n- Ì ¦ R4 W ë s V ê s? 0õ u § T Ó Þ X ¢ W ë s ¹ Å Ò Þ õ m Í m ¹ Å X ¢ W ë s Ç k ı n É ÞÞ Ã Å
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r 7HëH\"fH ¸ú ·ú9 ª 5Åx x9 ª f]X :x`¦ n- "é¶Ü¼Ð SX© %i. n- "é¶Ü¼Ð SX© ô
Ç áÔÐÐc+t[þtÉr 2- "é¶_ âĺü< q§ # {9ìøÍ$í`¦ {9t ·ú§¦ e6£§`¦ Ф. ¢¸ôÇ n- "é¶Ü¼Ð SX
©
H õ&ñ\"f \O)4R eH ©I[þt_ ½©gË:$í`¦ ¹1Ô`¦ ú e%3ܼ 9, s ½©gË:$íܼРK ĺoH ª
\Oj˵_ ?/ÂÒ ½¨¸\¦ SX½+É Ãº e%3.
PACS numbers: 03.67.Dd, 03.67.Hk
Keywords: ª 5Åx, îßôÇ ª f]X :x, ª \Oj˵
I. "e Â]Ø
&
³F V,o s6 x÷&¦ eH ñ rÛ¼%7_ îß$íÉr > í
ß
_ #Q9¹¡§\ Hçß`¦ ¿º¦ e. ÕªQ sכ Ér 8
É
r íß ·ú¦o7£§_ >hµ1Ï\ ]jHt L:|9 ú e6£§
`
¦ ìøÍ7£x H ´úsl¸ . sH ¸Zþtõ °ú Ér ¦¸_ &ñ
Ð :x r\ H D¥êøÍ`¦ 4R צ כ s. "f t F
K¸ ªôÇ ~½ÓZO_ DhÐîr ñ ·ú¦o7£§s >hµ1Ï÷&¦ e
. ª ñH sQôÇ Ãºכ¹\ ´úÓüt9 Ðîß\ DhÐîr
~
½
Ó¾Ó`¦ ]jr ¦ e.
ª
ñH 1984¸ Bennettü< Brassard\ _K %6£§ Ü
¼Ð è>h÷&%3 [1]. BB84¦ "î"î)a s áÔÐÐc+tÉr
ª
%i<Æ`¦ s6 x # v (key)\¦ ìrC H ~½Óds. s õ
&ñ\"f SXÒ¦&hܼР¸'õAü< °ú Ér ]j 3_ Óüt_ >rF
\
¦ SX½+É Ãº e. s ~½ÓZO sÊê ªôÇ ~½Ó_ s:r&h [2,6], z´+«>&h [7,9] ½¨ '÷&#Q M®o. Ðîß_ ª ô
Ç %ò%i ׿ BB84ü< °ú Ér áÔÐÐc+ts DhÐîr v ìrC ~½Ó d
`¦ ]jîßÙþ¡, ª \Oj˵ &³©`¦ s6 xôÇ :x GV,_ î
ß
`¦ ]jîß H áÔÐÐc+ts Beige et al\ _K ]jîß÷&
% 3
[10]. îßôÇ ª f]X 5Åxs¦ Ô¦oH s áÔ
ÐÐc+tÉr v ìrC õ&ñ \Os f]X&h GV, +þA$í`¦ :x
#
Bjrt\¦ §¨8 H ~½Óds. Beige_ ]jîß sÊê ´ú§
É
r ½¨ '÷&%3. Deng et al\ _ #, EPR`¦ s 6
xôÇ ¿º éß> ª f]X :x áÔÐÐc+ts ]jîß÷&%3ܼ 9
∗E-mail: [email protected]
[11], þjH\H ª 5Åx`¦ s6 x # îßôÇ f]X :x G
V,_ +þA$ís ]jîß÷&%3 [12].
EPR`¦ s6 xôÇ ª 5ÅxÉr Bennett et al\ _ # % 6
£
§Ü¼Ð ]jîß÷&%3 [13]. ª 5ÅxÉr \O)eH ôÇ ©_
©
Iü< ¦ :x GV,`¦ :x # ôÇ /BM\"f Ér /BMܼÐ
©
I\¦ s1lxrvH כ s. ´ú§Ér ½¨ '÷&¦ eܼ 9 [14,26], z´+«>&hܼи ´ú§Ér $í/BN`¦ sÀÒ%3 [27,28].
t
FKt @/ÂÒìr_ ª 5Åx x9 ª f]X :x\ @/ ô
Ç ½¨H Û¼2; 12 ` ØÔp:r rÛ¼%7 ¢¸H Û¼2; 1 ª r Û
¼%7ܼРëß[þt#Q Ç©c± (qubit) ©I\¦ @/©Ü¼Ð ¦ e
. ÕªQ z´]j&hܼР¦ _ ª rÛ¼%7s 0px
. "f :r 7HëH\"fH n- "é¶ ª ©I\¦ s6 x #
ª
5Åx x9 ª f]X :x ~½ÓZO`¦ e__ "é¶Ü¼Ð SX
©
# {9ìøÍo r(.
$ II©\"f Y> t l ñ\¦ {9ìøÍ$í`¦ {9t ·ú§¦ n-
"é¶Ü¼Ð SX© # F&ñ_Ùþ¡. ¢¸ôÇ III©\"fH n-
"
é
¶Ü¼Ð SX©ôÇ ª 5Åx ~½ÓZO`¦ ]jrÙþ¡. Õªo¦ ¢-a y
\O2³ n- "é¶ ª ©I\¦ s6 x # îßôÇ ª f]X :
x ~½ÓZO`¦ IV©\"f ]jrÙþ¡.
II. »M0nÉ
n- "é¶ Hilbert /BNçß\"f_ &ñ½©o)a ©I\¦ 6£§õ °ú s
³ð&³|¨c ú e.
|ψi =
n−1X
i=0
ci|ii, (1)
-74-
s
M: >ú ciH 6£§`¦ ëß7á¤ôÇ.
n−1X
i=0
|ci|2= 1. (2)
6£§Ü¼Ð l$ |ji\ 6 x H ¨8 H(n)\¦ 6£§õ °ú s
&ñ .
H(n)|ji = 1
√n
n−1X
k=0
ωjk|ki. (3)
d
(3)\"f, ω = e2πnis 9, s M: i = √
−1s. #l"f n = 2 {9 M:, H(n)H ôÇ >h_ Ç©c± \ 6 x H Hadamard
¨8 s. ÕªQÙ¼Ð, H(n)`¦ n- "é¶> Hadamard ¨8 s
¦ &ñ_ôÇ. s ¨8Ér Ä»m'o (unitary)s 9, 6
£
§_ ¸| `¦ ëß7á¤ôÇ.
H(n)n = I. (4)
6£§Ü¼Ð CN OT íß\¦ n "é¶Ü¼Ð SX©r CN OT(n)`¦ 6£§õ °ú s &ñ_ .
CN OT(n)|ki|li = |ki|l − ki. (5)
#
l"f A' nÉr n- "é¶ ª ©I\¦ _pôÇ.
"f CN OT(n)H unitarys 9, CN OT(n)n = I`¦ ëß7á¤ôÇ
.
6£§Ü¼Ð n- "é¶\"f_ Bell-states |ψkli`¦ 6£§õ °ú s
&ñ_ôÇ.
|ψkli = 1
√2
n−1X
j=0
ωjl|k + j, ji. (6)
0
A_ dÉr 6£§`¦ ëß7ᤠH n2- "é¶_ J$"f YL (tensor product) ª rÛ¼%7_ l$ (basis)\¦ ½¨$í ¦ e.
hψkl|ψk0l0i = δkk0δll0. (7)
\
V\¦ [þt,
|ψ00i = 1
√n(|0i|0i + |1i|1i + · · · + |n − 1i|n − 1i). (8)
t}ܼÐ, n2>h_ Ä»m'o ¨8 U(kl)`¦ 6£§õ °ú s
&ñ_ .
U(kl)|mi = ωlm|m + ki. (9) 0
A_ d (9)Ð ÂÒ', Bell-states |ψkliÉr ©I |ψ00i_ 'Í
P: "é¶è\¦ U(kl) ¨8 # %3#Q כ e`¦ ·ú ú e.
U(kl)(1)|ψ00i = |ψkli. (10)
#
l"f 0A' (1)Ér ¨8 U(kl) 'Í P: "é¶è\ %ò¾Ó`¦ Å
Ò%3H כ `¦ _pôÇ.
Fig. 1. The structure of quantum teleportation.
III. n- ̦R Syð8ý Wës ¹ÅÒÞ Uês0nÉ
0
A II©\"f &ñ_ôÇ l ñ\¦ s6 x #, n- "é¶>\"f_
ª
5Åx ~½ÓZO`¦ ]jrôÇ.
Aliceü< Bobs 6£§õ °ú Ér \O)eH ©I\¦ ìrCÙþ¡
¦ &ñ .
|ψ00iAB= 1
√n X
j
|jiA|jiB. (11)
#
l"f A' Aü< BH yy Aliceü< Bobs èÄ»
¦ e6£§`¦ _pôÇ. Alice n- "é¶> ª ©I |ψiA¯\¦ Bob\> 5Åx ¦ z·¦ &ñ .
|ψiA¯=
n−1X
j=0
cj|jiA¯. (12)
A' ¯AH Alice\ _K ]j#Q÷&¦ e6£§`¦ _pôÇ.
Fig. 1\"fü< °ú s, ^ rÛ¼%7Ér 3>h_ $íìr_ J$"f YL Ü
¼Ð sÀÒ#Q n- "é¶ ©Is. stage 1\"f, ^ ©I
H 6£§õ °ú .
|ψstage 1i = |ψiA¯|ψ00iAB
= 1
√n X
j,k
cj|jiA¯|kiA|kiB. (13)
6£§Ü¼Ð AliceH 'Í P:, ¿ºP: $íìr\ CN OT(n)íß
`
¦ 2[ôÇ. ÕªQ stage 2\"f, ^ ©IH 6£§õ °ú s
)a.
|ψstage 2i = 1
√n X
j,k
cj|jiA¯|k − jiA|kiB. (14)
6£§Ü¼Ð AliceH 'Í P: $íìr\ H(n)`¦ &h6 x #, 6£§ _
©I\¦ %3H.
|ψstage 3i = H(n)( ¯A)|ψstage 2i
= 1 n
X
j,k,l
cjωjl|liA¯|k − jiA|kiB. (15)
0
A dÉr 6£§õ °ú s ³ð&³|¨c ú e.
|ψstage 3i = 1 n
X
j,l,p
cjωjl|liA¯|piA|p + jiB. (16)
Fig. 2. The structure of quantum secure direct commu- nication.
t}ܼРAliceH \> 5ÅqôÇ ¿º >h_ ©I |ψiA¯,
|ψiA\¦ 8£¤&ñôÇ. ÕªQ stage 4_ ©IH 6£§õ °ú s Ô
æõ(collapse))a.
|ψstage 4i =X
j
cjωjl|liA¯|piA|p + jiB. (17)
Õ
ª õÐ Bobs t¦ eH ©IH 6£§õ °ú .
|ψstage 4iB =X
j
cjωjl|p + jiB. (18)
s
M: AliceH Bob\> _ 8£¤&ñ õ (l, p)\¦ ·ú9ïr
. t}ܼÐ, AliceÐ ÂÒ', 8£¤&ñ õ\¦ [þtÉr Bob Ér
_ Ç©c± \ ¨8 U(pl)−1`¦ 2[ ¦ 6£§õ °ú Ér õ\¦
% 3
> )a.
|ψstage 4iB =X
j
cj|jiB. (19)
0
A dܼÐÂÒ', Bobs þj7áx&hܼР%3Ér ©IH Alice_
"
é
¶A Bjrt |ψiA¯ü< °ú 6£§`¦ ·ú ú e.
IV. n- ̦R Syð8ý m¹ÅX¢ Wës ÇkınÉ ÞÞÃÅ U
ê s0nÉ
ë ß
Alice ¿º &ñú [kl]Ð ÂÒ ño)a Bjrt\¦ Bob\
>
Ð?/¦ z·¦ &ñ . ¢¸ôÇ Fig. 2ü< °ú s, Aliceü<
BobÉr ¿º >h_ n- "é¶_ ª ©I|ψ00iAB⊗ |ψ00iA ¯¯B\¦ /
B
NÄ» ¦ e¦ &ñ . yy_ ª ©I |ψ00iH II
©
\"f &ñ_ ÷&%31pws \O)e.
|ψstage 1i = |ψ00iAB|ψ00iA ¯¯B= 1 n
X
j,p
|jiA|jiB|piA¯|piB¯.
(20) 0
A\ ]jrôÇ 4>h_ ©I ׿\"f, 'Í P:, [j P: ©I_
A' Aü< ¯AH Alice\ _K ]j#Q÷&¦ e6£§`¦ _pôÇ
. 1lx{9 > ¿º P:, W1 P: ©I_ A' Bü< ¯BH
Bob\ _K ]j#Q÷&¦ e6£§`¦ _pôÇ. AliceH Bob\
>
[kl]sH Bjrt\¦ Ð?/U´ "é¶ôÇ. s\ AliceH
s èÄ» ¦ eH ©I |jiA\ U(kl)(A)`¦ íßôÇ.
"f ^ ©IH 6£§õ °ú s ³ð&³)a.
|ψstage 2i = U(kl)(A)|ψstage 1i
= 1 n
X
j,p
ωlj|j + kiA|jiB|piA¯|piB¯. (21)
#
l"f ¿º P:ü< [j P: $íìr_ 0Au\¦ 7ÅÒ, 6
£
§õ °ú s ³ð&³)a.
|ψstage 2i = 1 n
X
j,p
ωlj|j + kiA|piA¯|jiB|piB¯. (22)
6£§Ü¼Ð AliceH Bell-state |ψrqiÐ _ ©I |j + kiA|piA¯\¦ 8£¤&ñôÇ. ^ rÛ¼%7`¦ DhÐîr l$_ ½+Ë Ü
¼Ð ³ð&³÷&l 0AK"fH, l$ ¨8 ½©gË:`¦ ¹1Ô ôÇ
. 6£§Ér l$ ¨8 ½©gË:`¦ Ð#ïr.
|mi|pi =X
r,q
hψr,q|(|mi|pi)|ψr,qi. (23)
s
M: >ú[þtÉr 6£§õ °ú s ³ð&³|¨c ú e.
hψr,q|(|mi|pi) = 1
√nω(r−m)qδp+r,m, (24)
Õ
ªQÙ¼Ð, 6£§õ °ú Ér õ\¦ %3> )a.
|mi|pi = 1
√n X
q
ω−pq|ψm−p,qi. (25)
s
¨8 ½©gË:ܼРÂÒ', 6£§`¦ %3`¦ ú e.
|ψstage 2i = 1 n√ n
X
j,p,q
ωlj−pq|ψj+k−p,qiA ¯A|jiB|piB¯. (26) ë
ß
Alice_ 8£¤&ñ õ (α, β) ÷&%3 &ñ , 8
£
¤&ñ sÊê\ ^ rÛ¼%7Ér 6£§õ °ú s Ôæõ)a.
|ψstage 3i = 1
√n X
p
ω{l(α+p−k)−pβ}|ψα,βiA ¯A|α+p−kiB|piB¯. (27) d
(24)Ð ÂÒ', d (27)`¦ Bell-state\ @/ # >h ,
6£§õ °ú s ³ð&³)a.
|ψstage 3i = ωl(α−k)|ψα,βiA ¯A|ψα−k,l−βiB ¯B. (28) s
âĺ ω = e2πin ܼР&ñ , ωH 6£§õ °ú Ér $í|9 s
e:
n−1X
p=0
ωps= nδs,0. (29)