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Quantum Teleportation and Quantum Secure Direct Communication with n-dimensional Quantum States

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PACS numbers: 03.67.Dd, 03.67.Hk

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E-mail: [email protected]

[11], þjH\H €ªœ „5Åx`¦ s6 x #Œ îߖ„Ç f”]X :Ÿ G

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§Ü¼–Ð ]ߖ÷&%3 [13]. €ªœ „5ÅÉr \O)€e”H ôÇ Š©œ_



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t

FKt @Òìr_ €ªœ „5Åx x9 €ªœ f”]X :Ÿ\ @/ ô



Ç ƒ¨H Û¼—2; 12 `…ØÔp:r r¼%7› ¢¸H Û¼—2; 1 €ªœ r Û

¼%7›Ü¼–Ð ëߖ[þt# Ç©c± (qubit) ©œI\¦ @/©œÜ¼–Ð ¦ e

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. ÕªQ z´]j&h¼–Ð “¦ _ €ªœ r¼%7›s 0px 



. "f ‘:r 7H\"H n- "é¶ €ªœ ©œI\¦ s6 x #Œ

€ ª

œ „5Åx x9 €ªœ f”]X :Ÿ ~½ÓZO`¦ e”__ "é¶Ü¼–Ð SX‰



©

œ #Œ {9øÍo r(.

€



$ II©œ\"f Y> t lñ\¦ {9øÍ$í`¦ {9t ·ú§“¦ n-

"é¶Ü¼–Ð SX‰©œ #Œ F&ñ_þ¡. ¢¸ôÇ III©œ\"H n- 

"

é

¶Ü¼–Ð SX‰©œôÇ €ªœ „5Åx ~½ÓZO`¦ ]jrþ¡. Õªo¦ ¢- y

 \O2³ n- "é¶ €ªœ ©œI\¦ s6 x #Œ îߖ„Ç €ªœ f”]X :

Ÿ

 ~½ÓZO`¦ IV©œ\"f ]jrþ¡.

II. ƒ»M0nÉ

n- "é¶ Hilbert /Bߖ\"f_ &ñ½©)a ©œI\¦ 6£§õ °ú  s

 ³ð‰&³|¨c ú e”.

|ψi =

n−1X

i=0

ci|ii, (1)

-74-

(2)

s

 M: >º ciH 6£§`¦ ëߖ7á¤ôÇ.

n−1X

i=0

|ci|2= 1. (2)



6£§Ü¼–Ð l$ |ji\ Œ•6 x H 8Š H(n)\¦ 6£§õ °ú  s

 &ñ .

H(n)|ji = 1

n

n−1X

k=0

ωjk|ki. (3)

d

”

 (3)\"f, ω = enis 9, s M: i =

−1s. #Œl"f n = 2 {9 M:, H(n)H ôÇ >h_ Ç©c± \ Œ•6 x H Hadamard





8Š s. Õª¼–Ð, H(n)`¦ n- "é¶> Hadamard 8Š s

¦ &ñ_Ç. s 8Š“Ér Ä»m'o (unitary)s 9,  6

£

§_ ›¸| `¦ ëߖ7á¤ôÇ.

H(n)n = I. (4)



6£§Ü¼–Ð CN OT ƒߖ\¦ n "é¶Ü¼–Ð SX‰©œr CN OT(n)`¦ 6£§õ °ú s &ñ_ .

CN OT(n)|ki|li = |ki|l − ki. (5)

#

Œl"f A'‘ n“Ér n- "é¶ €ªœ ©œI\¦ _pÇ. 



"f CN OT(n)H unitarys 9, CN OT(n)n = I`¦ ëߖ7á¤ôÇ



.



6£§Ü¼–Ð n- "é¶\"f_ Bell-states |ψkli`¦ 6£§õ °ú  s

 &ñ_Ç.

kli = 1

2

n−1X

j=0

ωjl|k + j, ji. (6)

0

A_ d”Ér 6£§`¦ ëߖ7ᤠH n2- "é¶_ J$™"f YL (tensor product) €ªœ r¼%7›_ l$ (basis)\¦ ½¨$í ¦ e”.

klk0l0i = δkk0δll0. (7)

\

V\¦ [þ,

00i = 1

n(|0i|0i + |1i|1i + · · · + |n − 1i|n − 1i). (8)



t}Œ•ܼ–Ð, n2>h_ Ä»m'o 8Š U(kl)`¦ 6£§õ °ú  s

 &ñ_ .

U(kl)|mi = ωlm|m + ki. (9) 0

A_ d” (9)–Ð ÂÒ', Bell-states |ψklÉr ©œI |ψ00i_ 'Í





P: "鶙è\¦ U(kl) 8Š #Œ %3# כ e”`¦ ·ú˜ ú e”.

U(kl)(1)00i = |ψkli. (10)

#

Œl"f 0A'‘ (1)“Ér 8Š U(kl) 'Í P: "鶙è\ %ò†¾Ó`¦ Å

Ò%3H כ `¦ _pÇ.

Fig. 1. The structure of quantum teleportation.

III. n- ̦R S­Žyð8ý Wës ¹Å’ÒÞ Uês0nÉ

0

A II©œ\"f &ñ_Ç lñ\¦ s6 x #Œ, n- "é¶>\"f_

€ ª

œ „5Åx ~½ÓZO`¦ ]jrÇ.

Aliceü< Bobs 6£§õ °ú “Ér \O)€e”H ©œI\¦ ìrCþ¡



¦ &ñ .

00iAB= 1

n X

j

|jiA|jiB. (11)

#

Œl"f A'‘ Aü< BH yŒ•yŒ• Aliceü< Bobs ™èÄ» 

“

¦ e”6£§`¦ _pÇ. Alice n- "é¶> €ªœ ©œI |ψiA¯\¦ Bob\> „5Åx ¦ z·¦ &ñ .

|ψiA¯=

n−1X

j=0

cj|jiA¯. (12)



A'‘ ¯H Alice\ _K ]j#¦ e”6£§`¦ _pÇ.

Fig. 1\"< °ú s, „^‰ r¼%7›“Ér 3>h_ $íìr_ J$™"f YL Ü

¼–Ð sÒ# n- "é¶ ©œIs. stage 1\"f, „^‰ ©œI





H 6£§õ °ú .

stage 1i = |ψiA¯00iAB

= 1

n X

j,k

cj|jiA¯|kiA|kiB. (13)



6£§Ü¼–Ð AliceH 'Í P:, ¿ºP: $íìr\ CN OT(n)ƒߖ

`



¦ 2Ç. Õª stage 2\"f, „^‰ ©œIH 6£§õ °ú  s

 )a.

stage 2i = 1

n X

j,k

cj|jiA¯|k − jiA|kiB. (14)



6£§Ü¼–Ð AliceH 'Í P: $íìr\ H(n)`¦ &h6 x #Œ, 6£§ _

 ©œI\¦ %3H.

stage 3i = H(n)( ¯A)stage 2i

= 1 n

X

j,k,l

cjωjl|liA¯|k − jiA|kiB. (15)

0

A d”Ér 6£§õ °ú s ³ð‰&³|¨c ú e”.

stage 3i = 1 n

X

j,l,p

cjωjl|liA¯|piA|p + jiB. (16)

(3)

Fig. 2. The structure of quantum secure direct commu- nication.



t}Œ•ܼ–Ð AliceH \> 5ÅÇ ¿º >h_ ©œI |ψiA¯,

|ψiA\¦ 8£¤&ñôÇ. Õª stage 4_ ©œIH 6£§õ °ú s Ô



æõ (collapse))a.

stage 4i =X

j

cjωjl|liA¯|piA|p + jiB. (17)

Õ

ª Ð Bobs t¦ e”H ©œIH 6£§õ °ú .

stage 4iB =X

j

cjωjl|p + jiB. (18)

s

 M: AliceH Bob\> _ 8£¤&ñ  (l, p)\¦ ·ú˜r



. t}Œ•ܼ–Ð, Alice–Ð ÂÒ', 8£¤&ñ \¦ [þÉr Bob “Ér



_ Ç©c± \ 8Š U(pl)−1`¦ 2[ ¦ 6£§õ °ú “Ér \¦

% 3

> )a.

stage 4iB =X

j

cj|jiB. (19)

0

A d”¼–ÐÂÒ', Bobs þj7áx&h¼–Ð %3Ér ©œIH Alice_

"

é

¶A Bjrt |ψiA¯ü< °ú 6£§`¦ ·ú˜ ú e”.

IV. n- ̦R S­Žyð8ý Ž˜m¹Å”X¢ Wës ÇkınÉ —ÞÞÃÅ U

ê s0nÉ

ë ß

–€• Alice ¿º &ñú [kl]–Ð ÂÒ ñ)a Bjrt\¦ Bob\

>

 ˜Ð?¦ z·¦ &ñ . ¢¸ôÇ Fig. 2ü< °ú s, Aliceü<

Bob“Ér ¿º >h_ n- "é¶_ €ªœ ©œI|ψ00iAB⊗ |ψ00iA ¯¯B\¦ /

B

» ¦ e”¦ &ñ . yŒ•yŒ•_ €ªœ ©œI |ψ00H II



©

œ\"f &ñ_ ÷&%31pws \O)€e”.

stage 1i = |ψ00iAB00iA ¯¯B= 1 n

X

j,p

|jiA|jiB|piA¯|piB¯.

(20) 0

A\ ]jrÇ 4>h_ ©œI ׿\"f, 'Í P:, [j P: ©œI_



A'‘ Aü< ¯H Alice\ _K ]j#¦ e”6£§`¦ _pÇ



. 1lx{9 > ¿º P:, W1 P: ©œI_ A'‘ Bü< ¯H

Bob\ _K ]j#¦ e”6£§`¦ _pÇ. AliceH Bob\

>

 [kl]sH Bjrt\¦ ˜Ð?/U´ "é¶ôÇ. s\  AliceH



s ™èÄ» ¦ e”H ©œI |jiA\ U(kl)(A)`¦ ƒߖôÇ. 



"f „^‰ ©œIH 6£§õ °ú s ³ð‰&³)a.

stage 2i = U(kl)(A)stage 1i

= 1 n

X

j,p

ωlj|j + kiA|jiB|piA¯|piB¯. (21)

#

Œl"f ¿º P< [j P: $íìr_ 0Au\¦ 7ÅҀ,  6

£

§õ °ú s ³ð‰&³)a.

stage 2i = 1 n

X

j,p

ωlj|j + kiA|piA¯|jiB|piB¯. (22)



6£§Ü¼–Ð AliceH Bell-state |ψrqÐ _ ©œI |j + kiA|piA¯\¦ 8£¤&ñôÇ. „^‰ r¼%7›`¦ DÐîr l$_ ½+Ë Ü

¼–Ð ³ð‰&³÷&l 0AK"H, l$ 8Š ½©gË:`¦ ¹1Ô ôÇ



. 6£§“Ér l$ 8Š ½©gË:`¦ ˜Ð#Œïr.

|mi|pi =X

r,q

r,q|(|mi|pi)|ψr,qi. (23)

s

 M: >º[þÉr 6£§õ °ú s ³ð‰&³|¨c ú e”.

r,q|(|mi|pi) = 1

nω(r−m)qδp+r,m, (24)

Õ

ª¼–Ð, 6£§õ °ú “Ér \¦ %3> )a.

|mi|pi = 1

n X

q

ω−pqm−p,qi. (25)

s

 8Š ½©g˼–Ð ÂÒ', 6£§`¦ %3`¦ ú e”.

stage 2i = 1 n n

X

j,p,q

ωlj−pqj+k−p,qiA ¯A|jiB|piB¯. (26) ë

ß

–€• Alice_ 8£¤&ñ  (α, β) ÷&%3 &ñ , 8

£

¤&ñ sê\ „^‰ r¼%7›“Ér 6£§õ °ú s Ôæõ )a.

stage 3i = 1

n X

p

ω{l(α+p−k)−pβ}α,βiA ¯A|α+p−kiB|piB¯. (27) d

”

 (24)–Ð ÂÒ', d” (27)`¦ Bell-state\ @/ #Œ „>h ,



6£§õ °ú s ³ð‰&³)a.

stage 3i = ωl(α−k)α,βiA ¯Aα−k,l−βiB ¯B. (28) s

 âĺ ω = e2πin ܼ–Ð &ñ , ωH 6£§õ °ú “Ér $í|9 s

 e”:

n−1X

p=0

ωps= nδs,0. (29)

수치

Fig. 1. The structure of quantum teleportation.
Fig. 2. The structure of quantum secure direct commu- commu-nication.

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