修士論文.dvi

Size: px

Start display at page:

Download "修士論文.dvi"

しらんほがり
5 years ago
Views:

2 ( ) qubit qubit A 43 B 45 2

3 [],[2],[3] [4] 994 [3] ( ) [5] ( ) [6] [7] [2] [3] [8] 2.4 [9] [0] 3 3. EPR [] 3.2 [2] 3

4 3.3 [3][4] [6] ( 2-qubit ) 2-qubit ( ) 5 2 4

5 2. n n n [6] ρ ρ i p i ψ i >< ψ i () ψ i > p i 0 i p i = (2)(3) Trρ = (2) ρ = ρ 0 (3) A A = i p i ψ i A ψ i (4) A =Tr(ρA) (5) Trace u > u ρ u 0 (6) Trρ 2 (7) ρ jj ( ρ jj ) 2 = (8) j j (measurement) 5

6 : {M m } measurement operator m ψ > m p(m) = ψ M mm m ψ (9) M m ψ ψ M mm m ψ (0) measurement operator M mm m = I () m = m p(m) = m ψ M mm m ψ (2) POVM(Positive Operator Valued Measure)[] POVM operator E m = M mm m (3) operator ψ i > m p(m i) =tr( ψ i M m M m ψ i )=tr(m m M m ψ i ψ i ) (4) p(m) = i = i p(m i)p i p i tr(m mm m ψ i ψ i ) = tr(e m ρ) (5) : ρ (2) (3) : ρ t = ī [H,ρ] (6) h 6

7 : E m POVM(3) m p(m) =tr(e m ρ) (7) (quantum operation) ρ = E(ρ) (8) = k E k ρe k (9) E k POVM POVM CP ρ E(ρ) = k e k U[ρ e 0 e 0 ]U e k (20) E k = e k U e 0 (2) e l e 0 e 0 tre(ρ) = E k = tr(e(ρ)) (22) = tr( k = tr( k E k ρe k ) (23) E k E kρ) (24) E k E k = I (25) k 2 (Hadamard) (Controlled-not) 2 ( ) 7

8 ρ UρU U : ρ E(ρ) U ρ env 2: 3: 8

9 3 Hadamard ( ) H = 2 ( ) (26) Pauli X 0 X = 0 ( ) (27) Pauli Y 0 i Y = i 0 ( ) (28) 0 Pauli Z Z = (29) 0 ( ) 0 Phase S = (30) 0 i , 0 00, 0, 0 (3) C = (32) Pauli-X Polar decomposition A V U J, K A = UJ = KU (33) J, K J = A A, K = AA Singular value decomposition 9

10 A U, V D A = UDV (34) D singular value [7] φ AB A i A B i B φ = i λ i i A i B (35) λ i i λ2 i = [8] S(ρ) = tr(ρ log ρ) (36) ρ [9] S(ρ σ) = tr(ρ log ρ) tr(ρ log σ) (37) ρ σ S(ρ σ) 0 (38) 2.2 [4] EPR [] EPR Bell 0

11 Alice Bob J =0 SG magnet 4: EPR Experiment EPR ± 2 ( 39) EPR Φ = ( 0 0 ) (39) 2 up down Bell Bell [20] Bell (39) Φ ± = 2 ( 0 ± 0 ) Ψ ± = ( 00 ± ) 2 (40) Alice Bob ψ = a 0 + b (4) Alice Alice a b Ψ + (share ) Ψ AB =(a 0 + b )( 00 + )/ 2 (42) Bell Ψ AB = (a a 0 + b 00 + b )/ 2 = 2 [ Ψ + (a 0 + b )+ Ψ (a 0 b ) + Φ + (a + b 0 )+ Φ (a b 0 )] (43)

12 Alice Projective measurement Bell 40 Alice Φ + Φ + (a + b 0 ) (44) Bob Alice Φ + NOT 0, 0 Bob a 0 + b Alice Ψ + Bell Alice Bob 2.3 ( ) [8] N O x q x q f(x) (45) mod2 x q qubit qubit 0 x = x 0 f(x) = x 0 N M 2

13 qubit ( 0 )/ 2 O x ( 0 ) ( ) f(x) x ( 0 ) (46) 2 2 O x ( ) f(x) x (47) 0 n x 0 φ = N x (48) N /2 () O (2) H n (3) x=0 x ( ) δx0 x (49) (4) H n (3) I (2) (4) H n (2 0 0 I)H n =2 φ φ I (50) G =(2 φ φ I)O (5) M α φ = N M β M x/ solution x solution x (52) x (53) N M M N α + β N (54) 3

14 β G φ θ θ/2 θ/2 φ O φ α 5: Grover cos θ/2 = (N M)/N φ =cosθ/2 α +sinθ/2 β (55) G φ =cos 3θ 2 α +sin3θ β (56) 2 5 G k 2k + 2k + φ =cos( θ) α +sin( θ) β (57) 2 2 β k β G O( N/M) : O x q = x q f(x) f(x) =0,forall0 x<2 n x 0 f(x 0 )= x 0 O( 2 n ) O(). 0 n n x [ 0 ] 2 n 2 x=0 3. [(2 φ φ I)O] R 2 n x [ 0 ] 2 n 2 x 0 [ 0 2 ] 4. x 0 x=0 4

15 R π 2 n /4 O(N) O( N) 2.4 ( ) [2] ( ) (CPU) 0,,2 B δ(q 0, 0) = (q 0, 0,R) δ(q 0, ) = (q 0,,R) δ(q 0,B)=(q, 0,L) 5

16 δ(p, a) =(q, b, d) p a q b d R, L, N q (Q, Σ, Γ,δ,q 0,B,F) Q Γ ( ) B Γ Σ Γ {B} ( ) q 0 Q F Q ( ) δ : Q Γ Q Γ {R, L, N} δ : Q Γ P(Q Γ {R, L, N}) P(S) S ( ) k k-tm 93 M x x M (p, a a 2 a m,i) p Q a a 2 a m Γ, 0 i m Γ Γ i i M x α 0 =(q 0,x,0) M α n =(q f,y,i) M 6

17 M L(M) k-tm M T (n) n ω ω M T (n) k-tm M S(n) n ω ω M S(n) DTIME(T (n)) ={L: k-tm L O(T (n)) } NTIME(T (n)) ={L: k-tm L O(T (n)) } DSPACE(S(n)) ={L: k-tm L O(S(n)) } NSPACE(S(n)) ={L: k-tm L O(S(n)) } DL=DSPACE(log n): n k-tm log n NL=NSPACE(log n): n k-tm log n P= k DTIME(nk ): k-tm n NP= k NTIME(nk ): k-tm n PSPACE= k DSPACE(nk )= k NSPACE(nk ): k-tm k-tm n EXPTIME= k DTIME(2nk ): k-tm n DL NL P NP PSPACE EXPTIME (58) P = NP 7

18 QTM PTM QTM ( ) QTM QTM ( QTM) M =(Σ,Q,δ) Σ B Q q 0 q f q 0 δ δ : Q Σ C Σ Q {L,R} (59) C n 2 n DTM α C BPP x L QTM M L BQP:QTM EQP: QTM EQTime(T (n)): n T (n) QTM BQP( ): QTM 2 3 BQTime(T (n)): n T (n) QTM 2 3 EQP BQP P EQP, BPP BQP BQP PSPACE 3 8

19 φ b j = ± 2 j = ± 2 φ a φ a2 φ b2 6: Bell 3. EPR[] [20] 6 φ ai,φ bi,i=, 2 ± 2 ± ± a,a 2,b,b 2 ± a + b a b + a b 2 + a 2 b a 2 b 2 = ±2 (60) <a b > + <a b 2 > + <a 2 b > <a 2 b 2 > ±2 (6) a b E(a,b )=P ++ (a,b )+P (a,b ) P + (a,b ) P + (a,b ) (62) P ++ (a,b ) a,b ++ E(a,b ) = a b = ψ ( a σ)( b σ) ψ (63) 9

20 σ E(a,b )+E(a,b 2 )+E(a 2,b ) E(a 2,b 2 ) (64) 2 a = 2 a 2 2 = b = b 2 = 0 (65) 0 0 (65) 2 2 ± Ψ = ( 0 0 ) (66) 2 ρ AB ρ AB = p i σ Ai σ Bi (67) i p i i p i = ρ AB A B 3.2 [2] 20

21 Gisin [22] E(σ) = 0 (68) E(σ) =0 σ σ (U A U B ) E(σ) =E(U A U B σu A U B ) (69) σ p i σ i E(σ) p i E(σ i ) (70) σ i = A i B i σa i B i /p i p i =Tr(A i B i σa i B i ) A i B i A i B i LOCC Local Operations and Classical Comunication reduced von Neumann entropy E(σ) =S(tr A ρ)=s(tr B ρ) (7) additivity additivity additivity E(σ ρ) =E(σ)+E(ρ) (72) weakly additive E(σ σ) =2E(σ) (73) 2

22 ρ σ E(ρ) E(σ) weakly additive Popescu Rohrlich(997)[23] Vidal(2000)[24] Entanglement of formation[6] Entanglement of formation E(ρ) = min p i, Φ i pi S( Φ i ) (74) Φ i min S( Φ i ) S( Φ i )= Tr Φ i Φ i log( Φ i Φ i ) (75) Entanglement of formation qubit [25] concurrece C(ρ) R(ρ) = ρ ρ ρ (76) ρ = σ y σ y ρ σ y σ y λ,,λ 4 concurrence C(ρ) =max{0,λ λ 2 λ 3 λ 4 } (77) Entanglement of formation h E(ρ) =h( + C 2 ) (78) 2 h(x) = x log 2 x ( x)log 2 ( x) (79) 2-qubit [25] Entanglement of distillation[26] V A V B () S A S B (80) 22

23 S A V A ( ) S B V B (2) ( ) S A S A (8) A B (3)2 S A (82) S B (83) A B (4)Separable S S i S i (ρ) = j (A j B j )ρ(a j B j ) (84) A j,b j V A,V B (5)Positive-partial-transpose(PPT) Γ μ i ν j ρ Γ ω k χ l = μ i χ l ρ ω k ν i (85) Si Γ : ρ S i (ρ Γ ) Γ (86) Ψ + (V )= ii (87) dimv F (ρ) =Ψ + (V )ρψ + (V ) (88) i 23

24 ( ) Entanglement of distillation ρ C-distillable entanglement D C (ρ) (V A V B ) ni C T V ij V ij i n i p ij log n 2 dimv ij D C (ρ) (89) i j p ij ( F ij )log n 2 dimv ij 0 (90) i j Relative entropy of entanglement[27] E(ρ) =mins(ρ σ) (9) σ D S(ρ σ) = trρ log ρ trρ log σ (92) D E D E R E F (93) 3.3 ( [3]) [4] ρ A A 2 Ã Tr(Ã ρ) < 0, Tr(Ãσ) 0 (94) σ A A 2 Harn-Banach[28] 24

25 W,W 2 f α R w W, w 2 W 2 f(w ) <α f(w 2 ) (95) Ã g σ g( ρ) <β g(σ) (96) Ã g A g(ρ) =Tr(ρA) (97) I ρ, σ Tr(βIρ) =Tr(βIσ) = β Ã = A βi (98) ρ A A 2 Tr(Aρ) 0 (99) A Tr(AP Q) 0 P Q H H 2 ρ (99) ρ (99) A Tr(Aρ) < 0 σ Tr(Aσ) 0 P Q Tr(Aρ) < 0 (L) L(A, A 2 ) Λ S(Λ) = i E i Λ(E i) A A 2 (00) E i A Λ L(A, A 2 ) S(Λ) P A,Q A 2 Tr(S(Λ)P Q) 0 A H 25

26 {e l } P ij e l = δ jl e i {P ij } dimh i,j= (99) Tr[((I Λ) ij P ji P ij )ρ] 0 (0) Tr[((I ΛT ) ij P ji TP ij )ρ] 0 (02) T : A A TP ij = P ji T T 2 = I Λ :A A T Λ P 0 =(/d) ij P ji P ji (d =dimh ) (02) A A 2 ρ, (I ΛP 0 ) 0 (03) I Λ ρ, I ΛP 0 0 (04) I Λρ, P 0 =Tr[P 0 (I Λρ)] (05) Λ:A 2 A I Λρ Λ I Λρ Λ P 0 (05) ρ H H 2 ρ Λ:A 2 A I Λρ C 2 C 2 C 2 C 3 ρ ρ T2 ρ T2 = I Tρ (06) ρ ρ T2 Stromer Woronowicz[29][30] H = H 2 = C 2 H = C 3, H 2 = C 2 Λ:A A 2 Λ=Λ CP +Λ CP 2 T (07) 26

27 Λ CP i Λ i = I Λ CP i ρ T2 Λ ρ +Λ 2 ρ T2 ρ [3] ρ ρ =ΦΛΦ (08) λ i U =(U U 2 ) / 2 0 / 2 0 / 2 0 / 2 0 D φφ (09) U,U 2 D φ ρ = UρU E F (ρ )=f(max(0,λ λ 3 2 λ 2 λ 4 )) (0) f(c(ρ)) = H( + C 2 ) 2 () H(x) = x log x ( x)log( x) (2) C C max = max U U(4) (0,σ σ 2 σ 3 σ 4 ) (3) 27

28 σ i Singular value Q =Λ /2 Φ T U T SUΦΛ /2 (4) Φ,U,S C max max (0,σ σ 2 σ 3 σ 4 ) (5) V U(4) σ i Λ /2 V Λ /2 V Φ T U T SUΦ V (V = V T ) S S = S T S [32] S 0 0 S = i i 0 (6) i 0 0 i V V V T V U U = S OV Φ (7) V = V T V [33] Lemma A M n,r (C),B M r,m (C) k σ i (AB) i= k =, q =min{n, r, m} k σ i (A)σ i (B) (8) i= Wang Xi [34] Lemma 2A M n (C),B M n,m (C), i i k n k σ it (AB) t= k σ it (A)σ n t+ (B) (9) t= n =4 k = k =3,i =2,i 2 =3,i 3 =4 σ (AB) (σ 2 (AB)+σ 3 (AB)+σ 4 (AB)) σ (A)σ (B) σ 2 (A)σ 4 (B) σ 3 (A)σ 3 (B) σ 4 (A)σ 2 (B) (20) 28

29 A = Λ /2,B = V Λ /2 σ i (A) = σ i (B) = λ i (σ (σ 2 + σ 3 + σ 4 ))(Λ /2 V Λ /2 ) λ (2 λ 2 λ 4 + λ 3 ) (2) V V = (22) max (σ (σ 2 + σ 3 + σ 4 ))(Λ /2 V Λ /2 )=λ (2 λ 2 λ 4 + λ 3 ) (23) V U(4) V V = V T V V = 0 / 2 0 / (24) 0 i/ 2 0 i/ 2 U U = S OV D /2 φ Φ O SU(2) SU(2) = SO(4) (25) S (U U 2 )S SO(4) SO(4) Q Q = S (U U 2 )S O O(4) D φ U,U 2 SU(2) U = S OV D φ Φ =(U U 2 )S V D φ Φ (U U 2 ) / 2 0 / 2 0 / 2 0 / 2 0 D φφ (26) [35][36] 29

30 T S Alice Bob T S 7: BXOR 3.2 Entanglement of distillation 2-qubit [4][5] ρ U U ρ(u U ) du (27) W F = F Ψ Ψ + F 3 ( Ψ + Ψ + + Φ + Φ + + Φ Φ ) (28) F F Tr(ρ Ψ Ψ ) Ψ ±, Φ ρ n 7 BXOR[4] NOT [4] T F F ( F )2 F F ( F )+ 5 9 ( F (29) )2 F = bound bound PPT(Positive partial transposition) 30

31 distillable? bound PPT NPT 8:? bound undistillable bound bound NPT(Negative partial transposition) [36] 4 -qubit 2-qubit(2 2) 2-qubit 4. -qubit -qubit (??) 3

32 a a a : 32

33 a a a : ρ = ( + a σ) (30) 2 Trρ = Trρ 2 0 p p 0 ( ) E 0 = 0 p (3) 0 E = ( ) 0 p (32)

34 a a a : p =0.5 p p 0 p ( ) E 0 = 0 p (33) 0 E = ( ) 0 p (34) 0 34

35 4.2 2-qubit 4.2. ρ = 4 ( + i a i σ i + j b j σ j + ij c ij σ i σ j ) (35) Trρ = a i,b j c ij σ μ σ ν, (μ, ν =0, 4) σ 0 = σ i,i=, 2, 3 [38] N ( ) j α j x N j =det(x ρ) (36) j=0 α j!α = 2!α 2 = Trρ 2 3!α 3 = 3Trρ 2 +2Trρ 3 4!α 4 = 6Trρ 2 +8Trρ 3 +3(Trρ 2 ) 2 6Trρ 4 (37) ρ = 4 ( + pσ σ + qσ 2 σ 2 + rσ 3 σ 3 ) (38) [39] a i± σ i ± σ i (39) c ij± σ i σ j ± σ j σ i a, c (35) c ij i, j i j i, j =, 2, 3 35

36 2: 36

37 3: a 3+ 2 Ψ ± = 2 ( 0 ± 0 ) Φ ± = 2 ( 00 ± ) ( A) a 3+ 3 p q +p + q + a (p q)2 (40) a 3+ =0.5 a a 3+ =0.5 +p q 37

38 4: a 3+ 38

39 5: a 3 p + q + a (p + q)2 (4) + B a 3 5 +p q p + q a 2 3 +(p + q)2 (42) a 3 = p + q p q a 2 3 +(p q)2 (43) 39

40 6: a 3 40

41 a 3 a ± ± (p, q, r) (q, r, p) a 2± ± (p, q, r) (r, p, q) a 3± ± (p, q, r) not change c 2± ± (p, q, r) not change c 3± ± (p, q, r) (r, p, q) c 23± ± (p, q, r) (q, r, p) Hadamard Hadamard+ NOT NOT Hadamard Hσ H = σ 3,Hσ 2 H = σ 2 Hσ 3 H = σ (p, q, r) (r, q, p) σ 2 σ2 Γ = σ 2 σ 2 (p, q, r) (p, q, r) Hadamard+ NOT 00 NOT C p q r CH ρh C = p r q 4 q r p r q p (44) ρ = 4 ( + pσ 3 σ + qσ σ 3 rσ 2 σ 2 ) (45) 4

42 NOT +r 0 p q 0 CρC = 0 r 0 p + q 4 p q 0 +r 0 (46) 0 p + q 0 r ρ = 4 ( + pσ qσ σ 3 + r σ 3 ) (47) a i,b j a 3 = b 3 a = a 2 = b = b 2 =0 c ij

43 : a = b a b S 5 A ρ = 4 ( + pσ σ + qσ 2 σ 2 + rσ 3 σ 3 ) (48) 43

44 +r 0 0 p q ρ = 0 r p+ q p + q r 0 p q 0 0 +r (49) ( p q r) 4 ( + p + q r) 4 ( + p q + r) 4 ( p + q + r) (50) 4 r r = p q, +p+q, p+q, +p q (r p q, r +p + q, r p + q, r +p q) (p, q, r) =(,, ), (,, ), (,, ), (,, ) +r 0 0 p + q ρ = 0 r p q p q r 0 (5) p + q 0 0 +r ( + p q r) 4 ( p + q r) 4 ( p q + r) 4 ( + p + q + r) (52) 4 r p + q, r +p q, r +p + q, r p q 2 44

45 B a 3+ ρ = 4 ( + a 3+( σ 3 + σ 3 ) + pσ σ + qσ 2 σ 2 + rσ 3 σ 3 ) (53) +a + r 0 0 p q ρ = 0 r p+ q p + q r 0 p q 0 0 a + r (54) ( p q r) 4 ( + p + q r) 4 4 ( a (p q)2 + r) 4 ( + a (p q)2 + r) (55) r p q, r +p + q, r + a (p q)2,r a (p q)2?? +a + r 0 0 p + q ρ = 0 r p q p q r 0 (56) p + q 0 0 a + r ( + p q r) 4 ( p + q r) 4 4 ( a (p + q)2 + r) 4 ( + a (p + q)2 + r) (57) r +p q, r p + q, r + a (p + q)2,r + a (p + q)2 45

46 4 [] Nielsen and Chuang, Quantum Computation and Quantum Information Cambridge University Press, (2000). [2] D.Deutsch,Proc. R. Soc. Lond.,A (985) [3] P.W.Shor, Algorithms for Quantum Computation: Discrete Log and Factoring Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science,(994) [4] C.H.Bennett, G.Brassard, C.Crepeau, R.Jozsa, A.Peres, and K.Wotters Phys. Rev. Lett (993) [5] A.Peres, Quantum Theory: Concepts and Methods,Kluwer Acadedmic Publishers (995) [6] C.H.Bennett and G.Brassard, Quantum cryptography public key distribution and coin tossing Int. conf Computers, System & signal Processing Bangalore, India 75 (984) [7] A.Ekert and R.Josza,Reviews of Modern Physics,68, 733 (996) [8] L.Grover; e-print quant-ph/ [9] C.H.Papadimitriou, Computational Complexity Addison Wesley Longman (994) [0] (2002) [] A.Einstein, B.Podolsky and N.Rosen Phys. Rev (935) [2] V.Vedral e-print quant-ph/ [3] A.Peres Phys. Rev. Lett (996) [4] M.Horodecki, P.Horodecki, R.Horodecki Phys. Lett. A 223 (996) [5] C.H.Bennett, D.P.DiVincenzo, J.A.Smolin and W.K.Wotters Phys. Rev. A (996) [6] (999) 46

47 [7] E.Schmidt Math. annalen (907) [8] von Neumann Mathematische Grundlagen der Quantenmechanic Springer, Berlin (932) [9] H.Umegaki Kodai Math. Sem. Rep (962) [20] J.S.Bell, Speakable and Unspeakable in Quantum Mechanics Cambridge Univ. Press (987) [2] (997) [22] N.Gisin Phys. Lett A 20 5 (996) [23] S.Popescu and D.Rohrlich Phys. Rev. A 56 R339 (997) [24] G.Vidal J. Mod. Opt (2000) [25] W.K.Wotters Phys. Rev. Lett (998) [26] E.M.Rains Phys. Rev. A (999) [27] V.Vedral, M.B.Plenio K.Jacobs and P.L.Knight Phys. Rev. A (997) [28] F.Riesz and B.Sz.-Nagy Functional Analysis Dover (952) [29] E.Stromer Acta. Math (963) [30] S.L.Woronowicz Rep. Math. Phys (976) [3] F.Verstraete, K.Audenaert, T.D.Bie, B.D.Moor e-print quantph/000 [32] R.Horn and C.Johnson Matrix Analysis Cambridge University Press (985) [33] R.Horn and C.Johnson Topics in Matrix Analysis Cambridge University Press (99) [34] B.-Y.Wang and B.-Y.Xi Lin. Alg. Appl (997) [35] M.Horodecki, P.Horocecki, R.Horodecki Phys. Rev. Lett (998) [36] D.DiVincenzo, P.W.Shor, J.A.Smolin B.M.Terhal and A.V.Thapliyal Phys. Rev A (2000) [37] C.H.Bennett, G.Brassard, S.Popescu, B.Schumacher, J.A.Smolin, and W.K.Wootters Phys. Rev. Lett (996) 47

48 [38] G.Kimura Phys. Lett. A 339 (2003) [39] Kobayashi and Nomizu Foundations of Differential Geometry Volume John Wiley (963) 48

特集_03-07.Q3C

特集_03-07.Q3C 3-7 Error Detection and Authentication in Quantum Key Distribution YAMAMURA Akihiro and ISHIZUKA Hirokazu Detecting errors in a raw key and authenticating a private key are crucial for quantum key distribution