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Optics Communications 277 (2007) 219–222

Remote preparation of the N-particle GHZ state usingquantum statistics

Yan Xia *, Jie Song, He-Shan Song

School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China

Received 7 January 2007; received in revised form 27 April 2007; accepted 27 April 2007

Abstract

We present a remote preparation of the N-particle GHZ state protocol in which only the effects of quantum statistics of indistinguish-able particles are used. The N-particle GHZ state can be successfully prepared in the limit of N!1.� 2007 Elsevier B.V. All rights reserved.

PACS: 03.67.Hk; 03.65.Ud

Keywords: Remote state preparation; N-particle GHZ state; Quantum statistics

A principal goal of quantum information theory is tounderstand the resources necessary and sufficient for intacttransmission of quantum states. Obviously, the sender caneither physically send the particle or send a double infinityof bits of information across a classical channel to thereceiver. However, quantum teleportation process, origi-nally proposed by Bennett et al. [1], can transmit anunknown quantum state from a sender to a spatially dis-tant receiver via a quantum channel with the help of someclassical information. E.g., if Alice and Bob share an Ein-stein–Podolsky–Rosen (EPR) pair, Alice can teleport aqubit to Bob by first carrying out a Bell-state measurementon the qubit and one particle of the EPR pair, then sendingtwo bits of classical information to Bob, who in turn canperform a corresponding unitary operation on his particle(the other particle of the EPR pair) to get the state Alicewants to teleport. Recently, Lo [2] has presented an inter-esting new method to transmit pure known quantum stateusing a prior shared entanglement and some classical com-munication when the sender knows completely the trans-mitted state. This communication protocol is called

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.04.046

* Corresponding author. Tel.: +86 411 84706201.E-mail addresses: xia-208@163.com (Y. Xia), hssong@dlut.edu.cn

(H.-S. Song).

remote state preparation (RSP). The main differencebetween RSP and teleportation are in that: (1) in RSP Aliceknows the state that she wants Bob to prepare, in particu-lar, Alice need not own the state, but only know the infor-mation about the state, while in teleportation Alice mustown the teleported state, but she need not know the state;(2) in RSP, the required resource can be traded off betweenclassical communication cost and entanglement cost whilein quantum teleportation, two bits of forward classicalcommunication and one ebit of entanglement (an EPRpair) per teleported qubit are both necessary and sufficient,and neither resource can be traded off against the other[4,5].

RSP has attracted many attentions in recent years[2,3,6–9]. Pati [3] has found RSP protocol more economicalthan teleportation and requires only one classical bit fromAlice to Bob for some special ensembles. Bennett et al.have generalized RSP for arbitrary qubits, higher dimen-sional Hilbert spaces and also of entangled systems [4].Ye et al. [5] have considered the faithful remote state prep-aration using finite classical bits and a non-maximallyentangled state. Yu et al. have proposed a protocol [7]for remote preparation of a qudit using maximally entan-gled states of qubits. In addition, some authors have alsoinvestigated the RSP protocols via using different quantum

Fig. 1. This figure represents the setup for our remote preparationprotocol using quantum statistics. Initially, Alice and Bob, Alice andCharlie share two pairs of entangled systems, L and R. Each entangledpair is composed of two n-particle GHZ states. Alice can help Bob andCharlie remote preparation by performing a set of local operations usingonly standard 50/50 beam splitters, path measurements and one-wayclassical communication with Bob and Charlie.

220 Y. Xia et al. / Optics Communications 277 (2007) 219–222

channel such as partial EPR pairs [8] and three-particleGHZ entangled state [10].

Very recently, Dai et al. have proposed a protocol [6] forprobabilistic remote preparation of the four-particle GHZclass state using two partial entangled three-particle GHZentangled states as quantum channel. In Dai et al.’s paper[6], the successful total probability and classical communi-cation cost are calculated. The results indicate that suchremote preparation of the four-particle GHZ class staterequires 1 bit classical communication cost for the maxi-mally entangled quantum channel.

In this paper, we propose a protocol for remote prepa-ration of a N-particle (fermions) GHZ state using indistin-guishable particles quantum channel. The N-particle GHZstate can be successfully prepared if the N!1 limit, andclassical communication cost is greatly reduced.

Suppose that Alice (A) wants to help Bob (B) and Char-lie (C) remotely prepare a N-particle (N = 2n) GHZ state

j/in � 1ffiffiffi2p ðj 000 � � � 0|fflfflfflfflffl{zfflfflfflfflffl}

n

iBj111 � � � 1iC

þ j111 � � � 1iBj000 � � � 0iCÞ: ð1Þ

We look at the entanglement in the internal degrees offreedom of the particles, such as the spin in the case ofelectrons (fermions) or the polarization in the case of pho-tons (bosons), which have isomorphic Hilbert spaces. Oneof the differences between bosons and fermions is the factthat bosons are described by second-order differentialequations (e.g., Klein–Gordon equation), whereas ferm-ions are described by first-order differential equations(e.g., Dirac equation). Consequently, the properties ofthe associated conserved particle current j0(x) are substan-tially different for bosons and fermions. The most notabledifference is the fact that j0(x) cannot be negative forfermions, whereas it can be negative (as well as positive)for bosons.

We suppose that Alice knows |/in completely, but Boband Charlie do not know it at all. We also suppose thatthe quantum channel shared by Alice, Bob and Charlie iscomposed of two non-maximally entangled N-particleGHZ states as follows

j/in1 ¼ aj 000 � � � 0|fflfflfflfflffl{zfflfflfflfflffl}n

iAj111 � � � 1iB þ bj111 � � � 1iAj000 � � � 0iB;

ð2Þj/in2 ¼ cj 000 � � � 0|fflfflfflfflffl{zfflfflfflfflffl}

n

iAj111 � � � 1iC þ dj111 � � � 1iAj000 � � � 0iC;

ð3Þ

where jaj2 þ jbj2 ¼ 1 and jcj2 þ jdj2 ¼ 1. For the sake ofcompactness, we will rewrite Eqs. (2) and (3) as

j/in1 ¼ ajA0injB1in þ bjA1injB0in; ð4Þj/in2 ¼ cjA0injC1in þ djA1injC0in: ð5Þ

Particles A belong to Alice while particles B belong to Boband particles C belong to Charlie. In order to help Bob and

Charlie to remotely prepare the N-particle GHZ state ofEq. (1), Alice label j/in1 L and j/in2 R (see Fig. 1). Then,the total initial state is

qn ¼ qnL � qn

R ¼

a2c2 a2d�c b�ac2 b�ad�c

a2c�d a2d2 b�ac�d b�ad2

a�bc2 a�bd�c b2c2 b2d�c

a�bc�d a�bd2 b2c�d b2d2

26664

37775:ð6Þ

To write the total initial state in the form of Eq. (6), the fol-lowing bases are chosen as

ja1i � jA0inLjA0inRjB1inLjC1inR;ja2i � jA0inLjA1inRjB1inLjC0inR;ja3i � jA1inLjA0inRjB0inLjC1inR;ja4i � jA1inLjA1inRjB0inLjC0inR:

ð7Þ

Our protocol consists of bringing each of Alice’s parti-cles into a 50/50 beam splitter (particles i from the leftand the right systems go into beam splitter i, fori = 1, 2, . . ., n) and then making a specific measurementon the output particles using detectors Li and Ri (seeFig. 1). With this total initial state, next step we let the firstpair of particle (labelled by 1 in the circle) with Alice go tothe beam splitter 1. After all Alice’s particles from left hav-ing the specific unitary transformation Ui

p that flips theirspin or polarization.

Uip : jA "ili

! jA #ili; jA #ili

! jA "ili

Y. Xia et al. / Optics Communications 277 (2007) 219–222 221

The total state still takes the form of Eq. (6), we rewrite Eq.(6) in juin1 as follow:

juin1 ¼ ðU 1p � U 2

p � U 3p � � � �U n

pÞjuin0

¼ acjA1inLjA0inRjB1inLjC1inR þ adjA1inLjA1inRjB1inLjC0inRþ bcjA0inLjA0inRjB0inLjC1inR þ bdjA0inLjA1inRjB0inLjC0inR;

ð8Þ

where juin0 ¼ j/in1 � j/i

n2.

After above operations, Alice lets the first pair passthrough the beam splitter. The state of particles after thefirst pair having passed through the 50/50 beam splittercan be written in the same form as form in Eq. (6) in theHilbert space spanned by

ja1pi � jA1in�1L jA0in�1

R jAoutiL1R1jB1inLjC1inR;

ja2pi � jA1inLjA1inRjB1inLjC0inR;ja3pi � jA0inLjA0inRjB0inLjC1inR;ja4pi � jA0in�1

L jA1in�1R jAoutiL1R1

jB0inLjC0inR;

ð9Þ

where

jAoutiL1R1� 1ffiffiffi

2p ðjA0iL1

jA1iR1þ jA1iL1

jA0iR1Þ: ð10Þ

We first assume that these detectors do not absorb the par-ticles and do not disturb their internal degrees of freedom.Alice performs a path measurement on the first pair usingdetectors L1 and Rl (see Fig. 1). Discarding those particleswhich bunch, we arrive at

qn ¼ N 21

12a2c2 1ffiffi

2p a2d�c 1ffiffi

2p b�ac2 1

2b�ad�c

1ffiffi2p a2c�d a2d2 b�ac�d 1ffiffi

2p b�ad2

1ffiffi2p a�bc2 a�bd�c b2c2 1ffiffi

2p b2d�c

12a�bc�d 1ffiffi

2p a�bd2 1ffiffi

2p b2c�d 1

2b2d2

2666664

3777775 ð11Þ

where

N 1 ¼jacj2

2þ jbdj2

2þ jadj2 þ jbcj2

" #�ð1=2Þ

; ð12Þ

and the corresponding basis is the same as that in Eq. (9).The probability p1 of having state Eq. (11) is

p1 ¼jacj2

2þ jbdj2

2þ jadj2 þ jbcj2; ð13Þ

it arrives at th maximum 3/4 with a ¼ b ¼ 1=ffiffiffi2p

. Eq. (9)tells us that the bunching is only related to |a1pi and|a4pi, and these two terms have probability 1/2 of antibun-ching. Thus elements of the density matrix Eq. (11) differfrom the initial state upon the normalization factor 1=

ffiffiffi2p

or 1/2, depending on the total probability of antibunching.After obtaining the antibunching result, we can now let

the second pair of Alice’s particles (labelled by 2 in the cir-cle) pass through another 50/50 beam splitter and performthe same measurement, discarding again the bunchingresults. After repeating the same procedure n times, weobtain the final state

qn¼N 2n�

ð12Þna2c2 ð 1ffiffi

2p Þna2d�c ð 1ffiffi

2p Þnb�ac2 ð1

2Þnb�ad�c

ð 1ffiffi2p Þna2c�d a2d2 b�ac�d ð 1ffiffi

2p Þnb�ad2

ð 1ffiffi2p Þna�bc2 a�bd�c b2c2 ð 1ffiffi

2p Þnb2d�c

ð12Þna�bc�d ð 1ffiffi

2p Þna�bd2 ð 1ffiffi

2p Þnb2c�d ð1

2Þnb2d2

2666664

3777775;

ð14Þwith

Nn ¼jacj2

2n þjbdj2

2n þ jadj2 þ jbcj2" #�ð1=2Þ

; ð15Þ

on the bases

ja1f i � jAoutinL1R1jB1inLjC1inR;ja2f i � jA0inLjA0inRjB0inLjC1inR;ja3f i � jA1inLjA1inRjB1inLjC0inR;ja4f i � jAoutinL1R1jB0inLjC0inR:

ð16Þ

After applying the same procedure ‘‘infinitely many times’’(to an ‘‘infinitely large state’’), we obtain the state

ju1i � limn!1

1ffiffiffi2p ðjA0inLjA0inRjB0inLjC1inR

þ jA1inLjA1inRjB1inLjC0inRÞ: ð17Þ

The total probability is then p ¼ ðjacj2 þ jbdj2Þ=2. Nownote that even if Alice had, any kind of path detectors,our protocol would still work assuming Alice perform apath measurement on n particles. We would then have ab-sorbed all these n particles, and their state would be re-placed by the vacuum state |0i. In the n!1 limit thiswill factorize out to give the final maximally entangledstate:

qn!1BC ¼

12

12

12

12

" #; ð18Þ

on the bases of

jb1i �jB0injC1in;jb2i �jB1injC0in;

where ad = b c.In the protocols we have presented here, local opera-

tions are performed on one side only (Alice). Also, notethat Alice could perform the operations on the n particles,each one on their side, using either one beam splittersequentially or n � 1 beam splitters in parallel. The lattercase has a lower time complexity than the former, Butrequires more resources (higher space complexity). In thecase of bosons, the procedure is almost the same as forfermions, but this time we discard the antibunching resultsand keep the bunching ones.

In summary, we present a protocol for remote prepara-tion of the N-particle GHZ entangled state between twoparty (Bob and Charlie) using only the effects of quantum

222 Y. Xia et al. / Optics Communications 277 (2007) 219–222

statistics. The N-particle GHZ entangled state can be per-fectly prepared if in the n!1 limit, and Alice sends hisparticles pass though the 50/50 beam splitter and performpath measurement on these particles. From the point ofview of communication theory, our protocol is optimaland uses the minimum classical bits. Future work will com-prise investigating more extensive quantum informationprocessing procedures.

Acknowledgement

The project supported by National Natural ScienceFoundation of China under Grant No. 10575017.

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