11institutetext: Jiangsu Engineering Center of Network Monitoring, Nanjing
University of Information Science and Technology, Nanjing 210044, P. R. China
School of Computer and Software, Nanjing University of Information Science and
Technology, Nanjing 210044, P. R. China
School of Mathematics and Computing Science, Guilin University of Electronic
Technology, Guilin 541004, P. R. China
Division Department of Computer Science and Information Engineering, National
Dong Hwa University, Taiwan 97441, P. R. China
Corresponding author: Maojun Zhang 11email<EMAIL_ADDRESS>
# A novel quantum visual secret sharing scheme

Wenjie Liu 1122 Yinsong Xu 22 Maojun Zhang 33 Junxiu Chen 22 Ching-Nung Yang
4411223344

###### Abstract

Inspired by Naor et al.’s visual secret sharing (VSS) scheme, a novel $n$ out
of $n$ quantum visual secret sharing (QVSS) scheme is proposed, which consists
of two phases: sharing process and recovering process. In the first process,
the color information of each pixel from the original secret image is encoded
into an $n$-qubit superposition state by using the strategy of quantum
expansion instead of classical pixel expansion, and then these $n$ qubits are
distributed as shares to $n$ participants, respectively. During the recovering
process, all participants cooperate to collect these $n$ shares of each pixel
together, then perform the corresponding measurement on them, and execute the
$n$-qubit XOR operation to recover each pixel of the secret image. The
proposed scheme has the advantage of single-pixel parallel processing that is
not available in the existing analogous quantum schemes, and perfectly solves
the problem that in the classic VSS schemes the recovered image has the loss
in resolution. Moreover, its experiment implementation with IBM Q is conducted
to demonstrate the practical feasibility.

n$-qubitsuperpositionstate,$n$-qubit\textit{XOR}operation,quantumexpansion,quantumvisualsecretsharing,single-
pixelparallelprocessing,visualcryptography\par\par\par$

##  1 Introduction

In order to prevent the secret from being too concentrated and achieve the
purpose of spreading risk and tolerating intrusion, the idea of secret sharing
[1, 2] has been proposed. Secret sharing refers to methods for distributing a
secret amongst a group of participants, each of whom is distributed a share of
the secret. The secret can be reconstructed only when a sufficient number of
shares are combined together, but individual shares are of no use on their
own. This method provides an effective way for the security protection and
fair use of secret keys. As an important branch of modern cryptography, secret
sharing plays an important role in the preservation, transmission and
utilization of data.

Inspired by secret sharing, many experts and scholars have devoted themselves
to explore the research of visual cryptography, i.e., how to utilize the idea
of secret sharing to solve the problem of image encryption and decryption. In
1995, the first visual secret sharing (VSS) scheme based on pixel expansion
was proposed by Naor et al. [3], where a binary secret image is shared to
generate a plurality of noise-like shared images by dividing each of the
secret pixels into pixel blocks composed of $m$ sub-pixels, and the shared
image is $m$ times the secret image. Due to the presence of pixel expansion,
the visual quality of the recovered secret image may be not ideal. Since then,
most of the research has been carried out around reducing pixel expansion [4,
5, 6, 7, 8].

With the continuous development of quantum computing technology, many
researchers try to apply quantum mechanics in many fields, such as quantum key
distribution (QKD) [9, 10], quantum key agreement (QKA) [11, 12], quantum
secret sharing (QSS) [13, 14], quantum secure direct communication (QSDC) [15,
16], quantum remote state preparation (QRSP) [17, 18], quantum steganography
(QS) [19, 20, 21], delegating quantum computation (DQC) [22, 23], quantum
private query (QPQ) [25, 24] and even quantum machine learning [26, 27, 28].
Among them, QSS is an important research area [13, 14, 29, 30, 31, 32, 33,
34], and it can be viewed as the generalization of classical secret sharing to
the setting of quantum information. In 1999, Hillery et al. proposed the first
QSS scheme by using the Greenberger-Horne-Zeilinger (GHZ) state. In the
scheme, a GHZ triplet is split and each of the other two participants gets a
particle. Both participants are allowed to measure their particles in either
$x$ or $y$ direction, and their results are combined to determine the dealer’s
measurement result. This allows dealer to establish a joint key with two
participants which dealer can then use to send message. At the same year, a
threshold QSS scheme [14] was proposed by adopting quantum error correcting
code theory. Since then, various kinds of QSS schemes are constantly being
proposed [29, 30, 31, 32, 33, 34]. All these work can be divided into three
main categories, QSS of classical messages [29, 30], QSS of quantum
information [31, 32] where the secret is an arbitrary unknown state, and QSS
of both them [33, 34].

As far as we know, the literatures on how to use quantum mechanisms to solve
VSS schemes are rare [35, 36]. In 2014, Song et al. proposed a flexible
(${2^{k}}$, ${2^{k}}$) quantum image secret sharing (QISS) scheme [35], where
the whole secret image is repeatedly encoded into a quantum state, and then
split into sub-images as shares with multiple measurement operations. Although
the size of each share (the part of the original quantum image) becomes
smaller, it requires a mass of multi-qubit superposition states to produce
shares by measurement, and also loses the characteristic of single-pixel
parallel processing (i.e., one pixel as a unit for parallel processing) in
VSS. In order to solve the problem, we propose a novel $n$ out of $n$ quantum
visual secret sharing (QVSS) scheme based on Naor et al.’s scheme. In our
scheme, the color information of each pixel from the original secret image is
encoded into an $n$-qubit quantum superposition state, so the advantage of
single-pixel parallel processing can be preserved. Besides, we use the quantum
expansion strategy to perfectly solve the problem that the recovered image has
the loss in resolution in the classical VSS schemes, i.e., the recovered image
is as same as the original secret image.

The rest of this paper is organized as follows. Sect. 6 provides some
preliminary knowledge about quantum computation and Naor et al.’s VSS scheme.
The proposed ($n$, $n$) QVSS scheme, which consists of sharing process and
recovering process, is explicated in Sect. 3, and an example, ($3$, $3$) QVSS
scheme, is illustrated in Sect. 4. And then, the correctness is verified in
Sect. 5. We also compare and discuss with other analogous schemes in Sect. 6.
Moreover, its experiment implementation with IBM Q is conducted to demonstrate
the practical feasibility in Sect. 7. Finally, Sect. 8 gives the discussion
and conclusion of this paper.

##  2 Preliminaries

###  2.1 Quantum computation

As we know, the bit is the fundamental concept of classical information, and
has a state, either 0 or 1. Similar to the classical bit, the quantum bit
(called qubit) [37] is the basic unit of quantum information and has two
possible states $\left|0\right\rangle$ and $\left|1\right\rangle$, which is
often referred to as quantum superposition state,

$\left|\varphi\right\rangle=\alpha\left|0\right\rangle+\beta\left|1\right\rangle,$
(1)

where $\alpha$, $\beta$ are complex numbers, and
${\left|\alpha\right|^{2}}+{\left|\beta\right|^{2}}=1$. $\left|0\right\rangle$
and $\left|1\right\rangle$ can be represented by vectors,

$\left|0\right\rangle=\left[\begin{array}[]{l}1\\\ 0\end{array}\right],{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}\left|1\right\rangle=\left[\begin{array}[]{l}0\\\ 1\end{array}\right].$
(2)

Then, $\left|\varphi\right\rangle$ can be expressed in vector form
$\left|\varphi\right\rangle=\left(\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right)$.

Analogous to the way that a classical computer is built from an electrical
circuit containing wires and logic gates, a quantum computer is built from a
quantum circuit containing wires and elementary quantum gates to carry around
and manipulate the quantum information. Single-qubit gates, such as Pauli-X,
Pauli-Z, and H (Hadamard), are the simplest form of quantum gates, and they
can be described as $2\times 2$ unitary matrices as below,

$\textit{X}=\left[{\begin{matrix}0&1\\\ 1&0\end{matrix}}\right],{\kern
4.0pt}\textit{Z}=\left[{\begin{matrix}1&0\\\ 0&-1\\\
\end{matrix}}\right],{\kern
4.0pt}\textit{H}=\frac{1}{{\sqrt{2}}}\left[{\begin{matrix}1&1\\\
1&{-1}\end{matrix}}\right].$ (3)

Multi-qubit gates are also the important units in a quantum circuit. The
prototypical multi-qubit quantum logic gate is controlled-NOT (i.e., CNOT)
gate (shown in Fig. 1), which has two input qubits, known as the control qubit
and the target qubit, respectively. If the control qubit is set to 0, then the
target qubit is left alone. If the control qubit is set to 1, then the target
qubit is flipped.

Figure 1: Matrix representation and quantum circuit of CNOT gate.

Besides CNOT gate, Toffoli gate is another frequently used multi-qubit gate.
As illustrated in Fig. 2, Toffoli gate has three input bits and three output
bits: two of the bits are control bits that are unaffected by the action of
the Toffoli gate; the third bit is a target bit that is flipped if both
control bits are set to 1, and otherwise is left alone.

Figure 2: Truth table and quantum circuit of Toffoli gate.

###  2.2 Naor et al.’s Visual secret sharing scheme

The first visual secret sharing scheme was proposed by Naor et al. [3], where
the secret image consists of a collection of black and white pixels. As shown
in Fig. 3, each original pixel generates $n$ shares, and each share has a
collection of $m$ black and white subpixels (the process is named pixel
expansion). These $n\times m$ subpixels can be described by an $n\times m$
Boolean matrix $S=[{s_{ij}}]$, where $i\in\\{1,2,\cdots,n\\}$,
$j\in\\{1,2,\cdots,m\\}$, and ${s_{ij}}\in\\{0,1\\}$. If ${s_{ij}}=1$, the
$j$th subpixel in the $i$th share is black, otherwise, it white.

Figure 3: The pixel expansion process in Naor et al.’s VSS scheme

In the ($k$, $n$) VSS scheme, it mainly consists of two matrix sets ${C_{0}}$
and ${C_{1}}$ which are composed of $n\times m$ Boolean matrices $S$. To share
a white pixel, the dealer randomly chooses one $S$ in ${C_{0}}$, and to share
a black pixel, the dealer randomly chooses one $S$ in ${C_{1}}$. The chosen
matrix $S$ defines the colour of the $m$ subpixels in each one of the $n$
shares. The scheme is considered valid if the following two conditions are
met:

(1) $\forall{\kern 1.0pt}S{\kern 1.0pt}\in{\kern 1.0pt}{\kern 1.0pt}{C_{0}}$,
$\left\\{{{i_{1}},{i_{2}},\cdots,{i_{p}}}\right\\}\subseteq\left\\{{1,2,\cdots,n}\right\\}\left({p\geqslant
k}\right)$, vector $V=S[{i_{1}}]+S[{i_{2}}]+\cdots+S[{i_{k}}]$, so
$H(V)\leqslant d-\alpha m$, where ”$+$” means ”OR” logical operation, $S[i]$
indicates the $i$th row of $S$, $d$ and $\alpha$ are threshold and relative
difference respectively, and $H(V)$ represents the Hamming weight of $V$.
$\forall{\kern 1.0pt}S{\kern 1.0pt}\in{\kern 1.0pt}{\kern 1.0pt}{C_{1}}$,
$H(V)\geqslant d$.

(2)
$\forall\left\\{{{i_{1}},{i_{2}},\cdots,{i_{q}}}\right\\}\subseteq\left\\{{1,2,\cdots,n}\right\\}\left({q<k}\right)$,
the two collections of $q\times m$ matrices ${D_{t}}$ for $t\in\\{0,1\\}$
obtained by restricting each $n\times m$ matrix in ${C_{t}}$ ($t\in\\{0,1\\}$)
to rows ${i_{1}},{i_{2}},\cdots,{i_{q}}$ are indistinguishable in the sense
that they contain the same matrices with the same frequencies.

When the secret image needs to be recovered, any $k$ participants just print
their shares on transparencies respectively and stack their transparencies to
decrypt secret information by the human visual system. But any $k-1$ of them
gain no information about secret information.

As mentioned above, the pixel is treated as a separate basic unit throughout
the scheme, so this single-pixel based VSS scheme is well suitable for
parallel computing of images. In order to make individual participant get no
information, most VSS schemes use the strategy of pixel expansion to confuse
the information of subpixels. However, it may cause the loss in resolution
from the original image to the recovered one, i.e., the size of the shared
image is $m$ times the original secret image. Therefore, some researchers
tried to reduce pixel expansion [4, 5], and even implement non-expansion [6,
7, 8].

##  3 A novel ($n$, $n$) quantum visual secret sharing scheme

Suppose the dealer Alice wants to share her secret image, which consists of
$s$ black or white pixels, to $n$ participants $\left\\{{Bo{b_{1}},{\kern
1.0pt}Bo{b_{2}}{\kern 1.0pt},\cdots,{\kern 1.0pt}Bo{b_{n}}}\right\\}$, and the
black and white pixels are respectively represented by $|1\rangle$ and
$|0\rangle$. The secret image can be recovered by $n$ participants, but it
cannot be less than $n$ participants. The specific ($n$, $n$) QVSS scheme is
mainly composed of the sharing process and the recovering process.

###  3.1 Sharing process

During the sharing process of ($n$, $n$) QVSS scheme, the most important
operation is to encode the color information of each pixel from the secret
image, e.g., the $l$th pixel, into an $n$-qubit superposition state
${\left|{{C_{b}}}\right\rangle_{l}}$,

${\left|{{C_{b}}}\right\rangle_{l}}=\sum\limits_{i}^{\mathop{\oplus}\limits_{j=1}^{n}x_{i}^{j}=b}{\frac{{\left|{x_{i}^{1}x_{i}^{2}\cdots
x_{i}^{n}}\right\rangle}}{{\sqrt{{2^{(n-1)}}}}}},$ (4)

where $l\in\left\\{{1,2,\cdots,s}\right\\}$,
$i\in\left\\{{1,2,\cdots,{2^{n-1}}}\right\\}$,
$j\in\left\\{{1,2,\cdots,n}\right\\}$, $b\in\left\\{{0,1}\right\\}$ and
$x_{i}^{j}\in\left\\{{0,1}\right\\}$. $i$ stands for the $i$th possible case
where ${\mathop{\oplus}\limits_{j=1}^{n}x_{i}^{j}=b}$. $j$ represents $j$th
qubit and also corresponds to $j$th participant $Bo{b_{j}}$. The specific
process of encoding color information of $l$th pixel into
${\left|{{C_{b}}}\right\rangle_{l}}$ is shown in Fig. 4. If the $l$th pixel is
white, then $b=0$ (${\left|{{C_{0}}}\right\rangle_{l}}$); if the $l$th pixel
is black, then $b=1$ (${\left|{{C_{1}}}\right\rangle_{l}}$). The process of
encoding the color information of each pixel into a quantum superposition
state can be viewed as quantum expansion, which is analogous to pixel
expansion in most classical VSS schemes. Different from the pixel expansion,
quantum expansion not only makes the recovered image have no loss in
resolution, but also confuses the color information in each share (it makes it
impossible for an attacker to directly determine the color information of the
pixel).

Figure 4: The process of encoding color information into quantum superposition
state ${\left|{{C_{b}}}\right\rangle_{l}}$ ($b=0,1$)

The specific steps of the ($n$, $n$) QVSS sharing process are as follows (also
shown in Fig. 5).

Step 1: According to the previous context, Alice encodes the color information
of one pixel, e.g., the $l$th pixel, into an $n$-qubit superposition state
${\left|{{C_{b}}}\right\rangle_{l}}$.

Step 2: Alice distributes $n$ qubits $q_{l}^{1}$, $q_{l}^{2}$, $\cdots$,
$q_{l}^{n}$ which compose the state ${\left|{{C_{b}}}\right\rangle_{l}}$, as
shares to $Bo{b_{1}}$, $Bo{b_{2}}$, $\cdots$, $Bo{b_{n}}$, respectively.

Step 3: Alice repeats Step 1 and 2 to handle other $s$-1 pixels.

Figure 5: The whole sharing process of our proposed QVSS scheme

###  3.2 Recovering process

When all participants want to recover the secret image, we assume that
$Bo{b_{j}}$ is primarily responsible for performing specific operations. Then,
he needs to follow the steps below, which is shown in Fig. 7.

Step 1: $Bo{b_{j}}$ collects $n$ shares $q_{l}^{1}$, $q_{l}^{2}$, $\cdots$,
$q_{l}^{n}$ which correspond to the $l$th pixel, from remaining $n-1$
participants.

Step 2: $Bo{b_{j}}$ selects a measurement base
$\left\\{{\left|0\right\rangle,\left|1\right\rangle}\right\\}$ to measure the
$n$-qubit superposition state ${\left|{{C_{b}}}\right\rangle_{l}}$ which
consists of $q_{l}^{1}$, $q_{l}^{2}$, $\cdots$, $q_{l}^{n}$.
${\left|{{C_{b}}}\right\rangle_{l}}$ collapses into $n$-qubit certain state
$\left|{x_{i}^{1}x_{i}^{2}\cdots x_{i}^{n}}\right\rangle$.

Step 3: $Bo{b_{j}}$ performs XOR operation (its quantum circuit is illustrated
in Fig. 6) on the input quantum state $\left|{x_{i}^{1}x_{i}^{2}\cdots
x_{i}^{n}}\right\rangle$ and then get a result state $\left|{x_{i}^{1}\oplus
x_{i}^{2}\oplus\cdots\oplus x_{i}^{n}}\right\rangle$.

Figure 6: The $n$-qubit XOR circuit which is composed of multiple CNOT quantum
gates

Step 4: $Bo{b_{j}}$ makes judgment about the result state
$\left|{x_{i}^{1}\oplus x_{i}^{2}\oplus\cdots\oplus x_{i}^{n}}\right\rangle$,
if $\left|{x_{i}^{1}\oplus x_{i}^{2}\oplus\cdots\oplus
x_{i}^{n}}\right\rangle$ $=$ $\left|0\right\rangle$ , it means that the $l$th
pixel is white, otherwise, the $l$th pixel is black.

Step 5: $Bo{b_{j}}$ repeats Step 1 to 4 until the original secret image is
recovered.

Figure 7: The whole recovering process of our proposed QVSS scheme

In our scheme, different from pixel expansion, quantum expansion does not
bring about the actual expansion of the shared image size, i.e., the shared
image is the same size as the original secret image. So, our scheme does not
have loss in resolution and can completely recover the original secret image.

##  4 An example: (3, 3) QVSS scheme

Suppose Alice wants to share a secret image to $Bo{b_{1}}$, $Bo{b_{2}}$,
$Bo{b_{3}}$ and the secret image consists of 4 pixels, i.e., $n=3$ and $s=4$,
and the color information of each pixel is shown in Table 1.

Table 1: The color information of each pixel Pixel |  Color
---|---
1th pixel |  white
2th pixel |  black
3th pixel |  black
4th pixel |  white

During the sharing process, Alice encodes the color information of each pixel
into 3-qubit quantum superposition state ${\left|{{C_{b}}}\right\rangle_{l}}$.
Then, Alice distributes 3 shares $q_{l}^{1}$ (the first qubit),
$q_{l}^{2}$(the second qubit), and $q_{l}^{3}$ (the third one) which compose
${\left|{{C_{b}}}\right\rangle_{l}}$, to $Bo{b_{1}}$, $Bo{b_{2}}$,
$Bo{b_{3}}$, respectively, where $l\in\\{1,2,3,4\\}$. The quantum
superposition states and distributed shares of all pixels in the secret image
are listed in Table 2.

Table 2: The quantum superposition states and distributed shares of all pixel in the sharing process of (3, 3) QVSS scheme Pixel | ${\left|{{C_{b}}}\right\rangle_{l}}$ | Shares
---|---|---
1th pixel | ${\left|{{C_{0}}}\right\rangle_{1}}={1\over 2}(\left|{0_{1}^{1}0_{1}^{2}0_{1}^{3}}\right\rangle+\left|{0_{2}^{1}1_{2}^{2}1_{2}^{3}}\right\rangle+\left|{1_{3}^{1}0_{3}^{2}1_{3}^{3}}\right\rangle+\left|{1_{4}^{1}1_{4}^{2}0_{4}^{3}}\right\rangle)$ |  $q_{1}^{1}$, $q_{1}^{2}$, $q_{1}^{3}$
2th pixel | ${\left|{{C_{1}}}\right\rangle_{2}}={1\over 2}(\left|{1_{1}^{1}1_{1}^{2}1_{1}^{3}}\right\rangle+\left|{0_{2}^{1}0_{2}^{2}1_{2}^{3}}\right\rangle+\left|{0_{3}^{1}1_{3}^{2}0_{3}^{3}}\right\rangle+\left|{1_{4}^{1}0_{4}^{2}0_{4}^{3}}\right\rangle)$ |  $q_{2}^{1}$, $q_{2}^{2}$, $q_{2}^{3}$
3th pixel | ${\left|{{C_{1}}}\right\rangle_{3}}={1\over 2}(\left|{1_{1}^{1}1_{1}^{2}1_{1}^{3}}\right\rangle+\left|{0_{2}^{1}0_{2}^{2}1_{2}^{3}}\right\rangle+\left|{0_{3}^{1}1_{3}^{2}0_{3}^{3}}\right\rangle+\left|{1_{4}^{1}0_{4}^{2}0_{4}^{3}}\right\rangle)$ |  $q_{3}^{1}$, $q_{3}^{2}$, $q_{3}^{3}$
4th pixel | ${\left|{{C_{0}}}\right\rangle_{4}}={1\over 2}(\left|{0_{1}^{1}0_{1}^{2}0_{1}^{3}}\right\rangle+\left|{0_{2}^{1}1_{2}^{2}1_{2}^{3}}\right\rangle+\left|{1_{3}^{1}0_{3}^{2}1_{3}^{3}}\right\rangle+\left|{1_{4}^{1}1_{4}^{2}0_{4}^{3}}\right\rangle)$ |  $q_{4}^{1}$, $q_{4}^{2}$, $q_{4}^{3}$

Suppose $Bo{b_{2}}$ wants to recover the secret image in the recovering
process. In order to recover the first pixel, he should collect all shares
$q_{1}^{1}$, $q_{1}^{3}$ from $Bo{b_{1}}$ and $Bo{b_{3}}$. Then, he selects a
measurement base
$\left\\{{\left|0\right\rangle,\left|1\right\rangle}\right\\}$ to measure the
3-qubit superposition state ${\left|{{C_{0}}}\right\rangle_{1}}$ which is
composed of $q_{1}^{1}$, $q_{1}^{2}$, $q_{1}^{3}$. Suppose
${\left|{{C_{0}}}\right\rangle_{1}}$ collapses into
$\left|{0_{1}^{1}0_{1}^{2}0_{1}^{3}}\right\rangle$. After that, $Bo{b_{2}}$
performs XOR operation on $\left|{0_{1}^{1}0_{1}^{2}0_{1}^{3}}\right\rangle$
and get a result state $\left|{0_{1}^{1}\oplus 0_{1}^{2}\oplus
0_{1}^{3}}\right\rangle$. So, he can determine that the color of the first
pixel is white. Similarly, $Bo{b_{2}}$ can recover the remaining pixels in the
same way, and all the cases for $Bo{b_{2}}$ are listed in Table 3. By
comparing the initial pixel information before the sharing process (see Table
1) and the final pixel information after the recovering process (see the last
two columns in Table 3), it can be clearly found that our scheme can
completely recover the secret image.

Table 3: All the cases for $Bo{b_{2}}$ in the recovering process of (3, 3) QVSS scheme Shares | ${\left|{{C_{b}}}\right\rangle_{l}}$ | Collapsed state | Result state | Color | Pixel
---|---|---|---|---|---
$q_{1}^{1}$, $q_{1}^{2}$, $q_{1}^{3}$ | ${\left|{{C_{0}}}\right\rangle_{1}}$ | $\left|{0_{1}^{1}0_{1}^{2}0_{1}^{3}}\right\rangle$ | $\left|{0_{1}^{1}\oplus 0_{1}^{2}\oplus 0_{1}^{3}}\right\rangle=\left|0\right\rangle$ | white | 1th pixel
$q_{2}^{1}$, $q_{2}^{2}$, $q_{2}^{3}$ | ${\left|{{C_{1}}}\right\rangle_{2}}$ | $\left|{1_{1}^{1}1_{1}^{2}1_{1}^{3}}\right\rangle$ | $\left|{1_{1}^{1}\oplus 1_{1}^{2}\oplus 1_{1}^{3}}\right\rangle=\left|1\right\rangle$ | black | 2th pixel
$q_{3}^{1}$, $q_{3}^{2}$, $q_{3}^{3}$ | ${\left|{{C_{1}}}\right\rangle_{3}}$ | $\left|{1_{4}^{1}0_{4}^{2}0_{4}^{3}}\right\rangle$ | $\left|{1_{4}^{1}\oplus 0_{4}^{2}\oplus 0_{4}^{3}}\right\rangle=\left|1\right\rangle$ | black | 3th pixel
$q_{4}^{1}$, $q_{4}^{2}$, $q_{4}^{3}$ | ${\left|{{C_{0}}}\right\rangle_{4}}$ | $\left|{0_{2}^{1}1_{2}^{2}1_{2}^{3}}\right\rangle$ | $\left|{0_{2}^{1}\oplus 1_{2}^{2}\oplus 1_{2}^{3}}\right\rangle=\left|0\right\rangle$ | white | 4th pixel

##  5 Correctness analysis

Here, we mainly prove that the correctness of our scheme, which consists of
two criteria: (1) $n$ participants can cooperate to recover the original
secret image, (2) less than $n$ participants cannot recover the original one.

###### Theorem 5.1

The proposed ($n$, $n$) QVSS scheme can recover the original secret image when
all the participants work together.

Proof: Alice encodes each pixel’s color information into superposition state
${\left|{{C_{b}}}\right\rangle_{l}}$, and distributes $n$ qubits which compose
the ${\left|{{C_{b}}}\right\rangle_{l}}$ as shares to all participants
respectively. When all participants want to recover the original secret image,
they need to collect all $n$ shares corresponding to the same pixel and select
$Z$ basis to measure them. According to the definition of
${\left|{{C_{b}}}\right\rangle_{l}}$, there are ${2^{n-1}}$ possible cases
where make $\left|{x_{i}^{1}\oplus x_{i}^{2}\oplus\cdots\oplus
x_{i}^{n}}\right\rangle=\left|b\right\rangle$, that is, all possible cases go
through Step 3 of the recovering process and only one result state
$\left|b\right\rangle$ is obtained. It indicates that the color of the
recovered pixel is as same as the original one. So, with $n$ participants
involved in the recovering process, the scheme can recover the original secret
image.

###### Theorem 5.2

The proposed QVSS scheme does not work when any $k$ ($k<n$) participants
cooperate to recover the original secret image.

Proof: When $k$ participants want to recover the original secret image, we
assume that first $k$ participants collect their $k$ shares to recover the
$l$th pixel and $k$ is an even number. After Step 2 of the recovering process,
these $k$ shares collapse into $k$-qubit certain state
$\left|{x_{i}^{1}x_{i}^{2}\cdots x_{i}^{k}}\right\rangle$. The state
$\left|{x_{i}^{1}x_{i}^{2}\cdots x_{i}^{k}}\right\rangle$ has ${2^{k}}$
possible cases. Among these ${2^{k}}$ cases, there are
$C_{k}^{0}+C_{k}^{2}+\cdots+C_{k}^{k}$ cases that the number of $1$ among
$\left|{x_{i}^{1}x_{i}^{2}\cdots x_{i}^{k}}\right\rangle$ is even, and there
are $C_{k}^{1}+C_{k}^{3}+\cdots+C_{k}^{k-1}$ cases that the number of $1$
among $\left|{x_{i}^{1}x_{i}^{2}\cdots x_{i}^{k}}\right\rangle$ is odd. Then,
each of ${2^{k}}$ cases goes through Step 3 of recovering process. There are
$C_{k}^{0}+C_{k}^{2}+\cdots+C_{k}^{k}$ cases that $\left|{x_{i}^{1}\oplus
x_{i}^{2}\oplus\cdots\oplus x_{i}^{k}}\right\rangle$ $=$
$\left|0\right\rangle$, and there are $C_{k}^{1}+C_{k}^{3}+\cdots+C_{k}^{k-1}$
cases that $\left|{x_{i}^{1}\oplus x_{i}^{2}\oplus\cdots\oplus
x_{i}^{k}}\right\rangle$ $=$ $\left|1\right\rangle$. According to the nature
of combination number,
$C_{k}^{0}+C_{k}^{2}+\cdots+C_{k}^{k}=C_{k}^{1}+C_{k}^{3}+\cdots+C_{k}^{k-1}$.
So, the probabilities of getting $\left|0\right\rangle$ and
$\left|1\right\rangle$ are both equal to $\frac{1}{2}$. Similarly, when $k$ is
an odd number, the probabilities of getting $\left|0\right\rangle$ and
$\left|1\right\rangle$ are also equal to $\frac{1}{2}$. We can see that it is
impossible to determine whether the $l$th pixel is white or black.

Overall, we can see that with respecting the two criteria, the proposed scheme
is a well-defined ($n$, $n$) QVSS scheme.

##  6 Comparison and Discussion

In order to evaluate our scheme, we chose a classical VSS scheme [3] and two
quantum VSS schemes [35, 36] as references, and compare our QVSS scheme with
them from the following aspects: single-pixel parallel processing, pixel
expansion, and the loss in resolution. For a more intuitive representation,
the results of the comparison are shown in Table 4.

As mentioned in Sect. , the pixel is treated as a separate basic unit
throughout Naor et al.’s scheme. So, this scheme has the advantage of single-
pixel parallel processing. In order to make individual participant get no
information from shares, the scheme use the strategy of pixel expansion to
confuse the information of subpixels in each share. However, the size of the
shared image is $m$ times the original secret image and cause the loss in
resolution from the original image to the recovered one.

Then, we compare with two quantum VSS schemes, Song et al.’s scheme [35] and
Das et al.’s scheme [36]. In the Song et al.’s scheme, the whole secret image
is repeatedly encoded into a quantum state. So, the scheme does not retain the
advantage of single-pixel parallel processing, i.e., it is not suitable for
parallel computing of images. But, the scheme splits the original secret image
into sub-images as shares with multiple measurement operations. Therefore, it
does not need the strategy of pixel expansion. And the restored image is
exactly the same as the original secret image. Different from Song et al.’s
scheme, one pixel is treated as a unit for parallel processing in the Das et
al.’s scheme. However, the scheme still uses the strategy of pixel expansion
and apply the characteristics of quantum mechanics to determine the color of
each sub-pixel in each share, where each share has multiple sub-pixels. So,
the recovered image has the loss in resolution in the Das et al.’s scheme.

As same as Naor et al.’s scheme and Das et al.’s scheme [36], our scheme
retains the advantage of single-pixel parallel processing, i.e., the color
information of each pixel is encoded into $n$-qubit superposition state.
Different from them in other aspects, we use the strategy of quantum expansion
instead of pixel expansion. Therefore, the size of the shared image is as same
as the original one. In the recovering process, all participants cooperate to
measure the qubits they hold and execute the $n$-qubit XOR operation to
recover each pixel of the secret image. The recovered image is the same as the
original secret image and has no loss in resolution.

Table 4: Comparison among analogous quantum schemes, Naor et al.’s scheme and our scheme Schemes | Naor et al.’s scheme[3] | Song et al.’s scheme[35] | Das et al.’s scheme[36] | Our scheme
---|---|---|---|---
Single-pixel parallel processing | Yes | No | Yes | Yes
Pixel expansion | Yes | No | Yes | No
The loss in resolution | Yes | No | No | No

##  7 Experiment implementation with IBM Q

IBM Q [38] is an online platform that gives users in the general public access
to a set of IBM’s prototype quantum processors via the network. In this
section, we use IBM Q to demonstrate the practical feasibility of this scheme.

For the sake of brevity, we assume that $n=6$, i.e., the (6, 6) QVSS scheme is
the experimental object. In Step 1 of the sharing process, to share a white
pixel, we encode color information of the pixel into the 6-qubit superposition
state $\left|{{C_{0}}}\right\rangle$ which is composed of 6 qubits.
$\left|{{C_{0}}}\right\rangle$ is composed of the base states in Table 5, and
each base state in the quantum superposition state
$\left|{{C_{0}}}\right\rangle$ is with equal probability.

$\left|{{C_{0}}}\right\rangle=\sum\limits_{i}^{\mathop{\oplus}\limits_{j=1}^{6}x_{i}^{j}=0}{{{\left|{x_{i}^{1}x_{i}^{2}\cdots
x_{i}^{6}}\right\rangle}\over{\sqrt{32}}}}$ (5)

Since IBM Q is just a single quantum computer, which does not support the
transfer of quantum information between multiple nodes, so we ignore the
process of qubits transmission in our scheme (i.e., Step 2 of the sharing
process and Step 1 of the recovering process) in our quantum experiment, and
directly implement the reminder in IBM Q. In the sharing process, we firstly
design a quantum circuit (shown in Fig. 8) to construct
$\left|{{C_{0}}}\right\rangle$, and run it on the IBM Q platform. We measure
the state $\left|{{C_{0}}}\right\rangle$ and then get the probability of each
base state in $\left|{{C_{0}}}\right\rangle$, where the result is shown in
Fig. 9. We can find that the probability of each state is almost equal, and
the probability amplitudes of all base states float above and below
$\frac{1}{{\sqrt{32}}}$.

Table 5: All base states of the quantum superposition state $\left|{{C_{0}}}\right\rangle$ Number | Base state | Number | Base state | Number | Base state | Number | Base state
---|---|---|---|---|---|---|---
1 | $\left|{000000}\right\rangle$ | 9 | $\left|{010001}\right\rangle$ | 17 | $\left|{100001}\right\rangle$ | 25 | $\left|{101110}\right\rangle$
2 | $\left|{000011}\right\rangle$ | 10 | $\left|{010010}\right\rangle$ | 18 | $\left|{100010}\right\rangle$ | 26 | $\left|{110011}\right\rangle$
3 | $\left|{000101}\right\rangle$ | 11 | $\left|{010100}\right\rangle$ | 19 | $\left|{100100}\right\rangle$ | 27 | $\left|{110101}\right\rangle$
4 | $\left|{000110}\right\rangle$ | 12 | $\left|{010111}\right\rangle$ | 20 | $\left|{100111}\right\rangle$ | 28 | $\left|{110110}\right\rangle$
5 | $\left|{001001}\right\rangle$ | 13 | $\left|{011000}\right\rangle$ | 21 | $\left|{101000}\right\rangle$ | 29 | $\left|{111001}\right\rangle$
6 | $\left|{001010}\right\rangle$ | 14 | $\left|{011011}\right\rangle$ | 22 | $\left|{101011}\right\rangle$ | 30 | $\left|{111010}\right\rangle$
7 | $\left|{001100}\right\rangle$ | 15 | $\left|{011101}\right\rangle$ | 23 | $\left|{101101}\right\rangle$ | 31 | $\left|{111100}\right\rangle$
8 | $\left|{001111}\right\rangle$ | 16 | $\left|{011110}\right\rangle$ | 24 | $\left|{101110}\right\rangle$ | 32 | $\left|{111111}\right\rangle$

Figure 8: Implementation circuit of preparing the quantum superposition state
$\left|{{C_{0}}}\right\rangle$
Figure 9: The probability of each base state in the quantum superposition
state $\left|{{C_{0}}}\right\rangle$

After the state $\left|{{C_{0}}}\right\rangle$ is measured, the quantum
superposition state may collapse into one certain state from 32 base states.
We assume that $\left|{{C_{0}}}\right\rangle$ collapses into one certain state
$\left|{101000}\right\rangle$. Then, we perform XOR operation on the state
$\left|{101000}\right\rangle$ to get a result state on the IBM Q, where the
circuit is shown in Fig. 10 (a). The result state is $\left|0\right\rangle$
and its probability is shown in Fig. 10 (b). Finally, according to Step 4 of
the recovering process, we can determine that the pixel is white.

(a) Implementation circuit of XOR operation on $\left|{101000}\right\rangle$
(b) The probability of the result state

Figure 10:  The process of performing XOR operation on
$\left|{101000}\right\rangle$

To share a black pixel, the process of experiment is similar to the above. We
encode color information of the pixel into the quantum superposition state
$\left|{{C_{1}}}\right\rangle$. $\left|{{C_{1}}}\right\rangle$ can be composed
of each base state in Table 6. Each base state in the quantum superposition
state $\left|{{C_{1}}}\right\rangle$ is also with equal probability, so

$\left|{{C_{1}}}\right\rangle=\sum\limits_{i}^{\mathop{\oplus}\limits_{j=1}^{6}x_{i}^{j}=1}{{{\left|{x_{i}^{1}x_{i}^{2}\cdots
x_{i}^{6}}\right\rangle}\over{\sqrt{32}}}}.$ (6)

Table 6: All base states of the quantum superposition state $\left|{{C_{1}}}\right\rangle$ Number | Base state | Number | Base state | Number | Base state | Number | Base state
---|---|---|---|---|---|---|---
1 | $\left|{000001}\right\rangle$ | 9 | $\left|{010000}\right\rangle$ | 17 | $\left|{100000}\right\rangle$ | 25 | $\left|{101111}\right\rangle$
2 | $\left|{000010}\right\rangle$ | 10 | $\left|{010011}\right\rangle$ | 18 | $\left|{100011}\right\rangle$ | 26 | $\left|{110010}\right\rangle$
3 | $\left|{000100}\right\rangle$ | 11 | $\left|{010101}\right\rangle$ | 19 | $\left|{100101}\right\rangle$ | 27 | $\left|{110100}\right\rangle$
4 | $\left|{000111}\right\rangle$ | 12 | $\left|{010110}\right\rangle$ | 20 | $\left|{100110}\right\rangle$ | 28 | $\left|{110111}\right\rangle$
5 | $\left|{001000}\right\rangle$ | 13 | $\left|{011001}\right\rangle$ | 21 | $\left|{101001}\right\rangle$ | 29 | $\left|{111000}\right\rangle$
6 | $\left|{001011}\right\rangle$ | 14 | $\left|{011010}\right\rangle$ | 22 | $\left|{101010}\right\rangle$ | 30 | $\left|{111011}\right\rangle$
7 | $\left|{001101}\right\rangle$ | 15 | $\left|{011100}\right\rangle$ | 23 | $\left|{101100}\right\rangle$ | 31 | $\left|{111101}\right\rangle$
8 | $\left|{001110}\right\rangle$ | 16 | $\left|{011111}\right\rangle$ | 24 | $\left|{101111}\right\rangle$ | 32 | $\left|{111110}\right\rangle$

The process of constructing $\left|{{C_{1}}}\right\rangle$ and the probability
of each base state in $\left|{{C_{1}}}\right\rangle$ are shown in Fig. 11 and
Fig. 12, respectively. After the state $\left|{{C_{1}}}\right\rangle$ is
measured, we assume that $\left|{{C_{1}}}\right\rangle$ collapses into one
certain state $\left|{101010}\right\rangle$. The process of XOR operation and
the probability of the result state are shown in Fig. 13 (a) and (b)
respectively. Finally, according to Step 4 of the recovering process, we can
determine that the pixel is black.

Figure 11: Implementation circuit of preparing the quantum superposition state
$\left|{{C_{1}}}\right\rangle$
Figure 12: The probability of each base state in the quantum superposition
state $\left|{{C_{1}}}\right\rangle$

(a) Implementation circuit of XOR operation on $\left|{101010}\right\rangle$
(b) The probability of the result state

Figure 13: The process of performing XOR operation on
$\left|{101010}\right\rangle$

##  8 Conclusion

This paper presents a novel ($n$, $n$) quantum visual secret sharing scheme
based on Naor et al.’s scheme. In our scheme, we encode the color of each
pixel into $n$-qubit superposition state, to preserve the advantage of single-
pixel parallel processing. Moreover, our scheme uses the strategy of quantum
expansion instead of classical pixel expansion. This strategy solves the
problem that the recovered image has loss in resolution which caused by pixel
expansion. Moreover, compared with other classical VSS scheme and two quantum
VSS schemes, we can see that our scheme preserves the advantage of single-
pixel parallel processing and solves two problems, which are pixel expansion
and the loss in resolution. Besides, the proposed scheme is able to meet two
criteria (shown in Sect. 5) and has practical feasibility. Of course, ($t$,
$n$) scheme is a more application-oriented scheme. How to construct a
threshold scheme is one of our future work.

## Acknowledgment

The authors would like to express heartfelt gratitude to the anonymous
reviewers and editor for their comments that improved the quality of this
paper. And the support of all the members of the quantum research group of
NUIST is especially acknowledged, their professional discussions and advice
have helped us a lot.

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11institutetext: School of Astronomy, Institute for Research in Fundamental
Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran

# Testing Ghasemi-Nodehi-Bambi spacetime with continuum-fitting method

M. Ghasemi-Nodehi _email:_<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

The continuum-fitting method is the analysis of the thermal spectrum of the
geometrically thin and optically thick accretion disk around stellar-mass
black holes. A parametrization aiming to test the Kerr nature of astrophysical
black holes was proposed in Ghasemi-Nodehi and Bambi in EPJC 76: 290, 2016.
The metric contains 11 parameters in addition to the mass and spin parameters.
One can recover the Kerr case by setting all parameters to one. In this paper,
I study the continuum-fitting method in Ghasemi-Nodehi-Bambi background. I
show the impact of each of the parameters on the spectra. I then employ
$\chi^{2}$ studies and show that using the continuum-fitting method all
parameters of Ghasemi-Nodehi-Bambi spacetime are degenerate. However, the
parameter $b_{9}$ can be constrained in the case of a high spin value and the
Ghasemi-Nodehi-Bambi black hole as reference. This degeneracy means that the
spectra of the Kerr case cannot be distinguished from spectra produced in
Ghasemi-Nodehi-Bmabi spacetime. This is a problem as regards measuring the
spin of astrophysical black holes and constrain possible deviations from the
Kerr case of General Relativity.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

General Relativity (GR) was born more than a century ago and is still used for
describing the gravitational field and geometry of spacetime einstein1916 .
The theory has been extensively explored in the weak field regime, such as in
solar system experiments and radio pulsars observations will ; will2 ; will3 ;
will4 . There are many theoretical models that make the same prediction as
general relativity in the weak field regime but they deviate in strong field
regimes. The strong field regime can be either studied with gravitational
waves (GWs) or with electromagnetic radiation. Dynamical systems have a
signature on GW but an electromagnetic radiation method can be studied when
spacetime is static. Effects such as the deviation from geodesics motion can
be studied by electromagnetic radiation-related methods.

GR predicts astrophysical black holes (BHs) are Kerr BHs with spin and mass
parameters kerr1 ; kerr2 . There are different scenarios that deviate from the
Kerr case but there is degeneracy in measurement of their parameters and spin
parameters of Kerr spacetime. One way to test the Kerr hypothesis is to
parametrize the Kerr metric and try to constrain deviations from Kerr
spacetime. There are several parametrization in the literature p1 ; p2 ; p3 ;
p4 ; p5 ; p6 ; p7 ; p8 ; p9 ; p10 ; p11 ; p12 ; p13 ; gbma ; p14 . In this
paper, I consider the Ghasemi-Nodehi-Bambi (GB) metric, proposed in gbma . The
Kerr case would be recovered by setting all GB parameters to one. For an
electromagnetics-based test of the strong gravity regime, one can use X-ray
continuum spectra and iron line emission. One can also study BH shadow
observations with an event horizon telescope at mm/sub-mm wavelength. In spite
of large systematic errors of quasi-periodic oscillations (QPOs), one can use
QPO frequencies to probe strong field regimes as well. Iron line reverberation
mapping can be used as well.

Continuum spectra, iron lines and QPOs have already been observed by X-ray
missions including RXTE, Chandra, Nustar, ASCA, Suzaku, and XMM-Newton.

The shadow gbma , X-ray reflection spectroscopy gbiron , QPO qpoma  and iron
line reverberation qpoma  of the GB metric have already been studied. In the
current paper, I study the continuum-fitting method to check possible
constraints of the GB parametrization.

The continuum-fitting method is used to study the thermal spectrum of
geometrically thin and optically thick accretion disks confir . One can only
apply this method to stellar-mass BHs. This is because the spectrum of the
supermassive BHs is in the optical/UV band where dust absorption limits the
capability of accurate measurement. The temperature of the disk depends on the
mass of the compact object. The spectrum of the thin disks around stellar-mass
black holes is in the soft X-ray band. Currently, assuming the Kerr BH as
astrophysical BH, the spin parameter, $a_{*}=J/M^{2}$, of the BH can be
measured by this method. If one considers non-Kerr BHs, this technique can
measure deviations from the Kerr solution nk1 ; nk2 .

In this paper, I consider a GB background metric. I simulate thermal spectra
of the thin accretion disk and show the impact of the deformation parameters
on the spectra. By decreasing (increasing) the deformation parameter $b_{7}$
the spectra get softer (harder) and by decreasing (increasing) the parameters
$b_{4},b_{9},b_{10}$ the spectra get harder (softer). The impact of the
parameters $b_{2},b_{5},b_{8},b_{11}$ is very close to Kerr one. The impact of
the other parameters was studied before. References nk1 ; con1 ; con3  provide
the impact of all other parameters. By increasing (decreasing) the mass the
spectra get harder (softer). The mass accretion rate changes the temperature
of the disk. The distance of the source alters the normalization of the
spectrum. For higher (lower) inclination angle, the area of the observed
emitting region decreases (increases). If the spin parameter increases the
spectra get harder.

Then, in order to compare results with the Kerr case, I use a minimum
$\chi^{2}$ approach. I plot the related contour plots. The results show that
using continuum-fitting method the GB parameters are degenerate and cannot be
distinguished from the Kerr case. However, the parameter $b_{9}$ may be
constrained in the case of a high spin value and the GB BH as a reference.

The structure of the paper is as follows. In Sect. 2, I provide details of the
continuum-fitting method. Section 3 is devoted to a description of the metric
of the spacetime. Results and discussion are presented in Sect. 4. Finally, a
summary and conclusions are in Sect. 5.

##  2 The continuum-fitting method

The thermal spectrum can be written in terms of the photon flux number density
as measured by a distant observer nk1 ; con1 ; con3 ; Li ; boo . The photon
flux number density is given by

$\displaystyle N_{E_{\rm obs}}=\frac{1}{E_{\rm obs}}\int I_{\rm
obs}(\nu)d\Omega_{\rm obs}$ (1)

where $I_{obs}$ is specific intensity of radiation, $E_{obs}$ is the photon
energy, and $\nu$ is the photon frequency measured by a distant observer.
$d\Omega_{\rm obs}=dXdY/D^{2}$, $X$ and $Y$ are coordinate position of the
photon on the sky as seen by a distant observer, $D$ is the distance of the
source. In order to include all relativistic effects, one needs to compute the
photon trajectory from the emission point in the disk to the image plane of
the distant observer (detection point). We have

$\displaystyle N_{E_{\rm obs}}$ $\displaystyle=$ $\displaystyle\frac{1}{E_{\rm
obs}}\int I_{\rm obs}(\nu)d\Omega_{\rm obs}=\frac{1}{E_{\rm obs}}\int
w^{3}I_{\rm e}(\nu_{\rm e})d\Omega_{\rm obs}$ (2)

where $I_{e}$ is the local specific intensity of the radiation emitted by disk
and $w$ the redshift factor, which are

$\displaystyle I_{\rm e}(\nu_{\rm e})=\frac{2h\nu^{3}_{\rm e}}{f_{\rm
col}^{4}}\frac{\Upsilon}{\exp\left(\frac{h\nu_{\rm e}}{k_{\rm B}T_{\rm
col}}\right)-1}\,,$ (3) $\displaystyle w=\frac{E_{\rm obs}}{E_{\rm
e}}=\frac{\nu}{\nu_{\rm e}}=\frac{k_{\alpha}u^{\alpha}_{\rm
obs}}{k_{\beta}u^{\beta}_{\rm e}}\,.$ (4)

Regarding Eq. 3, since the disk is in thermal equilibrium the emission is
blackbody-like. One can define the effective temperature, $T_{\rm eff}$,
$\mathcal{F}(r)=\sigma T^{4}_{\rm eff}$, where $\sigma$ is the Stefan-
Boltzmann constant and $\mathcal{F}(r)$ is the time averaged energy flux from
Novikov-Thorne model nov1 ; nov2 . The Novikov-Thorne model can be used to
describe geometrically thin and optically thick accretion disks around the
black hole. The assumption is that the disk is in the equatorial plane. The
gas of the disk moves on a nearly geodesic circular orbit. $\mathcal{F}(r)$
can be written as

$\displaystyle\mathcal{F}(r)=\frac{\dot{M}}{4\pi\sqrt{-G}}F(r)\,\,\,,\,\,\,\sqrt{-G}=\sqrt{\alpha^{2}g_{rr}g_{\phi\phi}}$
(5)

where $\dot{M}$ is the mass accretion rate, and $F(r)$ is

$\displaystyle F(r)=\frac{\partial_{r}\Omega}{(E-\Omega
L_{z})^{2}}\int_{r_{\rm in}}^{r}(E-\Omega
L_{z})(\partial_{\rho}L_{z})d\rho\,.$ (6)

$E,L_{z},$ and $\Omega$ are, respectively, the conserved specific energy, the
conserved $z$-component of the specific angular momentum, and the angular
velocity for equatorial circular geodesics. $r_{\rm in}$ is assumed to be at
the innermost stable circular orbit (ISCO). Actually the disk’s temperature
near the inner edge is high and non-thermal effects are not negligible. For
this reason one can introduce a hardening factor or a color factor, $f_{\rm
col}$ and the color temperature is $T_{\rm col}(r)=f_{\rm col}T_{\rm eff}.$ In
Eq. 3, $\nu_{e}$ is the photon frequency, $h$ is Planck’s constant, $k_{B}$ is
the Boltzmann constant and $\Upsilon$ is a function of the angle between the
wavevector of a photon emitted by the disk and the normal of the disk surface.
Here I consider $\Upsilon=1$ for isotropic emission.
Regarding the $w$ in Eq. 4, $E_{e}=h\nu_{e},\nu$ being a photon frequency
measured by a distant observer, $k^{\alpha}$ is the 4-momentum of the photon,
$u^{\alpha}_{\rm obs}=(-1,0,0,0)$ is the 4-velocity of the distant observer,
and $u^{\alpha}_{\rm e}=(u^{t}_{\rm e},0,0,\Omega u^{t}_{\rm e})$ is the
4-velocity of the emitter. From Liouville’s theorem we have $I_{\rm
e}(\nu_{\rm e})/\nu_{\rm e}^{3}=I_{\rm obs}(\nu_{\rm obs})/\nu^{3}$.
Finally, the photon flux number density can be written as

$\displaystyle N_{E_{\rm obs}}$ $\displaystyle=$ $\displaystyle
A_{1}\left(\frac{E_{\rm obs}}{\rm
keV}\right)^{2}\int\frac{1}{M^{2}}\frac{\Upsilon
dXdY}{\exp\left[\frac{A_{2}}{wF^{1/4}}\left(\frac{E_{\rm obs}}{\rm
keV}\right)\right]-1}\,,$ (7)

where $A_{1}$ and $A_{2}$ are

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle\frac{2\left({\rm
keV}\right)^{2}}{f_{\rm col}^{4}}\left(\frac{G_{\rm N}M}{c^{3}hD}\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{0.07205}{f_{\rm
col}^{4}}\left(\frac{M}{M_{\odot}}\right)^{2}\left(\frac{\rm
kpc}{D}\right)^{2}\,{\rm\gamma\,keV^{-1}\,cm^{-2}\,s^{-1}}\,,$ $\displaystyle
A_{2}$ $\displaystyle=$ $\displaystyle\left(\frac{\rm keV}{k_{\rm B}f_{\rm
col}}\right)\left(\frac{G_{\rm
N}M}{c^{3}}\right)^{1/2}\left(\frac{4\pi\sigma}{\dot{M}}\right)^{1/4}$ (8)
$\displaystyle=$ $\displaystyle\frac{0.1331}{f_{\rm col}}\left(\frac{\rm
10^{18}\,g\,s^{-1}}{\dot{M}}\right)^{1/4}\left(\frac{M}{M_{\odot}}\right)^{1/2}\,.$

One finds

$\displaystyle u^{t}_{\rm e}=-\frac{1}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}\,,$ (9)

from the normalization condition $g_{\mu\nu}u^{\mu}_{\rm e}u^{\nu}_{\rm e}=-1$
and thus

$\displaystyle w=\frac{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}{1+\lambda\Omega}\,,$ (10)

where $\lambda=k_{\phi}/k_{t}$ is a constant of motion along the photon’s
path. All relativistic effects are encoded in the redshift factor $w$.

##  3 Ghasemi-Nodehi-Bambi spacetime

Reference gbma  proposed a new parametrization to the Kerr metric. The purpose
of this parametrization is to constrain possible deviations from a Kerr
solution of GR. The Kerr case would be recovered when all deformation
parameters are equal to one. While in other metrics the deformation parameters
are additive and reduce to the Kerr case for vanishing deformation parameters.
In this parametrization, 11 new parameters are introduced in front of any mass
and/or spin term. Any deviation from one deforms the spacetime more/less than
that of prediction of GR. The metric is as follows:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2b_{1}Mr}{r^{2}+b_{2}a^{2}\cos^{2}\theta}\right)dt^{2}$
(11)
$\displaystyle-\frac{4b_{3}Mar\sin^{2}\theta}{r^{2}+b_{4}a^{2}\cos^{2}\theta}dtd\phi+\frac{r^{2}+b_{5}a^{2}\cos^{2}\theta}{r^{2}-2b_{6}Mr+b_{7}a^{2}}dr^{2}$
$\displaystyle+\left(r^{2}+b_{8}a^{2}\cos^{2}\theta\right)d\theta^{2}$
$\displaystyle+\left(r^{2}+b_{9}a^{2}+\frac{2b_{10}Ma^{2}r\sin^{2}\theta}{r^{2}+b_{11}a^{2}\cos^{2}\theta}\right)\sin^{2}\theta
d\phi^{2}\,.$

Here we set $b_{1}=b_{3}=b_{6}=1$. $b_{1}$ is equal to one because it is the
coefficient of mass and $b_{3}=1$ in the same way; $b_{3}a$ is the asymptotic
specific angular momentum. $b_{6}$ is close to one from a solar system
experiment. I do not consider these three parameters in my continuum-fitting
method calculations.

##  4 Results and discussion

The results of my simulation of thermal spectra are shown in Figs.1-4. In Fig.
1 I have drawn the impact of deformation parameter $b_{i}$ on the thermal
spectra of the accretion disk for different values of the model parameters.
The values of the model parameters are $M=10M_{\odot}$, $\dot{M}=2\times
10^{18}$ g/s, $D=10$ kpc, $i=45^{\circ}$, $a_{*}=a/M=0.9,$. In each curve one
$b_{i}$ is equal to 15 and all others are equal to 1. In these simulations I
have assumed $f_{col}=\Upsilon=1$. As we see in the plot, the parameter
$b_{7}$ makes the spectra harder than the Kerr one on increasing. By
increasing the parameter $b_{4},b_{9}$ and $b_{10}$ the spectra get softer
than Kerr case. Parameters $b_{2},b_{5},b_{8},$ and $b_{11}$ are nearly
similar to Kerr one. The overall structure of the plots does not change for
different sets of parameters. The impact of all other parameters is studied in
nk1 ; con1 ; con3 .

Figure 1: Thermal spectra of accretion disks for different values of the
$b_{i}$ parameters. The values of the model parameters are $M=10M_{\odot}$,
$\dot{M}=2\times 10^{18}$ g/s, $D=10$ kpc, $i=45^{\circ}$, $a_{*}=0.9,$ and
one of $b_{i}=15$ in each case. In these simulations I have assumed
$f_{col}=\Upsilon=1$. See text for more details.

Now, in order to compare my results, I use the$\chi^{2}$ approach. I compare
the spectra produced in Kerr spacetime with one expected from the GB
background. The $\chi^{2}$ is defined as follows:

$\displaystyle\chi^{2}(a_{*},b_{j})=\frac{1}{n}\sum_{i=1}^{n}\frac{N_{i}^{GB}-N_{i}^{Kerr}}{\sigma_{i}^{2}}$
(12)

where summation is performed over $n$ sampling energies $E_{i}$.
$N_{i}^{Kerr}$ and $N_{i}^{GB}$ are the photon fluxes in the energy bin,
$[E_{i},E_{i}+\Delta E]$, for Kerr and GB metric. The error, $\sigma_{i}$, is
considered to be $15\%$ of the photon flux $N_{i}^{Kerr}$. For these results,
$M=10M_{\odot}$, $\dot{M}=2\times 10^{18}$ g/s, $D=10$ kpc, $i=45^{\circ}$ and
$a_{*}=0.6$. The contours are presented in Figs. 2 and 3. The brown, orange
and lighter orange are for $1\sigma,2\sigma$ and $3\sigma$ (68%, 95% and
99.7%) confidence level, respectively. If the contour plots would be closed,
this means we can constrain the parameters and there is no degeneracy, but,
here, the contours are not closed in Figs. 2 and 3. So we cannot constrain the
parameters. In other words, there is degeneracy between measurements of the
parameters $b_{i}$ and measurement of the spin parameter, so that the
parameters $b_{i}$ cannot be constrained. Degeneracy means one cannot
distinguish the properties of these spectra from spectra of a disk around a
Kerr BH.

Figure 2: Contour map of thermal spectra of accretion disks. Contour map of
parameter $b_{2}$ (top left panel), $b_{4}$ (top right panel), $b_{5}$ (bottom
left panel), $b_{7}$ (bottom right panel). The reference model is a Kerr BH
with spin 0.6. The other parameter values are $M=10M_{\odot}$,
$\dot{M}=2\times 10^{18}$ g/s, $D=10$ kpc, and $i=45^{\circ}$. See the text
for more details.

Figure 3: Contour map of thermal spectra of accretion disks. Contour map of
parameter $b_{8}$ (top left panel), $b_{9}$ (top right panel), $b_{10}$
(bottom left panel), $b_{11}$ (bottom right panel). The reference model is a
Kerr BH with spin 0.6. The other parameter values are $M=10M_{\odot}$,
$\dot{M}=2\times 10^{18}$ g/s, $D=10$ kpc, and $i=45^{\circ}$. See the text
for more details.

This degeneracy is a problem as regards measuring the spin of astrophysical
BHs. So far, spin measurement is based on the Kerr BH hypothesis but as we see
from the result of this study with the continuum-fitting method, the same
electromagnetic observation (thermal spectra here) can be reproduced by non-
Kerr spacetime as an example of GB spacetime with different spin and
deformation parameters.

I also consider the case for higher values of the spin, such as $0.9$, as
reference model but still the results are degenerate. Then, I consider the
reference BH not to be a Kerr BH. I consider a GB BH as reference BH in a
contour study with spin $0.9$ and $b_{i}=18$. In this case, only the parameter
$b_{9}$ may be constrained because the contour looks closed, as we see in Fig.
4. The parameter $b_{10}$ is harder to constrain is this case as well.

Figure 4: Contour map of thermal spectra for parameter $b_{9}$. The reference
model is a non-Kerr BH with $M=10M_{\odot}$, $\dot{M}=2\times 10^{18}$ g/s,
$D=10$ kpc, $i=45^{\circ}$, $a_{*}=0.9,$ and $b_{9}=18$. In these simulations
I have assumed $f_{col}=\Upsilon=1$. See text for more details.

As the result of other tests with electromagnetic radiation, considering
triplet frequency observations of GRO J1655-40, we find that the GB parameter
$b_{2}$ is $2.2_{-0.523}^{+0.395}$ but the other parameters cannot be
constrained qpoma . If we assume we have 10 observations in the future, we can
constrain more parameters, we find that $b_{2},b_{9},b_{10}$ and $b_{11}$ can
be well constrained, while $b_{4},b_{5},b_{7}$ and $b_{8}$ cannot. With X-ray
reflection spectroscopy, known as the iron line method, 200 ks observations
with future observational facilities such a LAD-eXTP can constrain all
parameters except for $b_{11}$. $b_{11}$ leaves very weak impact on the iron
line. A BH shadow method shows a weak impact on the parameters
$b_{2},b_{8},b_{9}$ and $b_{10}$. There is no impact on the shadow boundary by
the parameters $b_{4},b_{5},b_{7}$ and $b_{11}$. An iron line reverberation
mapping study of GB spacetime can constrain all parameters except $b_{11}$.
$b_{4}$ also is harder to constrain qpoma .
As we see the BH shadow can constrain some parameters but can not constrain
others. Iron line studies either with time-integrated analysis or with
reverberation can constrain all with exception of $b_{11}$. But a mock
observation of 10 BHs can constrain $b_{11}$ in addition to the other
parameters.
Unfortunately, due to degeneracy of the parameters in the continuum-fitting
method, we cannot further restrict GB parameters using this method. This might
be a general problem as regards measuring the spin of the BH.

A combination of different observational methods may help to constrain the
deformation parameters and break the degeneracy of the parameters.

##  5 Summary and conclusions

The continuum-fitting method is a method to study thermal spectra. The disk
should be geometrically thin and optically thick. Also this study is valid for
stellar-mass black holes.

Furthermore, parametrization of the Kerr BH hypothesis is one way to study
deviations from the predictions of GR and provide constraints on the deviation
from GR.

In this work I use the continuum-fitting method to check whether it can
constrain the GB metric parameters introduced in gbma . The GB metric contains
11 deformation parameters, which the Kerr case recovers when all deformation
parameters are set to one. I first compute the thermal spectra for different
$b_{i}$ parameters. My results show that by increasing parameter $b_{7}$ the
spectra get harder. Also by increasing the parameters $b_{4},b_{9},b_{10}$ the
spectra get softer and impact of the parameters $b_{2},b_{5},b_{8},$ and
$b_{11}$ is very close to the Kerr one.

I then consider my simulation as observational data and try to study possible
constraints of the deformation parameters of GB spacetime using the $\chi^{2}$
approach. My contour studies show that all deformation parameters are
degenerate. However, for the case of a high spin value with GB BH as
reference, the parameter $b_{9}$ may be constrained. This degeneracy means the
observation with specific spin and the deformation parameter can reproduce the
spectra of the Kerr case. This is a problem as regards of measuring the spin
of astrophysical black holes and constrain possible deviations from GR. Also
as a result of this degeneracy, in order to verify the Kerr hypothesis, any
deviation from the Kerr case should be vanish.

Different observational tests such as the BH shadow method, X-ray reflection
spectroscopy, iron line reverberation mapping and QPO introduce different
impacts of the parameters. Combination of different methods may help to break
the degeneracies.

Acknowledgments

This work is supported by School of Astronomy, Institute for Research in
Fundamental Sciences (IPM), Tehran, Iran.

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