting
that the last term, represented as $\frac{3G^{2}M^{2}}{c^{4}L^{2}}$ in the
International System of Units, is a first-order small quantity and would be
absent in the Newtonian limit Carroll2019 . Therefore, by expanding the exact
solution $u$ as $u\approx u_{0}+u_{1}$, where $u_{0}$ and $u_{1}$ are
respectively the zeroth-order and first-order approximations, satisfying
Carroll2019

$\displaystyle\frac{d^{2}u_{0}}{d\phi^{2}}+\frac{u_{0}}{1-\ell}-1$
$\displaystyle=$ $\displaystyle 0,$ (34)
$\displaystyle\frac{d^{2}u_{1}}{d\phi^{2}}+\frac{u_{1}}{1-\ell}$
$\displaystyle=$ $\displaystyle\frac{3M^{2}}{L^{2}}u_{0}^{2}.$ (35)

The solution of zeroth-order equation (34) is given by

$u_{0}=(1-\ell)\left[1+e\cos\left(\frac{\phi}{\sqrt{1-\ell}}\right)\right],$
(36)

where the integration constants are chosen such that the initial value is
$\phi_{0}=0$ and the orbital eccentricity is $e$. If $\ell=0$, this solution
describes a closed ellipse, which is precisely the orbit predicted by
Newtonian mechanics. However, due to the appearance of the Lorentz-violating
parameter $\ell$ in the phase, the elliptical orbit is not closed any more.

By substituting the zeroth-order solution into Eq. (35), it is straightforward
to check that the first-order solution is given by

$u_{1}=\frac{3(1-\ell)^{3}M^{2}}{L^{2}}\left[\left(1+\frac{e^{2}}{2}\right)-\frac{e^{2}}{6}\cos\left(\frac{2\phi}{\sqrt{1-\ell}}\right)+\frac{e\phi}{\sqrt{1-\ell}}\sin\left(\frac{\phi}{\sqrt{1-\ell}}\right)\right].$
(37)

It is noted that the first term in the square bracket is simply a constant
displacement, and the second term oscillates around zero. However, the third
term accumulates over successive orbits and contributes the important effect
during rotation. We therefore only consider the contribution of this term in
the first-order solution.

Combining the zeroth-order and first-order solutions, we finally arrive at

$u\approx(1-\ell)\left[1+e\cos\left(\frac{\phi}{\sqrt{1-\ell}}\right)+\frac{3(1-\ell)^{\frac{3}{2}}M^{2}}{L^{2}}e\phi\sin\left(\frac{\phi}{\sqrt{1-\ell}}\right)\right].$
(38)

For convenience, it can be further approximated and rewritten in the form of
an elliptical equation,

$u\approx(1-\ell)\left[1+e\cos\left[\left(1-\frac{3(1-\ell)^{2}M^{2}}{L^{2}}\right)\frac{\phi}{\sqrt{1-\ell}}\right]\right].$
(39)

The period of $\phi$ of the orbits reads

$\Phi=\frac{2\pi\sqrt{1-\ell}}{1-\frac{3(1-\ell)^{2}M^{2}}{L^{2}}}\approx
2\pi\sqrt{1-\ell}\left(1+\frac{3(1-\ell)^{2}M^{2}}{L^{2}}\right).$ (40)

During each orbit of the planet, the perihelion advances by an angle
$\Delta\Phi=\Phi-2\pi$, which is obtained by taking the lowest order in the
expansion of $\ell$, yielding

$\Delta\Phi\approx\frac{6\pi M^{2}}{L^{2}}-\ell\pi.$ (41)

The first term $\Delta_{\text{GR}}=6\pi\frac{M^{2}}{L^{2}}$ is exactly the
estimation derived from GR, while the second term
$\Delta_{\text{LV}}=-\ell\pi$ represents the contribution arising from the
effect of LSB.

For Mercury, the predicted value of its perihelion precession in GR is
$42.9814^{\prime\prime}/\text{century}$, while the observed value is measured
as $\Delta\Phi_{\text{Ex}}=(42.9794\pm 0.0030)^{\prime\prime}/\text{century}$
Casana2018 . The precession value can be converted into the precession per
orbit by using the period of Mercury, which is approximately 87.969 days. By
requiring that the theoretical prediction of the present theory is consistent
with the experimental value, i.e.,
$\Delta\Phi_{\text{Ex}}^{\text{Min}}\leq\Delta\Phi\leq\Delta\Phi_{\text{Ex}}^{\text{Max}}$,
we obtain the constraints on the Lorentz-violating parameter as $-3.7\times
10^{-12}\leq\ell\leq 1.9\times 10^{-11}$.

###  IV.2 Deflection of light

For photons with $\eta=0$, the orbital equation can be obtained from Eqs.
(29), (30), and (31), giving by

$\displaystyle\left[\frac{d}{d\phi}\left(\frac{1}{r}\right)\right]^{2}=\frac{E^{2}}{L^{2}}-\frac{A(r)}{r^{2}}.$
(42)

By redefining $u={1}/{r}$ and differentiating the equation with respect to
$\phi$, we have

$\frac{d^{2}u}{d\phi^{2}}+\frac{u}{1-\ell}=3Mu^{2}.$ (43)

Following the approach in the previous subsection, we employ a perturbative
method to solve the equation and expand the exact solution $u$ as $u\approx
u_{0}+u_{1}$. Once again, the last term, represented as
$\frac{3GM}{c^{2}}u^{2}$ in the International System of Units, is a small
first-order quantity and therefore can be neglected in the zeroth-order
calculation. As a result, the zeroth-order solution satisfies a homogeneous
equation, i.e.,

$\frac{d^{2}{{u}_{0}}}{d\phi^{2}}+\frac{{u}_{0}}{1-\ell}=0.$ (44)

The solution is given by

${u}_{0}=b^{-1}\sin\left[\frac{\phi}{\sqrt{1-\ell}}\right],$ (45)

where the integration constants have been chosen appropriately so that $b$
represents the impact parameter and the incident angle is $\phi_{0}=0$. When
$\ell=0$, the solution describes a straight line, which is exactly the
Newtonian prediction. However, in the presence of Lorentz-violating effect, it
is evident that the exit angle of the light is no longer $\phi=\pi$, but is
instead given by $\phi=\sqrt{1-\ell}\pi\approx\pi-\frac{\pi\ell}{2}$.
Consequently, the Lorentz-breaking effect will induce an additional deflection
angle of $\delta_{\text{LV}}=-\frac{\pi\ell}{2}$.

By substituting the zeroth-order solution into the first-order equation

$u_{1}^{\prime\prime}+\frac{u_{1}}{1-\ell}=3Mu_{0}^{2},$ (46)

we obtain the solution

${u}_{1}=\frac{(1-l)M}{b^{2}}\left[1+\cos^{2}\left(\frac{\phi}{\sqrt{1-\ell}}\right)\right].$
(47)

Therefore, we arrive at the final solution

$u=\frac{1}{b}\sin\left[\frac{\phi}{\sqrt{1-\ell}}\right]+\frac{(1-\ell)M}{b^{2}}\left[1+\cos^{2}\left(\frac{\phi}{\sqrt{1-\ell}}\right)\right].$
(48)

When a ray of light is incident from infinity and exits to infinity, we have
$r\to\infty$ or $u\to 0$. Then, by solving the above equation with $u=0$, we
obtain the expression

$\sin\left[\frac{\phi}{\sqrt{1-\ell}}\right]=\frac{b-\sqrt{b^{2}+8(1-\ell)^{2}M^{2}}}{2(1-\ell)M}.$
(49)

Taking into account $\ell\ll 1$ and $M\ll 1$, the incident angle at lowest
order can be approximated as

$\phi_{\text{in}}=-\frac{2M}{b}.$ (50)

Furthermore, the exit angle at the lowest order can be expressed as

$\phi_{\text{ex}}=\pi+\frac{2M}{b}-\frac{\pi\ell}{2}.$ (51)

Here, the second term precisely represents the Lorentz-violating effect
observed in the zeroth-order solution, as discussed earlier.

So the total angle of deflection is obtained as

$\delta=\frac{4M}{b}-\frac{\pi\ell}{2}.$ (52)

The first term, $\delta_{\text{GR}}=\frac{4M}{b}$, arises from the prediction
of GR, while the second term, $\delta_{\text{LV}}=-\frac{\pi\ell}{2}$, stems
from the effect of LSB.

For a light ray that grazes the surface of the Sun, with $M=M_{\odot}$ and
$b=R_{\odot}$, GR predicts $\delta_{\text{GR}}=1.7516687^{\prime\prime}$. The
observed value of light deflection on the solar surface is given by
$\theta=\frac{1}{2}(1+\gamma)1.7516687^{\prime\prime}$ with $\gamma=0.99992\pm
0.00012$ Lambert2011 . Finally, the constraints on $\ell$ is given by
$-1.1\times 10^{-10}\leq\ell\leq 5.4\times 10^{-10}$.

###  IV.3 Time delay of light

The Shapiro time-delay effect is observed by measuring the travel time of a
light ray between two stations located at large distances from a massive
object, such as the Sun. To simplify the analysis, we consider the motion of
light on the equatorial plane ($\theta=\pi/2$) of a central celestial object.
Since the light is moving along a null geodesic, we have $ds^{2}=0$, i.e.,

$-A(r)dt^{2}+A(r)^{-1}dr^{2}+r^{2}d\phi^{2}=0.$ (53)

After applying the solution in Eq. (48) and performing some simple algebra, we
obtain the relation

$r^{2}d\phi^{2}=\frac{b^{2}}{(1-\ell)\left(r^{2}-b^{2}\right)A(r)^{2}}dr^{2}.$
(54)

Substituting it into Eq. (53) and considering that the Lorentz-violating
parameter $\ell$ and $M/r$ (represented as $\frac{2GM}{c^{2}r}$ in the
International System of Units) are both small quantities, we have

$dt\approx\pm\frac{1-\ell}{\sqrt{r^{2}-b^{2}}}\left(1+\frac{2M}{r}+\frac{b^{2}M}{r^{3}}\right)rdr.$
(55)

where the plus (minus) sign refers to the outgoing (infalling) light ray.

Considering a light ray (or radar signal) that travels from an emitter
situated at $r_{\text{E}}$ to a receiver at $r_{\text{R}}$, the time taken for
the travel can be calculated as

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(1-\ell\right)\Bigg{[}\left(\sqrt{r_{\text{E}}^{2}-b^{2}}+\sqrt{r_{\text{R}}^{2}-b^{2}}\right)+M\left(\frac{\sqrt{r_{\text{R}}^{2}-b^{2}}}{r_{\text{R}}}+\frac{\sqrt{r_{\text{E}}^{2}-b^{2}}}{r_{\text{E}}}\right)$
(56) $\displaystyle+$ $\displaystyle
2M\left(\log\left(\frac{\sqrt{r_{\text{E}}^{2}-b^{2}}+r_{\text{E}}}{b}\right)+\log\left(\frac{\sqrt{r_{\text{R}}^{2}-b^{2}}+r_{\text{R}}}{b}\right)\right)\Bigg{]}.$

It is evident that when the Lorentz-violating parameter $\ell$ is absent, this
result reduces to the prediction of GR. The first term inside the square
bracket represents the time required for the light to propagate along a
straight line from $r_{\text{E}}$ to $r_{\text{R}}$ in a flat spacetime, while
the remaining terms represent Shapiro gravitational time delay.

As the most precise test of time delay currently comes from the data obtained
by the Cassini mission during its journey to Saturn Bertotti2003 , we consider
the scenario where a radar signal is emitted from the Earth at
$r_{\text{E}}=r_{\oplus}$, then grazes near the Sun and travels to the
spacecraft located at $r_{\text{R}}$ before returning to Earth along the same
path. Then, due to the impact parameter $b\ll r_{\oplus},r_{\text{R}}$, the
total round-trip time delay $T_{\text{tol}}=2T$ can be approximated as

$\displaystyle T_{\text{tol}}$ $\displaystyle\approx$
$\displaystyle\left(1-\ell\right)\left[2\left(\sqrt{r_{\oplus}^{2}-b^{2}}+\sqrt{r_{\text{R}}^{2}-b^{2}}\right)+4M_{\odot}\left(\log\left(\frac{4r_{\oplus}r_{\text{R}}}{b^{2}}\right)+1\right)\right]$
(57) $\displaystyle\equiv$ $\displaystyle\left(1-\ell\right)\left(T_{0}+\delta
T_{\text{GR}}\right),$

where $T_{0}$ represents the round-trip time in flat spacetime and $\delta
T_{\text{GR}}$ represents the round-trip time delay caused by the curved
spacetime. By noticing that $\ell T_{0}\ll T_{0}$ and $\delta T_{\text{GR}}\ll
T_{0}$, the total excess time delay is given by

$\delta T_{\text{tol}}=T_{\text{tol}}-T_{0}\approx\delta
T_{\text{GR}}-lT_{0}.$ (58)

Therefore, the time delay resulting from Lorentz violation is $\delta
T_{\text{LV}}=-lT_{0}$.

For the 2002 superior conjunction of Cassini, the spacecraft was at
$r_{\text{R}}\approx 8.43$AU from the Sun and the radar grazes near the Sun
with an impact parameter $b\approx 1.6R_{\odot}$. Using Eq. (57), we obtain
$T_{0}=9411.2$s and $\delta T_{\text{GR}}=2.8\times 10^{-4}$s. The measured
value of the post-Newtonian parameter $\gamma$ in the formula $\delta
T=2(1+\gamma)M_{\odot}\log\left(\frac{4r_{\oplus}r_{\text{R}}}{b^{2}}\right)$
is reported as $\gamma=1+\left(2.1\pm 2.3\right)\times 10^{-5}$ Bertotti2003 .
Therefore, we obtain the constraints on the Lorentz-violating parameter $\ell$
as $-6.1\times 10^{-13}\leq\ell\leq 2.8\times 10^{-14}$.

##  V Conclusions

The static and spherically symmetric spacetime was investigated in the context
of a gravity theory featuring LSB induced by a nonzero VEV of KR filed. By
considering both the presence and absence of the cosmological constant, we
derived the exact solutions that exhibit new properties resulting from the
Lorentz-violating effect. Furthermore, the thermodynamic properties of the
AdS-Schwarzschild-like black hole were examined, revealing that the standard
first law of thermodynamics and Smarr formula hold as usual. Interestingly, it
was observed that the Lorentz-violating parameter $\ell$ does not impact the
critical size, which determines the local and global stability of the black
holes.

Solar Tests | Constraints
---|---
Mercury precession | $-3.7\times 10^{-12}\leq\ell\leq 1.9\times 10^{-11}$
Light deflection | $-1.1\times 10^{-10}\leq\ell\leq 5.4\times 10^{-10}$
Shapiro time-delay | $-6.1\times 10^{-13}\leq\ell\leq 2.8\times 10^{-14}$
Table 1: Constraints on the Lorentz-violating parameter $\ell$ from Solar
System tests.

In order to investigate the physical implications of the obtained solutions,
we analyzed several classical gravitational experiments within the Solar
System, including the perihelion precession of Mercury, deflection of light,
and Shapiro time delay. Our analysis revealed that the Lorentz-violating
effect does contribute to the corrections observed in these experiments. By
utilizing the measured results from these experiments, we were able to
constrain the value of the Lorentz-violating parameter $\ell$. The resulting
constraints are listed in Tab. 1. It shows that the Shapiro time-delay imposes
the most stringent constraints on the Lorentz-violating effect, with the range
of $-6.1\times 10^{-13}\leq\ell\leq 2.8\times 10^{-14}$. These stringent
constraints indicate that the possible Lorentz-violating effect is highly
restricted in the nature.

## ACKNOWLEDGMENTS

We thank Yu-Xiao Liu, Shao-Wen Wei, Xiao-Long Du, Peng Cheng and Si-Jiang Yang
for helpful discussions. Ke Yang also acknowledges the generous hospitality
during a visit to the Lanzhou Center for Theoretical Physics, where part of
this work was completed. This work was supported by the National Natural
Science Foundation of China under Grant No. 12005174.

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# Quantum tomography of structured light patterns from simple intensity
measurements

M. Gil de Oliveira  A. L. S. Santos Junior  Instituto de Física, Universidade
Federal Fluminense, 24210-346 Niterói, RJ, Brazil  P. M. R. Lima  Departamento
de Física, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG,
Brazil  A. C. Barbosa  B. Pinheiro da Silva  Instituto de Física, Universidade
Federal Fluminense, 24210-346 Niterói, RJ, Brazil  S. Pádua  Departamento de
Física, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG,
Brazil  A. Z. Khoury  Instituto de Física, Universidade Federal Fluminense,
24210-346 Niterói, RJ, Brazil

###### Abstract

We study the tomography of spatial qudits encoded on structured light photons.
While direct position measurements with cameras do not provide an
informationally complete Positive Operator Valued Measure (POVM) in the space
of fixed order modes, we complement this POVM with an astigmatic
transformation. The enlarged POVM is informationally complete, allowing full
characterization of the spatial quantum state from simple intensity
measurements in both the intense and in the low photocount regimes. For
intense light, the standard technique of linear inversion is used. For the low
photocount regime, we employ Bayesian mean inference, and study how the
quality of the tomographic reconstruction behaves as we increase the
photocounts. In both cases, we also perform the tomography using a
convolutional neural network, which displays an increased flexibility in
exchange for a slightly lower quality reconstruction in some of the cases.
These methods will be useful for classical and quantum communication with
structured light.

Introduction \- The field of structured light has found promising applications
in areas such as optical communication Cozzolino  _et al._ (2019); Gibson  _et
al._ (2004); Wang  _et al._ (2012), optical tweezers Melo  _et al._ (2020);
Almeida  _et al._ (2023) and quantum cryptography Mirhosseini  _et al._
(2015); Sit  _et al._ (2018); Vallone  _et al._ (2014). Central to them is the
transverse spatial structure of light, and, therefore, its characterization is
extremely important, which is reflected by a multitude of approaches that have
been proposed in recent years. Methods in which a Spatial Light Modulator
(SLM) is part of the measurement apparatus are commonly used Hiekkamäki  _et
al._ (2019); Jia  _et al._ (2023); Bent  _et al._ (2015). Such methods have
the disadvantage that, in each reflection, the beam loses power, so they might
not be suitable for very faint sources. Methods involving machine learning
techniques have also been employed da Silva  _et al._ (2021); Zia  _et al._
(2023); Guo  _et al._ (2022). These techniques have been successfully applied
to the classification of structures that have undergone atmospheric turbulence
Krenn  _et al._ (2014); Xiang  _et al._ (2022). In Zhou  _et al._ (2021), a
technique based on using the polarization to find the density matrix in
position basis was also introduced while, in Rambach  _et al._ (2021), self-
guided tomography was used to perform accurate tomography in high dimensional
spaces.

Here, we approach this problem using the quantum state tomography formalism,
allowing us to employ well established techniques in this subject Paris and
Řeháček (2004); Heinosaari and Ziman (2011). By focusing our attention in
finite dimensional spaces generated by Hermite-Gaussian modes, we
mathematically specify the kind of measurements performed by cameras, and come
to the conclusion that these measurements are not informationally complete in
the considered spaces. We then show how this problem is simply remedied with
the use of a single spherical lens acting as a variable angle mode converter
Beijersbergen  _et al._ (1993); Vaity  _et al._ (2013); Buono  _et al._
(2022). The use of cylindrical lens analyzis from the phase space perspective
has also been proposed in Raymer  _et al._ (1994) to tackle this sort of
problem. The advantage of our approach is that it gives complete and simple
methods to perform the state tomography of structured light qudits, showing
great performance in the experimental tests and suitability for practical
purposes.

This work will deal with two regimes: in the first, we treat intense beams,
which are equivalent to measurements on many copies of a quantum system and,
therefore, one has reliable estimates for the probabilities of each
experimental outcome. Then, we explore the low photocount regime, which is
more subtle, as one cannot determine with certainty the state being observed.
This fact is taken into account by some tomographic techniques, such as
maximum likelihood Smolin  _et al._ (2012); Shang  _et al._ (2017) or the
variational tomography method Maciel and Vianna (2009); Maciel  _et al._
(2011). Here we will employ Bayesian inference Granade  _et al._ (2016);
Lukens  _et al._ (2020); Landa  _et al._ (2022); Lohani  _et al._ (2023);
Blume-Kohout (2010), where one assigns a probability for each possible state.
This probability is updated with each new observation, being initially broad
when one has performed a low amount of measurements, and then concentrating
around a particular state as more and more data is used to update our
knowledge.

Taking into account the huge amount of research in this area which is
conducted with the use of machine learning, we also found it important to try
this method for both intense and single photon regimes. This comparison may
reveal valuable insights regarding the strengths and weaknesses of machine
learning approaches. All trained machine learning models, codes Gil de
Oliveira (2024a); marcsgil (2024a, b) and the experimental datasets Gil de
Oliveira (2024b) used in this work are freely available.

Formulation of the problem \- We wish to perform the tomography of the spatial
structure of the electromagnetic field by performing measurements with a
camera. We will assume that only modes of a fixed polarization and frequency
are occupied, so that the only degree of freedom will be the spatial modes,
which are characterized by a set of orthonormal basis functions
$\\{u_{k}(\mathbf{r})\\}$. We will also assume that the single photons are in
a state described by a density operator of the form

$\rho=\sum_{j,k}\rho_{jk}\ket{1}_{k}\bra{1}_{j}\,,$ (1)

where $\ket{1}_{k}$ is a single photon Fock state of a spatial mode
characterized by the function $u_{k}(\mathbf{r})$, in such a way that
$a_{j}\ket{1}_{k}=\delta_{jk}\ket{0}_{k}$, where, $a_{j}$ is the annihilation
operator corresponding to mode $u_{j}(\mathbf{r})\,$. Our task will be to
determine the matrix elements $\rho_{jk}$ from the measurements of an
observable which we describe in what follows. The negative frequency part of
the electric field operator is defined by the expression

$E^{-}(\mathbf{r})\propto\sum_{n}a_{n}^{\dagger}u_{n}(\mathbf{r}),$ (2)

while the positive frequency part is given by
$E^{+}=\left(E^{-}\right)^{\dagger}$. The normalized correlation function in
normal order is then

$\Gamma(\mathbf{r},\mathbf{r}^{\prime})=\dfrac{\expectationvalue{E^{-}(\mathbf{r})E^{+}(\mathbf{r}^{\prime})}}{\int_{\mathbb{R}^{2}}\expectationvalue{E^{-}(\mathbf{r})E^{+}(\mathbf{r})}d\mathbf{r}}=\sum_{j,k}\rho_{jk}u_{j}^{*}(\mathbf{r}^{\prime})u_{k}(\mathbf{r})\,.$
(3)

We will assume that a camera, which will be our measurement apparatus,
registers the photocount probability $P(\mathcal{R})$ at the pixel which
covers region $\mathcal{R}$ according to Loudon (2000)

$P(\mathcal{R})=\int_{\mathcal{R}}\Gamma(\mathbf{r},\mathbf{r})d\mathbf{r}\,.$
(4)

Formally, we may regard $\Gamma$ as a density matrix in position basis, living
in the Hilbert space $\mathcal{L}^{2}(\mathbb{R}^{2};\mathbb{C})$ formed by
the set of square integrable functions from $\mathbb{R}^{2}$ to $\mathbb{C}$,
with the standard inner product. The hypothesis
$\eqref{eq:measurement_condition}$ can be translated as the probability of a
measurement described by the operator Hobson (2013)

$F(\mathcal{R})=\int_{\mathcal{R}}\outerproduct{\mathbf{r}}{\mathbf{r}}d\mathbf{r}\,,$
(5)

which we call a position measurement.

From the practical point of view, quantum state tomography in
$\mathcal{L}^{2}(\mathbb{R}^{2};\mathbb{C})$ is not feasible - it is a
infinite dimensional space, which would require an infinite amount of
measurements in order to be completely characterized. It would also need that
a measurement apparatus with infinite spatial extension and resolution. Here,
we will restrict ourselves to finite subspaces of
$\mathcal{L}^{2}(\mathbb{R}^{2};\mathbb{C})$, spanned by the Hermite-Gaussian
modes

$u^{HG}_{mn}\left(\boldsymbol{r}\right)=\mathcal{N}_{mn}H_{m}\left(x\right)H_{n}\left(y\right)e^{-(x^{2}+y^{2})/2},$
(6)

where we have chosen units such that the waist $w=\sqrt{2}$ and the
normalization constant $\mathcal{N}_{mn}=\left[2^{(m+n)}\pi
m!n!\right]^{-1/2}$. In particular, we will work in the subspaces of fixed
order $N=m+n$, which have dimension $N+1$. This choice simplifies the
measurement process, because modes of fixed order propagate with the same Gouy
phase, so the the intensity pattern is invariant with respect to the
measurement plane, up to a scaling constant. For each order $N$, we will use
the basis functions $u_{n}\equiv u^{HG}_{N+1-n,n-1}$ ($n=1,\ldots,N+1$).

Quantum State Tomography \- In order to perform the tomography, and to be able
to decide if the measurement of (5) is enough to characterize our state, we
discuss some theoretical results regarding quantum state tomography. Let
$\operatorname{Her}(\mathcal{H})\subset\mathcal{H}$ be the set of hermitian
operators acting on a Hilbert space $\mathcal{H}$. Elements of
$\operatorname{Her}(\mathcal{H})$ may be written as double kets:
$|F\rangle\\!\rangle$. Such a space is also a real Hilbert space under the
inner product

$\langle\\!\langle E|F\rangle\\!\rangle=\Tr E^{\dagger}F=\Tr EF\,.$ (7)

The state of a quantum system is represented by an element $\rho$ of the set
of positive semi-definite operators
$\operatorname{Pos}(\mathcal{H})\subset\operatorname{Her}(\mathcal{H})$ such
that $\tr\rho=1$. Observables are represented by elements
$A\in\operatorname{Her}(\mathcal{H})$ so that its expectation value is given
by $\expectationvalue{A}=\langle\\!\langle A|\rho\rangle\\!\rangle\,$. A
Positive Operator Valued Measure (POVM) is a set of observables
$\\{F_{m}\\}\subset\operatorname{Pos}(\mathcal{H})$ with the property that
$\sum_{m}F_{m}=I$, where $I$ is the identity operator. These operators model
the possible outcomes of an experiment: outcome $m$ happens with probability
$p(m)=\langle\\!\langle F_{m}|\rho\rangle\\!\rangle$. The POVM conditions
ensure that $p(m)\geq 0$ and $\sum_{m}p(m)=1$. The POVM formalism has been
used for minimum quantum tomography of polarization states Ling  _et al._
(2006), qubits and qudits in path states Pimenta  _et al._ (2010, 2013);
Cardoso  _et al._ (2019) or in vector vortex beams Al Khafaji  _et al._
(2022).

One easily checks that the position measurement satisfies
$[F(\mathcal{R})]^{2}=F(\mathcal{R})$, so it is a projector, and therefore,
positive. We then conclude that, given a set of disjoint regions
$\\{\mathcal{R}_{m}\\}$ such that $\cup_{m}\mathcal{R}_{m}=\mathbb{R}^{2}$,
the set of operators $\\{F(\mathcal{R}_{m})\\}$ is a POVM. We call such a set
a position POVM.

The simplest method which solves the problem of quantum state tomography is
linear inversion, which we describe in what follows. A POVM with $M$ elements
induces a linear map $T:\operatorname{Her}(\mathcal{H})\to\mathbb{R}^{M}$
defined by the expression

$T|\Omega\rangle\\!\rangle=\left(\langle\\!\langle
F_{1}|\Omega\rangle\\!\rangle,\ldots,\langle\\!\langle
F_{M}|\Omega\rangle\\!\rangle\right).$ (8)

In order to be suitable for tomography, we want that the measurement
probabilities $\mathbf{p}=T|\rho\rangle\\!\rangle$ uniquely determine the
state $|\rho\rangle\\!\rangle$. This is assured if the transformation $T$ is
injective, or, equivalently, if there is no nonzero $|\Omega\rangle\\!\rangle$
such that $T|\Omega\rangle\\!\rangle=\boldsymbol{0}$. A POVM such that the
associated transformation $T$ is injective is said to be informationally
complete.

By choosing a basis $\\{\Omega_{n}\\}\subset\operatorname{Her}(\mathcal{H})$,
we can specify an arbitrary state $|\rho\rangle\\!\rangle$ by a list of
coefficients $\mathbf{x}=(x_{1},\ldots,x_{N})$ such that
$|\rho\rangle\\!\rangle=\sum_{n}x_{n}|\Omega_{n}\rangle\\!\rangle$. Then,
denoting by $\mathbb{T}$ the matrix of $T$ with respect to the canonical basis
of $\mathbb{R}^{M}$ and $\\{\Omega_{m}\\}$, which has entries
$\mathbb{T}_{mn}=\langle\\!\langle F_{m}|\Omega_{n}\rangle\\!\rangle$, we have
$\mathbf{p}=\mathbb{T}\,\mathbf{x}$. When $\mathbb{T}$ is injective,
$\mathbb{T}^{\dagger}\mathbb{T}$ is invertible, and then we can explicitly
write the solution of the above equation as

$\mathbf{x}=(\mathbb{T}^{\dagger}\mathbb{T})^{-1}\mathbb{T}^{\dagger}\mathbf{p},$
(9)

which, in theory, solves the tomography problem.

Tomography of the spatial structure \- The space of fixed order is spanned by
the basis $\\{\ket{u_{1}},\ldots,\ket{u_{N+1}}\\}\,$, which induces the
orthonormal basis of the space of the hermitian operators defined by the
expressions

$\displaystyle|W_{j}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle\outerproduct{u_{j}}{u_{j}}\,,$
$\displaystyle|X_{jk}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\outerproduct{u_{j}}{u_{k}}+\outerproduct{u_{k}}{u_{j}}\right)\,,$
(10) $\displaystyle|Y_{jk}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\left(\outerproduct{u_{j}}{u_{k}}-\outerproduct{u_{k}}{u_{j}}\right)\,,$

where $j=1,\ldots N+1$ and $k=1,\ldots j-1$. We then have

$\displaystyle\langle\\!\langle F(\mathcal{R})|W_{j}\rangle\\!\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathcal{R}}\absolutevalue{u_{j}(\mathbf{r})}^{2}d\mathbf{r}\,,$
$\displaystyle\langle\\!\langle F(\mathcal{R})|X_{jk}\rangle\\!\rangle$
$\displaystyle=$
$\displaystyle\sqrt{2}\real\int_{\mathcal{R}}u_{j}^{*}(\mathbf{r})u_{k}(\mathbf{r})d\mathbf{r}\,,$
(11) $\displaystyle\langle\\!\langle F(\mathcal{R})|Y_{jk}\rangle\\!\rangle$
$\displaystyle=$
$\displaystyle\sqrt{2}\imaginary\int_{\mathcal{R}}u_{j}^{*}(\mathbf{r})u_{k}(\mathbf{r})d\mathbf{r}\,,$

which allows us to assemble the matrix $\mathbb{T}$. Unfortunately, this also
allows us to conclude that no position POVM is informationally complete in the
space of fixed order, because $\langle\\!\langle
F(\mathcal{R})|Y_{jk}\rangle\\!\rangle=0$ for any region $\mathcal{R}$.
Therefore, according to (8), $T|Y_{jk}\rangle\\!\rangle=\boldsymbol{0}$ and
$T$ is not injective.

To remedy this incompleteness problem, one may augment a POVM $\\{F_{m}\\}$ by
performing an unitary transformation $U$ on the state before measuring it,
which generates the larger POVM $\\{F_{m}/2,U^{\dagger}F_{m}U/2\\}$. This is a
standard practice, for example, in polarization tomography, where the actual
measurements are projections on the horizontal and vertical polarization basis
$\\{\ket{H},\ket{V}\\}\,$, but the informationally complete POVM can be
obtained by preceding the polarization projection with unitary transformations
performed by half and quarter wave plates.

Of particular interest to the tomography of the spatial structure of light is
the unitary

$U_{\theta}=\sum_{m,n=0}^{\infty}e^{i(m-n)\theta}\outerproduct{u^{HG}_{mn}}{u^{HG}_{mn}},$
(12)

which is physically realized by an astigmatic mode converter made of
cylindrical or tilted spherical lenses Beijersbergen  _et al._ (1993); Vaity
_et al._ (2013); Buono  _et al._ (2022). The parameter $\theta$ can be
adjusted experimentally by tuning the propagation distance and the tilt angle
of the lens sup . Denoting
$F_{\theta}(\mathcal{R})=U^{\dagger}_{\theta}F(\mathcal{R})U_{\theta}\,$, we
have that

$\displaystyle\langle\\!\langle
F_{\theta}(\mathcal{R})|W_{j}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle\langle\\!\langle F(\mathcal{R})|W_{j}\rangle\\!\rangle\,,$
$\displaystyle\langle\\!\langle
F_{\theta}(\mathcal{R})|X_{jk}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}\cos\left[(j-k)\theta\right]\langle\\!\langle
F(\mathcal{R})|X_{jk}\rangle\\!\rangle\,,$ (13)
$\displaystyle\langle\\!\langle
F_{\theta}(\mathcal{R})|Y_{jk}\rangle\\!\rangle$ $\displaystyle=$
$\displaystyle-\sqrt{2}\sin\left[(j-k)\theta\right]\langle\\!\langle
F(\mathcal{R})|X_{jk}\rangle\\!\rangle\,.$

Therefore, for this augmented POVM, we can avoid having $\langle\\!\langle
F_{\theta}(\mathcal{R})|Y_{jk}\rangle\\!\rangle=0$ by properly choosing a
value for $\theta$.

The possibility of performing tomography of pure states of fixed order using
this technique was demonstrated in da Silva  _et al._ (2021), where the
standard value $\theta=\pi/2$ was used. In that work, the tomography was
performed using machine-learning, but there was no explanation of why the
measurements performed do indeed carry enough information. The theory
presented here explains why and allows the extension of the method to mixed
states, what is of key importance for actual applications in quantum
communication protocols. First, let us consider a pure state
$\ket{\psi}=\sum_{j}c_{j}\ket{u_{j}}$. The knowledge of the main diagonal
elements $\rho_{jj}=\langle\\!\langle W_{j}|\rho\rangle\\!\rangle$ already
allows the determination of $\absolutevalue{c_{j}}$, while the first secondary
diagonal $\rho_{j,j+1}=(\langle\\!\langle
X_{j,j+1}|\rho\rangle\\!\rangle+i\langle\\!\langle
Y_{j,j+1}|\rho\rangle\\!\rangle)/\sqrt{2}$ allows the determination of the
relative phases between each successive $c_{j}$, which completely
characterizes the state. Note that $\langle\\!\langle
F_{\pi/2}(\mathcal{R})|Y_{jk}\rangle\\!\rangle$ will still be zero when $j-k$
is even, so that this augmented POVM is still informationally incomplete,
except in the case of order $N=1$, when the only possible value for $j-k$ is
$1\,$, and the $\theta=\pi/2$ mode conversion will be enough to provide
complete information even for mixed states. This gives the hint for achieving
informational completeness for arbitrary orders: let us restrict ourselves to
any order strictly smaller then $N$ and choose an angle $\theta=\pi/N\,$.
Then, one can verify that $\langle\\!\langle
F_{\theta}(\mathcal{R})|Y_{jk}\rangle\\!\rangle\neq 0$ for every allowed value
of $j,k$, which allows the POVM to be informationally complete. Therefore,
from the measurement of the direct and converted images, one can retrieve the
spatial density matrix of our light beam, as demonstrated by the experiment we
describe next.

Experimental verification with intense mixed states \- To verify the theory
just described, we used the setup illustrated in Fig. 1. The beam of a HeNe
laser at 633nm wavelength is sent to a spatial light modulator (SLM). After
the SLM, the beam passes through the spatial filter formed by two spherical
lenses, $L_{1}$ and $L_{2}\,$, with an iris ($I$) in between. After passing
through the spatial filter, the beam is split in two halves by a half-
waveplate ($\lambda/2$) and a polarizing beam splitter (PBS). Half of the beam
is sent to a spherical lens $L_{D}\,$, which directly projects the image onto
a CCD camera. The other half is sent to a spherical tilted lens ($L_{T}$),
tuned to generate a $\pi/6$ mode converter, which projects the converted image
onto the camera. To synthesise the mixed states, we use the technique
described in Cañas  _et al._ (2022); Marques  _et al._ (2015), where a movie
is played in the SLM, and the captured intensities are summed in post
processing.

Figure 1: Experimental setup for tomography of spatial photonic qudits. After
reflection on the spatial light modulator (SLM), the beam is sent to a spatial
filter composed by the lenses $L_{1}$ and $L_{2}\,$, and an iris $I\,$. The
transmitted beam is split in two halves by a half-waveplate ($\lambda/2$) and
a polarizing beam splitter ($PBS$). One half is directly imaged on the camera
($CAM$) by the lens $L_{D}\,$, and the other one is transmitted through a
tilted lens $L_{T}\,$, which projects the converted image on the camera.

In this regime, we will compare the linear inversion method with a machine-
learning based one. For the linear inversion method, we have to be able to
calibrate the grid of our camera with respect to our measured modes, in order
to be able to determine their offset from the origin and their waist. This was
done by sending a Gaussian reference beam with the same waist as the produced
modes. The machine learning method, similar to what is described in da Silva
_et al._ (2021), utilizes a convolutional neural network, which is a function
that accepts as input our two images, and outputs a prediction of the state
that produced these images. The network has a set of tunable internal
parameters, which are adjusted during a training phase. More specifically, we
show the network a series of simulated images, for which we know the
corresponding density matrix. The network will, at first, make a random
prediction, which is compared with the true value using the Hilbert-Schmidt
distance $D(\rho,\sigma)=\Tr\left[(\rho-\sigma)^{2}\right]$. The parameters of
the network are then adjusted in order to minimize this distance. After this
training phase, the network is ready to perform predictions of real
experimental data. For each order, we have to train a different network, as
the number of outputs is different in each case sup . One important
improvement of the present method when compared with the one of da Silva  _et
al._ (2021) is in the nature of the training data. In reference da Silva  _et
al._ (2021), the network was trained with experimental data, while here we
used simulated noisy data Zhang  _et al._ (2021). There are a few trade-offs
when comparing these two approaches: first, although the training with
experimental data yields better predictions, we can’t be sure if the network
has indeed learned a general method to deal with the problem or if it has only
learned the experimental imperfections of our particular setup. Second, there
may be situations where it is impractical to acquire the required amount of
training data experimentally, which is of the order of $10^{4}$ or $10^{5}$
different modes.

In order to compare the two tomography methods, we use the fidelity
$\mathcal{F}(\rho,\sigma)=\left(\Tr\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2}$
between the true state $\rho$ and the predicted state $\sigma$ as a figure of
merit. Then, for each order, we perform the tomography of $100$ random mixed
states sampled from the distribution described in Zyczkowski  _et al._ (1998)
and calculate the average fidelity obtained for each method, as shown in Fig.
2. Both methods achieve a reasonably high fidelity, with a minimum of almost
$0.97$ even for order $5$. In this regime, we can see that the linear
inversion method achieves the highest fidelity. Notice, nonetheless, that the
machine learning method does not require a reference beam, and can be an
alternative when such a calibration is not possible.

Figure 2: Mean fidelity over 100 intense mixed states as a function of the
order.

Experimental verification for low photocounts \- We will also show that it is
possible to perform the tomography even in the low photocount regime. For that
end, we use an experimental setup similar to the previous case. There are only
two differences: first, the camera is now an intensified charge coupled device
(ICCD), preceded by a filter to reduce the incoming power and allow single
photocounts to be isolated. Second, we use a $\pi/2$ mode converter, as we
will now only deal with pure states. Some experimental images are shown in
Fig. 3.

Figure 3: Evolution of the images as one increases the photocounts for a
superposition of order $4$. The left and middle images are experimental
results with $128$ and $2048$ photocounts. The right image is a simulation of
the corresponding pattern in the intense regime. For this order, we already
achieve a $0.9$ fidelity when using as low as $512$ photocounts.

To perform the tomography, we will also utilize the machine learning method,
training once again with simulated data, but this time we directly use the
infidelity as our loss function. We also train a different neural network for
each combination of order and photocount. Notice that, as now we have a low
number of photocounts, we cannot reliably estimate the probabilities of each
outcome, and, therefore, the linear inversion method cannot be used, so we use
Bayesian mean inference instead. In this method, one notes that the state
$\rho$, which determine the likelihood $p(\boldsymbol{\mathcal{R}}|\rho)$ of a
series of photocounts happening at the regions
$\boldsymbol{\mathcal{R}}=(\mathcal{R}_{m_{1}},\ldots,\mathcal{R}_{m_{k}})$,
cannot be known with certainty, so we assign it a probability distribution
$p(\rho)$, called a prior, which represents our ignorance before seeing any
observations. Then, one uses Bayes’ rule
$p(\rho|\boldsymbol{\mathcal{R}})\propto
p(\boldsymbol{\mathcal{R}}|\rho)p(\rho)$ to calculate the posterior
distribution $p(\rho|\boldsymbol{\mathcal{R}})$, which represents our updated
knowledge about the state $\rho$ after performing the observations
$\boldsymbol{\mathcal{R}}$. Then, one proves that the optimal point prediction
with the given amount of information is the mean $\expectationvalue{\rho}$
over the posterior distribution Blume-Kohout (2010). To counteract the fact
that the measurements are not informationaly complete for orders larger than
$1$, we apply the knowledge that we are producing only pure states by taking
as our prediction the pure state corresponding to the eigenvector of
$\expectationvalue{\rho}$ with the largest eigenvalue. Just as in Linear
Inversion, we also need an intense Gaussian calibration beam to serve as a
reference for our pixel grid. For more information on the computational
details, see sup .

In order to compare these two methods, we performed measurements of $50$
random Haar pure states and compared the average fidelity as a function of the
photocounts for orders $1$ to $4$. For each amount of photocounts, we train a
new neural network. The results can be seen in Figure 4. As expected, the
fidelities rise as we increase the photocounts, and, with as low as $128$
photocounts, we are already able do achieve $0.99$ fidelity for order $1$ when
using the Machine Learning. This method appears to be the winner for a low
number of photocounts, but Bayesian inference catches up when the counts are
increased and, for orders 1 and 4, even surpasses the Machine Learning for
high enough counts.

Figure 4: Fidelity as a function of the number of photocounts for different
orders. The markers joined by full lines represent the Machine learning, while
the ones joined by dashed lines represent the Bayesian inference.

Conclusion \- We formulate the recognition of the transverse structure of
light as a problem of quantum state tomography and propose an informationally
complete POVM based on two simple image measurements: a direct image and
another one after an astigmatic transformation. Then, we experimentally
demonstrate the efficacy of the protocol in both the intense and in the low
photocount regime. In order to actually perform the tomography, we tested
three methods, each with its own strengths and weaknesses. For the intense
regime, linear inversion seems to be the most suitable choice, as it achieves
the highest fidelity and is the fastest method sup , only requiring simple
linear algebra operations. For the photocount regime, we have a less clear
scenario, with either machine learning or Bayesian inference achieving the
best results depending on the order and photocount. Machine learning has the
overall advantage of requiring no calibration beam. The results and methods
reported here will be certainly useful for optical communication protocols
both in classical and quantum domains.

###### Acknowledgements.

Acknowledgments \- Funding was provided by Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Instituto
Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ
465469/2014-0) and Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP, grant 2021/06823-5).

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