QUANTUM SECRET SHARING VIA FOUR PARTICLE ASYMMETRIC ENTANGLED STATE

Transcription

1 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN QUANTUM SECRET SHARING VIA FOUR PARTICLE ASYMMETRIC ENTANGLED STATE Pn-ru Zho, Yun-jing Zhou, Jin-wei Zho, Ling-shn Xu, Yun-hong To Deprtment of Mthemtis, College of Siene, Ynbin University, Ynji, Jilin ABSTRACT A generl sheme for quntum seret shring of n rbitrry single-qubit stte with four-qubit symmetri entngled stte is proposed. In this sheme, the sender performs Bell mesurement on his prtile, the two ollbortors nd the reeiver perform joint unitry opertion on the rest prtiles. Then the two ollbortors perform mesurement on their prtile using mutully orthogonl bses. Thus the reeiver n suessfully reonstrut the single prtile with ertin probbility. Keywords: Quntum seret shring; symmetri entngled stte; joint unitry opertion; Bell mesurement. Introdution Quntum entnglement, s physil resoure, plys n importnt role in ll spets of quntum informtion, for instne quntum teleporttion[ 6], quntum dense oding[7 8], quntum seure shring[9 ], et. Up to now, there re mny hievements bout quntum entnglement in theory nd experiment, whih n be seen in the literture[ ]. In 999, Hillery et l.[9] proposed the first quntum seure shring protool using three or four prtile entngled GHZ lss s quntum hnnel. In, Zhng Qunyong et l.[] proposed quntum seret shring sheme of rbitrry single-qubit stte with three prtile symmetri entngled stte. In 6, Zhng Jinzhong et l.[] proposed sheme of quntum seret shring bsed on four-qubit symmetri entngled stte. In 7, we proposed sheme for ontrolled quntum teleporttion of n unknown single prtile[], using speil four prtile symmetri entngled stte s quntum hnnel. In this pper, we intend to disuss quntum seret shring of n rbitrry single-qubit stte. This work is supported by Nturl Siene Foundtion of Chin under number 665. Progressive Ademi Publishing, UK Pge 7

2 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN Corresponding uthor: Yun-hong To, E-mil: toyunhong@6.om The quntum hnnel tht linking the prties is four prtile symmetri entngled stte where the oeffiient is rbitrry. There inludes four prties, one sender, three gents, where ny gent n ooperte with other gents to reonstrut the unknown single-qubit stte. We will use method whih is more simple nd onvenient thn tht in [] to present its whole proess nd we will lulte the totl suess probbility of the sheme. Quntum seret shring Sheme Suppose there re four legitimte prtiipnts: Alie, Bob, Chrlie nd Dvid. Among them, Alie is the sender of the messge, Bob, Chrlie nd Dvid re the gents of the messge, ny gent n ooperte with other gents to reonstrut Alie's messge.now, Alie wnts to send n unknown single-prtile stte to Bob, Chrlie or Dvid, Where, re omplex numbers, nd. Assume Alie, Bob, Chrlie nd Dvid shred the following four prtile symmetri stte b Where, b, re rel numbers, b Alie, prtile,, belongs to Bob, Chrlie nd Dvid, respetively. The totl stte of the system n be denoted s follows In order to teleport stte nd b. Prtile belongs to b, firstly, Alie performs Bell mesurement on prtile nd. Then the stte will be projeted into ny one of the following sttes b b b Progressive Ademi Publishing, UK Pge 8

3 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN b Where,. Depending on the differene of the gent who reonstruts the Alie messge, we nlyze the sheme in the following three ses. Cse. Alie ppoints Dvid to reonstrut quntum stte informtion. Bsed on the bove disussion, if Bob nd Chrlie gree to help Dvid, then they three perform joint unitry trnsformtion on prtile,,. If the stte of the prtile,, is or,the joint unitry trnsformtion n be written s D It s mtrix is æ ö - W= D - ± ± ç è Thus the stte of or beome the following stte b Then Bob nd Chrlie perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Dvid n reonstrut the stte, nd the probbility of suess Progressive Ademi Publishing, UK Pge 9

4 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN is. If the stte of the prtile,, is or,the joint unitry trnsformtion n be written s D It s mtrix is æ ö W= D ± ç è Thus the stte of or beome the following stte b Then Bob nd Chrlie perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Dvid n reonstrut the stte is., nd the probbility of suess So the totl probbility of suess is. Cse. Alie ppoints Bob to reonstrut quntum stte informtion. If Chrlie nd Dvid gree to help Bob, then they three perform joint unitry trnsformtion Progressive Ademi Publishing, UK Pge

5 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN on prtile,,. If the stte of the prtile, is or,the joint unitry trnsformtion n be written s B It s mtrix is æ ö ± W= B - ç - ± è Thus the stte of or beome the following stte b Then Chrlie nd Dvid perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Bob n reonstrut the stte is., nd the probbility of suess If the stte of the prtile,, is or,the joint unitry trnsformtion n be written s B Progressive Ademi Publishing, UK Pge

6 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN It s mtrix is æ ö ç - W= B ç - - ç ± è Thus the stte of or beome the following stte b Then Chrlie nd Dvid perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Bob n reonstrut the stte is., nd the probbility of suess So the totl probbility of suess is. Cse. Alie ppoints Chrlie to reonstrut quntum stte informtion. If Bob nd Dvid gree to help Chrlie, then they three perform joint unitry trnsformtion on prtile,,. If the stte of the prtile,, is or,the joint unitry trnsformtion n be written s + C It s mtrix is Progressive Ademi Publishing, UK Pge

7 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN æ ö ± W= C - ç - ± è Thus the stte of or beome the following stte b Then Bob nd Chrlie perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Chrlie n reonstrut the stte is., nd the probbility of suess If the stte of the prtile,, is or,the joint unitry trnsformtion n be written s C + It s mtrix is Progressive Ademi Publishing, UK Pge

8 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN æ ö ç ç ± è W= C ç Thus the stte of or beome the following stte b Then Bob nd Chrlie perform the mesurement on prtile, under the bses,,,. From the bove eqution, we n see tht only the mesurement result of the prtile, is, Chrlie n reonstrut the stte is., nd the probbility of suess So the totl probbility of suess is. Conlusion This pper meliorted the four-prtile symmetri entngled stte hnnel in [], its oeffiients n be ny rel, stisfying +b + =. And we nlyze the sheme in three situtions, one of Dvid, Bob nd Chrlie is the reeiver, nd the other is the ontroller. It is found tht the suessful probbility in eh se is. Tht is, lthough the reipients re different, the suessful probbility is equl. The sheme n be generlized to quntum seret shring of multi-prtile. REFERENCES [] C. H. Bennett, G. Brssrd, C. Crepeu, et l. Teleporting n Unknown Quntum Stte vidul Clssil nd Einstein-Podolsky-Rosen Chnnels[J]. Phys. Rev. Lett, 99, 7(): [] X. Q. Su, G. C. Guo. Quntum Teleporttion[J]. Progress in Physis,, (): [] H. Tng, J. X. Fng, X. M. Qin. Entnglement Teleporttion in Two Quntum Chnnels[J]. Progressive Ademi Publishing, UK Pge

9 Europen Journl of Mthemtis nd Computer Siene Vol. No., 7 ISSN Journl of Jingsu Tehers University of Tehnology, 6, (6): 5-. [] L. B. Chen, R. B. Jin. Reliztion of Two-destintion Teleporttion by Using Two Prtilly Entngled Sttes[J]. Journl of Foshn University(Nturl Siene Edition), 8, 6(): -. [5] H. Liu, F. L. Yn. Teleporttion of n Unknown Single-prtile Stte[J]. Journl of Hebei Norml University(Nturnl Siene Edition), 7, (): -. [6] X. M. Xiu, L. Dong, Y. J. Go. Probbilisti Teleporttion of n Unknown One-Prtile Stte by Generl W Stte[J]. Commun. Theor. Phys, 7, 7(): [7] K. Mttle, H. Weinfurter, P. G. Kwit, A. Zeilinger. Dense oding in experimentl quntum ommunition[j]. Physil Review Letters, 996, 76(5): [8] Z. Y. Xue, Y. M. Yi, Z. L. Co. Quntum dense oding dense oding sheme vi vity dey[j]. Journl of Modern Optis, 5, 5(8): [9] M. Hillery, V. Buzek, A. Berthium. Quntum seret shring[j]. Physil Review A, 999, 59(): [] Q. Y. Zhng. Quntum Seret Shring of Single-qubit Stte vi Triprtite Entngled Sttes[J]. Chinese Journl of Quntum Eletronis,, 9(): -6. [] J. Z. Zhng, W. H. Zhng. Two Quntum Seret Shring Sheme Bsed on Four-qubit Entngled Sttes[J]. Applition Reserh of Computers, 6, (): 5-8. [] P. R. Zho, H. X, Y. H. To. Controlled Quntum Teleporttion vi Four Prtile Asymmetri Entngled Stte[J]. IOSR Journl of Applied Physis, 7, 9(): -7. Progressive Ademi Publishing, UK Pge 5