# The wronskian solution of the constrained discrete KP hierarchy

Maohua Li1,2,3, Jingsong He1∗ 1\. Department of Mathematics, Ningbo
University, Ningbo, 315211 Zhejiang, China
2\. School of Mathematical Sciences, USTC, Hefei, 230026 Anhui, China
3\. Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

From the constrained discrete KP (cdKP) hierarchy, the Ablowitz-Ladik lattice
has been derived. By means of the gauge transformation, the Wronskian solution
of the Ablowitz-Ladik lattice have been given. The $u_{1}$ of the cdKP
hierarchy is a Y-type soliton solution for odd times of the gauge
transformation, but it becomes a dark-bright soliton solution for even times
of the gauge transformation. The role of the discrete variable $n$ in the
profile of the $u_{1}$ is discussed.

∗ Corresponding author

Keywords: Constrained discrete KP hierarchy, Gauge transformation, Wronskian
solution

Mathematics Subject Classification (2000): 37K10, 37K40, 35Q51, 35Q55.

##  1\. Introduction

In the past few years, lots of attention have been given to the study of
Kadomtsev-Petviashvili (KP) hierarchy [1, 2] in the field of integrable
systems. The Lax pairs, Hamiltonian structures, symmetries and conservation
laws, the $N$-soliton, tau function, the gauge transformation, reductions etc.
of the KP hierarchy and its sub-hierarchies have been discussed. There are
several sub-hierarchies of the KP by considering different reduction
conditions on the Lax operator $L$. One of them is called constrained KP (cKP)
hierarchy [3, 4, 5] by setting the Lax operator as
$L=\partial+\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$. The cKP hierarchy
contains a large number of interesting soliton equations. The basic idea of
this procedure is so-called symmetry constraint [3, 4, 5]. The negative part
of the Lax operator of the constrained KP, i.e.
$\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$, is a generator [2] of the
additional symmetry [6] of the KP hierarchy. And the additional symmetry of
BKP hierarchy and CKP hierarchy have been given [7, 8]. Very recently, by a
further modification of the additional flows, the additional symmetries of the
constrained BKP and constrained CKP hierarchies are given in references [9,
10].

It is well known that a continuous integrable system has a discrete analogue
in general. The famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. There are several different kinds of the discrete hierarchies
including differential-difference KP (dKP) hierarchy [11, 12], semi-discrete
integrable systems, full discrete equations and so on. The differential-
difference KP hierarchy, defined by the difference operator $\Delta$, is one
interesting object of the discrete integrable systems. Note that, the
additional symmetry of dKP hierarchy and it’s Sato Bäcklund transformations
have been given in reference [13]. Moreover, gauge transformation is one kind
of powerful method to construct the solutions of the integrable systems for
both the continuous KP hierarchy [14, 15, 16, 17, 18, 19, 20, 21, 22] and the
dKP hierarchy [23, 24]. It is discussed to reduce the gauge transformation of
the dKP hierarchy to the constrained discrete KP(cdKP) hierarchy [25]. And the
algebraic structure of the additional symmetry of the cdKP hierarchy also has
been found [26], which is same for the cKP hierarchy [8].

A crucial observation [12] about the KP hierarchy and the dKP hierarchy is
that the $\tau$ function of the discrete KP hierarchy can be constructed by
shift of the $t$ of the $\tau$ function of the continuous KP hierarchy. It is
an interesting question to find any other difference among the two
hierarchies. In this direction, the correspondence between the solutions of
discrete and continuous hierarchy can be used to explore the difference
between them. In particular, a key step is to demonstrate how the discrete
variable $n$ affects the profile of the solutions of the dKP hierarchy.

The purpose of this paper is to find the the correspondence between the
solutions of the KP hierarchy and the dKP hierarchy by means of the multi-
channel gauge transformation. The paper is organized as follows. Some basic
results of the dKP hierarchy and the cdKP hierarchy are summarized in Section
2. The main theorem about the solution of cdKP hierarchy are give in Section
3. An example is give in section 4. We find that the odd kinds of gauge
transformation of cdKP hierarchy can change to a new profile of solution of
the cdKP hierarchy. Section 5 is devoted to conclusions and discussions.

##  2\. the cdKP hierarchy

Let $L$ be a general first-order pseudo difference operator(PDO)

$L(n)=\Delta+\sum_{i=1}^{\infty}u_{i}(n)\Delta^{-i},$ (2.1)

the cdKP hierarchy [26] is defined by the following Lax equation

$\frac{\partial L}{\partial t_{l}}=[B_{l},L],B_{l}:=(L^{l})_{+},l=1,2,\cdots,$
(2.2)

associated with a constrained Lax operator

$L^{l}_{-}=\sum_{i=1}^{m}q_{i}(t)\Delta^{-1}r_{i}(t),$ (2.3)

which is $m$-components Lax operator of the cdKP hierarchy. It has relation
between the dynamical variables $q_{i},r_{i}$ and $u_{i}$. Specially,
$u_{1}=q_{1}\Lambda^{-1}(r_{1})$, where $\Delta=\Lambda-I$. The eigenfunction
and adjoint eigenfunction $q_{i}(t),r_{i}(t)$ are important dynamical
variables in the cdKP hierarchy. It can be checked that the Lax equation (2.2)
is consistent with the evolution equations of the eigenfunction (or adjoint
eigenfunction)

$\displaystyle\begin{cases}q_{i,t_{m}}=B_{m}q_{i},\\\
r_{i,t_{m}}=-B_{m}^{*}r_{i},\quad B_{m}=(L^{m})_{+},\forall m\in
N.\end{cases}$ (2.4)

Therefore the cdKP hierarchy in eq.(2.2) is well defined.

From the Lax equation (2.2), we get the first nontrival $t_{2}$ flow equations
of the cdKP hierarchy for $m=1,l=2$ as

$\displaystyle\begin{cases}q_{1,t_{2}}=\Delta^{2}q_{1}+2q_{1}^{2}r_{1}=q_{1}(n+2)-2q_{1}(n+1)+q_{1}(n)+2q_{1}^{2}r_{1},\\\
r_{1,t_{2}}=-{\Delta^{*}}^{2}r_{1}+2q_{1}r_{1}^{2}=r_{1}(n)-2r_{1}(n-1)+r_{1}(n-2)+2q_{1}(n)r_{1}(n)^{2}.\end{cases}$
(2.5)

It is nothing but the Ablowitz-Ladik lattice [27]. It can be reduced to the
discrete non-linear Schrödinger (DNLS) equation [28] by letting
$r_{1}=q_{1}^{*}$ and a scaling transformation $t_{2}=it_{2}$.

The Lax operator in eq.(2.3) can be generated by the dressing action

$L=W\circ\Delta\circ W^{-1},$ (2.6)

with a dressing operator

$W(n;t)=1+\sum^{\infty}_{j=1}w_{j}(n;t)\Delta^{-j}.$ (2.7)

Further the flow equation (2.2) is equivalent to the so-called Sato equation,

$\partial_{t_{l}}W=-(L^{l})_{-}\circ W.$ (2.8)

Denote the exponential function as following

$Exp(n;t,z)=(1+z)^{n}exp(\sum_{i=1}^{\infty}t_{i}z^{i})=exp(\sum_{i=1}^{\infty}(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i}),$
(2.9)

then

$\Delta Exp(n;t,z)=zExp(n;t,z),\Delta^{*}Exp^{-1}(n;t,z)=zExp^{-1}(n;t,z).$
(2.10)

There are the wave function $w(n;t,z)$ and the adjoint wave function
$w^{*}(n;t,z)$ for the dKP hierarchy as the following forms:

$w(n;t,z)=W(n;t)Exp(n;t,z)=(1+\frac{w_{1}(n;t)}{z}+\frac{w_{2}(n;t)}{z^{2}}+\cdots)exp(\sum_{i=1}^{\infty}(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i})$
(2.11)

and

$\displaystyle w^{*}(n;t,z)$ $\displaystyle=$
$\displaystyle(W^{-1}(n-1;t))^{*}Exp^{-1}(n;t,z)$ (2.12) $\displaystyle=$
$\displaystyle(1+\frac{w_{1}^{*}(n;t)}{z}+\frac{w_{2}^{*}(n;t)}{z^{2}}+\cdots)exp(\sum_{i=1}^{\infty}-(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i}).$

There also exists a $\tau$ function $\tau(n;t)$ for the dKP hierarchy [12]
such that the wave function is expressed by

$\displaystyle w(n;t,z)=\frac{\tau(n,t-[z^{-1}])}{\tau(n,t)}Exp(n;t,z),$
(2.13)

and the adjoint wave function is expressed by

$\displaystyle
w^{*}(n;t,z)=\frac{\tau(n,t+[z^{-1}])}{\tau(n,t)}Exp^{-1}(n;t,z),$ (2.14)

where $[z]=(z,z^{2}/2,x^{3}/3,\cdots).$

The difference $\Delta-$Wronskian [24]

$\tau_{\Delta}(n)=W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m})=\left|\begin{array}[]{cccc}q_{1}&q_{2}&\cdots&q_{m}\\\
\Delta q_{1}&\Delta q_{2}&\cdots&\Delta q_{m}\\\
\vdots&\vdots&\ddots&\vdots\\\
\Delta^{m-1}q_{1}&\Delta^{m-1}q_{2}&\cdots&\Delta^{m-1}q_{m}\end{array}\right|$
(2.15)

is a $\tau$ function of dKP hierarchy. In this section, we will reduce
$\tau_{\Delta}(n)$ in (2.15) to a $\tau$ function of the constrained discrete
KP hierarchy.

Now we consider a chain of gauge transformation operator of multi-channel
difference type $T_{d}$ [19, 21, 25] starting from the initial $m$-component
Lax operator $L^{(0)}=L=L_{+}+\sum_{i=1}^{m}q_{i}(t)\Delta^{-1}r_{i}(t)$,

$L^{[0]}\xrightarrow{T_{d}^{[1]}(q_{1}^{[0]})}L^{[1]}\xrightarrow{T_{d}^{[2]}(q_{2}^{[1]})}L^{[2]}\rightarrow\dots\rightarrow
L^{[n-1]}\xrightarrow{T_{d}^{[n]}(q_{n}^{[n-1]})}L^{[n]}.$ (2.16)

Here the index $i$ in the gauge transformation operator
$T_{d}^{[i]}(q_{j}^{[j-1]})$ $(j>i)$means the $i$-th gauge transformation, and
$q_{j}^{[j-1]}$ (or $r_{j}^{[j-1]}$) is transformed by $(j-1)$-steps gauge
transformations from $q_{j}$ (or $r_{j}$), $L^{[k]}$ is transformed by
$k$-steps gauge transformations from the initial Lax operator $L$.

Now we firstly consider successive gauge transformations in (2.16). We define
the operator as

$T_{m}=T_{d}^{[m]}(q_{m}^{[m-1]})\circ\dots\circ T_{d}^{[2]}(q_{2}^{[1]})\circ
T_{d}^{[1]}(q_{1}^{[0]}),$ (2.17)

in which

$\displaystyle q_{i}^{[j]}=T_{d}^{[j]}(q_{j}^{[j-1]})\circ\dots\circ
T_{d}^{[2]}(q_{2}^{[1]})\circ T_{d}^{[1]}(q_{1}^{[0]})q_{i},i,j=1,\cdots,m;$
(2.18) $\displaystyle
r_{k}^{[j]}=((T_{d}^{[j]})^{-1})^{*}(q_{j}^{[j-1])})\circ\dots\circ((T_{d}^{[2]})^{-1})^{*}(q_{2}^{[1]})\circ((T_{d}^{[1]})^{-1})^{*}(q_{1}^{[0]})r_{k},j,k=1,\cdots,m.$
(2.19)

It means that $q_{i}^{[0]}=q_{i}$, $r_{i}^{[0]}=r_{i}$. We shall find another
criterion for the Wronskian entries $f_{1},f_{2},\cdots,f_{n}$ leading to cdKP
flows. The following theorem can be easily got from the Ref. [25].

######  Theorem 2.1.

The gauge transformation operator $T_{m}$ and $T_{m}^{-1}$ have the following
determinant representation:

$\displaystyle T_{m}$ $\displaystyle=$ $\displaystyle
T_{d}^{[m]}(q_{m}^{[m-1]})\circ\dots\circ T_{d}^{[2]}(q_{2}^{[1]})\circ
T_{d}^{[1]}(q_{1}^{[0]})$ (2.25) $\displaystyle=$
$\displaystyle\frac{1}{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m})}\left|\begin{array}[]{ccccc}q_{1}&q_{2}&\cdots&q_{m}&1\\\
\Delta q_{1}&\Delta q_{2}&\cdots&\Delta q_{m}&\Delta\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\Delta^{m-1}q_{1}&\Delta^{m-1}q_{2}&\cdots&\Delta^{m-1}q_{m}&\Delta^{m-1}\\\
\Delta^{m}q_{1}&\Delta^{m}q_{2}&\cdots&\Delta^{m}q_{m}&\Delta^{m}\end{array}\right|,$

and

$\displaystyle T_{m}^{-1}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}q_{1}\circ\Delta^{-1}&\Lambda(q_{1})&\Lambda(\Delta
q_{1})&\cdots&\Lambda(\Delta^{m-2}q_{1})\\\
q_{2}\circ\Delta^{-1}&\Lambda(q_{2})&\Lambda(\Delta
q_{2})&\cdots&\Lambda(\Delta^{m-2}q_{2})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
q_{m}\circ\Delta^{-1}&\Lambda(q_{m})&\Lambda(\Delta
q_{m})&\cdots&\Lambda(\Delta^{m-2}q_{m})\end{array}\right|\frac{(-1)^{m-1}}{\Lambda(W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m}))}$
(2.30) $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\phi_{i}\circ\Delta^{-1}b_{i}$ (2.31)

with

$b_{i}=(-1)^{m+i}\Lambda(\frac{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},\hat{i},q_{i+1},\dots,q_{m})}{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},q_{i},q_{i+1},\dots,q_{m})}).$
(2.32)

Here $\hat{i}$ means that the column containing $q_{i}$ is delete from
$W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},q_{i},q_{i+1},\dots,q_{m})$ and the
last row is also deleted. Here the determinant of $T_{m}$ is expanded by the
last column and collecting all sub-determinants on the left side of the
$\Delta^{i}$ with the action $"\circ"$. And $T_{m}^{-1}$ is expanded by the
first column and all the sub-determinants are on the right side with the
action $"\circ"$.

##  3\. Wronskian solution of constrained discrete KP hierarchy

Similar to reference [29], it has [26]

$(K\circ q\circ\Delta^{-1}\circ r)_{-}=K(q)\circ\Delta^{-1}\circ
r,(q\circ\Delta^{-1}\circ r\circ K)_{-}=-q\circ\Delta^{-1}\circ K^{*}(r),$
(3.1)

for a pure-difference operator $K$ and two arbitrary smooth functions ($q,r$).

An important fact is that there exist two $m$-th order $\Delta$-differential
operators

$A=\Delta^{m}+a_{m-1}\Delta^{m-1}+\dots+a_{0},B=\Delta^{m}+b_{m-1}\Delta^{m-1}+\dots+b_{0},$
(3.2)

such that $AL^{l}$ and $L^{l}B$ are differential operators. From
$(AL^{l})_{-}=0$ and $(L^{l}B)_{-}=0$, we get that $A$ and $B$ annihilate the
functions $q^{i}$ and $r_{j}$ , i.e., $A(q_{1})=\dots=A(q_{m})=0,$
$B^{*}(r_{1})=\dots=B^{*}(r_{m})=0$, that implies $q_{i}\in Ker(A)$, $r_{i}\in
Ker(B^{*})$, $i=1,\dots,m$. The dimension of $Ker(A)$ is $m$.

The following theorem provides a criterion for reducing the $\Delta$-Wronskian
$\tau$ function in (2.15) of dKP hierarchy to the $\Delta$-Wronskian $\tau$
function of the cdKP hierarchy defined by (2.2).

######  Theorem 3.1.

The constrained discrete KP hierarchy has a solution
$L=(L)_{+}+\sum_{j=1}^{m}f_{j}\circ\Delta\circ g_{j}$ generated by the $\tau$
function $\tau_{\Delta}(n)=W_{m}^{\Delta}(f_{1},\cdots,f_{m})\neq 0$ satisfies
the $k$-constrained with some suitable functions $q_{1},q_{2},\cdots,q_{M}$
and $r_{1},r_{2},\cdots,r_{M}$ if and only if

$W_{m+M+1}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{i_{1}},\cdots,\Delta^{k}f_{i_{M+1}})=0$
(3.3)

for all $(M+1)$ indices $1\leq i_{1}<i_{2}<\cdots<i_{M+1}\leq m,$ which is
equivalent to

$\mbox{}W_{M+1}^{\Delta}(\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{2}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\cdots,\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{M+1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})})=0.$
(3.4)

Here $f_{i}$ satisfied linear $\Delta-$difference equations

$\frac{\partial f_{i}}{\partial t_{k}}=\Delta^{k}f_{i},\quad
i=1,2,\cdots,m;k=1,2,\cdots.$ (3.5)

###### Proof.

Similar to case of KP hierarchy [30] and $q$-KP hierarchy [31], there has the
following Wronskian identity

$\displaystyle
W_{m+M+1}^{\Delta}(\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},$
$\displaystyle\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{2}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\cdots,\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{M+1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})})$
(3.6)
$\displaystyle=W_{m}(f_{1},\cdots,f_{m},\Delta^{k}f_{i_{1}},\cdots,\Delta^{k}f_{i_{M+1}})$

which provides the equivalence between (3.3) and (3.4).

Using Theorem 2.1 and the relation of operator identities in (3.1), one finds

$(L^{k})_{-}=(W\circ\Delta^{k}\circ
W^{-1})_{-}=\sum_{j=1}^{m}W(\Delta^{k}f_{j})\Delta^{-1}g_{j}$ (3.7)

As it was pointed out in the beginning of this section, there exists an $m$-th
order differential operator $A$ such that $AL^{k}$ is a difference operator.
Application to $W(f_{j})=0$ yields

$0=AL^{k}(W(f_{j}))=AW\circ\Delta^{k}(f_{j})=AW(\partial_{t_{k}}(f_{j})).$
(3.8)

So,

$W(\partial_{t_{k}}(f_{j}))=\frac{W_{m}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{j})}{W_{m}^{\Delta}(f_{1},\cdots,f_{m})}\in
Ker(A).$ (3.9)

Since the kernel of $A$ has dimension $m$, at most $m$ of these functions
$\Delta^{k}f_{j}$ can be linearly independent. So, (3.4) is deduced.

Conversely, if (3.4) holds, then there exists one $M$-component of cdKP
$(M<m)$ constrained from (3.7). The equation (3.4) implies that at most $M$ of
functions $W(\Delta^{k}(f_{j}))$ are linearly independent, here $f_{j}$
satisfy (3.5). Then we can find suitable $M$ functions
${q_{1},q_{2},\dots,q_{M}}$, which are linearly independent, to express
functions $W(\Delta^{k}(f_{j}))$ as

$W(\partial_{t_{k}}(f_{j}))=\frac{W_{m}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{j})}{W_{m}^{\Delta}(f_{1},\cdots,f_{m})}=\sum_{i=1}^{M}c_{ij}q_{i},j=1,\dots,m.$
(3.10)

with some constant $c_{ij}$. Taking it back into the (3.7), it becomes

$(L^{k})_{-}=\sum_{j=1}^{m}(\sum_{i=1}^{M}c_{ij}q_{i})\circ\Delta^{-1}\circ
g_{j}=\sum_{i=1}^{M}q_{i}\circ\Delta^{-1}\circ(\sum_{j=1}^{m}c_{ij}g_{j})=\sum_{i=1}^{M}q_{i}\circ\Delta^{-1}\circ
r_{i},$ (3.11)

then a $m$-component cdKP hierarchy is reduced to a $M$-component cdKP
hierarchy. ∎

Remark: This theorem is a difference version of the corresponding theorem of
the Ref. [30].

The Wronskian solution of the cdKP hierarchy can be got by the Theorem 3.1
under the gauge transformation. If the initial Lax operator of the constrained
discrete KP hierarchy is a ”free” operator $\Delta$, then $L=\Delta$ means
that the initial $\tau$ function $\tau_{\Delta}$ is $1$.

##  4\. Example of reducing dKP hierarchy to cdKP hierarchy

In this section, we use the method in Theorem 3.1 to find the solution of the
multi-component cdKP hierarchy. We discuss the cdKP hierarchy generated by
$T_{i}\mid_{i=2}$, possesses a $\tau$ function

$\tau_{\Delta}^{(2)}=W_{2}^{\Delta}(f_{1},f_{2})=\left|\begin{array}[]{cc}f_{1}&f_{2}\\\
\Delta f_{1}&\Delta f_{2}\end{array}\right|=f_{1}\circ\Delta
f_{2}-f_{2}\circ\Delta f_{1},$ (4.1)

with

$f_{1}=f_{11}(z_{1},n,t)+f_{12}(z,n,t),f_{2}=f_{21}(z_{2},n,t)+f_{22}(z_{3},n,t).$
(4.2)

Here

$\displaystyle
f_{11}(z_{1},n,t)=(1+z_{1})^{n}e^{\xi_{1}},f_{12}(z,n,t)=(1+z)^{n}e^{\xi},$
$\displaystyle
f_{21}(z_{2},n,t)=(1+z_{2})^{n}e^{\xi_{2}},f_{22}(z_{3},n,t)=(1+z_{3})^{n}e^{\xi_{3}},$

where $\xi_{i}=c_{i}+z_{i}t_{1}+z_{i}^{2}t_{2}+z_{i}^{3}t_{3},i=1,2,3$ and
$\xi=d+zt_{1}+z^{2}t_{2}+z^{3}t_{3},$ $c_{i}$ and $d$ are arbitrary constants.
These functions $f_{1}$ and $f_{2}$ satisfy linear equations (3.5) for
$k=1,2,3$. By (3.5), the cdKP hierarchy generated by $T_{2}$ is in the form of

$\displaystyle L^{l}$ $\displaystyle=$
$\displaystyle(L^{l})_{+}+(T_{2}(\Delta^{k}f_{1}))\circ\Delta^{-1}\circ
g_{1}+(T_{2}(\Delta^{k}f_{2}))\circ\Delta^{-1}\circ g_{2}$ (4.3)
$\displaystyle\stackrel{{\scriptstyle constraint}}{{====}}$
$\displaystyle(L^{l})_{+}+q_{1}\circ\Delta^{-1}\circ r_{1},$ (4.4)

where $q_{1},r_{1}$ are undetermined, which can be expressed by $f_{1}$ and
$f_{2}$ as follows. $\tau_{\Delta}^{(2)}$ possesses a form as

$\displaystyle\tau_{\Delta}^{(2)}$ $\displaystyle=$ $\displaystyle
W_{2}^{\Delta}(f_{1},f_{2})$ (4.5) $\displaystyle=$
$\displaystyle(z_{2}-z_{1})(1+z_{1})^{n}(1+z_{2})^{n}e^{\xi_{1}+\xi_{2}}+(z_{2}-z)(1+z)^{n}(1+z_{2})^{n}e^{\xi+\xi_{2}}$
$\displaystyle+(z_{3}-z_{1})(1+z_{1})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{3}}+(z_{3}-z)(1+z)^{n}(1+z_{3})^{n}e^{\xi+\xi_{3}}.$

According to (3.3) in Theorem 3.1, the restriction for $f_{1}$ and $f_{2}$ to
reduce (4.3) to (4.4) is given by

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
W_{4}^{\Delta}(f_{1},f_{2},f_{1}^{(k)},f_{2}^{(k)})$ (4.6) $\displaystyle=$
$\displaystyle(z^{k}-z_{1}^{k})(z_{2}^{k}-z_{3}^{k})V(z_{1},z_{2},z_{3},z)F(n;z_{i},t),$

with Vandermonde determinant

$V(z_{1},z_{2},z_{3},z)=\left|\begin{array}[]{cccc}1&1&1&1\\\
z_{1}&z_{2}&z_{3}&z\\\ z_{1}^{2}&z_{2}^{2}&z_{3}^{2}&z^{2}\\\
z_{1}^{3}&z_{2}^{3}&z_{3}^{3}&z^{3}\end{array}\right|,$ (4.7)

and

$F(n;z_{i},t)=(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}(1+z)^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}+\xi}.$
(4.8)

Obviously, $f_{1}$ and $f_{2}$ satisfy (4.6) by setting $z=z_{2}$ and
$d=c_{2}$. Then the $\tau$ function of a single component $k$-constrained cdKP
hierarchy defined by

$\displaystyle\tau_{cdKP}^{\Delta}$ $\displaystyle=$
$\displaystyle(z_{2}-z_{1})(1+z_{1})^{n}(1+z_{2})^{n}e^{\xi_{1}+\xi_{2}}+(z_{3}-z_{1})(1+z_{1})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{3}}$
(4.9)
$\displaystyle+(z_{3}-z_{2})(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{2}+\xi_{3}},$

which is deduced by (4.5) with $\xi_{2}=\xi$. It means that we indeed reduced
the $\tau$ function $\tau_{\Delta}^{(2)}$ in (4.5) of the dKP hierarchy to the
$\tau$ function $\tau_{cdKP}^{\Delta}$ of the $1$-component cdKP hierarchy.

We would like to get the explicit forms of $q_{1}$ and $r_{1}$ of cdKP
hierarchy in (4.4). With the determinant representation of $T_{2}$ and
$T_{2}^{-1}$, one can have

$\displaystyle f_{1}^{\Delta}$ $\displaystyle\triangleq
T_{2}(\Delta^{k}f_{1})=\frac{(z_{1}^{k}-z_{2}^{k})V(z_{1},z_{2},z_{3})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}$
(4.10a) $\displaystyle f_{2}^{\Delta}$ $\displaystyle\triangleq
T_{2}(\Delta^{k}f_{2})=\frac{(z_{3}^{k}-z_{2}^{k})V(z_{1},z_{2},z_{3})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}$
(4.10b) $\displaystyle g_{1}^{\Delta}$
$\displaystyle\triangleq(T_{2}^{-1})^{*}(\Delta^{k}g_{1})=-\Lambda(\frac{f_{2}}{W_{2}^{\Delta}(f_{1},f_{2})})=-\Lambda(\frac{(1+z_{2})^{n}e^{\xi_{2}}+(1+z_{3})^{n}e^{\xi_{3}}}{\tau_{cdKP}^{\Delta}})$
(4.10c) $\displaystyle g_{2}^{\Delta}$
$\displaystyle\triangleq(T_{2}^{-1})^{*}(\Delta^{k}g_{1})=\Lambda(\frac{f_{1}}{W_{2}^{\Delta}(f_{1},f_{2})})=\Lambda(\frac{(1+z_{1})^{n}e^{\xi_{1}}+(1+z_{2})^{n}e^{\xi_{2}}}{\tau_{cdKP}^{\Delta}})$
(4.10d)

with the Vandermonde determinant

$V(z_{1},z_{2},z_{3})=\left|\begin{array}[]{ccc}1&1&1\\\ z_{1}&z_{2}&z_{3}\\\
z_{1}^{2}&z_{2}^{2}&z_{3}^{2}\end{array}\right|.$ (4.11)

It is clearly

$(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}=(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}.$

So the $q_{1}$ in (4.4) is

$\displaystyle q_{1}\mbox{}$
$\displaystyle\triangleq(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}=(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}$
(4.12) $\displaystyle=$
$\displaystyle\mbox{}\frac{(z_{3}^{k}-z_{2}^{k})(z_{1}^{k}-z_{2}^{k})(z_{2}-z_{1})(z_{3}-z_{1})(z_{3}-z_{2})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}.$

And the (4.3) is reduced to

$\displaystyle(L^{l})_{-}$ $\displaystyle=$ $\displaystyle
f_{1}^{\Delta}\circ\Delta^{-1}\circ
g_{1}^{\Delta}+f_{2}^{\Delta}\circ\Delta^{-1}\circ g_{2}^{\Delta}$ (4.13)
$\displaystyle=$
$\displaystyle(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}\circ\Delta^{-1}\circ\frac{g_{1}^{\Delta}}{(z_{3}^{k}-z_{2}^{k})}+(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}\circ\Delta^{-1}\circ\frac{g_{2}^{\Delta}}{z_{1}^{k}-z_{2}^{k}}$
$\displaystyle=$ $\displaystyle q_{1}\circ\Delta^{-1}\circ r_{1},$

where

$\displaystyle r_{1}$ $\displaystyle\triangleq$
$\displaystyle\frac{g_{1}^{\Delta}}{(z_{3}^{k}-z_{2}^{k})}+\frac{g_{2}^{\Delta}}{z_{1}^{k}-z_{2}^{k}}$
(4.14) $\displaystyle=$
$\displaystyle\Lambda(\frac{(z_{3}^{k}-z_{2}^{k})(1+z_{1})^{n}e^{\xi_{1}}+(z_{3}^{k}-z_{1}^{k})(1+z_{2})^{n}e^{\xi_{2}}+(z_{2}^{k}-z_{1}^{k})(1+z_{3})^{n}e^{\xi_{3}}}{(z_{3}^{k}-z_{2}^{k})(z_{1}^{k}-z_{2}^{k})\tau_{cdKP}^{\Delta}}).$

For simplicity, denote $t_{1}=x;t_{2}=y$. In particular, choosing $z_{1}=z$,
$z_{2}=0$, $z_{3}=-z$, $c_{1}=c$, $c_{2}=0$, $c_{3}=-c$ then $\xi_{2}=0$,
$\xi_{3}=-\xi_{1}$ and

$q_{1}=\frac{(-1)^{k+2}z^{2k+2}(1+z)^{n}(1-z)^{n}e^{2z^{2}y}}{(1+z)^{n}(1-z)^{n}+\frac{(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}}{2}e^{z^{2}y}}.$
(4.15)

Base on above choice,

$r_{1}=-\frac{1}{z^{k+1}}\Lambda(\frac{\frac{[(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}]}{2}+e^{-z^{2}y}}{\frac{[(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}]}{2}+(1-z^{2})^{n}e^{z^{2}y}})$
(4.16)

and if $k$ is odd, and

$r_{1}=-\frac{1}{z^{k+1}}\Lambda(\frac{\frac{(1+z)^{n}e^{\eta}-(1-z)^{n}e^{-\eta}}{2}}{\frac{(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}}{2}+(1-z^{2})^{n}e^{z^{2}y}})$
(4.17)

if $k$ is even. Here $\eta\triangleq c+zx+z^{3}t_{3}.$

So the dynamical variable $u_{1}=q_{1}\Lambda^{-1}(r_{1})$ of the Lax operator
$L$ of the cdKP hierarchy

$u_{1}=2z^{k+1}\frac{(1-(-1)^{k})+e^{-(c+zx+z^{3}t_{3})+z^{2}y}(1-z)^{n}-e^{(c+zx+z^{3}t_{3})+z^{2}y}(-1)^{k}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{c+zx+z^{3}t_{3}}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-(c+zx+z^{3}t_{3})}}{(1+z)^{n}})^{2}}.$
(4.18)

An example is

$u_{1}=2z^{k+1}\frac{(1-(-1)^{k})+e^{-zx+z^{2}y}(1-z)^{n}-e^{zx+z^{2}y}(-1)^{k}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},$
(4.19)

by setting $t_{3}=0,c=0.$ For this case

$\displaystyle
u_{1}=2z^{k+1}\frac{2+e^{-zx+z^{2}y}(1-z)^{n}+e^{zx+z^{2}y}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},\text{$k$
is odd},$ (4.20) $\displaystyle
u_{1}=2z^{k+1}\frac{e^{-zx+z^{2}y}(1-z)^{n}-e^{zx+z^{2}y}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},\text{$k$
is even.}$ (4.21)

Remark: Actually, (4.6) also can be satisfied by other two choices $z=z_{1}$
or $z=z_{3}$. But $u_{1}$ (4.18) will be only one-soliton solution because the
$f_{1}^{\Delta}=0$ in (4.10a) or $f_{2}^{\Delta}=0$ in (4.10b) separately.

The graph of $q_{1}=q_{1}(x,y,n),r_{1}=r_{1}(x,y,n),u_{1}=u_{1}(x,y,n)$ were
plotted in below for fixed $k=1,2$. We shall discuss the function of the gauge
transformation for the the cdKP hierarchy to emphasize two sides about the
discrete variable $n$ of it and the times variable $k$ of the gauge
transformation of it. The profile of $q_{1},r_{1},u_{1}$ are plotted according
to the value of discrete variable $n$ from $0$ to $2$ and the value of time
$k$ of the gauge transformations from $1$ to $2$. The five conditions of the
profile of $q_{1},r_{1},u_{1}$ are $\\{n=0,k=1\\}$,$\\{n=1,k=1\\}$,
$\\{n=2,k=1\\}$, $\\{n=0,k=2\\}$, $\\{n=1,k=2\\}$ and $\\{n=2,k=2\\}$ as
following.

(1).The profile of $q_{1}$ are plotted with $k=1,n=0,1,2$ in Figure 1, Figure
2 and Figure 3 respectively.

(2).The profile of $r_{1}$ are plotted with $k=1,n=0,1,2$ in Figure 4, Figure
5 and Figure 6 respectively.

(3).The Y-type soliton profile of $u_{1}$ are plotted with $k=1,n=0,1,2$ in
Figure 7 , Figure 8 and Figure 9 respectively.

(4).The bright-dark soliton profile of $u_{1}$ are plotted with $k=2$ and
$n=0,1,2$ in Figure 10, Figure 11 and Figure 12 respectively.

From the graphs of the solution of cdKP hierarchy, it can be found that:

(1) The profile of the solution $q_{1}$ of the cdKP hierarchy is decreasing to
the one of the classical KP hierarchy in Ref. [30] when $n\to 0$ see Figure 1
($k=1$). For $r_{1},u_{1}$ of the cdKP hierarchy, the profile of its are also
decreasing the analogues of the classical KP hierarchy (see Figure 4 and
Figure 7).

(2)When the times $k$ of gauge transformation is an odd number, the profiles
of $u_{1}$ become the Y-type soliton, see Figure 7, Figure 8 and Figure 9.

(3)When the times $k$ of gauge transformation is an even number, the profiles
of $u_{1}$ become bright-dark soliton, see Figure 10, Figure 11 and Figure 12.

For the end of showing more detail about dependence of $u_{1},q_{1}$ on $n$,
it is necessary to define $n$-effect quantity $\Delta
u_{1}(z,x,y,n)=u_{1}(z,x,y,n)-u_{1}(z,x,y,n=0)=u_{1}(n)-u_{1}(0)$ for fixed
$z=0.5$. Figure 13 are plotted for the $\Delta u_{1}(z,x,y,n)$ where $n=1,2,3$
respectively, which shows the dependence of $u_{1}$ on $n$. It was obviously
they are decreasing to almost zero when $n$ goes from $3$ to $1$ with fixed
$z=0.5$. They also demonstrate that discretization of the cdKP hierarchy keeps
the profile of the soliton though it has discrete variable $n$. These figures
give us again an opportunity to observe the role of discrete variable $n$ in
the Wronskian solution of the cdKP hierarchy.

##  5\. Conclusions

In this paper, the Wronskian solutions of the equation in the cdKP hierarchy
have been given by means of the multi-channel gauge transformation. Based on
the results of our previous papers [25, 26], Theorem 3.1 provides a necessary
and sufficient condition of the $k$-constrained discrete KP hierarchy with $m$
components. As an example, the reduction from $2$-cdKP hierarchy to $1$-cdKP
hierarchy is presented. It can be found that the profiles of solution $u_{1}$
of cdKP hierarchy can be the Y-type solition by the odd number times gauge
transformation, but the solution of cdKP hierarchy becomes bright-dark
solition by even times gauge transformation. From these profiles, it can be
find that the solution $u_{1}$ of the cdKP hierarchy is decreasing to the
analogues of the classical KP hierarchy when $n\to 0$.

Acknowledgments This work is supported by the National Natural Science
Foundation of China under Grant Nos.11271210 and 11201251, K.C.Wong Magna Fund
in Ningbo University, the Natural Science Foundation of Zhejiang Province
under Grant No. LY12A01007 and Science Fund of Ningbo University
(No.XYL14028). One of the authors (MH) is supported by Erasmus Mundus Action 2
EXPERTS III and would like to thank Prof. Antoine Van Proeyen for many helps.

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Figure 1.  The profile of the solution $q_{1}$(4.15) of equation (2.5) (left)
and its density plot (right)with $c=0,t_{3}=0,z=0.5,k=1$ and $n=0$ .

Figure 2. The profile of the solution of $q_{1}$(4.15) of equation (2.5)
(left) and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$
and $n=1$.

Figure 3. The profile of the solution of $q_{1}$(4.15) of equation (2.5)
(left) and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$
and $n=2$.

Figure 4.  The profile of the solution $r_{1}$ (4.16) (left) of equation (2.5)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and
$n=0$. The vertical axis $r$ denotes the $r_{1}$.

Figure 5. The profile of $r_{1}$ (4.16) (left) and its density plot (right)
with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$. The vertical axis $r$
denotes the $r_{1}$.

Figure 6. The profile of $r_{1}$ (4.16) (left) and its density plot (right)
with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=2$. The vertical axis $r$
denotes the $r_{1}$.

Figure 7. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=0$. The
vertical axis $u$ denotes the $u_{1}$.

Figure 8. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$. The
vertical axis $u$ denotes the $u_{1}$.

Figure 9. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=2$. The
vertical axis $u$ denotes the $u_{1}$.

Figure 10. The profile of bright-dark type soliton of $u_{1}$ (4.21) (left)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=0$
. The vertical axis $u$ denotes the $u_{1}$.

Figure 11. The profile of bright-dark type soliton of $u_{1}$ (4.21) (left)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=1$
. The vertical axis $u$ denotes the $u_{1}$.

Figure 12.  The profile of bright-dark soliton of $u_{1}$ (4.21) (left) and
its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=2$.
The vertical axis $u$ denotes the $u_{1}$.

(a) (b) (c)

Figure 13.  $\Delta u_{1}(z,x,y,n)$ with $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$ in
(a), $2$ in (b) and $3$ in (c).


∎

11institutetext: Dintomon Joy 22institutetext: Supin P Surendran
33institutetext: Sabir M 44institutetext: Department of Physics, Cochin
University of Science and Technology, Kochi - 682 022, India
44email<EMAIL_ADDRESS>
# Efficient Deterministic Secure Quantum Communication protocols using
multipartite entangled states

Dintomon Joy  Supin P Surendran  Sabir M

(Received: date / Accepted: date)

###### Abstract

We propose two deterministic secure quantum communication (DSQC) protocols
employing three-qubit GHZ-like states and five-qubit Brown states as quantum
channels for secure transmission of information in units of two bits and three
bits using multipartite teleportation schemes developed here. In these
schemes, the sender’s capability in selecting quantum channels and the
measuring bases leads to improved qubit efficiency of the protocols.

###### Keywords:

Deterministic Secure Quantum Communication GHZ-like state Five-qubit Brown
state Qubit efficiency

##  1 Introduction

The field of quantum cryptography which offers unconditional security in
communication between legitimate users has emerged as an important area of
research in this information age. In 1984, Bennett and BrassardBennett
proposed the first Quantum Key Distribution (QKD) protocol for secure
communication. Following this, several protocolsEkert Long were suggested all
of which required a unique secret key to be shared between the users before
communication. Later developments showed how secure quantum communication
could be achieved without the need for shared secret keys. Among the two group
of protocols, the deterministic secure quantum communication (DSQC)
protocolsLo  the receiver can read out the message only after the exchange of
at least one bit of classical information per qubit. But, in quantum secure
direct communication(QSDC) the message is read out directly. Compared with
QSDC, the DSQC protocols are more secure in that the message-carrying qubits
need not be transmitted through external channels. Both protocols, however,
require classical communication during the error checking and eavesdropping
detection processes.

In 1999, Shimizu and ImotoShimizu  proposed the first DSQC protocol using EPR
pairs and Bell measurements. Later, Beige et.al.Beige  proposed a scheme based
on single photons. In 2004, Yan and ZhangYan  found the first secure DSQC
protocol based on teleportation. Following this work, several schemes
employing multipartite entangled channels and teleportation were proposedHJCao
DXG08 XGC09 QCY13 Gaot Cao Zhang  Many DSQC protocols based on entanglement
swappingGaoswap Man Xiu Gaoepr ManZ  and order rearrangement of particlesZhu
LiXH  were also suggested. The advantage of teleportation based protocols is
that these are more secure even in noisy channelHassan  and are suitable for
quantum error correctionLi .

In this paper we propose two new DSQC protocols one of these using the three-
qubit GHZ-like state and the other using the five-qubit Brown states for
secure transmission of information in units two bits and three bits employing
multipartite teleportation techniques. In these schemes, it is the sender who
decides which entanglement channel and measurement basis are to be used. The
receiver and the possible eavesdroppers are ignorant of the choices. The
receiver performs unitary operations based on the classical information
obtained from the sender and finally performs joint measurements to read out
the secret message.

This paper is organized as follows. In section 2, we present the teleportation
scheme of a special class of two particle state using GHZ-like state and a
special class of three particle state using five-qubit Brown state in 3. The
details of our new DSQC protocols using GHZ-like state and Brown state are
given in 3. In section 4, we discuss the possible eavesdropping attacks. In
section 5, we discuss and compare the efficiency of our protocol with other
existing protocols. The final section contains our conclusions.

##  2 Teleportation of special classes of two and three particle states

###  2.1 Teleportation of a two particle state using GHZ-like state

In 2014, Nandi and MazumdarNandhi  proposed a scheme for teleportation of a
special form of two particle state,
$\left|\tau\right\rangle_{12}=\alpha(\left|00\right\rangle+\left|11\right\rangle)_{12}+\beta(\left|01\right\rangle+\left|10\right\rangle)_{12}$
using GHZ-like state

$\left|\Phi_{G}\right\rangle_{345}=\frac{1}{\sqrt{2}}\\{\left|0\right\rangle_{3}\left|\psi^{+}\right\rangle+\left|1\right\rangle_{3}\left|\phi^{+}\right\rangle_{45}\\}$
(1)

as entanglement channel. Here, the states
$\left|\phi^{\pm}\right\rangle=\frac{1}{\sqrt{2}}\\{\left|01\right\rangle\pm\left|10\right\rangle\\}$
and
$\left|\psi^{\pm}\right\rangle=\frac{1}{\sqrt{2}}\\{\left|00\right\rangle\pm\left|11\right\rangle\\}$
represent the Bell basis states. A three particle joint measurement is
performed in the measurement basis:
$\left|\zeta^{\pm}\right\rangle=\frac{1}{2}\\{(\left|001\right\rangle\left|111\right\rangle)\pm(\left|010\right\rangle-\left|100\right\rangle)\\}$
and
$\left|\eta^{\pm}\right\rangle=\frac{1}{2}\\{(\left|000\right\rangle-\left|110\right\rangle)\pm(\left|011\right\rangle-\left|101\right\rangle)\\}$,
to achieve this teleportation. In the DSQC protocol proposed in section:3, we
need a teleportation scheme for a related two particle state.

$\left|\tau^{\prime}\right\rangle_{12}=\alpha(\left|00\right\rangle-\left|11\right\rangle)_{12}+\beta(\left|01\right\rangle-\left|10\right\rangle)_{12}\\\
$ (2)

We show, here, that this can be done by utilizing the Nandi-Mazumdar method by
choosing a different GHZ-like state

$\displaystyle\left|\Phi^{\prime}_{G}\right\rangle_{345}=\frac{1}{\sqrt{2}}\\{\left|0\right\rangle_{3}\left|\phi^{-}\right\rangle+\left|1\right\rangle_{3}\left|\psi^{-}\right\rangle_{45}\\}$
(3)

as the entanglement channel. Then the total wave function is
$\Gamma_{12345}=\left|\tau^{\prime}\right\rangle_{12}\otimes\left|\Phi^{\prime}_{G}\right\rangle_{345}$
and Alice performs a three particle joint measurement on particles in her
possession $(1,2,3)$ in a new measurement basis:
$\left|\zeta^{\prime\pm}\right\rangle=\frac{1}{2}\\{(\left|001\right\rangle-\left|111\right\rangle)\pm(\left|010\right\rangle-\left|100\right\rangle)\\}$
and
$\left|\eta^{\prime\pm}\right\rangle=\frac{1}{2}\\{(\left|000\right\rangle-\left|110\right\rangle)\pm(\left|011\right\rangle-\left|101\right\rangle)\\}$.
After getting the information about measurement result from Alice, Bob
performs unitary operations on his particles $(4,5)$ as given in Table:1 and
recovers the teleported state. It may be noted that the table:1 coincides with
the table given in Nandhi  except for the changes in labels representing
measurement result and the state of Bob’s particles (4,5).

Alice’s | Classical | State of Bob’s particles (4,5) | Unitary
---|---|---|---
measurement | information |  | operation
result |  |  |
$\left|\zeta^{\prime+}\right\rangle$ | 00 | $\alpha(\left|00\right\rangle-\left|11\right\rangle)+\beta(\left|01\right\rangle-\left|10\right\rangle)$ | $(I\otimes I)$
$\left|\zeta^{\prime-}\right\rangle$ | 01 | $\alpha(\left|00\right\rangle-\left|11\right\rangle)-\beta(\left|01\right\rangle-\left|10\right\rangle)$ | $(\sigma_{z}\otimes\sigma_{z})$
$\left|\eta^{\prime+}\right\rangle$ | 10 | $\alpha(\left|01\right\rangle-\left|10\right\rangle)+\beta(\left|00\right\rangle-\left|11\right\rangle)$ | $(I\otimes\sigma_{x})$
$\left|\eta^{\prime-}\right\rangle$ | 11 | $\alpha(\left|01\right\rangle-\left|10\right\rangle)-\beta(\left|00\right\rangle-\left|11\right\rangle)$ | $(\sigma_{z}\otimes i\sigma_{y})$
Table 1: Classical information corresponding to Alice’s measurement result,
state of particles (4,5) and unitary operations performed by Bob.

###  2.2 Teleportation of a three particle state using five-qubit Brown state

We extend the two-qubit teleportation scheme given above for the teleportation
of a three particle state

$\displaystyle\left|\psi\right\rangle_{123}$ $\displaystyle=$
$\displaystyle\\{\alpha(\left|000\right\rangle-\left|111\right\rangle)+\beta(\left|001\right\rangle+\left|110\right\rangle)$
$\displaystyle+$
$\displaystyle\gamma(\left|010\right\rangle+\left|101\right\rangle)+\delta(\left|011\right\rangle-\left|100\right\rangle)\\}_{123}$

where $\alpha,\beta,\gamma$ and $\delta$ are unknown coefficients satisfying
the relation $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1$.

The quantum channel is chosen as the maximally entangled five-qubit Brown
stateBrown . Brown et.al., initially obtained this state via a numerical
optimization procedure and later Muralidharan and PanigrahiMurali  proposed a
simple theoretical method for the physical realization of this state. The
five-qubit Brown state is given by

$\left|\Psi\right\rangle_{45678}=\frac{1}{2}\\{\left|001\right\rangle\left|\phi^{-}\right\rangle+\left|010\right\rangle\left|\psi^{-}\right\rangle+\left|100\right\rangle\left|\phi^{+}\right\rangle+\left|111\right\rangle\left|\psi^{+}\right\rangle\\}_{45678}$
(5)

This can be rewritten in equivalent form in the following order as given in
Lin .

$\left|\Psi\right\rangle_{67\>458}=\frac{1}{2}\\{\left|00\right\rangle\left|G_{010}\right\rangle-\left|01\right\rangle\left|G_{111}\right\rangle+\left|10\right\rangle\left|G_{001}\right\rangle-\left|11\right\rangle\left|G_{100}\right\rangle\\}_{67\>458}\\\
$ (6)

Here, the three-qubit states defined by

$\left|G_{ijk}\right\rangle=\frac{1}{\sqrt{2}}\\{\left|0\right\rangle\left|j\right\rangle\left|k\right\rangle+(-1)^{i}\left|1\right\rangle\left|j\oplus
1\right\rangle\left|k\oplus 1\right\rangle\\}$ (7)

where $i,j,k$ takes values $\\{0,1\\}$ and $\oplus$ represents addition modulo
2, are the GHZGHZ  states. The equation:2.2 can be expressed in terms of these
states as

$\left|\psi\right\rangle_{123}=\sqrt{2}(\alpha\left|G_{100}\right\rangle+\beta\left|G_{001}\right\rangle+\gamma\left|G_{010}\right\rangle+\delta\left|G_{111}\right\rangle)_{123}$

Alice, now, performs a five-particle joint measurement on her particles
$(1,2,3,6,7)$ in the measurement basis
$\\{\left|\Phi_{1,2...16}\right\rangle\\}$ given below:

$\displaystyle\left|\Phi_{1,2}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|00\right\rangle-\left|G_{111}\right\rangle\left|01\right\rangle\pm\left|G_{001}\right\rangle\left|10\right\rangle\mp\left|G_{100}\right\rangle\left|11\right\rangle\\}$
(8)
$\displaystyle\left|\Phi_{3,4}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|00\right\rangle+\left|G_{111}\right\rangle\left|01\right\rangle\pm\left|G_{001}\right\rangle\left|10\right\rangle\pm\left|G_{100}\right\rangle\left|11\right\rangle\\}$
$\displaystyle\left|\Phi_{5,6}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|01\right\rangle-\left|G_{111}\right\rangle\left|00\right\rangle\pm\left|G_{001}\right\rangle\left|11\right\rangle\mp\left|G_{100}\right\rangle\left|10\right\rangle\\}$
$\displaystyle\left|\Phi_{7,8}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|01\right\rangle+\left|G_{111}\right\rangle\left|00\right\rangle\pm\left|G_{001}\right\rangle\left|11\right\rangle\pm\left|G_{100}\right\rangle\left|10\right\rangle\\}$
$\displaystyle\left|\Phi_{9,10}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|10\right\rangle-\left|G_{111}\right\rangle\left|11\right\rangle\pm\left|G_{001}\right\rangle\left|00\right\rangle\mp\left|G_{100}\right\rangle\left|01\right\rangle\\}$
$\displaystyle\left|\Phi_{11,12}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|10\right\rangle+\left|G_{111}\right\rangle\left|11\right\rangle\pm\left|G_{001}\right\rangle\left|00\right\rangle\pm\left|G_{100}\right\rangle\left|01\right\rangle\\}$
$\displaystyle\left|\Phi_{13,14}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|11\right\rangle-\left|G_{111}\right\rangle\left|10\right\rangle\pm\left|G_{001}\right\rangle\left|01\right\rangle\mp\left|G_{100}\right\rangle\left|00\right\rangle\\}$
$\displaystyle\left|\Phi_{15,16}\right\rangle=\frac{1}{2}\\{\left|G_{010}\right\rangle\left|11\right\rangle+\left|G_{111}\right\rangle\left|10\right\rangle\pm\left|G_{001}\right\rangle\left|01\right\rangle\pm\left|G_{100}\right\rangle\left|00\right\rangle\\}$

Alice then communicates her measurement result to Bob via four bits of
classical information. In order to recover the teleported state, appropriate
unitary operations are performed by Bob on his particles $(4,5,8)$ as given in
table: 2.

Alice’s result | State of qubits (6,7,8) | Unitary
---|---|---
and Classical |  | operation
information |  |
$\left|\Phi_{1}\right\rangle 0000$ | $\sqrt{2}(\alpha\left|G_{100}\right\rangle+\beta\left|G_{001}\right\rangle+\gamma\left|G_{010}\right\rangle+\delta\left|G_{111}\right\rangle)$ | $I\otimes I\otimes I$
$\left|\Phi_{2}\right\rangle 0001$ | $\sqrt{2}(-\alpha\left|G_{100}\right\rangle-\beta\left|G_{001}\right\rangle+\gamma\left|G_{010}\right\rangle+\delta\left|G_{111}\right\rangle)$ | $\sigma_{z}\otimes\sigma_{z}\otimes I$
$\left|\Phi_{3}\right\rangle 0010$ | $\sqrt{2}(-\alpha\left|G_{100}\right\rangle+\beta\left|G_{001}\right\rangle+\gamma\left|G_{010}\right\rangle-\delta\left|G_{111}\right\rangle)$ | $\sigma_{z}\otimes I\otimes\sigma_{z}$
$\left|\Phi_{4}\right\rangle 0011$ | $\sqrt{2}(\alpha\left|G_{100}\right\rangle-\beta\left|G_{001}\right\rangle+\gamma\left|G_{010}\right\rangle-\delta\left|G_{111}\right\rangle)$ | $I\otimes\sigma_{z}\otimes\sigma_{z}$
$\left|\Phi_{5}\right\rangle 0100$ | $\sqrt{2}(-\alpha\left|G_{001}\right\rangle-\beta\left|G_{100}\right\rangle-\gamma\left|G_{111}\right\rangle-\delta\left|G_{010}\right\rangle)$ | $I\otimes I\otimes\sigma_{x}$
$\left|\Phi_{6}\right\rangle 0101$ | $\sqrt{2}(\alpha\left|G_{001}\right\rangle+\beta\left|G_{100}\right\rangle-\gamma\left|G_{111}\right\rangle-\delta\left|G_{010}\right\rangle)$ | $\sigma_{z}\otimes\sigma_{z}\otimes\sigma_{x}$
$\left|\Phi_{7}\right\rangle 0110$ | $\sqrt{2}(\alpha\left|G_{001}\right\rangle-\beta\left|G_{100}\right\rangle-\gamma\left|G_{111}\right\rangle+\delta\left|G_{010}\right\rangle)$ | $\sigma_{z}\otimes I\otimes i\sigma_{y}$
$\left|\Phi_{8}\right\rangle 0111$ | $\sqrt{2}(-\alpha\left|G_{001}\right\rangle+\beta\left|G_{100}\right\rangle-\gamma\left|G_{111}\right\rangle+\delta\left|G_{010}\right\rangle)$ | $I\otimes\sigma_{z}\otimes i\sigma_{y}$
$\left|\Phi_{9}\right\rangle 1000$ | $\sqrt{2}(\alpha\left|G_{111}\right\rangle+\beta\left|G_{010}\right\rangle+\gamma\left|G_{001}\right\rangle+\delta\left|G_{100}\right\rangle)$ | $I\otimes\sigma_{x}\otimes I$
$\left|\Phi_{10}\right\rangle 1001$ | $\sqrt{2}(-\alpha\left|G_{111}\right\rangle-\beta\left|G_{010}\right\rangle+\gamma\left|G_{001}\right\rangle+\delta\left|G_{100}\right\rangle)$ | $\sigma_{z}\otimes i\sigma_{y}\otimes I$
$\left|\Phi_{11}\right\rangle 1010$ | $\sqrt{2}(-\alpha\left|G_{111}\right\rangle+\beta\left|G_{010}\right\rangle+\gamma\left|G_{001}\right\rangle-\delta\left|G_{100}\right\rangle)$ | $\sigma_{z}\otimes\sigma_{x}\otimes\sigma_{z}$
$\left|\Phi_{12}\right\rangle 1011$ | $\sqrt{2}(\alpha\left|G_{111}\right\rangle-\beta\left|G_{010}\right\rangle+\gamma\left|G_{001}\right\rangle-\delta\left|G_{100}\right\rangle)$ | $I\otimes i\sigma_{y}\otimes\sigma_{z}$
$\left|\Phi_{13}\right\rangle 1100$ | $\sqrt{2}(-\alpha\left|G_{010}\right\rangle-\beta\left|G_{111}\right\rangle-\gamma\left|G_{100}\right\rangle-\delta\left|G_{001}\right\rangle)$ | $\sigma_{x}\otimes I\otimes I$
$\left|\Phi_{14}\right\rangle 1101$ | $\sqrt{2}(\alpha\left|G_{010}\right\rangle+\beta\left|G_{111}\right\rangle-\gamma\left|G_{100}\right\rangle-\delta\left|G_{001}\right\rangle)$ | $i\sigma_{y}\otimes\sigma_{z}\otimes I$
$\left|\Phi_{15}\right\rangle 1110$ | $\sqrt{2}(\alpha\left|G_{010}\right\rangle-\beta\left|G_{111}\right\rangle-\gamma\left|G_{100}\right\rangle+\delta\left|G_{001}\right\rangle)$ | $i\sigma_{y}\otimes I\otimes\sigma_{z}$
$\left|\Phi_{16}\right\rangle 1111$ | $\sqrt{2}(-\alpha\left|G_{010}\right\rangle+\beta\left|G_{111}\right\rangle-\gamma\left|G_{100}\right\rangle+\delta\left|G_{001}\right\rangle)$ | $\sigma_{x}\otimes\sigma_{z}\otimes\sigma_{z}$
Table 2: Alice’s measurement results and corresponding operations performed by
Bob

In a similar fashion another three-qubit state
$\left|\psi^{\prime}\right\rangle_{123}=\\{\alpha(\left|000\right\rangle+\left|111\right\rangle)+\beta(\left|001\right\rangle-\left|110\right\rangle)+\gamma(\left|010\right\rangle-\left|101\right\rangle)+\delta(\left|011\right\rangle+\left|100\right\rangle)\\}_{123}$
needed in the DSQC protocol proposed in section:3 can be teleported. The
appropriate five-qubit channel $\left|\Psi^{\prime}\right\rangle$ and the
measurement basis $\left|\Phi^{\prime}_{1,2...16}\right\rangle$ are obtained
by making the replacements
($\left|G_{100}\right\rangle\rightarrow\left|G_{000}\right\rangle$,
$\left|G_{001}\right\rangle\rightarrow\left|G_{101}\right\rangle$,
$\left|G_{010}\right\rangle\rightarrow\left|G_{110}\right\rangle$ and
$\left|G_{111}\right\rangle\rightarrow\left|G_{011}\right\rangle$) in
equations: 6 and 8 respectively. i.e., $\left|\Psi\right\rangle_{67458}$
changes to $\left|\Psi^{\prime}\right\rangle_{67458}$ and
$\left|\Phi_{1,2...16}\right\rangle$ changes to
$\left|\Phi^{\prime}_{1,2...16}\right\rangle$. The recovery operations to be
performed by Bob, corresponding to Alice’s classical information, are same as
in table:2with the labels representing measurement results and state of Bob’s
particles changed accordingly.

##  3 New Deterministic Secure Quantum Communication protocols

Applying the teleportation schemes in the preceding section, we propose two
novel DSQC protocols for communication of information in units of 2 bits or 3
bits.

###  3.1 DSQC protocol using GHZ-like state

Our protocol for sending information in units of 2 bits makes use of the GHZ-
like state as the quantum channel and involves the following steps.

1\. Preparation of GHZ-like states: The sender, Alice, prepares copies of two
orthogonal GHZ-like states $\left|\Phi_{G}\right\rangle$ and
$\left|\Phi^{\prime}_{G}\right\rangle$, their number depending on the size of
secret messages to be sent. The ordered sequence of three particle GHZ-like
state $\left|\Phi_{G}\right\rangle_{345}$ is given by
$[(P_{3}^{1},P_{4}^{1},P_{5}^{1}),(P_{3}^{2},P_{4}^{2},P_{5}^{2}),....(P_{3}^{n},P_{4}^{n},P_{5}^{n})]$
and the sequence of $\left|\Phi^{\prime}_{G}\right\rangle_{345}$ is given by
$[(P_{3}^{{}^{\prime}1},P_{4}^{{}^{\prime}1},P_{5}^{{}^{\prime}1}),(P_{3}^{{}^{\prime}2},P_{4}^{{}^{\prime}2},P_{5}^{{}^{\prime}2}),....(P_{3}^{{}^{\prime}n},P_{4}^{{}^{\prime}n},P_{5}^{{}^{\prime}n})]$.
The entangled states in each of these sets are given a unique number, with the
superscripts representing the order in the sequence and the subscript
representing particle labels.

2\. Splitting Home Block and Travel Block: The block of all the first
particles in the sequences $\left|\Phi_{G}\right\rangle$ and
$\left|\Phi^{\prime}_{G}\right\rangle$,
$[P_{3}^{1},P_{3}^{{}^{\prime}1},P_{3}^{2},P_{3}^{{}^{\prime}2},...P_{3}^{n},P_{3}^{{}^{\prime}n}]$
are kept at sender’s location and we refer to them as the home block. The
block of all second and third particles from both sequences
$[(P_{4}^{1},P_{5}^{1}),(P_{4}^{2},P_{5}^{2}),....,(P_{4}^{n},P_{5}^{n})],$
and
$[(P_{4}^{{}^{\prime}1},P_{5}^{{}^{\prime}1}),(P_{4}^{{}^{\prime}2},P_{5}^{{}^{\prime}2})$
$....,(P_{4}^{{}^{\prime}n},P_{5}^{{}^{\prime}n})]$ are referred to as the
travel block.

3\. Arrangement of particles for Error checking and Message transmission:
Alice selects $\frac{N}{2}$ channels for error checking, where N represents
the even number of secret information to be transmitted. The order and
position of corresponding particles in the home block are noted by Alice and
the leftover particles gets reserved for message transmission. In order to
send information bits 00 or 01 Alice uses the channel
$\left|\Phi_{G}\right\rangle$ and for sending 10 or 11 she uses
$\left|\Phi^{\prime}_{G}\right\rangle$ . Alice arranges all the particles in
both travel and home block according to the particular message to be sent.
Alice sends the travel block to Bob and retains the home block with herself.

4\. Error checking: On receiving the travel block, Bob sends a signal back to
Alice confirming the reception. Alice, then, initiates the error checking
process by performing single particle measurement on her particles initially
chosen for error checking. After the measurement, Alice announces the
positions of particles that were chosen for error checking. Bob performs
either Bell measurement or single particle measurement on the corresponding
particles in his possession according to the information received from Alice
and he, in turn, announces his measurement outcomes. Alice, now, checks for
correlations in the measurement results by comparing it with equations: 1 and
3. If the error rate is high they abort the communication. Otherwise they
proceed to the next step.

5\. Teleportation: In the scheme of communicating in units of two bit the
users make an agreement to encode the bits $00,01,10$ and $11$ in the Bell
states $\left|\psi^{+}\right\rangle$, $\left|\phi^{+}\right\rangle$,
$\left|\psi^{-}\right\rangle$, and $\left|\phi^{-}\right\rangle$ respectively.
With appropriate choice the coefficients $\alpha$ and $\beta$ in
$\left|\tau\right\rangle_{12}$ and $\left|\tau^{\prime}\right\rangle_{12}$
(effectively reducing these to the Bell states) information can be conveyed
securely using the teleportation scheme described earlier. For instance, if
Alice wants to send 10 or 11, she choose the state
$\left|\tau^{\prime}\right\rangle_{12}$ and channel
$\left|\Phi^{\prime}_{G}\right\rangle_{345}$. The coefficient $\alpha$ or
$\beta$ is made zero depending on whether the information is 11 or 10. Alice,
then, follows the teleportation procedure given in section:2. Finally, Bob
performs a Bell measurement to read out the secret information.

Our protocol works efficiently when the sender prepares the secret message
beforehand. For example, consider the case where Alice wants to communicate a
four bit secret information ‘0011’ to Bob. She prepares two channels,
$\left|\Phi_{G}\right\rangle$, $\left|\Phi^{\prime}_{G}\right\rangle$ and
arrange them in the same order as that of the messages i.e.,
$\left|\Phi_{G}\right\rangle$ taken first and
$\left|\Phi^{\prime}_{G}\right\rangle$ second. The steps 2, 3 and 4 are
followed. Finally, the state $\left|\tau\right\rangle_{12}$ with coefficient
$\beta=0$ or $\left|\psi^{+}\right\rangle$ and
$\left|\tau^{\prime}\right\rangle_{12}$ with $\alpha=0$ or
$\left|\phi^{-}\right\rangle$, corresponding to information 00 and 11, are
teleported using the appropriate schemes given in section:2. In this scheme,
it may be noted that the participants (receiver and eavesdropper) are not
aware of the different entanglement channels and measurement basis employed by
the sender. The receiver performs unitary operations based on the classical
information obtained from sender and read out the secret messages by
performing Bell measurements.

###  3.2 DSQC protocol using five-qubit maximally entangled Brown states.

By a straight foreword generalization of the procedure given above, Alice can
transmit information in units of 3 bits by using the five-qubit Brown states
as quantum channels and employing the teleportation scheme given in
section:2.2. Before transmitting messages Alice and Bob agree that the states
$\left|G_{ijk}\right\rangle$, where ${i,j,k}$ take values $\\{0,1\\}$
correspond to 8 different messages $(000,001,010,011,100,101,110,111)$
respectively. For instance, if Alice wants to transmit a six bit secret
information ‘100 110’ to Bob, she chooses two channels
$\left|\Psi\right\rangle$ and $\left|\Psi^{\prime}\right\rangle$ and then
arrange them in the order of messages i.e., $\left|\Psi\right\rangle$ taken
first and $\left|\Psi^{\prime}\right\rangle$ second.
$\left|G_{100}\right\rangle$ and $\left|G_{110}\right\rangle$ are the
corresponding messages and they are prepared by setting the coefficient
$\alpha=1$, other coefficients to zero in equation:2.2 and $\gamma=1$, other
coefficients to zero in its conjugate equation
$\left|\psi^{\prime}\right\rangle$ respectively. Alice follows the steps
discussed in the previous section and finally teleport the states as given in
section:2. Here, 8 bits of classical information is used up in this process.

##  4 Security Analysis

In teleportation-based DSQC protocols, the information carrying qubits do not
travel and thus the security of information solely depends on the security of
quantum channel shared between Alice and Bob. The most common attack
strategies of an eavesdropper for DSQC using GHZ-like is discussed below.

Measure and resend Attack: In this type of attack, the eavesdropper Eve may
capture the travel block from Alice and measure all the qubits in either Bell
basis or perform single particle measurement on them and resend it to Bob.
After this process, neither Alice-Bob pair nor any Alice-Eve pair shares
entanglement. Therefore it is impossible for Eve to gain any useful
information from Alice’s side using this type of attack, as long as she
employs teleportation procedure to transmit the secret information. Further,
in the first error checking process the correlation results itself will reveal
the presence of Eve and the communication is abandoned.

Intercept and resend Attack: The eavesdropper may try to intercept the travel
block and replace it with particles (6, 7) which is a part of random GHZ-like
states prepared by Eve and resend them to Bob. Now, Eve establishes an
entanglement relation with Alice’s particle and may receive the teleported
message. But, on checking the correlations given in equations: 1 and 3, Alice
detects the presence of Eve. If the error rate is high, she informs Bob to
stop the entire process.

Entanglement Attack:  In this attackYan Gaot , the eavesdropper may capture
the travel block and entangle the ancilla particles (6,7) in the state
$\left|00\right\rangle_{67}^{n}$ with the $n^{th}$ order of particles (4,5) in
the travel block, where $n$ represents the order of entangled GHZ-like
particles in the travel block. Eve uses CNOT gate to entangle ancilla qubits
with travel block. The particles (4,5) of GHZ-like state acts as control bits
and (6,7) as target bits respectively. This changes the total state to

$\displaystyle\left|\Psi\right\rangle_{34567}$ $\displaystyle=$ $\displaystyle
CNOT(5;7)CNOT(4;6)\left|\Phi_{G}\right\rangle_{345}\otimes\left|00\right\rangle_{67}$
$\displaystyle=$
$\displaystyle\frac{1}{2}[\left|0\right\rangle_{3}(\left|\phi^{+}\right\rangle_{45}\left|\phi^{+}\right\rangle_{67}+\left|\phi^{-}\right\rangle_{45}\left|\phi^{-}\right\rangle_{67})$
$\displaystyle+$
$\displaystyle\left|1\right\rangle_{3}(\left|\psi^{+}\right\rangle_{45}\left|\psi^{+}\right\rangle_{67}+\left|\psi^{-}\right\rangle_{45}\left|\psi^{-}\right\rangle_{67})]$

The same operation changes the state of
$\left|\Phi^{\prime}_{G}\right\rangle_{345}$ to

$\displaystyle\left|\Psi^{\prime}\right\rangle_{34567}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\left|0\right\rangle_{3}(\left|\phi^{+}\right\rangle_{45}\left|\phi^{-}\right\rangle_{67}+\left|\phi^{-}\right\rangle_{45}\left|\phi^{+}\right\rangle_{67})$
$\displaystyle+$
$\displaystyle\left|1\right\rangle_{3}(\left|\psi^{+}\right\rangle_{45}\left|\psi^{-}\right\rangle_{67}+\left|\psi^{-}\right\rangle_{45}\left|\psi^{+}\right\rangle_{67})]$

Equation: 4 shows, the state of (4,5) and (6,7) are correlated and equation: 4
shows that they are anti-correlated. By entangling ancillary particles to
travel block Eve gets access to Alice’s information carrying qubits. But,
during the error checking process, Alice can detect the presence of Eve 50% of
times by checking relations given in equations 1 and 3. Alice informs Bob to
continue the process if the error rate is less than 50% and otherwise to
abandon. In the case where Eve goes undetected, the probability of secret
message getting leaked is still $50\%$ due to the superposition of states
existing between particles $(4,5,6,7)$.

##  5 Efficiency Analysis

In 2000, CabelloCabello  put forward a simple definition to find the qubit
efficiency of a given protocol. Here the qubit efficiency $\eta_{2}$ depends
on the ratio of the number of messages transmitted $b_{s},$ to the sum of
total number of classical bits $b_{t}$ utilized to decode the message and the
total number of qubits $q_{t}$ used in the protocol. Note that, here the decoy
qubits and the classical bits used for eavesdrop checking are not included to
calculate the efficiency.

$\eta_{2}=\frac{b_{s}}{q_{t}+b_{t}}$ (11)

In 2012 Banerjee and PathakAA  proposed a modified measure of efficiency which
considers the decoy qubits used for eavesdrop checking. Since the number of
classical bits used in eavesdrop checking varies linearly with number of decoy
qubits used, inclusion of the number of decoy qubits in efficiency
calculations seems more appropriate. Using the definition of AA (including
decoy qubits) and the one by Cabello  (excluding decoy qubits), we have
calculated the qubit efficiency of different teleportation based protocols Yan
HJCao DXG08 XGC09 QCY13  and have tabulated them in table: 3. We find that the
qubit efficiency of these protocols are $\leq 20\%$. In these calculations we
have taken the number of decoy states as equal to the number of transmitted
qubits following suggestion in AA

In our protocol with GHZ-like state as quantum channel, to communicate a four
bit message $(b_{s}=4)$ we require two GHZ-like channels or 6 qubits
i.e.,$(q_{t}=6)$ and four bits of classical information $(b_{t}=4)$. As the
classical signal from Bob after the receipt of travel block becomes
asymptotically(N large) insignificant, it is not included in $b_{t}$. By
definition of Cabello , this leads to a qubit efficiency of $\eta$ = 40%. But,
in order to ensure security, we used $\frac{N}{2}$ GHZ-like channels for
eavesdrop checking in the protocol. In our example, we used two GHZ-like
states for eavesdrop checking. Hence the total number of qubits consumed
becomes $(q_{t}=12)$. The efficiency calculated according to AA  gives $\eta$
= 25%. In the case of sending a six bit message $(b_{s}=6)$ given in
section:3.2, using two five qubit channels $(q_{t}=10)$ and 8 bits of
classical information $(b_{t}=8)$ the qubit efficiency is $\eta$ = 33.33%
according to Cabello . Using the definition of AA , it becomes 21.43%.

DSQC |  $\eta_{2}=$ Efficiency(%) |  | Entanglement
---|---|---|---
protocol | Without decoy | With decoy AA  | Channel
| qubits Cabello  | qubits |
YZ04 Yan  | 25.00 | 16.67 | Bell pairs
CS06 HJCao  | 16.67 | 09.52 | W-State
DXG08 DXG08  | 20.00 | 12.50 | W-State
XGC09 XGC09  | 30.00 | 18.75 | Six-particle
QCY13 QCY13  | 25.00 | 16.67 | Four-qubit Cluster
PP-1 | 40.00 | 25.00 | GHZ-like State
PP-2 | 33.33 | 21.43 | Five-Qubit Brown
Table 3: Comparing efficiency of Teleportation-based DSQC protocols. PP-
Proposed protocol

##  6 Conclusion

We have proposed an extension of Nandi-Mazumdar scheme for teleportation of a
special class of two particle states and have also developed a method for the
teleportation of special class of three-particle state using five-qubit Brown
state as entanglement channel. The different teleportation schemes introduced
here are then combined effectively to build novel DSQC protocols. These works
provide new insight into the way in which teleportation can be used for
efficient DSQC protocols. The comparison of qubit efficiency shows that the
proposed protocol is better than the other DSQC protocols based on
teleportation Yan HJCao DXG08 XGC09 QCY13 . This increase in efficiency
results from the sender’s capability of choosing different channels and
measurement basis independently. Even though our protocol requires the sender
to prepare the secret message beforehand, the efficiency factor gives it an
edge over other existing protocols. We suggest that the efficiency of using
five-qubit Brown state can be further increased if the sender employs more
entanglement channels and appropriate measurement basis.

##  7 Acknowledgements

We thank Kerala State Council for Science, Technology and Environment for
providing the financial support for this work. The authors also thank Kishore
Thapliyal and Rishi Dutt Sharma for their suggestions regarding efficiency
analysis.

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