Memoirs of the Faculty of Science Kochi University (Information Science) Vol. 26 (2005), No. 6 ( ) ( 2 k-subsystem  total ) semi-separability (TSS) k–subsystem partial semi-separability ( PSS ) 1994 1 1.1 |0} |1) |0〉  |1〉 2ヒ1 2 k–subsystem total semi-separability (TSS) k–subsystem partial semi-separability ( PSS ) 1 1 [3] n
|φί⟩ (1) 0 1 (qubit) 2 1 2 ( |形〉( ) (Paul Dirac) ( | 〈ψ| = (|ψ)† Dirac
|ψ) = α|0) + β|1) α, β |α|² + |β|² = 1 |1) | β |²
0 2.1.1 1 ) 2n 2.1 2n 1 2.1 2.2 3 (wave function) ψ (x,y,z) t ψ 2 |ψ(r,t)|² = ψ*(r,t)ψ(r,t) (2) ψ(r,t)
ψ 〈ゆ〉＝ 1 (3) 2.1 2 ( 1 r 1 t r = 2.2.1 ib
ñ
U(Δt)
Ĥ
(20) E
t2
ify
ib to → t = to + ∆ t → t' = t + ∆ t
U(t',to) = U(t',t)U(t,to) U(t',to) f (t) ñ¢(r) = E¢(r) (22) ) E Ĥ φ(r) (4) - i E L ψ(r,t) = φ(r)e (23) ( ) 2 t 1 2.2 (steady state) t r t to (quantum correlation)
B
A B
|1 A 1 B ⟩
|0 A 0 B } |0 A 0 B ⟩ (a) (b)
B |1 B ⟩ 1935 EPR (entanglement) 2.3.2 A 2.3.1 B 2.3 A B | ψ ⟩ = Z (|0 A 0 B ) + |1 A 1 B }) 2.3.4 2.4 2.3 2.3 |0 A 0 B }
|0 A 0 B } |1 A 1 B } 2.4 2.3.3 A B 2 √3(|0 A 1 B ⟩ + |1 A 0 B ⟩ + |1 A 1 B ⟩ ) 2 A B β|0 A 1 B } |0 A 1 B } α|0 A 0 B ⟩ + |0 A 0 B ⟩ 2 |1 A 1 B ⟩ A |0 A ⟩ α|0 A 0 B ⟩ + β |0 A 1 B ⟩ = |0 A ⟩ (α |0 B ⟩ + β |1 B ⟩ ) 1
2 α|0 A 0 B ⟩ + β |1 A 1 B ⟩ 2.4 2.4.1 AND, NOT, OR, NOR NOT or 2n × 2n 1 2 NOT
0 1 [1] AND U ≡ (24)
2.6 2.6)[1]
2.5 NOT |0}
NOT
2.4.2
n 2.7 NOT 2.7 A |0 A ⟩
2.9 NOT ( 1)
1 NOT 2.9 U NOT NOT NOT (Controlled Not Gate) |B> 2.10 NOT
2
|B}' lo)
CNOT Unitary Gate) (Controlled 2 1 0
|A) = |0) → |A)' = |A) = |0) ,  |B)' = |B) |A) = |1) → |A)' = |A) = |1) ,  |B)' = U|B) 2 1
0
|B>’ |A>’ (28) (Hadamard Gate) 2
NOT |A) |A)' = |A) A
2.8 CNOT 2.8 0 ( 2)
|1 A } |A>
3 
 
 NOT 2.12 3 2 NOT 2.4.3 2 1 |1) 2 NOT 2.12 3 0 1 0 0 1 0
0 0
0
1
0 0
2 2.11 2.11 0 1 1
(32) B (
4.しNOT

 

|A),  |B),  |C) |0 A 0 B 0 C \ . .  |1 A 1 B 1 C \ ( 4)
|A) 2.14) ( (n − 1) n 1 √2 3) √2 A | A > ' 3.2 3.2.1 β β 1 | ψ ⟩
(35)
(36) 2.14 n 3 3.1 β2 | ψ ⟩ λ j λ j = (mixed state) (pure state) 3.2.2 p | ψ ⟩
A, B 2
1 1 0 1 0
0 0 0 (37) 0 1 2
9 1
4.1) 4.1 2 (40) (41) P2 β 4 4.1 2 2 2 2 β |0 A 0 B ⟩ + P1 4.2 TSS PSS 2 (TSS) (PSS)[5]   2 n 2 l: li k: (PSS) (semi-separability) (45) |sub1} } {sub2} } · · } {subk} , {subk+1} k: PSS k 4.3 n k: (n > k) k: k: (TSS) (44) |sub1}۞|sub2}۞· · ·۞|subk} 4.3 k: k: 4.3 TSS 4.2 2 l: |ψ) A H A B H B H AB H A {|0 A }, |1 A }} {|0B}, |1B}} H B H AB {|0 A 0 B }, |0 A 1 B }, |1 A 0 B }, |1 A 1 B }} n 4.2 k |0 A 0 B ⟩ = |0 A ⟩ ⊗ |0 B ⟩ ,  |0 A 1 B ⟩ = |0 A ⟩ ⊗ |1 B ⟩ |1 A 0 B ⟩ = |1 A ⟩ ⊗ |0 B ⟩ ,  |1 A 1 B ⟩ = |1 A ⟩ ⊗ |1 B ⟩ (46)
| ψ } H AB |ψ} = C 0 | 0 A 0 B ⟩ +C 1 | 0 A 1 B ⟩ +C 2 | 1 A 0 B ⟩ +C 3 | 1 A 1 B ⟩ (47)
(48) |ゆ⟩ = α0|0 A ⟩ ⊗ |0 B ⟩ + α1 |1 A ⟩ ⊗ |1 B ⟩ |0 A 0 B ⟩ , |1 A 1 B ⟩ |1 A 0 B ⟩ , |0 A 1 B ⟩ |0 A 0 B ⟩ ,  |0 A 1 B ⟩ |1 A 0 B ⟩ ,  |1 A 1 B ⟩ (|0 A 0 B ),  |0 A 1 B }) |ψ) = βo|0 A ⟩ ⊗ |1 B ⟩ + β1 |1 A ⟩ ⊗ |0 B ⟩
αρ ≥ 0, αι ≥ [6]    α,    α, 0, α 2 + α 2 = 1  β 0, β 1 βο ≥ 0, β1 ≥ | ψ ⟩ 4.4 0, β2 + β2 = 1 4.4 4.5 C1|0 A 1 B ⟩ C0|0A0B} + C1|0A1B} √|Co|² + |C1|²
√|Co|² + |C1|²|0 A 1 B | C2|1A0B}    C3|1A1B} C2|1 A 0 B ⟩ + C3|1 A 1 B ⟩ √|C2|² + |C3|² |1 A 0 B } √[C2]² + [C3]²|1 A 0 B } |1 A 1 B ⟩ Col0A0B\
TSS
TSS TSS(A,B,CD) TSS(A,B,CD) |Ψ ABCD ⟩ = |Ψ A ⟩ ⊗ |Ψ B ⟩ ⊗ |Ψ CD ⟩ |ΨCD} = |ΨC} $ |ΨD}
A |1 A } H AB 4
|ゆcd) = α|ゆc) ⑧ |ゆd) + β|ゆc)' ⑧ |ゆd)'   (51) D (52) TSS(A, BC, D) |Ψ ABCD ⟩ = |Ψ A ⟩ ⊗ |Ψ BC ⟩ ⊗ |Ψ D ⟩ (53) (52) |ゆcd} = {ゆc} $ {ゆd} |Ψ ABCD〉＝ |Ψ A〉 S |Ψ B〉 S |Ψ C〉 S |Ψ D〉 |Ψ ABC } |Ψ ABC ⟩ =  |Ψ A ⟩ ∅ |Ψ B ⟩ ∅ |Ψ C ⟩ TSS(A,BC) : |Ψ ABC ⟩ = |Ψ A ⟩ ⊗ |Ψ BC ⟩ TSS(B, AC) : |Ψ ABC ⟩ = |Ψ B ⟩ S |Ψ AC ⟩ TSS(C, AB) : |Ψ ABC ⟩ = |Ψ C ⟩ ∅ |Ψ AB ⟩ 3 
 
 TSS
3 3. TSS TSS
A, B, C, D 4 |Ψ ABCD }
PSS
(64) A     D A     D PSSB(A,C,D), PSSC(A,B,D) PSS PSSB(A,C,D), PSSC(A,B,D)
A, B, C, D
k:
|Ψ ABCD ⟩ = |Ψ A ⟩ ∅ |Ψ B ⟩ ∅ |ゆc} & |ゆd} 3 |Ψ ABC } PSS |Ψ ABC } |Ψ〉 =  |Ψ A ⟩ ⑧ |Ψ B ⟩ ⑧ |Ψ C ⟩ |Ψ A ⟩ $ |Ψ B ⟩ } B ⟩ $ |Ψ C ⟩ |ψ A PSSc(A,B), PSSa(B,C), PSSb(A,C) PSSc(A,B), PSSa(B,C), PSSb(A,C) 3 PSS 6. PSS PSS TSS 13 ( 2)
1 PSS
5. PSS PSS
ρ = π[(|0 A ){0 A |) ⊗ (|Ψ bc ){Ψ bc |)+ (|1 A ){1 A |) $ (|Ψ BC ){Ψ BC |)] 2 {|0 A \{0 A | & (|0 B 1 C \{0 B 1 C | {1 B 0 C | = +|1 B0c \{0 B1c | + |1 B0c \{1 B0c |}}) + 2 {|1 A }{1 A } $ {|0 B 1 C }{0 B 1 C }{1 B 0 C } −|1 B 0 C ⟩⟨ 0 B 1 C ⟩ + |1 B 0 C ⟩⟨ 1 B 0 C | ⟩} ) 1
| {|0 A 0 B 1 C } {0 A 0 B 1 C } {0 A 1 B 0 C } +|0 A 1 B 0 C ⟩⟨ 0 A 0 B 1 C ⟩ + 0 A 1 B 0 C ⟩⟨ 0 A 1 B 0 C ⟩ } 1 {|1 A 0 B 1 C } {1 A 0 B 1 C } {1 A 1 B 0 C } −|1 A 1 B 0 C ⟩⟨ 1 A 0 B 1 C ⟩ + 1 A 1 B 0 C ⟩⟨ 1 A 1 B 0 C |} (66)
) 0 1 0 0 0 0 0
A 2 | V BC | B, C =|√ BC ) ž|¥BC}, ž|¥BC}
0 0 0
C PSSc(A, B) PSS
A, B 1 1 C
PSS PSS PSS PSS PSS PSS 5) PSS PSS 8. PSS PSS 16 6 9. 1 7. PSS A, B A B B C C A A A, C A
B, C A, B, C |0 B 1 C ⟩ + |1 B 0 C ⟩ |0 A 0 B 1 C ⟩ + |1 A 1 B 0 C ⟩ 10. 3 TSS 3 1 2 ; TSS A, B, C 3 TSS(AB,C)  TSS(A,BC) |Ψ ABC } |ψ ABC } TSS(AB,C) (73) |Ψ ABC ⟩ = |Ψ AB ⟩ ⊗ |Ψ C ⟩ |Ψ AB ⟩ = |Ψ A ⟩ ∅ |Ψ B ⟩ | ψ ab ⟩ {|ψв}, |ψв}'} |Ψ ABC ⟩ = α(|Ψ A ⟩ ⊗ |Ψ B ⟩ ) ⊗ |Ψ C ⟩ + β(|ψ a)′ ✆ |ψ b)′) ✆ |ψ c) = α|ψ a ⟩ ⊗ (|ψ b ⟩ ⊗ |ψ c ⟩ ) + β|ψ a ) ′ $ (|ψ b ) ′ $ (ψ c ) ) (75) |ψ A TSS(A, BC) |Ψ ABC ⟩ = |Ψ A ⟩ ∅ |Ψ BC ⟩ (76) |ψ a ⟩ |Ψ AB ⟩ = |Ψ A ⟩ ∅ |Ψ B ⟩ |Ψ ABC ⟩ = |Ψ A ⟩ ⊗ |Ψ B ⟩ ⊗ |Ψ C ⟩ TSS 2 3 3 2 TSS 3 2 3 TSS TSS 11. 4 1 3 2 TSS 4 A, B, C, D B, C, D |ψ ABCD TSS(BC,D), TSS(BD,C) B, C, D A |Ψ ABCD ⟩ ＝ |Ψ A ⟩ ∅ |Ψ BCD ⟩ |Ψ AB ⟩ = α |Ψ A ⟩ ⑧ |Ψ B ⟩ + β |Ψ A ⟩ ′ ⑧ |Ψ B ⟩ ′ （74) |Ψ ABCD ⟩ |ゆBCD} 10) 3 α ≥ 0, β ≥ 0, α² + β² = （ TSS A 2 1 {|ψ a }, |ψ a }'} 1 |ψ BCD }
gcd( x r / 2 _ n |ΨBCD\ = |ΨB\ S |ΨC\ S |ΨD\ (77)
|Ψ ABCD ⟩ = |Ψ A ⟩ ∅ |Ψ B ⟩ ∅ |Ψ C ⟩ ∅ |Ψ D ⟩ (78) Shor
|reg1}|reg2} (1) (7)
(2) 1 1
1 2
5 5.1 Shor n Shor 21 n f n (a) = x α  mod n n
.r
r
(79) 2
(3) 2 T log(q)
1: [3]
|k} 1 x α mod n = k a
c'/q λ/r
1
2 (bell states)
5.2 1 gcd( x r /2 — 1, n ) gcd( x r /2 + 1, n ) 1 r n λ i T B ( A
|φ⟩ B A � 5.1 2
(88) |ξab}
)
1 [3,4]
4
A A B 1 |Φ†⟩ ↔ α|0⟩ + β|1⟩,   |Φ¯⟩ ↔ α|0⟩ − β|1⟩ |Ψ +) ↔ α|1) + β|0),   |Ψ −) ↔ α|1) − β|0)   (92) 2 4
|α|² + | β|² = 1 |ψ3} = α|0 A 0 B } + β } 1 A 1 B } α ≠ β
[1] [2] [3] C.P. S.H. [4] "Multi- [5] Gilles Brassard and Tal Mor, particle Entanglement via Two-Particle Entan- glement", C.P. Williams (Ed.):  QCQC'98, LNCS (Lecture Notes in Computer Science) 1509, pp.1- 9 (1999) [6] Ayman F. Abouraddy, Bahaa E. A. Saleh, "Degree of entanglement for two qubits", Physi- cal review A, volume 64, 050101(R)