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# An Extended Quantum Process Algebra (eQPAlg) Approach For Distributed
Quantum Systems

Salman Haider
Department of Computer Science
Government College University
Anarkali Bazaar 54000 Lahore, Punjab, Pakistan
<EMAIL_ADDRESS>
&Dr. Syed Asad Raza Kazmi
Department of Computer Science
Government College University
Anarkali Bazaar 54000 Lahore, Punjab, Pakistan
<EMAIL_ADDRESS>

###### Abstract

In this work, we have expounded the communication procedure of quantum systems
by means of process algebra. The main objective of our research effort is to
formally represent the communication between distributed quantum systems. In
this new proposed communication model we have ameliorated the existing rules
of Lalire’s quantum process algebra QPAlg. We have brought some important
modification in QPAlg by introducing the concept of formally specifying the
Quantum teleportation protocol. We have further introduced the formal
description of protocol by using programs that best explains its working and
satisfies the specification. Examples have been provided to describe the
working of the improved algebra that formally explain the sending and
receiving of both classical as well as quantum data, keeping in mind the
principal features of quantum mechanics.

Keywords: Quantum computing, Quantum mechanics, Quantum teleportation
protocol, Process algebra

##  1 Introduction

Quantum information theory is an advanced approach and is basically an amalgam
of computer science, physics, and mathematics. It uses the principles of
quantum mechanics. In quantum mechanics, the state of the system is expressed
by a wave function. The fundamental laws of quantum mechanics are widely
adopted in computation as well as in communication. As a comparatively modern
computational model, the quantum computing[29][15] brings the dawn of solving
the so-called NP problem because of the strong parallel computation power of
quantum computing. Many of the principles of quantum mechanics, such as
quantum no-cloning theorem, uncertainty principle, and entanglement provide
quantum communication protocols the provable security. We first introduce the
key aspects of quantum mechanics in computer science such as quantum
computation and quantum communication. The novel concepts of process algebra
in the light of formal methods have been explained further with a brief
overview of previous researches.

###  1.1 Quantum Computation & Communication

Quantum Computing is a new computational theory in computer science. The
machine which carries out this computation is referred to as Quantum Computer.
We know in digital computing, computations are performed by Standard Computers
that consist of transistors. Such types of standard computers are able to
interpret data only in the binary format that consist of either 0 or 1 state.
On the other hand, the novel Quantum computation uses quantum bits (qubits)
which can be in a superposition of the two states. Quantum computers could
possess incredibly large processing power and could process much more amount
of information at a given time. The amount of information flow quantum
mechanical computer can measure would be: $2^{N}$ where N = Number of qubits.
For larger qubits, if we are given 300 qubits then $2^{300}$ would be huge
enough that it might be equal to the total number of particles present in the
observeable universe.

A classical computer can take millions of years to find the prime factor of
2048 bits number whereas using a quantum computer it will take only a few
minutes. The processing power of quantum computer when we are given with 30
qubits would be equal to the classical computer that computes at 10 Teraflops.
These computers took advantages of quantum superposition to reduce T(n):
Number of steps requires to get the computational results and can only perform
faster operations only if we are using some particular algorithms as we can
perform a number of computational operations in an exponentially smaller time.
Nowadays, communication devices and protocols are based on the principles of
classical physics where information is communicated in the form of bits. In
addition to quantum computing, the important aspects of quantum mechanics [24]
such as quantum teleportation[4], no-cloning theorem and entanglement [22]
needs to be studied (For details, one should refer to these [24][22][29][15].
Quantum computing actually offers the option of an entirely new paradigm for
information processing, with the alluring possibility of having the capacity
to break existing crypto-systems.

In recent years substantial progress has been made in the area of quantum
communication. Moreover, the laws of quantum mechanics have made it possible
to accomplish the communication tasks that are not possible using classical
principles. Some of these tasks include dense coding [3]; unconditionally
secure quantum key distribution [24] and quantum teleportation (QT). The basic
protocols of quantum communication tasks were propounded from 1984 to 1993.
For example

  * •

In 1984, Bennett and Brassard put forward the very first protocol of QKD known
as BB84 protocol. In this protocol, there is a sender that relays a random
sequence of bits and then distributes it to a remote receiver using quantum
means.

  * •

Moving to the concept of dense coding, aka super dense coding was proposed
first time by Bennett [14] in 1992. In this, a sender can communicate two
classical bits of information to the receiver by sending only one qubit of
information with the condition that the sender and receiver must share a prior
entanglement.

  * •

In 1993, Bennett introduced the quantum teleportation (QT) scheme [4][22][2]
in which a sender transmits an unknown quantum state to a distant receiver
using two bits of classical communication along with an entangled state that
has already been shared by both the sender and receiver.

After the publication of these fundamental research works regarding quantum
protocols, several new quantum communication protocols have been coined. Some
of which do not need security as these are already secured such as quantum
teleportation and super dense coding and some of these require security proof
such as protocols of quantum key distribution (QKD), Deterministic secure
quantum communication (DSQC), Quantum secret sharing (QSS), quantum dialogue
(QD) and Quantum secure direct communication (QSDC) [27][5].

###  1.2 Algebra of Quantum Processes

In the algebra of quantum processes, formal methods are used to provide
systematic techniques for modeling [4], analysis and verification of quantum
systems. It is normally used to model the behavior of systems that are created
with both classical and quantum information. There are many formal languages
available pertinent to specification and verification. There has been greater
than before interest in process algebra [16] for the specification and
modeling of concurrent systems. It is appealing to unite quantum computing and
classical computing under the same process algebra framework as most quantum
communication protocols involve quantum information and classical information,
quantum computing and classical computing. In the last few years, serious
efforts have been made to link the computer science areas like programming
languages and formal verification with quantum information processing.
The scheme of devising a robust computational machine based on quantum
mechanical laws dates back to 1980 [31] and [32]. Then, Feynman [6] propounded
that a quantum mechanical system becomes useful for computing, a great deal of
attention was paid over quantum computing based on quantum information theory.
Quantum computing (as discussed earlier) can provide a great speedup as
compared to its classical analogue after taking advantage of the
characteristics of being in a superposition of quantum states. [8] [9] [26].
In order to offer an approach of examining the computational difficulties in a
more logical way, some researchers started studying the semantics and design
of quantum programming languages (QPL). Knill composed a set of fundamental
transition rules for describing the quantum pseudo-codes [3], whereas Omer
proposed the very 1st real-time programming language for quantum systems known
as QCL[8] [9]. In the meantime, different programming languages have been
proposed that would possibly be working on quantum computers (Refer to this
survey by Gay [7]), among which is qGCL which resembles Dijkstra guarded
commands language was formulated by Zuliani et al. in [23] [34]. An extended
version of C++ with principles of quantum systems, which is being proposed as
a C++ library, was proposed by Bettelli [21]. But the first breakthrough was
when Selinger [25] proposed the quantum language (QPL) which was first
function-based programming language. QPL combines high-level classical design
with its basic operations on quantum information. A short time ago, Selinger
et al. [19] further enhanced this function-based programming language with the
same quantum information paradigm along with classical control where the
information is being shared, the same idea has been extended in the calculus
being introduced by Arrighi et al. [18].

Van [30][31] presented a quantum $\lambda$ calculus collaborating higher-order
logic. Up till now the languages in hand, mostly serve the sequential quantum
computing, where no transition rules have been presented to model the
communication between distant observers. The formulation of such programming
languages has now started that would possibly describe the concurrent quantum
systems along with the principles of communication between them.
Quantum cryptography [20], is able to provide an utmost level of security even
if it is vulnerable to attacks of intended eave-droppers. Rapid progress was
made in quantum cryptographic systems that now these have become commercially
available [1]. Now we are in direct need of a language which can describe a
concurrent system more instantly both on the basis of quantum and classical
computation. Additionally, a system for formulating the quantum concurrent
activities is needed that can provide methods to verify the various properties
of that specific system such as security, secrecy, and correctness of any
cryptography-based quantum protocol. Jorrand et al. [10], and Gay et al. [11]
independently made the effort of designing a generalized framework of modeling
concurrent quantum systems where process algebra has been extended to the
quantum setting. Jorrand et al. [10] formulated an algebra for quantum systems
which could describe both quantum and classical data sharing. After that,
Lalire proposed a probabilistic branching bisimulation for their previously
formulated language which was able to identify quantum systems connected to
their process graph with similar branching structure[12]. While Gay et al.
[28] described a process algebraic language called Communicating Quantum
Processes (CQP) which is an instance of calculus that was introduced by Lalire
et al. [12], which gave both the primitives for unitary transformations and
measurements for the analysis of quantum communication systems.

###  1.3 Overview of this paper

This paper is organized as follows: In Section 2, we review some basic notions
from linear algebra and quantum mechanics which will be used in this paper.
The syntax of eQPAlg is presented in Section 3. Section 4 and 5 are the major
contribution of the present paper. First, we write the formal specifications
of the well-known quantum teleportation protocol with eQPAlg then show that it
indeed teleports any qubit from one party to another with a cost of sending
only 2 classical bits. Secondly, programs are written that could satisfy the
specification and these programs can also be implemented in the form of
computer code for simulation. The improved algebraic transition rules for
communication between two quantum processes are defined and examples are given
for supporting these transition rules. Section 6 is the concluding section in
which we outline the research work and point out some problems for further
study.

##  2 Preliminaries

For reader’s understanding, we introduce some fundamental concepts about
linear algebra and quantum mechanics (Reader is intended to read [29][15] and
[33] for details). The syntax and structural operational semantics of the
former quantum process algebra can be found here ([14][10] and [12] for
detailed study).

###  2.1 Linear Algebra: Vector Spaces and Operators

Study of vector spaces and linear transformations in linear algebra. Hilbert
Space, where quantum mechanics are formulated, is an significant vector space
in mathematics and physics. First we give fundamental definitions and notes,
then officially define Hilbert Space.

####  2.1.1 Qubit

The fundamental unit of representing information in quantum computation is
called quantum bit or qubit[13]. Alike bit, a qubit can be in one of the two
states. We will write these states by $\ket{0}$ and $\ket{1}$. Anything
enclosed using this notation $\ket{\,}$ is known as state, vector or a ket. A
classical bit can only be in one state, it can be 0 or 1. A qubit can exist in
$\ket{0}$ or $\ket{1}$ and it can also occurs in both states and this state is
called superposition. If we have a state $\ket{\psi}$ then the superposition
state will be:

$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$

$\alpha$ and $\beta$ are the complex coefficients. While measuring
$\ket{\psi}$, we can have the probability of finding $\ket{\psi}$ in one of
the two states which is calculated by modulus squared $\alpha$, $\beta$ such
that:

$|\alpha|^{2}$: Gives the probability of finding $\ket{\psi}$ in $\ket{0}$
$|\beta|^{2}$: Gives the probability of finding $\ket{\psi}$ in $\ket{1}$

####  2.1.2 Vector Spaces

The term vector is used for representing the quantum state of a physical
system in a complex vector space. An important vector space in quantum
computation is $\mathbb{C}^{n}$ consisting of "n-tuples" of different complex
numbers. We can name the components of $\mathbb{C}^{n}$ using
$\ket{a}\,\ket{b}\,\ket{c}$. One can denote each part of this type of vector
space in the form of n-dimensional column vector such as:

$\displaystyle\ket{a}=\begin{pmatrix}a_{1}\\\ a_{2}\\\ .\\\ .\\\ .\\\ a_{n}\\\
\end{pmatrix}$

We use such notations to denote qubits. A qubit in Hilbert space
$\mathbb{C}^{2}$ (which we will discuss later) can be written in column vector
format:

$\displaystyle\ket{\psi}=\begin{pmatrix}\alpha\\\ \beta\\\ \end{pmatrix}$

####  2.1.3 Basis and Dimension

If we have a group of vectors spanning the space of a vector V and linearly
independent to each other, we can call this group as a basis and the No. of
elements in this basis set is known as the dimension of V [13]. Basis set of
qubits states $\ket{0}$ and $\ket{1}$ for $\mathbb{C}^{2}$ with:

$\displaystyle\ket{0}=\begin{pmatrix}1\\\ 0\\\
\end{pmatrix},\;\ket{1}=\begin{pmatrix}0\\\ 1\\\ \end{pmatrix}$

####  2.1.4 Inner Product

The generalized form of dot products that can be along with simple vectors in
the Euclidean space. In dot product we have two vectors mapped into a real
number whereas inner product get two vectors from $\mathbb{C}^{2}$ and do the
mapping into complex number so the inner product is normally considered as
complex number. We have two vectors $\ket{u}$, $\ket{v}$ and the inner product
can be represented by $\innerproduct{u}{v}$. To calculate inner product among
two vectors, we have to first compute the hermitian conjugate of one of the
two vectors.

$\displaystyle(\ket{u})^{\dagger}=\bra{u}$

If we have

$\displaystyle\innerproduct{u\,}{\,v}=0$

then it is stated that $\ket{u}$, $\ket{v}$ will be orthogonal to each other
and if the norm of a vectors is unity such that

$\displaystyle\innerproduct{u}{v}=1$

then $\ket{u}$, $\ket{v}$ are normalized. If we have a vector which isn’t
normalized then we get a normalized vector by calculating the norm (that can
be a real number) this way,

$||u||=\sqrt{\innerproduct{u}{u}}$

and dividing the vector by it. If every component of a group of vectors is
orthogonal as well as normalized to one another then this set is said to be
the orthonormal.

####  2.1.5 Bra-ket Notation

Most commonly written notation for representing the states of a quantum system
in the orthonormal basis is Bra-ket. If we have vectors $\ket{u}$, $\ket{v}$
then we will call it a ket or column vector and we can find its bra or dual
vector by calculating the hermitian conjugate. So, the bras corresponding to
kets will be $\bra{u}$, $\bra{v}$.

####  2.1.6 The Trace of an operator

If we have an operator in the form of matrix then we can compute the trace of
this operator by summing up its diagonal elements. Such as,

$\displaystyle Z=\begin{pmatrix}w&x\\\ y&z\\\ \end{pmatrix},\quad Tr(Z)=w+z$

In case, the operator can be in outer product form then we can calculate the
trace by adding the inner products with its the basis states. If we have our
vector in basis $\ket{v_{i}}$, then trace of operator will be

$Tr(Z)=\sum_{i=0}^{n}\expectationvalue{v_{i}}{Z}$

####  2.1.7 Hilbert Space

An Hilbert Space[13] is a two dimensional complex vector space and a
generalization of Euclidean space where a qubit resides. It is represented by
$\mathcal{H}$.

####  2.1.8 Tensor Product

If $H_{1}$, $H_{2}$ are 2-Hilbert spaces where
$\ket{\phi_{1}}\,\epsilon\,H_{1}$ $\ket{\phi_{2}}\,\epsilon\,H_{2}$ are the
two vectors belongs to $H_{1}$ and $H_{2}$ then the tensor product for these
vectors can construct a larger Hilbert space H represented by state vector
$\ket{\psi}$ such as

$\ket{\psi}=\ket{\phi_{1}\otimes\phi_{2}}$

###  2.2 Quantum Mechanics

Now we provide a review on the basics of quantum mechanics that is needed to
understand the paper.

####  2.2.1 Quantum Gates

In the context of quantum mechanics, data processing id done through gates
also known as unitary operators[13]. A quantum gate or a logic gate is the
basic element of quantum computation consisting of small quantum registers and
connected with wires to build a circuit. Quantum algorithm is actually the
number of gates which are connected together and performing different
operations on quantum states to get the desired measurement. They can also be
represented by unitary matrices. Quantum gates can perform reversible
computation whereas in most of the classical gates only perform irreversible
computation. For examples, a classical AND gate performs operation on two
different bits and the measurement results will not be reverted back to get
the original bits. However, in reversible computing such as NOT gate, one can
measure the input bit from the resulting bit. Hadamard Gate: Hadamard gate[13]
can perform actions on a single qubit. This plot the basic state $\ket{0}$ to
$\frac{\ket{0}+\ket{1}}{\sqrt{2}}$ and $\ket{1}$ to
$\frac{\ket{0}-\ket{1}}{\sqrt{2}}$ that states that the measurement to be
performed will have equal probabilities of 0 or 1. It is represented as the 2
X 2 matrices form[13]:

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$

As, $HH^{*}=I$, here I is the identity matrix and hence H is a unitary
operator.

CNOT Gate: CNOT gate operates on a pair of qubits where the first qubit is the
control qubit whereas the other qubit is the target qubit[13]. NOT Gate or
Pauli-X operation will be applied to target qubit with respect to control
qubit values. If the value of control qubit is $\ket{1}$ then it flips the
state of target qubit to $\ket{0}$ otherwise leaves it unchanged. The matrix
representation of a 2-qubit CNOT operator can be given by

$\displaystyle CNOT=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{pmatrix}$

Phase-Shift Gates This a group of single-qubit gates that changes the basic
state $\ket{1}$ to $e^{\iota\phi}\ket{1}$ and leaves the basic state $\ket{0}$
unchanged. The phase-shift is represented by $\phi$

$\displaystyle R_{\phi}=\begin{pmatrix}1&0\\\ 0&e^{\iota\phi}\\\
\end{pmatrix}$

The examples of phase-shift gates are Pauli-Z gate where $\phi=\pi$, the phase
gate where $\phi=\frac{\pi}{2}$ and $\frac{\pi}{8}$ gate where
$\phi=\frac{\pi}{4}$

####  2.2.2 Pauli Operators

Pauli operators are of fundamental importance in quantum computation.
Different symbolic representations are used for Pauli operators, sometimes
represented by $\sigma_{0}\,,\sigma_{1}\,,\sigma_{2}\,$ , $\sigma_{3}\,$ and
sometimes I, X, Y or Z, or $\sigma_{0}\,,\sigma_{x}\,,\sigma_{y}\,$ ,
$\sigma_{z}\,$ where,
$\sigma_{0}=I\,,\sigma_{1}=\sigma_{x}=X\,,\sigma_{2}=\sigma_{y}=Y\,$ and
$\sigma_{3}=\sigma_{z}=Z\,$ [13]. As the name suggests, applying identity on
any state will leaves the state unchanged:

$I\ket{\psi}=\ket{\psi}\quad\text{such that}\quad
I\ket{0}=\ket{0},\>I\ket{1}=\ket{1}$

The next operator is known as bit-flip and is denoted by X. It operates the
same as NOT operator:

$X\ket{0}=\ket{1},\;\;X\ket{1}=\ket{0}$

Third operator is abbreviated by Y and it operates as follows:

$Y\ket{0}=-\iota\ket{1},\;\;Y\ket{1}=\iota\ket{0}$

Lastly, we have Z operator sometimes called phase-flip and is denoted by Z. It
acts as follows:

$Z\ket{0}=\ket{0},\;\;Z\ket{1}=-\ket{1}$

####  2.2.3 Hermitian Operator

In quantum theory, one special type of operator is Hermitian. Operators which
represents the physical observables are hermitian. An operator A is said to be
hermitian if and only if

$A=A^{\dagger}$

The Pauli operators are hermitian.

####  2.2.4 Unitary Operator

A linear operator represented by U also known as unitary, if the adjoint of U
equals its inverse if $UU^{{\dagger}}\,=\,U^{{\dagger}}U\,=\,I$, here I
represents the identity operator and $U^{{\dagger}}$ denotes the conjugate
transpose[13].

####  2.2.5 Density Operators

For describing quantum state to vector we can use use Density operators.
Density operators represents the mixed(uncertain) state of the system in a
compact way. There are basically two formulations of density operators, first
one formulation uses probabilities and other uses trace-preserving matrices. A
density operator is a linear operator in Hilbert space $\mathcal{H}$ and is
represented by $\varrho$. Density operator satisfies following conditions:

  1. 1.

A density operator is Hermitian, that is, $\varrho$ = $\varrho^{\dagger}$.

  2. 2.

$\varrho$ must be positive, only if, $\expectationvalue{\psi}{\varrho}\geq 0$.

  3. 3.

Trace of density operator $\varrho$ is equal to one, that is, $Tr(\varrho)=1$.

$\mathcal{D}(\mathcal{H})$ represents the group of all positive density
operators exists in $\mathcal{H}$.

####  2.2.6 Quantum Measurement

Quantum measurement contains of a group of measurement operators which is
$M_{m}$, m represents the measurement results and satisfying the following:

$\sum_{m}M^{\dagger}_{m}M_{m}=\mathcal{I}_{\mathcal{H}}$

##  3 Algebra for Distributed Quantum Systems - eQPAlg

Motivated from classical process algebras, which offer a framework for
modeling cooperating computations, a process algebraic notation has been
outlined, named QPAlg for Quantum Process Algebra, that provides a uniform
style to formal descriptions of synchronous and distributed computations
comprising each quantum and classical elements.

On the quantum aspect, QPAlg provides quantum variables, operations on quantum
variables (unitary operators and measure observables), likewise as new sorts
of communications involving the quantum world [14].
The idea of this paper is to model the quantum operations involved in
distributed quantum systems into the algebra of Lalire [27] with a slight
extension of writing specifications. The extended algebra also provides the
support not only to formally analyze the data flow in quantum communication
protocols that involve classical data, but also we can write down their
specifications. The current work introduces an easy to understand and a short
description of classical and quantum components as well as communication
between distributed quantum systems.

###  3.1 The Quantum Process Algebra - eQPAlg

Syntax:

$\begin{split}process::=nil\,|\,end\,|\,action.process\,|\,process;process\,|\,process\backslash\\{gate\\_list\\}\\\
|\,\textbf{[}var\\_decl\\_list.process\textbf{]}\,|\,process\textbf{[}var\\_list\textbf{]}\,|\,process\parallel
process\,|\,process\parallel_{\\{var\\}}process\end{split}$

$\begin{split}action\,\,::=\,\,com\,|\,unit\\_transf|\,measure\end{split}$

$\begin{split}com\,\,::=\,\,gate!exp\,|\,gate!measure|\,gate?variable\end{split}$

$\begin{split}unit\\_transf\,\,::=\,\,unitary\\_operator\textbf{[}var\\_list\textbf{]}\end{split}$

$\begin{split}measure\,\,::=\,\,observable\textbf{[}var\\_list\textbf{]}\end{split}$

$\begin{split}var\\_decl\,\,::=\,\,variable:var\\_type\end{split}$

$\begin{split}proc\\_def\,\,::=\,\,process\\_name\,\xrightarrow{def}\,process\end{split}$

$\begin{split}Spec\,\,::=\,\,\big{(}Var,Op,Eq,State\big{)}\end{split}$

Where:

$Op\,\,::=\,\,\wedge\,|\,\vee\,|\,\Rightarrow\,|\,+\,|\,\geqslant\,|\,\coloneqq\,|\,\equiv$

Var ::= Any variable of classical or quantum type
Eq ::= Any equation or expression
State ::= Any quantum state represented by Dirac Notation

######  Theorem 1 (Specification)

A specification is a statement written using mathematical expressions that is
used to describe a system during the system analysis, system design and
requirement analysis. A specification is generally not an executable program.
A specification is said to be implementable if there is at least one output
state produced for each input state of the specification. They are used to
describe the what, not the how. Normally, in writing specifications various
process calculi are used. However, we will be using standard logical
expressions such as conjunction $\wedge$ for joining expressions, disjunction
$\vee$ in place of OR, implication $\Rightarrow$ for implication, equivalence
$\equiv$ for approximately equal to.

######  Theorem 2 (Program)

To specify the behavior of a computer a program is used or A program is an
implementable form of specification.

###  3.2 Representing Distributed Quantum Systems

Let we have two quantum processes $P$ and $Q$ that represent two quantum
systems in the state $\ket{\psi}$ and $\ket{\phi}$ then $P\parallel Q$
represents the distributed quantum systems that may have entanglement between
them then we have:

  * •

p and q are integer variables representing the variables associated to $P$ and
$Q$.

  * •

$Var_{p}$ and $Var_{q}$ are the set of variables used by $P$ and $Q$.

  * •

If we have $n+m$ quantum system in state $\psi$ then first nqubits represents
quantum system $P$ and mqubits represents quantum system $Q$ then

$P=\psi_{0...n}\quad Q=\psi_{n...m}$

If we have

$\psi=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$

as an entangled quantum system then $P$ owns the first qubit of state $\psi$
and $Q$ owns the second qubit of state $\psi$ and we can say that the
entangled system $\psi$ has been distributed as well as shared between $P$ and
$Q$.

To represent such entangled quantum systems, we use parallelism:

$P\,\parallel_{\psi}\,Q=(P\otimes Q)_{\psi}$

where

$P\,\parallel_{\psi}\,Q=\psi_{0},p\otimes\psi_{1},q$

So,

$P\,\parallel_{\psi}\,Q=Var_{p}\otimes Var_{q}$

###### Theorem 3

Distributed Quantum Systems are both Distributed and Shared
Shared: If $P$ performs any operation to its state $\psi_{0}$ then it will
affect the results of $Q$’s $\psi_{1}$ state.
Distributed: $P$ and $Q$ can only perform the operation to its own qubit and
can access their own set of variables.

####  3.2.1 Performing Operations on Quantum Systems

The representation of unitary operations is written as:

$P=\psi_{0...n}\quad Q=\psi_{n...m}$

Then

$P=\cup_{{P}_{\psi_{0...n}}}\quad Q=\cup_{{Q}_{\psi_{n...m}}}$

For Distributed Quantum Systems:
To apply a unitary operation on two distributed quantum systems, we write it
as follows:

$\begin{split}P\,\parallel_{\psi}\,Q=(\cup_{{P}_{\psi_{0...n}}}\otimes\cup_{{Q}_{\psi_{n...m}}})\\\
=(\cup_{{P}}\otimes\cup_{{Q}})_{\psi}\quad\quad\quad\;\\\
\text{where}\quad\psi=(\psi_{0...n}\otimes\psi_{n...m})\quad\quad\end{split}$

####  3.2.2 Performing Measurement on Quantum Systems

The representation of Quantum measurement is written as:

$\begin{split}P=measure\;P_{\psi,p}\quad Q=measure\;Q_{\psi,q}\end{split}$

For Distributed Quantum Systems:
To apply measurement on two distributed quantum systems, we write it as
follows:

$\begin{split}P\,\parallel_{\psi}\,Q=measure\;{P}_{\psi_{0},p}\otimes
measure\;Q_{\psi_{1},q}\\\ \psi_{0}=\psi_{0...n}\,\text{qubit system,}\\\
\psi_{1}=\psi_{n...m}\,\text{qubit system}\end{split}$

##  4 Formal Analysis of Quantum Teleportation

We have used Quantum teleportation which is a quantum communication protocol
initially described by Bennett et. al. ([2]), Now this protocol has become so
famous and is widely used as an integral part of some sophisticated quantum
communication protocols.We have benefited from this advanced protocol because
it helps to achieve successful passing of quantum data using a classical
channel along with a pair of entangled qubits, which implies that no qubits
could be transferred during this process.
Informal description of Quantum teleportation protocol[2] is as follows: In a
state $\ket{\phi}$ = $\frac{\ket{00}+\ket{11}}{\sqrt{2}}$ , Alice and Bob have
a common pair of entangled qubits where Alice having some qubits with some
unidentified state that she wants to establish communication with Bob’s
quantum state

$\ket{\psi}=(\alpha\ket{0}+\beta\ket{1})$

. Alice initiates it by associating the qubit with the first half of the
entangled pair that she actually intends to teleport. To do so, Alice first
has to perfom CNOT, then Hadamard Transformation and apply measurement
operator on the qubits she has. Alice then shares the results of this
measurements to Bob. Here the measurement result will be the 2-bits. Bob then
gets these 2bits and starts performing identity operator to his qubit. Bob
ultimately retrieves the state that Alice wants to teleport to him.
The informal description of working of quantum teleportation can be seen in
the Figure 4.1. The description that has been presented above is not enough
because it just defines how the quantum system evolves.

Figure 1: Informal description of Protocol

On the other hand, the protocol has been described informally in plain
English. The revolutions of the previous century have simplified certain
algebras. Several non-simple algebras still wait to be simplified such as
Algebra for modelling Quantum Teleportation protocol. In previous approaches,
algebraic approach for teleportation was a little complex and demands a
thorough understanding of the Algebra they use to model the teleportation
protocol. Our research effort is to formalize the protocol in an easy to
understand and simplified way. This would also helps us in verifying the
correctness of quantum teleportation protocol using ePAlg syntax. The previous
approaches mentioned in literature (e.g. [34]) have tried to formulate
teleportation as a program which can be used as a tool to implement a
specification in this way $\psi^{{}^{\prime}}=\phi$. We have observed that the
given specification can also be tested with a program which requires
transferring the qubit through a quantum channel that needs not be the
teleportation. Moreover, the specification has not explained the sending or
receiving of the two classical bit that is the integral part in defining the
teleportation process. For this communication, we also require a pair of
maximally entangled qubit.
As an illustration of the algebra, we give details of the teleportation
protocol of [17]. We write down the specification for Quantum Teleportation
protocol and then write programs to satisfy these specifications.

###  4.1 Specification

The formal specification of quantum teleportation has been given as:

$\begin{split}S\coloneqq
var_{Alice}\,\wedge\,var_{Bob}\,\wedge\,\ket{\phi}\,\wedge\,\ket{\psi}\,\wedge\,\text{1
ebit}+\text{2 cbits}\geqslant\text{1 qubit}\end{split}$

The specification shows that the teleportation protocol works in a way that
instead of using 1 qubit, we will be using 1 maximally entangled bit (1 ebit)
and 2 classical bits (2 cbit) with a cost of sending 2 cbits from Alice to Bob
for obtaining the Bob’s z qubit. We have two processes $P^{Alice}$ and
$Q^{Bob}$, and $\ket{\phi}$ = $\frac{\ket{00}+\ket{11}}{\sqrt{2}}$ is the
entangled state that is partitioned among Alice and Bob as the first qubit
$\ket{\phi_{0}}$ owns by Alice whereas the second qubit $\ket{\phi_{1}}$
belongs to Bob. Now Alice owns the two qubits, one from entangled state and
other that needs to be sent to Bob $\ket{\psi}$ =
$(\alpha\ket{0}+\beta\ket{1})$. Moreover, pq are the classical variables of
type integer belongs to Alice that will store the classical outcomes obtained
after applying measurement and will be sent on Channel and rs variables
belongs to Bob which will receive the classical data from Alice and then apply
unitary operations. And the, $var_{Alice}=\psi\phi_{0},pq$ and
$var_{Bob}=\phi_{1},rs$ will be the set of variables that both of these
processes own.

###  4.2 Algebraic Programs

We now describe the programs $P^{Alice}$ and $Q^{Bob}$ that represents
respectively, the sender as well as the receiver of teleportation protocol are
represented by the program Teleport.

####  4.2.1 Program : $P^{Alice}$

$\begin{split}P^{Alice}\coloneqq\bigg{[}x:Qubit,\,\,y:Qubit\\\
x=\ket{\phi_{0}},y=\ket{\psi}\\\ CNOT[var_{Alice}]\\\ \Rightarrow CNOT[x,y]\\\
\Rightarrow CNOT[\ket{\phi_{0}},\ket{\psi}];\\\ H[x]\Rightarrow
H[\ket{\phi_{0}}\\\ \big{[}p:Integer,q:Integer\\\ measure[var_{Alice},pq]\\\
\Rightarrow measure[\\{x,y\\},pq]\\\ \Rightarrow
measure[\ket{\phi_{0}\psi},pq];\\\ \big{]}\\\ &\bigg{]}\end{split}$

Figure 2: Behaviour of Alice

####  4.2.2 Program : $Q^{Bob}$

$\begin{split}Q^{Bob}\coloneqq\Bigg{[}z:Qubit\\\
\;\;\;\;\;\;z=\ket{\phi_{1}}\\\ \bigg{[}r:Integer,\,\,s:Integer\\\
\;\;\;\;\;\;\;\ket{\phi_{1}}\Rightarrow Z[r].X[s].\ket{\phi_{1}}\\\
\bigg{]}\\\ &\Bigg{]}\\\ \end{split}$

Figure 3: Behaviour of Bob

####  4.2.3 Program : BuildEPR

$\begin{split}BuildEPR\coloneqq\Bigg{[}\ket{\Psi}:Qubit\\\
\;\;\;\;\bigg{[}a:Qubit,\;b:Qubit\\\ \;\;\;\;\;\;a=\ket{0},\;b=\ket{0}\\\
H[a]\;\otimes\;I[b]\\\ \;\;\;\;\;\;\Rightarrow H\ket{0}\otimes I\ket{0};\\\
CNOT[\;H[a]\;\otimes\;I[b]\;]\\\ \Rightarrow CNOT[\;H\ket{0}\otimes
I\ket{0}\;]\\\ \Rightarrow\ket{\Psi}\bigg{]}\\\ &\;\;\;\Bigg{]}\\\
\end{split}$

The above program creates EPR pair by applying Hadamard gate on first quantum
state after applying CNOT gate with two $\ket{0}$ states as an input. The
output state $\ket{\Psi}$ will be the EPR pair which is a classical example of
an entangled pair.

Figure 4: Behaviour of BuildEPR

####  4.2.4 Program : $QCOM$

$\begin{split}P^{Alice}\;\parallel_{\phi}\;Q^{Bob}\coloneqq\;Alice_{\phi_{0}\psi,pq}\;\parallel\;Bob_{\phi_{1},rs}\end{split}$

####  4.2.5 Program : Teleport

$\begin{split}Teleport\coloneqq\Bigg{[}\ket{\Psi}:Qubit\\\
\bigg{[}a:Qubit,\;b:Qubit\\\ \ket{\Psi}=BuildEPR[a,b]\\\
\ket{\Psi}=\ket{\phi_{0}}\;\otimes\;\ket{\phi_{1}}\\\ QCom\Rightarrow
P^{Alice}\;\parallel_{\phi}\,\,Q^{Bob}\\\ \bigg{]}\\\ &\Bigg{]}\\\
\end{split}$

The program Teleport will also be able to demonstrate that teleportation can
get the z-qubit of Bob, if it has the real state of x-qubit of Alice, by
sending only 2-bits from Alice towards Bob. [14].

##  5 Formally Modelling of Quantum Communication

Our main objective is to improve some algebraic representations for modeling
the communication between two quantum processes. Before modeling the
communication between quantum processes, we must accept that the quantum
algorithms are made up of classical and quantum parts. They require
cooperation between quantum and classical computation. Also, we know that
quantum computation is probabilistic by nature and the results of these
computations are checked and verified by the classical part. In case the
classical result of the quantum computation is incorrect, we have to repeat
this computation until we get the correct results. Quantum teleportation best
exhibits the cooperation between classical and quantum computation. Through
our upgraded process algebra, we intend to improve the previous version of
communication transition rules between classical and quantum systems just to
model this cooperation.

###  5.1 Improved Transition Rules for Modelling Communication

This section is subjected to extend some of the existing transition rules of
communication between two quantum processes defined in [14][12]. For
communication, we have rewritten quantum and classical input, output
transition rules here again with reasons on why previously written transition
rules need improvement.

###  5.2 Classical Communication Transition Rules

In order to transfer classical data between two quantum processes, we have
extended the syntax of classical value-passing [17] and also modeled the
transition rules for communication between two systems.

####  5.2.1 C-IN

$\frac{}{g?x.P\textbackslash C\quad\xrightarrow{g?v}\quad P\textbackslash
C^{{}^{\prime}}}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$C^{{}^{\prime}}=\langle\;s,q=\rho,f\cup\\{x\rightarrow{v}\\}\;\rangle$

  * •

$x\;\epsilon\;Var(s)$ , $x\;\epsilon\;$ Nat(Integer type) and $v\epsilon N$ N:
Set of Natural numbers

####  5.2.2 C-OUT:

$\frac{}{g!v.P\textbackslash C\quad\xrightarrow{g!v}\quad P\textbackslash C}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$v\epsilon N$ N: Set of Natural numbers

####  5.2.3 C-COM:

$\frac{P\textbackslash C\,\xrightarrow{g?v}\quad P^{{}^{\prime}}\textbackslash
C\quad\quad Q\textbackslash C\,\xrightarrow{g!v}\quad
Q^{{}^{\prime}}\textbackslash C}{P\parallel Q\textbackslash
C\,\xrightarrow{\tau}\quad P^{{}^{\prime}}\parallel
Q^{{}^{\prime}}\textbackslash C}$

$\frac{P\textbackslash C\,\xrightarrow{g!v}\quad P^{{}^{\prime}}\textbackslash
C\quad\quad Q\textbackslash C\,\xrightarrow{g?v}\quad
Q^{{}^{\prime}}\textbackslash C}{P\parallel Q\textbackslash
C\,\xrightarrow{\tau}\quad P^{{}^{\prime}}\parallel
Q^{{}^{\prime}}\textbackslash C}$

where context of both systems remains unchanged. These transition rules are
similar to those defined in communicating data between two classical systems
in classical process algebra[17]. Contexts do not change in these transition
rules as the information being modified is classical by nature and and also
the quantum state of the accompanying systems remains unchanged.

###  5.3 Quantum Communication Transition Rules

Communication between two quantum systems defines cooperation by classical
information and quantum information. We have improved some important input and
output transition rules here. Using these transition rules, we have written a
rule that can define the communication between two quantum systems using
parallel composition operator from process algebra [17].

####  5.3.1 Q-IN1:

$\frac{}{g?x.P\textbackslash C\quad\xrightarrow{g?v:\sigma}\quad
P\textbackslash C^{{}^{\prime}}}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$C^{{}^{\prime}}=\langle\;s,x.q=\rho\otimes\sigma,f\;\rangle$

  * •

$\sigma\;\epsilon\;\mathcal{D(H}_{2})$

  * •

$x\;\epsilon\;Var(s)$ , $x\;\epsilon\;q$ and $v\notin q$

####  5.3.2 Q-IN2:

$\frac{}{g?x.P\textbackslash C\quad\xrightarrow{g?v}\quad P\textbackslash C}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$x\;\epsilon\;Var(s)$ , $x\;\epsilon\;q$ and $v\epsilon q-\\{g?x.P\\}$

In previous work [14][12], the quantum input rule (Q-IN1) has been defined in
a similar way, but that rule works only when the quantum system being input
(here quantum variable v is representing input system) must not be in
entangled state or is not correlated classically with the quantum system
represented by x. As the entanglement is a physical phenomenon where the state
of different quantum systems can be found entangled in such a way that the
state of any of the two quantum systems cannot be measured independently of
the state of other system even if they are physically light years away. The
below mentioned rule clearly models the statement by this process algebra:

$\frac{}{g?x.P\textbackslash C\quad\xrightarrow{g?v:\rho^{{}^{\prime}}}\quad
P\textbackslash C^{{}^{\prime}}}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$C^{{}^{\prime}}=\langle\;s,q=\sigma,f\;\rangle$

  * •

$\sigma\;\epsilon\;\mathcal{D(H}_{2})$

  * •

$x\;\epsilon\;Var(s)$ , $x\;\epsilon\;q$ and $v\notin q$

  * •

$C^{{}^{\prime}}=\langle\;s,Tr_{q}(\sigma)=\rho^{{}^{\prime}},Tr_{v}(\sigma)=\rho,f\;\rangle$

We can model any kind of quantum data being input using this rule as no such
change has been applied to the new quantum state other than this:
$Tr_{v}\sigma=\rho$ that says: the initial state of quantum systems will not
be changed. The rule we derived above seems little absurd as it does not
retain the property of being image finite [17] from its previous states
$g?x.P\textbackslash C$ where context of this system is $C=\langle
x,q=\rho,f\rangle$ and the action it applies $g?v:\rho^{{}^{\prime}}$, there
are never-ending numerous determined configurations fulfilling this rule.
Taking in view the above discussion, we have introduced these two quantum
input transition rules which explain the situations where the qubit is being
inputted from outside the context of the system and within the context,
separately. The point to be noted here again is that the context of the system
remains unchanged after applying quantum input rule-2.

The input system has already been explained within the context of that
specific process. And the above-mentioned rule doesn’t modify the quantum
state of this entire system (stored in x) as the process is only referencing
the quantum system.

####  5.3.3 Q-OUT:

$\frac{}{g!x.P\textbackslash C\quad\xrightarrow{g!x}\quad P\textbackslash C}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$x\;\epsilon\;Var(s)$ and $x\;\epsilon\;q$

The output rule proposed in [14][12] is as follows:

$\frac{}{g!x.P\textbackslash C\quad\xrightarrow{g!x}\quad P\textbackslash
C^{{}^{\prime}}}$

where

  * •

$C=\langle\;s,q=\rho,f\;\rangle$

  * •

$C^{{}^{\prime}}=\langle\;s\textbackslash\\{x\\},q\textbackslash\\{x\\}=Tr_{q/\\{x\\}}(\rho),f\;\rangle$

  * •

$x\;\epsilon\;Var(s)$ and $x\;\epsilon\;q$

The instinct behind this old version of output rule is that once the quantum
state of the system is sent out (output) to another system then this state
(represented by variable name x) can be removed from the quantum sequence list
as defined in this part ($q\textbackslash\\{x\\}$ and also from the stack of
variables (\\{x}) of running process. Further, $Tr_{q\\{x\\}}(\rho)$ is used
to obtain the quantum state by applying partial trace on $\rho$ over the qubit
value in x. This transition rule also implies that the context of the quantum
system being output is also changed.
For justification of our improved transition rule, we assume that the system
being sent out is correlated or entangled with the systems remaining in the
context and once the state is removed from the context then this cannot be
referenced in future. Now, after some time, if we input the same system again
which has just been removed from the context then this can be problematic as
we do not have this system in our context and hence we cannot measure the
entangled state anymore. So, the Q-Output rule presented here can help us
prevent this problem to happen since we do not wish remove the state being
sent out from the context, so we keep that information. In this way, the
context of the current system would be untouched.

####  5.3.4 Q-COM:

$\frac{P\textbackslash C\,\xrightarrow{g?v}\quad P^{{}^{\prime}}\textbackslash
C\quad\quad Q\textbackslash C\,\xrightarrow{g!v}\quad
Q^{{}^{\prime}}\textbackslash C}{P\parallel Q\textbackslash
C\,\xrightarrow{\tau}\quad P^{{}^{\prime}}\parallel
Q^{{}^{\prime}}\textbackslash C}$

$\frac{P\textbackslash C\,\xrightarrow{g!v}\quad P^{{}^{\prime}}\textbackslash
C\quad\quad Q\textbackslash C\,\xrightarrow{g?v}\quad
Q^{{}^{\prime}}\textbackslash C}{P\parallel Q\textbackslash
C\,\xrightarrow{\tau}\quad P^{{}^{\prime}}\parallel
Q^{{}^{\prime}}\textbackslash C}$

This is a simple rule representing the communication between two quantum
systems outside of the context (using Q-IN2), but this is not completely
quantum by nature as quantum state of the system is neither sent nor any
measurements have to be applied so the context of these two systems remains
unchanged.

###  5.4 Examples

Some examples are presented below to elaborate the transition rules formulated
in last section. In the below mentioned theorems, U will be the set of unitary
transformation:

$U=\\{H,CNot\\}$

where H will be the Hadamard transformation and CNot will be the "Controlled-
Not" operation. In the following theorems, we perform the Hadamard
transformation by using our transition rules and explain how quantum systems
are passed between two quantum processes even if they are entangled. After
that, we see how the quantum systems work when we perform the unitary
transformation.

###### Theorem 4

If

$P=g?y.P^{{}^{\prime}}\quad\,,\,Q=g!x.Q^{{}^{\prime}}$

and

$R\quad=\quad(P\parallel Q)\textbackslash g\;\text{where}\;y\notin
q-(g!x.Q^{{}^{\prime}})$

Then

$T_{1}\,=\,R\textbackslash
C\quad\xrightarrow{\tau}\quad(P^{{}^{\prime}}\\{x/y\\}\parallel
Q^{{}^{\prime}})\textbackslash g,C$

Proof: In the above transition $T_{1}$, the quantum system x is being sent out
from the process Q to process P through quantum channel g. Note that, the
context of the system remains unchanged as density operator $\rho$ only
represents the quantum state of the output system x and doesn’t store any
other quantum information such as the position of the system. So, the above
transition is valid and support the correctness of our communication rule.

###### Theorem 5

If

$P_{1}=g?y.H[y].P_{1}^{{}^{\prime}}\quad,\quad Q_{1}=(P_{1}\parallel
Q)\textbackslash g$

and

$\rho\,=\,\ket{1}_{x}\bra{1}\otimes\rho^{{}^{\prime}}$ $\text{where}\quad\quad
y\notin
q\;-\;(g!x.P_{1}^{{}^{\prime}}),\;\;\rho^{{}^{\prime}}\,\in\,\mathcal{D(H}_{Var-\\{x\\}})$

Then

$T_{2}\,=\,Q_{1}\textbackslash
C\quad\xrightarrow{\tau}\quad(H[x].P_{1}^{{}^{\prime}}\\{x/y\\}\parallel
Q^{{}^{\prime}})\textbackslash g,C$
$\quad\quad\quad\xrightarrow{H[x]}\quad(P_{1}^{{}^{\prime}}\\{x/y\\}\parallel
Q^{{}^{\prime}})\textbackslash g,C^{{}^{\prime}}$

Now the new state will be

$\ket{-}_{x}\bra{-}\otimes\rho^{{}^{\prime}}$

Proof:

In this transition $T_{2}$, the context of the system is changed as the
quantum system x is being sent out from $Q$ to $P_{1}$ and then the Hadamard
transformation is applied on state $\rho$ of the quantum system x and state
$\rho$ of the quantum system x is changed from $\ket{1}$ to $\ket{-}$.

###### Theorem 6

If

$P_{2}=g?y.CNOT[y,z].P_{2}^{{}^{\prime}}\quad,\quad Q_{2}=(P_{2}\parallel
Q)\textbackslash g$

and

$\gamma\,=\,\ket{-}_{x}\bra{-}\otimes\ket{1}_{z}\bra{1}\otimes\gamma^{{}^{\prime}}$
$\text{where}\quad\quad y\notin
q\;-\;(g!x.P_{2}^{{}^{\prime}}),\;\;\gamma^{{}^{\prime}}\,\in\,\mathcal{D(H}_{Var-\\{x,z\\}})$

Then

$T_{3}\,=\,Q_{2}\textbackslash
C\quad\xrightarrow{\tau}\quad(CNOT[x,z].P_{2}^{{}^{\prime}}\\{x/y\\}\parallel
Q^{{}^{\prime}})\textbackslash g,C$
$\quad\quad\quad\xrightarrow{CNOT[x,z]}\quad(P_{2}^{{}^{\prime}}\\{x/y\\}\parallel
Q^{{}^{\prime}})\textbackslash g,C^{{}^{\prime}}$

Now the new state will be

$\ket{\alpha_{11}}_{x}\bra{\alpha_{11}}\otimes\gamma^{{}^{\prime}}$

Proof:

In this transition $T_{3}$, again the context of the system is changed as the
system x is being sent out from Q to $P_{2}$ then unitary operation CNOT
applied on quantum state

$\gamma$

of the system x along with the system z and quantum state of the physical
systemxz will be entangled. As, we can see before applying CNOT, we have the
quantum system x and quantum system z as separable states but once we applied
CNOT operation we get the entanglement between these two quantum systems.

##  6 Discussion and Conclusion

Complete formal descriptions of quantum algorithms that uses the principles of
quantum mechanics must uses both classical as well as quantum computational
components and utilize these to make the communication and cooperation work.
Furthermore, modeling of distributed and concurrent quantum systems, quantum
to quantum communication and quantum communication protocols that physically
transfers the qubits from one location to another must also be taken into
account.

In this paper, we have discussed the theory of formal modeling of concurrent
and distributed quantum processes and provided with some improved transition
rules for checking the correctness of communication between two quantum
processes. We have discussed the limitation of some of the communication
transition rules of the previous notation algebraic notation of quantum
processes.

Motivated by Milner’s classical process algebras, which give an extended
framework to formally model the cooperation between processes, Lalire have had
introduced an algebraic notation for communicating processes which is named as
Quantum Process Algebra (QPAlg). This Quantum Process Algebra gives a
homogeneous style to formal specification and modeling of distributed and
concurrent quantum systems consisting of both classical as well as quantum
data. On the quantum part, QPAlg gives quantum variables, the operations on
quantum variables such as applying quantum gates, pauli operators and the
measurement observables, and also new types of communications including the
quantum world. The operational semantics ensures that these quantum systems,
their operations and the communication between these systems work as indicated
by the postulates of quantum mechanics.

We have introduced an improved version of some algebraic representations for
modelling the communication between two quantum processes and proved the
formal model as a tool to check the correctness of practical concurrent
quantum systems. Also, we have presented a direct approach for two quantum
systems to communicate. We have identified a different approach to model the
communication between distant quantum systems.We have further introduced the
support of writing the specifications of working of distributed quantum
systems. As an example, we formally specify the working of Quantum
teleportation protocol and also provided the programs that best satisfies
these specifications. Diagrams have also been added to best describe the flow
and formal working of protocol. Moreover, we have introduced the concept of
Entanglement based instant messaging and mathematically perform the
calculations.

Further, in our examples we assumed to have two quantum systems and modeled
two different examples of classical and quantum processes for checking the
level of correctness of the communication between them. We have also applied
our transition rules for some test examples where two processes are using the
communication primitives of Quantum Process Algebra (QPAlg) provided that the
communication is entirely based of the principles of quantum mechanics. We
have elaborated the examples to describe the concurrent communication
activities between these modeled systems.
On the theoretical side, there is a need to prove if the given program can
prove the formal specification of teleportation protocol. We also aim to
develop an approach for the development of a easy to understand quantum
programming language and use our extended form of eQPAlg for modelling and
specifying quantum superdense coding and BB84 protocol. On the practical side,
there is a need to work on more practical examples for modeling cryptographic
systems. Possibly a quantum cryptographic protocol can be modeled using the
cryptographic protocol simulators.

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# Natural flavor conservation in a three Higg-doublet Model

A. C. B. Machado<EMAIL_ADDRESS>Instituto de Física Teórica–Universidade
Estadual Paulista
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil  Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Santo André-SP, 09210-170
Brazil.  V. Pleitez<EMAIL_ADDRESS>Instituto de Física
Teórica–Universidade Estadual Paulista
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil

(03/08/13)

###### Abstract

We consider an extension of the electroweak standard model with three Higgs
doublets transforming under the $S_{3}$ symmetry as a doublet and a singlet.
Because of the $S_{3}$ symmetry and an appropriate vacuum alignment the mass
matrices in all the scalar sectors have the same form and are diagonalized by
the same matrix which is of the tribimaximal type. Moreover, independently of
the fermion representations under $S_{3}$, there is no flavor changing neutral
currents at tree level, although in order to obtain realistic mass matrices,
quarks and leptons must transform as singlets under $S_{3}$. Two of the scalar
doublets are fermiophobic and they are mass degenerate unless soft terms
violating the $S_{3}$ symmetry are added. The third Higgs doublet which is
singlet under $S_{3}$, correspond to the standard model one in the sense that
its mass and couplings to fermions and bosons are exactly the same of the SM
at tree level.

###### pacs:

12.60.Fr 12.15.-y

##  I Introduction

The resonance with mass of approximately 125 GeV that was discover at LHC is
in agreement, within the experimental error, with the standard model (SM)
Higgs field cmsatlas . Hence, it implies that all possible new physics
scenarios have to include such scalar field. The SM symmetry breaking is the
simplest and economic but nothing prevent the model to have more scalars
multiplets transforming in several representations of $SU(2)$: more doublets,
singlets and even triplets. All of then have already been considered in
literature as extension of the SM. It means that the source of electroweak
symmetry breaking may be more complicated than in the SM. Moreover, if there
exist more scalar fields the SM Higgs couplings to fermion and gauge bosons
are usually reduced. In fact, this case is still possible with the current LHC
data, see for instance Ref. holdom .

Here we will show an extension of the SM with three Higgs doublets
transforming under $S_{3}$ as a doublet $(D)$and a singlet ($S$) and having an
appropriate vacuum alignment. An interesting feature of our model is that
unlike other multi-Higgs models, in which the SM-like Higgs boson mixes with
the others and this is why its coupling to fermion and gauge boson are
reduced, this does not happens in the present model because of the $S_{3}$
symmetry and the vacuum alignment: the combination of the three doublets
transforming as singlet under $S_{3}$ ($S$) is exactly the SM Higgs with all
its couplings the same as in the SM with one Higgs doublet and it does not mix
with the other $SU(2)$ doublets. The latter ones are in a doublet of $S_{3}$
there is no mixing between them too. The difference of the singlet $S$ and the
SM Higgs doublet is that the former has trilinear and quartic interactions
with the members of the doublet $D$.

On the other hand, it is well known, from experiments, that processes which
change flavor through the effects of neutral currents are quite suppressed
fcncexp . It is also well known, since the seminal paper of Glashow and
Weinberg gw , that the suppression of flavor changing neutral currents (FCNCs)
is natural if it depends only on the symmetry and the representation content
of the model. In general multi-Higgs models give rise to unwanted FCNCs.
Notwithstanding in these models it is possible to have natural flavor
conservation if the fermion masses, of a given charge, are generated by a
single source or, as in the standard electroweak model, a unique Higgs doublet
generates all fermion masses. In order to obtain natural flavor conservation
discrete Abelian symmetries like $Z_{2}$ gw , continuous Abelian u1 , and non-
Abelian continuos su3  or discrete a4  family symmetries are usually imposed
or, to assume the alignment in flavor space of the Yukawa matrices in the so
called A2HDM pich . In particular, the two-Higgs doublet models are the most
considered in literature, see for instance Ref. branco . Discrete symmetries
have also been used to gain some predictivity in the flavor problem, usually
to reproduce ansatze of mass matrix textures, and they origin may be related
to more fundamental (new) physics nilles . However, in all these cases the
goal was to explain the texture of the fermion mass matrices, for instance,
for getting the so called tribimaximal mixing matrix in the leptonic sector
tbm , in particular the $S_{3}$ symmetry was considered since many years ago
s3 . Things become more complicated when there are more than two Higgs
doublets. Thus, larger symmetries may be needed in order to simplify the
analysis of the scalar potential which may be or not related to the fermion
mass matrices. Discrete symmetries for three-Higgs doublet model, without
references to these matrices, were classified in Ref. ivanov .

Here we will consider a three-Higgs doublet model (model A) in which the mass
square matrices of all the scalar fields at the tree level have the same form
and for this reason, they are diagonalized with the same unitary matrix, $U$,
and there is mass degenerated states in the $S_{3}$ symmetry if it is not
softly broken. The matrix $U$ is, in some cases, of the tribimaximal type.
This is a consequence of the representation content of the discrete symmetry,
$S_{3}$, imposed to the model, one doublet $D$ and a singlet $S$ as well an
appropriate vacuum alignment, and it also implies that the model naturally
suppresses FCNC mediated by neutral scalars in the quark and lepton sectors,
if we extend the criterion for having natural flavor conservation to include
an appropriate vacuum alignment with the minimum of the scalar potential being
global and stable, at least at tree level. In a different version (model B)
the mass square are matrices are all diagonal at tree level if the $S_{3}$
symmetry is not softly broken.

The outline of this paper is as follows. In Sec. II we give the most general
scalar potential invariant under the gauge and $S_{3}$ symmetries. In Sec. III
we consider in details the scalar sector of the model A (see below) when all
VEVs are equal to each other, i.e., $v_{1}=v_{2}=v_{3}$; the same is doing in
Sec. IV for the model B (see below) but now with $v_{1}=v_{SM}$ and
$v_{2}=v_{3}=0$. The Yukawa interactions are the same in both models and are
briefly discussed in Sec. V. Our conclusions are in Sec. VII and in the
Apendices we show the constraint equations for arbitrary VEVs for model A in
the Appendix A and model B in the Appendix. B.

##  II Three Higgs-scalar doublet models

The model is an extension of the electroweak standard model with three Higgs
scalars, all $SU(2)$ doublets having $Y=+1$ and the most general scalar
potential invariant under $SU(2)\otimes U(1)_{Y}\otimes S_{3}$ symmetry is
given by:

$\displaystyle V(D,S)$ $\displaystyle=$
$\displaystyle\mu^{2}_{s}S^{\dagger}S+\mu^{2}_{d}[D^{\dagger}\otimes
D]_{1}+\lambda_{1}([D^{\dagger}\otimes
D]_{1})^{2}+\lambda_{2}[(D^{\dagger}\otimes D)_{1^{\prime}}(D^{\dagger}\otimes
D)_{1^{\prime}}]_{1}$ (1) $\displaystyle+$
$\displaystyle\lambda_{3}[(D^{\dagger}\otimes D)_{2}(D^{\dagger}\otimes
D)_{2}]+\lambda_{4}(S^{\dagger}S)^{2}+\lambda_{5}[D^{\dagger}\otimes
D]_{1}S^{\dagger}S+\lambda_{6}[[S^{\dagger}D]_{2}[S^{\dagger}D]_{2}]_{1}$
$\displaystyle+$ $\displaystyle\\{\lambda_{7}S^{\dagger}[D\otimes
D^{\dagger}]_{1}S+\lambda_{8}[(S^{\dagger}\otimes D)_{2}(D^{\dagger}\otimes
D)_{2}]_{1}+H.c.\\}$

We recall that denoting $\textbf{2}=(x_{1},x_{2})$, we have
$\textbf{2}\otimes\textbf{2}=\textbf{1}\oplus\textbf{1}^{\prime}\oplus\textbf{2}^{\prime}$
where $\textbf{1}=x_{1}y_{1}+x_{2}y_{2}$,
$\textbf{1}^{\prime}=x_{1}y_{2}-x_{2}y_{1}$,
$\textbf{2}^{\prime}=(x_{1}y_{2}+x_{2}y_{1},x_{1}y_{1}-x_{2}y_{2}$), and
$\textbf{1}^{\prime}\otimes\textbf{1}^{\prime}=\textbf{1}$ ishimori .

We have two possibilities to build the doublet $D$ and the singlet $S$ if the
three scalar doublets are in the reducible triplet representation of $S_{3}$,
say, $\textbf{3}=(H_{1},H_{2},H_{3})$ where
$H_{i}=(H^{+}_{i}\,H^{0}_{i})^{T}$. This reducible representation is broken
down to the irreducible singlet and doublet ones, i.e.,
$\textbf{3}~{}=~{}\textbf{2}+\textbf{1}\equiv D+S$, where:

$\displaystyle S=\frac{1}{\sqrt{3}}(H_{1}+H_{2}+H_{3})\sim\textbf{1},$
$\displaystyle
D\equiv(D_{1},D_{2})=\left[\frac{1}{\sqrt{6}}(2H_{1}-H_{2}-H_{3}),\frac{1}{\sqrt{2}}(H_{2}-H_{3})\right]\sim\textbf{2}.$
(2)

We shall call this model A. On the other hand, the one of the $SU(2)$ doublet,
say $H_{1}$, is a singlet of $S_{3}$ and the other two, say $H_{2},H_{3}$
transform as the irreducible doublet of $S_{3}$:

$S=H_{1}\sim\textbf{1},\quad D=(H_{2},H_{3})\sim\textbf{2}.$ (3)

Here, we shall call this model B.

Both sort of models have been already consider in literature. For instance,
model A was considered in Ref.kubo . Model B was considered a long time ago
pakvasa  and more recently in Refs. mondragon , and in a five-Higgs doublets
supersymmetric model in kaneko . All the previous articles have used the
$S_{3}$ symmetry to addressed the texture of the fermion mass matrices.
However, here we shall give up this issue and concentrate on the extension of
the standard model with three Higgs doublets, using the $S_{3}$ symmetry just
to simplifying the analysis of the scalar sector and suppress flavor changing
neutral currents at the tree level. Thus, fermions transform trivially under
$S_{3}$ and the mass matrices are of the general form but there is no flavor
violation mediated by neutral scalars. Models A and B seem very similar
specially when we choose two vacuum alignment which apparently lead one model
into another, since they could be related by a change of weak basis in the
representation of $S_{3}$ ishimori . However, they have different mass
matrices and trilinear and quartic interactions are also different.

##  III Model A: $v_{1}=v_{2}=v_{3}$

We make as usual, the decomposition of the symmetry eigenstates as
$H^{0}_{i}=(1/\sqrt{2})(v_{i}+\eta^{0}_{i}+iA^{0}_{i}),\;i=1,2,3$, and assume,
for the sake of the simplicity, all VEVs being real.

The case when all VEVs are different from zero was considered in Ref. kubo
and the general constraint equations for this case are given in the Appendix
A. Here let us consider the case when $v_{1}=v_{2}=v_{3}=v$, and the
constraint equations in Eq. (35) become

$t_{1}=t_{2}=t_{3}=v(\mu^{2}_{s}+3\lambda_{4}v^{2}),$ (4)

and $t_{i}=0$ implies $\mu^{2}_{s}=-3\lambda_{4}v^{2}<0,\;\lambda_{4}>0$.

All scalar mass matrices has the form

$M^{2}_{n}=\left(\begin{array}[]{ccc}a_{n}&b_{n}&b_{n}\\\ b_{n}&a_{n}&b_{n}\\\
b_{n}&b_{n}&a_{n}\end{array}\right),$ (5)

where $a_{n},b_{n}>0$ (or $a_{n},b_{n}<0$ ) and $n$ denotes the scalar sector:
$n=h,a,c$ for the scalar, pseudoscalar and charged scalar fields.

This matrix is diagonalized as follows:
$U^{T}_{TBM}M^{2}_{n}U_{TBM}=\textrm{diag}(a_{n}+2b_{n},a_{n}-b_{n},a_{n}-b_{n})$,
hence $a_{n}-2b_{n}\geq 0$ and $a_{n}+b_{n}\geq 0$, with

$U_{TBM}=UP=(e_{1}\,e_{2}\,e_{3})=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&-\sqrt{\frac{2}{3}}&0\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{2}}\end{array}\right),$
(6)

where we have denoted the eigenvectors by $e_{1},e_{2},e_{3}$, and the
matrices $U$ and $P$ in (6) are give by

$U=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&0&0\\\
\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right),\quad
P=\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right).$ (7)

In the case of CP-even neutral real scalars, we have
$3a_{h}=2\mu_{d}^{2}+(8\lambda_{4}+3\bar{\lambda}^{\prime})v_{SM}^{2}$, and
$6b_{h}=-2\mu_{d}^{2}+(16\lambda_{4}-3\bar{\lambda}^{\prime})v_{SM}^{2}$,
where $\bar{\lambda}^{\prime}=(\lambda_{5}+\lambda_{6}+2\lambda_{7})$, and the
eigenvalues are the following:

$\displaystyle m^{2}_{h_{1}}=\lambda_{4}v^{2}_{SM},\qquad
m^{2}_{h_{2}}=m^{2}_{h_{3}}\equiv
m^{2}_{h}=\mu^{2}_{d}+\frac{3}{2}\bar{\lambda}^{\prime}v^{2}_{SM},$ (8)

where we have used $v=v_{SM}/\sqrt{3}$. Denoting as $h^{0}_{i}$ the mass
eigenstates, we have $\eta^{0}_{i}=(U_{TBM})_{ij}h^{0}_{j}$, where $U_{TBM}$
is given in (6). The scalar $h^{0}_{1}$ can be identified with the standard
model Higgs scalar.

In the CP-odd neutral scalars sector, the mass matrix is given as in (5) but
now with
$3a_{a}=2\mu_{d}^{2}+(2\lambda_{4}+3\bar{\lambda}^{\prime\prime})v_{SM}^{2}$
and
$6b_{a}=-2\mu_{d}^{2}+(4\lambda_{4}-3\bar{\lambda}^{\prime\prime})v_{SM}^{2}$,
where $\bar{\lambda}^{\prime\prime}=(\lambda_{5}+\lambda_{6}-2\lambda_{7})$
and in this case we obtain the following masses:

$\displaystyle m^{2}_{a_{1}}=0,\qquad m^{2}_{a_{2}}=m^{2}_{a_{3}}\equiv
m^{2}_{a}=\mu^{2}_{d}+\frac{3}{2}\bar{\lambda}^{\prime\prime}v^{2}_{SM}$ (9)

Denoting $a^{0}_{i}$ the pseudo-scalar mass eigenstates, we have
$A^{0}_{i}=(U_{TBM})_{ij}a^{0}_{j}$.

Similarly in the charged scalars sector we use (5) with
$6a_{c}=2\mu_{d}^{2}+(2\lambda_{4}+3\lambda_{5})v_{SM}^{2}$ and
$12b_{c}~{}=~{}-2\mu_{d}^{2}+(4\lambda_{4}-3\lambda_{5})v_{SM}^{2}$ and in
this case we obtain the following masses:

$\displaystyle m^{2}_{c_{1}}=0,\qquad m^{2}_{c_{2}}=m^{2}_{c_{3}}\equiv
m^{2}_{c}=\frac{1}{4}(2\mu^{2}_{d}+\lambda_{5}v^{2}_{SM}).$ (10)

Finally, if $H^{+}_{i}$ denote the charged scalar symmetry eigenstates and
$h^{+}_{i}$ the respective mass eigenstates, we have
$H^{+}_{i}=(U_{TBM})_{ij}h^{+}_{j}$. The mass degeneracy is due to a residual
symmetry as we will see below.

The $\mu^{2}_{d}$ parameter appears in Eqs. (8), (9) and (10). This parameter
may be $>0$ or $<0$, and since it is not protected by any symmetry, it may be
larger than the electroweak scale. On one hand, if $\mu^{2}_{d}>v^{2}_{SM}$ or
$|\mu^{2}_{d}|<v^{2}_{SM}$ the masses of the scalar $h^{0}_{2,3}$,
pseudoscalar $a^{0}_{2,3}$ and the charged scalar $h^{\pm}_{2,3}$ are heavier
than $h^{0}_{1}$, independently of the values of the $\lambda$’s and $v_{SM}$.
On the other hand, if $\mu^{2}_{d}<0$ and
$\bar{\lambda}^{\prime},\bar{\lambda}^{\prime\prime}>0,\lambda_{5}>0$, all
these particles may be lighter than $h^{0}_{1}$.

Using the mixing matrix in Eq. (6), we can write the Higgs scalars doublet $D$
and the singlet $S$, in terms of the mass eigenstates, $h^{0}_{i},a^{0}_{i}$
and $h^{\pm}_{i}$, as

$\displaystyle S\equiv h_{1}=\left(\begin{array}[]{c}h^{+}_{1}\\\
\frac{1}{\sqrt{2}}(3v+h^{0}_{1}+ia^{0}_{1})\end{array}\right),\;D\equiv-(h_{2},h_{3}),\;h_{k}=\left(\begin{array}[]{c}h^{+}_{k}\\\
\frac{1}{\sqrt{2}}(h^{0}_{k}+ia^{0}_{k})\end{array}\right),$ (15)

where $k=2,3$ and we see that the would-be Goldstone bosons are all in the
singlet $S$. Notice that the doublet $S=h_{1}$ is the only source of fermion
masses, thus there is no flavor violation at tree level (see Sec. V). This is
independent of the way that fermions transform under $S_{3}$, however as we
will show below, in order to obtain realistic mass matrices at tree, fermions
must transform as singlet under $S_{3}$. Notice that in the unitary gauge
$h^{+}_{1}$ and $a_{1}$ are the two would be Goldstone bosons required by the
SM.

Compare the situation in Eq. (15) with the case of two doublets in which the
mass eigenstates neutral scalar are denoted usually as $h_{1}$ and $h_{2}$,
and the pseudoscalar $A$, while $H_{u},H_{d},$ and $A_{u},A_{d}$ denote the
symmetry eigenstates, we have

$h_{1}=\sqrt{2}(H_{d}\sin\alpha-
H_{u}\cos\alpha),\;\;h_{2}=-\sqrt{2}(H_{d}\cos\alpha+H_{u}\sin\alpha),\;\;A=\sqrt{2}(A_{d}\sin\beta-
A_{u}\cos\alpha)$ (16)

$\alpha$ is a mixing angle between the two neutral scalars and where
$\tan\beta=v_{u}/v_{d}$. In this case the SM Higgs is given by
$h_{SM}=h_{1}\sin(\alpha-\beta)-h_{2}\cos(\alpha-\beta)$, and only if
$\sin(\alpha-\beta)=1$ it is identical with the SM branco .

The potential in Eq. (1), which is written in terms of the $SU(2)$ scalar
doublets with their component being symmetry eigenstates in Eq. (2), can be
written with $SU(2)$ scalar doublet with their components being the mass
eigenstates given in Eq. (15):

$\displaystyle V(h_{i})$ $\displaystyle=$ $\displaystyle
3\lambda_{4}v^{2}h^{\dagger}_{1}h_{1}+\mu^{2}_{d}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})+\lambda_{1}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})^{2}+\lambda_{2}(h^{\dagger}_{2}h_{3}-h^{\dagger}_{3}h_{2})^{2}$
(17) $\displaystyle+$
$\displaystyle\lambda_{3}[(h^{\dagger}_{2}h_{3}+h^{\dagger}_{3}h_{2})^{2}+(h^{\dagger}_{2}h_{2}-h^{\dagger}_{3}h_{3})^{2}]+\lambda_{4}(h^{\dagger}_{1}h_{1})^{2}+\lambda_{5}h^{\dagger}_{1}h_{1}(h^{\dagger}_{2}h_{2}+h^{\dagger}_{3}h_{3})$
$\displaystyle+$
$\displaystyle\lambda_{6}[|h_{1}^{\dagger}h_{2}|^{2}+|h_{1}^{\dagger}h_{3}|^{2}]+\\{\lambda_{7}[(h^{\dagger}_{1}h_{2})^{2}+(h^{\dagger}_{3}h_{1})^{2}]+\lambda_{8}[h^{\dagger}_{1}h_{2}(h^{\dagger}_{2}h_{3}+h^{\dagger}_{3}h_{2})$
$\displaystyle+$ $\displaystyle
h^{\dagger}_{1}h_{3}(h^{\dagger}_{3}h_{3}-h^{\dagger}_{2}h_{2})]+H.c.\\}.$

Notice from the scalar potential in Eq. (17), that there is still a residual
$S_{2}\equiv Z_{2}$ symmetry: it is invariant under the exchange of the
doublets $h_{2}\leftrightarrow h_{3}$. The mass degeneracy is due to this
unbroken $S_{2}$ symmetry. That is described mathematically by the following
operator

$E=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\end{array}\right).$ (18)

If the eigenvectors of the mass square matrix are also eigenvectors of $E$ we
can distinguish the three eigenvectors $e_{1},\,e_{2},\,e_{3}$, with which the
tribimaximal matrix in Eq. (6) is obtained, even if $e_{2}$ and $e_{3}$ are
mass degenerated: $Ee_{2}=e_{2}$ and $Ee_{3}=-e_{3}$.

We will show later on under which conditions the potential in Eq. (1) is
bounded from below. For the moment, just notice that when $v_{1}=v_{2}=v_{3}$,
if $\lambda_{4}>0$. Under these conditions the minimum of the scalar potential
($V_{min}=-\lambda_{4}v^{4}_{SM}$) is global and stable as long as the masses
square, given in (8), (9), and (10), are all positive and, with the conditions
for the $\lambda$’s given in Sec. VI are satisfied. However, the stability of
the solution $v_{1}=v_{2}=v_{3}$ under radiative corrections will be studied
elsewhere.

The residual $S_{2}$ symmetry can be broken, if necessary, to avoid the mass
degeneracy and also the domain wall problem, and this can be done by quantum
corrections and/or by soft terms in the scalar potential. As an illustration,
here we break the $S_{2}$ symmetry by adding the following quadratic terms
$\mu^{2}_{nm}H^{\dagger}_{n}H_{m}$, $n,m=2,3$ to the scalar potential in (1).
The mass matrices in all the scalar sectors are now of the form

$M^{2}_{n}=\left(\begin{array}[]{ccc}a_{n}&b_{n}&b_{n}\\\
b_{n}&a_{n}+\mu^{2}_{22}&b_{n}+\mu^{2}_{23}\\\
b_{n}&b_{n}+\mu^{2}_{23}&a+\mu^{2}_{33}\end{array}\right),$ (19)

where $\mu^{2}_{nm}$ are naturally small and real for the sake of simplicity
(however see below). Although even when $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$
and $\mu^{2}_{23}=\mu^{2}$, the matrix above is diagonalized by the
tribimaximal matrix, as the neutrinos masses altarelli ), this is not possible
with scalars fields: in there is no Goldstone bosons. In order to have the
correct number of these bosons we have to impose that
$\mu^{2}_{22}=\mu^{2}_{33}=-\mu^{2}_{23}\equiv\mu^{2}$. In this case the
matrix in (19) is still diagonalized by tribimaximal matrix in Eq. (6), and
the eigenvalues are now $(2a_{n}+b_{n},a_{n}-b_{n},a_{n}-b_{n}+\mu^{2})$ and
we still have $S=h_{1}$ and $D=-(h_{2},h_{3})$, as in the previous case, and
there is no FCNC and two of the doublets are inert too.

After the discussion above, it seems reasonable to include the vacuum
alignment as part of the mechanism of natural flavor conservation.
Independently of the neutral current issue, the model also implement inert
scalar doublets: $h_{2}$ and $h_{3}$ do not couple to matter and do not
acquire vacuum expectation value, they only have electroweak, trilinear and
quartic interactions inert1 ; inert2 . In this case, is the lightest scalar is
the neutral ones and fermiophobic, they are stable and good candidates for
dark matter dark .

##  IV Model B: $v_{1}\equiv v_{SM}$, $v_{2}=v_{3}=0$

As we discussed in Sec. II, another representation content of the three Higgs
doublets that has been considered in literature is that in Eq. (3) and the
scalar potential is the same as in model A, see Eq. (1). The constraint
equations are given in the Appendix B. Notice from Eq. (36) the $\lambda_{8}$
term is the one which breaks the symmetry between $v_{2}$ and $v_{3}$, hence
only when $\lambda_{8}=0$ we can have the solutions: (i) with $v_{1}=v_{SM}$
and $v_{2}=v_{3}=0$. This can be accomplished by using a $Z_{2}$ symmetry
under which $D\to-D$ and all the other fields are even; (ii): all VEVs equals
$v_{1}=v_{2}=v_{3}\equiv v$. However, the latter solution is not the most
interesting since it implies extra massless scalars. Here we shall consider
only the solution (i), that seems similar to the model A with
$v_{1}=v_{2}=v_{3}$ since in both cases the $S_{3}$ doublet $D=(D_{1},D_{2})$
has no VEVs. However, the models are phenomenological different since the
trilinear are different the decays of the fermiophobic scalars are different
in model A and B.

The constraint equations in Eq. (36) implies, with the vacuum aligmnet given
above, $\mu^{2}_{s}=-\lambda_{4}v^{2}_{SM}$ and the mass square matrices are
all diagonal: there is no mixing among the fields in each charge sector. At
tree level the masses are the same as in model A, see Eqs. (8) -(10), but due
to the difference in the interactions, quantum corrections must be different
in both models. The mass eigenstates doublets of $SU(2)$ are denoted, as
before, by $h_{1,2,3}$. In this case the scalar potential in terms of the mass
eigenstates is given also in Eq. (17), but with $\lambda_{8}=0$. Unlike the
model A, there is no mixing among the mass eigenstate scalar fields therefore
these fields are in the irreducible representations of $S_{3}$ too:
$S=h_{1}\equiv H_{1}$ and $D=(h_{2},h_{3})\equiv(H_{2},H_{3})$. The symmetry
$h_{2}\leftrightarrow h_{3}$ is now another $S_{3}$ symmetry. This symmetry,
again if it is necessary, can be softly broken by adding terms like
$\mu^{2}_{nm}H^{\dagger}_{n}H_{m}$, $n,m=2,3$ ($\mu^{2}$’s are also consider
real for simplicity). Assuming that $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$, and
$\mu^{2}_{23}=\mu^{2}$, the mass matrices are given by

$M^{2}_{n}=\left(\begin{array}[]{ccc}m^{2}_{n_{1}}&0&0\\\
0&m^{2}_{n_{2}}+\nu^{2}&\mu^{2}\\\
0&\mu^{2}&m_{n_{2}}+\nu^{2}\end{array}\right),$ (20)

where $n=h,a,c$, for scalar, pseudoscalar and charged scalar field,
respectively. The mixing now is only in the fermiophobic sector and the masses
square are

$\displaystyle\bar{m}^{2}_{h_{1}}=m^{2}_{h_{1}},\;\;\bar{m}^{2}_{h_{2}}=m^{2}_{h}+2\mu^{2}-\nu^{2},\;\;\bar{m}^{2}_{h_{3}}=m^{2}_{h}+2\mu^{2}+\nu^{2},$
$\displaystyle\bar{m}^{2}_{a_{1}}=0,\;\;\bar{m}^{2}_{a2}=m^{2}_{h}+2\mu^{2}-\nu^{2},\;\;\bar{m}^{2}_{a3}=m^{2}_{h}+2\mu^{2}+\nu^{2},$
$\displaystyle\bar{m}^{2}_{c_{1}}=0,\;\;\bar{m}^{2}_{c_{2}}=m^{2}_{c}+\mu^{2}-\frac{1}{2}\nu^{2},\;\;\bar{m}^{2}_{c_{3}}=m^{2}_{c}+\mu^{2}+\frac{1}{2}\nu^{2}$
(21)

The mass matrices of the form in (20) are diagonalized by the orthogonal
matrix

$U=\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right),$ (22)

and the mixing between $h_{2}$ and $h_{3}$ sector is maximal. Notice that
there are still mass degeneracy between $CP$-even ad $CP$-odd scalars.

Thus, in terms of the mass eigenstate fields, the scalar doublets of $SU(2)$
are written as $S=h_{1}$ and
$D\equiv-(D_{1},D_{2})=-(-h_{2}+h_{3},h_{2}+h_{3})$, where $h_{i}$ ar the
$SU(2)$ doublets written in terms of the mass eigenstate fields. Explicitly

$\displaystyle S\equiv h_{1}=\left(\begin{array}[]{c}h^{+}_{1}\\\
\frac{1}{\sqrt{2}}(3v+h^{0}_{1}+ia^{0}_{1})\end{array}\right),\;$ (25)
$\displaystyle
D_{1}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(-h^{+}_{2}+h^{+}_{3})\\\
\frac{1}{2}[-h^{0}_{2}+h^{0}_{3}+i(-a^{0}_{2}+a^{0}_{3})]\end{array}\right),D_{2}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}(h^{+}_{2}+h^{+}_{3})\\\
\frac{1}{2}[h^{0}_{2}+h^{0}_{3}+i(a^{0}_{2}+a^{0}_{3})]\end{array}\right).$
(30)

##  V The Yukawa sector

If in the lepton and quark sectors all fields transform, in both models, as
singlet under $S_{3}$, and for this reason they only interact with the singlet
$S$ as following:

$-\mathcal{L}_{yukawa}=\bar{L}_{iL}(G^{l}_{ij}l_{jR}S+G^{\nu}_{ij}\nu_{jR}\tilde{S})+\bar{Q}_{iL}(G^{u}_{ij}u_{jR}\tilde{S}+G^{d}_{ij}d_{jR}S)+H.c.,$
(31)

$\tilde{S}=i\tau_{2}S^{*}$.

As we said before, the fermion masses arise only through the VEV of the
singlet $S$ which is the only linear combination with a non-zero VEV, see Eqs.
(15) or (30). Hence, there is no FCNC in the lepton and quark sectors because
there is just one source of the fermion masses which are given by
$M^{f}=(v_{SM}/\sqrt{2})G^{f}$, $f=l,\nu,u,d$ and where $v_{SM}=246$ GeV. The
neutral interactions are
$(\sqrt{2}/v_{SM})\bar{f}_{L}\hat{M}^{f}f_{R}h^{0}_{1}$, $\hat{M}^{f}$ is the
diagonal mass matrix in the $f$-sector. These mass matrices are general enough
to accommodate a realistic $V_{PMNS}$ and $V_{CKM}$ mixing matrices. However,
since the right-handed neutrinos may have a Majorana mass term we can have a
type-I seesaw mechanism. Fermions are mass degenerated at tree level, if their
left-handed and right-handed components transform as a triplet 3 under
$S_{3}$, for this reason the choice that all fermions transform as singlets
under $S_{3}$ is to guarantee realistic mass and mixing matrices. Moreover,
this choice implies that the triplet of the scalar reducible triplets has to
be broken into the doublet and singlet.

##  VI Analysis of the scalar potential

In order to have a potential bounded from below we consider only the quartic
terms in Eq. (17) by defining as in gunion :

$\displaystyle\chi_{1}=h_{1}^{\dagger}h_{1}~{};~{}\chi_{2}=h_{2}^{\dagger}h_{2}~{};~{}\chi_{3}=h_{3}^{\dagger}h_{3}$
(32)
$\displaystyle\chi_{4}=\Re(h_{1}^{\dagger}h_{2})~{};~{}\chi_{5}=\Im(h_{1}^{\dagger}h_{2})~{};~{}$
$\displaystyle\chi_{6}=\Re(h_{1}^{\dagger}h_{3})~{};~{}\chi_{7}=\Im(h_{1}^{\dagger}h_{3})~{};~{}\chi_{8}=\Re(h_{2}^{\dagger}h_{3})~{};~{}\chi_{9}=\Im(h_{2}^{\dagger}h_{3}),$

it is possible now rewrite the quartic terms in the scalar potential given in
Eq. (17) as follows:

$\displaystyle V(D,S)_{4}$ $\displaystyle=$
$\displaystyle[\sqrt{\lambda_{4}}\,\chi_{1}-\sqrt{\lambda_{1}+\lambda_{3}}\,(\chi_{2}+\chi_{3})]^{2}$
$\displaystyle+$
$\displaystyle[\lambda_{5}+2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}][\chi_{1}(\chi_{2}+\chi_{3})-\chi_{4}^{2}-\chi_{5}^{2}-\chi_{6}^{2}-\chi_{7}^{2}]$
$\displaystyle+$ $\displaystyle
2[\lambda_{6}+\lambda_{5}+2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}](\chi_{4}^{2}+\chi_{6}^{2})$
$\displaystyle+$ $\displaystyle[2\Re
e\lambda_{7}-2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}-\lambda_{5}-\lambda_{6}][\chi^{2}_{4}+\chi^{2}_{6}-\chi^{2}_{5}-\chi^{2}_{7}]$
$\displaystyle-$ $\displaystyle
4\lambda_{3}\,\chi_{1}+4(\lambda_{3}-\lambda_{2})\chi^{2}_{8}+4(\lambda_{3}-\lambda_{2})\chi^{2}_{9}$
$\displaystyle+$ $\displaystyle 4\Im
m\lambda_{8}[\chi_{5}(\chi_{3}-\chi_{2})+2\chi_{7}\chi_{8}]$

The constrains on the parameters $\lambda_{i}$ such that the potential with
the assumed vacuum alignment is bounded from below, are given by:

$\displaystyle\lambda_{1}+\lambda_{3}>0,~{}\lambda_{4}>0,~{}\lambda_{3}<0,~{}\lambda_{5}>-2\sqrt{(\lambda_{1}+\lambda_{3})\lambda_{4}}$
(34) $\displaystyle
2\Re(\lambda_{7})-\lambda_{5}-\lambda_{6}>2\sqrt{\lambda_{4}(\lambda_{1}+\lambda_{3})}.$

Notice that although this is a scalar potential with three Higgs scalars
doublets under $SU(2)$ it is as simple as the two doublet case in Ref. gunion
.

##  VII Conclusions

In general, multi-Higgs doublet models have complicated scalar potentials. In
particular, if the Higgs sector also has, like the fermion sectors, three
generations, we should expect the existence of extra symmetries to make simple
the interactions and mass spectra in the scalar sectors. The symmetry, if any,
would be discrete for not to have extra Goldstone bosons. In literature, this
sort of models has been consider to get a given texture of the fermion mass
matrix. However, the main concern in this paper is to use the $S_{3}$ symmetry
to have three-Higgs doublet models in which the scalar mass spectra and the
mixing among scalar fields are simple, despite the three Higgs doublets, and
the models remain closer to the standard model than a three-Higgs doublets
without $S_{3}$ symmetry. Moreover, like multi-Higgs models with no flavor
changing neutral currents mediated by neutral scalars, the only mixing
parameters appearing in the Yukawa interactions are the CKM and PMNS angles
and phases.

We have presented two models in which the three scalar doublets of $SU(2)$
transform in different ways under the irreducible representations of $S_{3}$
which have two fermiophobic doublets. The main difference between both models
is that in model B all mass matrices are diagonal and there is no mixing in
these sectors. When the soft terms are added, in model B unlike the model A
there is now mixing between the inert doublets. Hence, in the former model
there is at least one extra mixing angle. The cases considered here is that of
maximal mixing. In both models, as in the two-doublet model pich , there is
also an alignment of the Yukawa matrices, but in model A this is achieved even
without using a global $SU(2)$ transformation in order to define a basis in
which only one of the doublet gain a nonzero VEV, but because the $\langle
D_{1,2}\rangle=0$ where $D_{i}$ are the components of the $S_{3}$ doublet in
Eq. (2).

The neutral component of singlet $S$ can be identified with the standard model
Higgs scalar. But the inert doublet shall be produced in weak photon and
vector fusion and it implies that the $\gamma\gamma$, $WW$ and $ZZ$ decay
modes have an enhancement in both models. See Ref. 2gamma  for details. The
present model also implies a Higgs boson with SM couplings but additional
invisible decay modes if the inert doublets are lighter than the SM-like
scalar invisible . This depend on the value of $\mu^{2}_{d}$ in Eqs. (8)-(10)
in model A when the $S_{3}$ symmetry is not softly broken. The same in Model
B. The issue of $CP$ violation in the present context is as follows. If we
would like to maintain the dark scenario of the doublet $D$ even when the VEVs
are complex, we should expect that
$v_{1}e^{i\theta_{1}}=v_{2}e^{i\theta_{2}}=v_{3}e^{i\theta_{3}}=Ve^{i\Theta}$
was a stable minimum of the scalar potential. However assume that this is the
case the phase $\Theta$, which appears only in the singlet $S$, can be
transformed away with a global $U(1)$ transformation as it happens in the
standard model. Thus, in the situation there is no spontaneous $CP$ violation
through the VEVs. However, it is still possible to have soft $CP$ violation
through the quadratic non-diagonal term in the scalar potential
$\mu^{2}h^{\dagger}_{2}h_{3}$ assuming that $\mu^{2}$ is complex, as in Ref.
wu . Note that the $\lambda_{6}$ term has a form
$\lambda_{6}([S^{\dagger}D]_{2})^{2}+\lambda^{*}_{6}([D^{\dagger}S]_{2})^{2}$,
assuming a global phase rotation $S\rightarrow Se^{ia_{S}}$ and $D\rightarrow
De^{ia_{D}}$, and considering that the content of the theory cannot change by
this global phases we have that $\lambda_{6}=|\lambda_{6}|e^{i\alpha}$, is
possible we chose $a_{S}-a_{D}=\alpha/2+\pi/2$, so, is possible that
$\lambda_{6}=-|\lambda_{6}|$, and we have the same analyses for $\lambda_{8}$.
It is interesting that there are dark matter candidates in this model but this
deserve a detailed study zapata . Finally, we would like to mention that
depending on the masses of the dark scalars the SM-like neutral scalar may
have invisible decays invisible .

We would like to stress that the existence of two fermiophobic doublets, the
tribimaximal mixing in model A and no mixing at all in model B, and the flavor
conservation in the neutral currents mediated by scalars are consequences of
three ingredients: i) the $S_{3}$ symmetry, ii) with the representation
content of the fermion and scalar multiplets, and, iii) the vacuum alignment.
For instance, a three Higgs doublet model without any additional symmetry but
with the same vacuum alignment does not have the features above.

###### Acknowledgements.

The authors would like to thank to CAPES (ACBM) for fully support and to CNPq
and FAPESP (VP) for partial support.

##  Appendix A Constraint equations in model A

From (1) we obtain the constraint equations are written explicitly as

$\displaystyle
18t_{1}=6\mu^{2}_{d}(2v_{1}-v_{2}-v_{3})+6\mu^{2}_{s}V+(2(4\lambda_{1}+4\lambda_{3}+\lambda_{4}+2(-2\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime}))v_{1}^{3}$
$\displaystyle-3(4\lambda_{1}+4\lambda_{3}-2\lambda_{4}-\sqrt{2}\lambda_{8}-\bar{\lambda}^{\prime})v_{1}^{2}(v_{2}+v_{3})+6v_{1}((\lambda_{4}+2(\lambda_{1}+\lambda_{3}+\sqrt{2}\lambda_{8}))v_{2}^{2}$
$\displaystyle+(2\lambda_{4}-\lambda_{5}-\lambda_{6}-2(\lambda_{7}+\sqrt{2}\lambda_{8}))v_{2}v_{3}+(\lambda_{4}+2(\lambda_{1}+\lambda_{3}+\sqrt{2}\lambda_{8}))v_{3}^{2})$
$\displaystyle+(v_{2}+v_{3})((-4\lambda_{1}-4\lambda_{3}+2\lambda_{4}+\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime})v_{2}^{2}$
$\displaystyle+(4\lambda_{1}+4\lambda_{3}+4\lambda_{4}-4\lambda_{5}-4\lambda_{6}-8\lambda_{7}-7\sqrt{2}\lambda_{8})v_{2}v_{3}$
$\displaystyle+(-4\lambda_{1}-4\lambda_{3}+2\lambda_{4}+\sqrt{2}\lambda_{8}+\bar{\lambda}^{\prime})v_{3}^{2}))$
(35)

where $V=v_{1}+v_{2}+v_{3}$, and
$\bar{\lambda}^{\prime}=\lambda_{5}+\lambda_{6}+2\lambda_{7}$. We have denoted
$\langle H^{0}_{i}\rangle=v_{i}/\sqrt{2}$. We have verified that the solution
$v_{1}=v_{2}=v_{3}$ produce a global and stable minimum for a width range of
the parameters $\lambda$s and $\mu^{2}_{d}$ and the potential is bounded from
below.

##  Appendix B Constraint equations in model B

With the representation in Eq. (3), the constrain equation are

$\displaystyle 2t_{1}$ $\displaystyle=$ $\displaystyle
v_{1}\left[2\mu^{2}_{s}+2\lambda_{4}v^{2}_{1}+\bar{\lambda}^{\prime}(v^{2}_{2}+v^{2}_{3})-\frac{\lambda_{8}}{v_{1}}\left(v^{3}_{2}+v_{2}v^{2}_{3}\right)\right],$
$\displaystyle 2t_{2}$ $\displaystyle=$ $\displaystyle
v_{2}\left[2\mu^{2}_{d}+\bar{\lambda}^{\prime}v^{2}_{1}+(\lambda_{1}+\lambda_{3})(v^{2}_{2}+v^{2}_{3})-3\lambda_{8}\left(v_{1}v_{2}-\frac{v_{1}v^{2}_{3}}{v_{2}}\right)\right],$
$\displaystyle 2t_{3}$ $\displaystyle=$ $\displaystyle
v_{3}[2\mu^{2}_{d}+\bar{\lambda}^{\prime}v^{2}_{1}+2(\lambda_{1}+\lambda_{3})(v^{2}_{2}+v^{2}_{3})+6\lambda_{8}v_{1}v_{2}],$
(36)

and we see that even in the general case when $v_{1}\not=v_{2}\not=v_{3}$ they
are different from the respective equations in model A, see Eq. (35).

Notice that the $\lambda_{8}$ term avoid the zero solution for $v_{1}$ and
$v_{2}$. If this term is forbidden with a $Z_{2}$ symmetry under which
$D\to-D$ and all the other fields being even under this symmetry, we can have
the solution $v_{1}=v_{SM}$ and $v_{2}=v_{3}=0$. However, without the
$\lambda_{8}$ term it is easy to see from (1) that the scalar potential is
invariant under $S_{3}\otimes S^{\prime}_{3}$. With the soft terms there are
the following contributions, $4\mu^{2}_{22}v_{2}$ and $\mu^{2}_{23}v_{3}$ in
$t_{2}$, and $\mu^{2}_{33}v_{3}$ and $\mu^{2}_{23}v_{2}$ in $t_{3}$. The soft
terms, with $\mu^{2}_{22}=\mu^{2}_{33}=\nu^{2}$, and $\mu^{2}_{23}=\mu^{2}$,
break one of the $S_{3}$ while the other is trivially realized by the scalar
singlet that does not mix with the others two.

The neutral component of singlet $S$ can be identified with the standard model
Higgs scalar. But the inert doublet shall be produced in weak photon and
vector fusion and it implies that the $\gamma\gamma$, $WW$ and $ZZ$ decay
modes may have an enhancement in both models. See Ref. 2gamma  for details.

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