TẠP CHÍ KHOA HỌC SỐ 31/2019
85
POWER OF THE CONTROLLER IN CONTROLLED JOINT REMOTE
STATE PREPARATION OF AN ARBITRARY QUBIT STATE
Nguyen Van Hop
Faculty of Physics, Hanoi National University of Education
Abstract:
The
type
of
four-particle
partially
entangled
state
which
is
suitable
for
controlled joint remote state preparation of an arbitrary qubit state is designed. With the
controller’s assistance, the protocol is perfect as both its fidelity and total success
probability are equal to one. In the opposite case, the analytical expression of the
minimal averaged controller’s power is calculated. Furthermore, the dependence of the
minimal
averaged
controller’s
power
on
the
parameter
of
the
quantum
channel
is
analyzed and the values of the parameter of the quantum channel to the controller is
powerful are pointed out.
Keywords: Controlled joint remote state preparation, four-particle partially entangled
channel, controller’s power.
Email: hopnv@hnue.edu.vn
Received 25 March 2019
Accepted for publication 25 May 2019
1. INTRODUCTION
Quantum entanglement [1] that has been recognized as a spooky feature of quantum
machinery plays
a vital
role as
a potential
resource
for quantum
communication
and
quantum
information
processing.
Quantum
teleportation
(QT)
[2],
which
was
firstly
suggested
by
Bennett
et
al.,
is
one
of
the
most
important
applications
of
shared
entanglement for securely and faithfully transmitting a quantum state from a sender to a
spatially distant receiver without directly physical sending that state but only by means of
local operation and classical communication. After the first appearance of QT, this method
has not only attracted much attention in theory [3] but also obtained several significant
results in experiment [4]. As an obvious extension of teleportation scheme, two modified
versions to be established are remote state preparation (RSP) scheme [5 -10] which the
sender knows the full identity of the to be prepared state and joint remote state preparation
(JRSP) scheme [11-18] which each sender allowed to know only a partial information of
the state to be prepared. In RSP, the sender completely knows the information of the state,
while in QT, neither the sender nor the receiver has any knowledge of the state to be
q q
1
86
TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI
transmitted. The catch of RSP is that the full identity of the prepared state is disclosed to
the sender, who can reveal the information to outside. To overcome this drawback, joint
remote state preparation, was proposed. In JRSP, the information of the initial state is
secretly shared by two or more senders, located at distant sites, in such a way that none of
the senders can know the full. In contrast to general RSP, in JRSP, the receiver can
remotely reconstruct the original state only under the collective cooperation of all the
senders. As a matter of fact, JRSP protocols are probabilistic with probabilities of less than
one.
However, for practical purposes, it is often required to control the overall mission.
This can be accomplished by the present of a controller in the protocol, who at the last
moment decides ending of a mission after carefully judging all the related situations.
Controlled joint remote state preparation [19] (CJRSP) have been studied. The controller,
to be able to perform his role, has to share beforehand with the senders as well as the
receiver a quantum channel which is general considered as a maximal entangled state.
In this paper, we use a partially entangled quantum channel but in case the controller
agrees to cooperate, both protocols are perfect (the average fidelity and success probability
are equal to unit). The problem of the role of the controller in the task is dedicated. What is
the role of the controller in preventing unwanted situation from happening such as the
receiver exposes information. This question is not considered in Ref. [19] but will be
answered in our paper by virtue of quantitatively calculation of power of the controller in
various situations. The results show that if the controller doesn’t cooperate, i.e., he does
nothing, the receiver cannot obtain with certainty a state with quality better than that
obtained classically.
2. CONTENT
2.1. The working quantum channel
Suppose that Alice 1 and Alice 2 are in a task to help the Bob remotely prepare an
arbitrary one-particle state under control of controller Charlie, the arbitrary state can be
expressed as
y
= cos 2 0 +sin 2eif 1 ,
(1)
in which q,f are real.
To manipulate the task of CJRSP, we use the partially entangled state as the quantum
channel
Q(b) 1234 =
2 (sinb 0111 +cosb 1111 1000 )1234 ,
(2)
1
2
2
1
1
( )
U q =
,
2
e
2
1
TẠP CHÍ KHOA HỌC SỐ 31/2019
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which can be generated from the GHZ state
GHZ 1234 = 21/2 (1)j
j=0
j, j1, j1, j1 1234
as follows
Q(b) 1234 = R1 (b)CNOT 1R1 (b) GHZ 1234 ,
(3)
where CNOT 1 is a controlled-NOT gate acting on two-qubit as
CNOT21 = H1 H2CNOT2H1 H2
with
CNOT2 k,l 12 =
k,k l 12
and
H x
1/
2 0

1x 1 .
R1 (b)
is
a
rotation gate acting on a single-qubit state as
R1 (b) k 1 = cos(b / 2) k 1 (1)k sin(b / 2) k 1 1 .
In this nonlocal resource, Alice 1 holds qubit 2 and the information about q , Alice 2 holds
qubit 3 and the information about f, Bob holds qubit 4 and Charlie holds qubit 1. This
state is characterized by angle b
whose values to be known by only the controller. We are
now in the position to employ the above-listed quantum channel for our purpose.
2.2. The controlled joint remote state preparation of an arbitrary qubit state
In order to realize the controlled joint remote state preparation of an arbitrary qubit
state, our protocol is performed by four steps as follows.
In the first step, the Alice 1 acts on her qubit the unitary operator
cosq/ 2
sinq/ 2
sinq/ 2
cosq/ 2
(4)
then she measures her qubit on the basis {k 2 ;k{0,1}}.
In the second step, depending on the result of the Alice 1, the Alice 2 exerts different
operators on her qubit. If the Alice 1 obtains the result 0 and announces them to the Alice
2, respectively, the Alice 2 will apply to her qubit the unitary operator
V(0) (f)=
1 1
2 1
if
eif .
(5)
If the Alice 1 obtains the result 1 and announces them to the Alice 2, respectively, the
Alice 2 will apply to her qubit the unitary operator
V(1) (f)=
1 eif
2 eif
1.
(6)