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2 () 4 () 5 -BB84 5 -a)bb84 5. BB84 5. BB BB BB84 -b)bb c) d)bb B9 36 -a)b b)b E9 47 -a)e b)e9 5
3 - 5 -a)gv GV (3) a) b) a) b) (4) a) b) a)
4 -b) a) b) (5)
5 () 4
6 ( BB84 BB84 BB84 BB84 BB84 One-Time Pad S. Wiesner Wiesner Bennett Brassard Bennett 98 BB BennettBessetteBrassardSalvailSmolin Experimentalquantum cryptography Journal of Cryptology 99 EPR E9 Ekert 99 B9 BB84 BB84 Exclusive OR(XOR) One-Time Pad 5
7
8
9 8
10 Alice Bob A Parity:(0) Parity:() B () () C () (0) D E ()- - a)-. 9
11 n m f Ζ a Ζ ( n > m) f m y f f ( y ) nm y x f ( x) = y Ρ(y) m y Ρ(y) m f x t x i = c L, x, i t = c t Ρ y x = c, L, x = c ) t i i t ( i i t x = c L, x, t = c f ( x) = y f x y, c, c,, c, L Ρ y x = c, L, x = c ) = Ρ( ) ( i t y it n x t f m f ( x) = y Ρ( y) = t f m m m f f t 0
12 m Ρ L ) = ( y xi = c,, x c i t = t n x t f n m xn n x n n x x n y = f (x) m m m t t m n t m m m m =,,3 f : Ζ n a Ζ t n f : Ζ a Ζ n
13 n t 3 f : Ζ a Ζ n 3 4n t n (mod 7) 7 4n t n = (mod 7) [] C. H. Bennett and G. Brassard, Quantum Cryptography: Public key Distribution and Coin Tossing, Proc. of IEEE int. Conf. On Comp. Sys. And Signal Proc., Bangalore, India, 984. [] C. H. Bennett, Quantum Cryptography Using Any Two Nonorthogonal States, Phys. Rev. Lett., Vol.68, No., 99. [3] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, J. Smolin, Experimental Quantum Cryptography, Journal of Cryptology, Vol.5, pp.3-8, 99. [4] A. K. Ekert, Quantum Cryptography Based on Bell s Theorem, Phys. Rev. Lett., Vol.67, No.6, 99.
14 [5] S. Wiesner, S., "Conjugate coding", Sigact News, vol. 5, no., 983, pp [6] C. H. Bennett, G. Brassard, S. Breidbart and S. Wiesner, "Quantum cryptography, or unforgeable subway tokens", Advances in Cryptology: Proceedings of Crypto 8, August 98, Plenum Press, pp [7] C. H. Bennett, G. Brassard and J-M. Robert, "Privacy amplification by public discussion", SIAM Journal on Computing, vol. 7, no., April 988, pp [8] G. Brassard and L. Salvail, "Secret-key reconciliation by public discussion", Advances in Cryptology Eurocrypt '93 Proceedings, May 993, to appear. [9] C. H. Bennett, G. Brassard, C. Crépeau and U. M. Maurer, "Generalized privacy amplification", to appear in IEEE Transactions on Information Theory, 995. [0] P. W. Shor, "Algorithms for Quantum Computation: Discrete Log and Factoring", Proc. of the 35th Annual IEEE Symposium on Foundations of Computer Science, 994. B. Chor, O. Goldreich, J. Hastad, J. Freidmann, S. Rudich and R. Smolensky, The bit extraction problem or t-resilient functions, 6th IEEE Symp. Foundations of Computer Science, 985,
15 4
16 5
17 i i E i i U 0, 90 E i U 90 E E0 0 E i U 0 E E 0 E ij i 6
18 7 i E ij ij E i 35, E 00 E E U i E 0 E E U i = E E E E E = E E E E E = E E E E E = E E E E E
19 U i j E E ij ij E ij E ij i i = Tr photon[( U E 90 )( U E = E00 E00 + E0 E i i 0 ρ ρ 0 = Tr photon[( U E 0 )( U E 0 = E0 E0 + E E i i ρ ρ 0 i ρ 0 ρ 8
20 9 i i ε )) ( )log ( log ( ε ε ε ε = n r
21 ε 0.9 r(x) []C. H. Bennett and G. Brassard, "Quantum cryptography: Public-key distribution and coin tossing", Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, December 984, pp []C. H. Bennett, "Quantum cryptography using any two nonorthogonal 0
22 states", Physical Review Letters, vol. 68, no., 5 May 99, pp [3]E. Biham, M. Boyer, P. O. Boykin, T. Mor and V. Roychowdhury, A proof of the security of quantum key distribution, In Proc. of the Thirty-Second Annual ACM Symposium on Theory of Computing. ACM Press, New York, 999. arxiv:quant-ph/ [4]A. K. Ekert, Quantum Cryptography Based on Bell s Theorem, Phys. Rev. Lett., Vol.67, No.6, 99. [5]A. Peres, Quantum Theory: Concepts and Methods,Kluwer Academic Publishers, Boston, 993. [6]C. H. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, "Experimental quantum cryptography", Journal of Cryptology, Vol. 5, pp. 3 8, 99. [7]C. H. Bennett, G. Brassard, S. Breidbart and S. Wiesner, Qunatum cryptography, or unforgeable subway tokens. In Advances in Cryptology: Proceedings of Crypto 8, pp , Plenum Press, 98. [8]C. H. Bennett, G. Brassard, S. Breidbart and S. Wiesner, "Eavesdrop-detecting quantum communications channel", IBM Technical Disclosure Bulletin, vol. 6, no. 8, January 984, pp [9]B. Chor, O. Goldreich, J. Hastad, J. Freidmann, S. Rudich and R. Smolensky, The bit extraction problem or t-resilient functions. In 6th IEEE Symp. Foundations of Computer Science, pp , 985. [0]C. H. Bennett, G. Brassard and J-M. Robert, "Privacy amplification by public discussion", SIAM Journal on Computing, vol. 7, no., April 988, pp []U. M. Maurer, Secret key agreement by public discussion from common information, IEEE Transactions on Information Theory, vol. 39, no. 3, May 993, pp []G. Brassard and L. Salvail, Secret-key reconciliation by public discussion. Advances in Cryptology, Eurocrypt '93 Proceedings, 993 [3]C. H. Bennett, G. Brassard, C. Crépeau and U. M. Maurer, "Generalized privacy amplification", IEEE Transactions on Information Theory, 995. [4]C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, Phys. Rev. A. 54, pp (996) [5]D. Bouwmeester, A. Ekert and A. Zeilinger, editors. The Physics of Quantum Information, Springer, 000.
23 [6]H. K. Lo and H. F. Chau. Unconditional security of quantum key distribution over arbitrarily long distances., Science, Vol. 83, pp , 999. arxiv: quant-ph/ [7]D. Mayers. Unconditional security in quantum cryptography, arxiv: quant-ph/ [8]P. W. Shor and J. Preskill. Simple proof of security of the bb84 quantum key distribution protocol, arxiv: quant-ph/ , 000. [9]E. Biham, M. Boyer, G. Brassard, J. van de Graaf, and T. Mor, Security of quantum key distribution against all collective attacks, arxiv: quant-ph/9800, 998.
24 ()--c) ()--c)-. BB84 [] [ ] [ ] Si-APD APD(Si, Ge, InGaAs) ()--c)-. 3dB/km 0.3dB/km % 0, 3
25 [ ] / 3/ ()--c) / 0 / 4
26 ()--c) / ()--c)-3. Alice 0,45,90,35 Bob Alice 4 0,45,90,35 Bob =8 Alice 0 Bob Bob 0-90 Alice Bob
27 Bob 0 Alice 0 Bob Bob Alice Bob or 35 Bob / BB Bob Bob BB84 Alice Bob OK NG NG OK OK NG OK OK 0 0 ()--c)-4. BB84 6
28 ()--c)-3. Mach-Zehnder Mirror 0 Beam Splitter Beam Splitter φ Mirror ()--c)-5. Mach-Zehnder 0 Beam Splitter /3/ / 7
29 Mach-Zehnder BB84 Mirror Alice Bob φ B 0, / Beam Splitter φ A Beam Splitter 0, /,,3/ Mirror ()--c)-6. Mach-Zehnder BB84 [Mach-Zehnder BB84 ]. Alice Bob 0,. Alice Bob 3. Alice bit 0 / 3/ / 3/ 4. Bob bit 0 / Bob Beam Splitter 0 6. Alice Alice (0, )(/, 3/) Bob Bob 0 / Alice 0 8
30 BB84 Alice Bob A φ A φ B ()--c)-7. BB84 S-S S-L L-S L-L t AliceBob Alice Bob 4 []C.H.Bennett, G.Brassard: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, Indiea(IEEE, New York, 984) 75. 9
31 30
32 3
33 3
34 33 Path i Path u + =
35 u = e iφ Path + i Path []C. H. Bennett and G. Brassard, Quantum Cryptography: Public key Distribution and Coin Tossing, Proc. of IEEE int. Conf. On Comp. Sys. And Signal Proc., Bangalore, India, 984. []C. H. Bennett, Quantum Cryptography Using Any Two Nonorthogonal States, Phys. Rev. Lett., Vol.68, No., 99. [3]A. K. Ekert, Quantum Cryptography Based on Bell s Theorem, Phys. Rev. Lett., Vol.67, No.6, 99. [4]L. Goldenberg and L. Vaidman, Quantum Cryptography Based on Orthogonal States, Phys. Rev. Lett., Vol.75, No.7, 995. [5]B. Huttner, N. Imoto, N. Gisin and T.Mor, Quantum cryptography with coherent states, Phys. Rev. A, Vol.5, No.3, 995. [6]M. Koashi and N. Imoto, Quantum Cryptography Based on Split Transmission of One-Bit Information in Two Steps, Phys. Rev. Lett., 34
36 Vol.79, No., 997. [7]K. Shimizu and N. Imoto, Quantum Cryptography Based on Split Transmission of One-Bit Information in Two Steps, Phys. Rev. Lett. Vol.79, p.383, 997. [9]T. Hirano, T. Konishi and R. Namiki, Quantum cryptography using balanced homodyne detection, quant-ph/ ,
37 ()-B9 ()--a)b9 ()--a) BB84 [] 989 [] EPR E9 [3] BB84 Bennet BB84 E9 4 B9 Alice Bob Alice Bob Alice Bob ()--a)- B9 Alice u, u 0 [5] ( ) u, u 0 36
38 u 0 u0 u u, u0 u,, u u u 0 u0 = u u = u 0 u = u u0 0 * 0 0 u0 u Bob POMV( ) P 0 u u, P u u [ P 0, P ] 0 P 0 u = 0, P u0 = P0 u P u 0 P0 u 0 u0 P0 P0 u0 = u0 u < 0 P0 u u0 37
39 u 0 P ()--a)-..alice 0 u0 u Bob u u 0 u u u 0 u u 0 u 0 u 0 u. Bob 0 P0 P P 0 P 0 P 0 P P 0 P P P 0 P P 3. Bob Alice Alice Bob ()--a)-3. ( :=0. ) ( ) 0,,, K 38
40 ( 0 0 ) α( α ) α α / = e n= 0 α n n! n α 0 + α α + + O( α 3 ) ( α << ) [6] α α 0. α 0 α, α α α = exp( α ) 0 ()--a)- 4. [4] Alice Bob Mirror Mirror Mirror Mirror PSA PSB UBS A UBS B UBS UBS ()--a)-. B9 UBS PSAPSB UBS Alice Bob PSAPSB 4 PSAPSB 39
41 t ()--a)-3. PSA 0 PSB 0 ()--a)-. PSA PSB 0 u 0 0 P 0 0 u 0 u P - 0 P 0 - u P
42 []C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 984), pp.75. []C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, J. Cryptology 5 (99), pp.3. [3]A. Ekert, Phys. Rev. Lett. 67,(99)pp.66. [4]C. H. Bennett, Phys. Rev. 68,(99)pp.3. [5] -3 (983). [6] (996). 4
43 ()--b)b9 ()--b)- no-cloning [] BB84 []E9 [3] no-cloning B9 [4] B9 BB84 B9 ()--a)b Eve Bob Alice Alice Bob Alice Eve Mirror Mirror Mirror Mirror PSA PSE UBS A UBS UBS E UBS Bob Mirror Mirror Mirror Mirror PSE PSB UBS E UBS UBS ()--b)-. B9 Eve B UBS Eve Alice (<0.) 4
44 0 µ P( 0 ) = P( ) = µ Eve Alice / Eve P ( ) = µ << Eve µ P ( 0 ) + = µ ( + µ ) ( µ ) µ ( µ ) Eve Alice Bob Eve Bob Alice Bob Eve B9 no-cloning ( ) no-cloning u 0 u < 43
45 no-cloning u u 0 ()--b)-. ()--b)- B9 BB84 0 B9 0 ()--b)- R [5] R = qµνη η t d q / ν Hz ηt ηd R B9 BB84 44
46 µ = 0. [6] B9 η d n dark ( 00%) 830nm 50% 550nm APD( ) % n dark B9 Alice Bob (Visibility) 0.5% [6] 45
47 []W. K. Wooters and W. H. ZurekNature 99, (98),pp.80. []C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 984), pp.75. [3]A. Ekert, Phys. Rev. Lett. 67,(99)pp.66. [4]C. H. Bennett, Phys. Rev. 68,(99)pp.3. [5]H. Zbinden, H. Bechmann-Pasquinucci, N. Gisin, G. Ribordy, Appl. Phys. B67, (998),pp.743. [6]H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and, W. Tittel, Electron Lett. 33,(997), pp
48 ()-E9 ()--a)e9 ()--a)-. E9 EPR Einstein-Podolsky-Rosen EPR Source Alice Bob 0 0 EPR Alice Bob 47
49 [ ] EPR φ = ( b b ) Alice Bob EPR Source ()--a)-. EPR E9 E9 [E9 ] ) source (pair of entangled photons) ) Alice Bob 3) Alice Bob 4) 5) 6) tampering 7) Alice Bob 8) base 9) BB84 privacy amplification 48
50 ()--a)-. E9 φ = ( b b ) [ ] step. source Alice Bob step. 3 Alice Bob a a a Alice z φ = 0 φ = π / 4 φ 3 = π / 8 b Bob z = 0 b b φ φ = π / 8 φ 3 = π/ 8 a b E( φ, φ ) i j Alice φ a i Bob φ b j a b a b a b a b E( φ i, φj ) = P+ + ( φi, φj ) + P ( φi, φj ) P+ ( φi, φ j ) P + ( φ a i, φ b j ) E( φ a i, φ b j a b ) = cos[( φ φ )] i j step3. S Alice Bob a b a b a b a b S = E φ, φ ) + E( φ, φ ) + E( φ, φ ) E( φ, φ ) ( 3 3 S Clauser, Horne, Shimony, Holt Bell S CHSH inequality S = step4. Alice Bob 49
51 step5. first group second group step7. Alice Bob step8. Alice Bob first group step9. Alice Bob S disturb S = step0. (legitimate users)second grou (anti-correlated) privacy amplification []A.K.Ekert: Phys. Rev. Lett. 67 (99) 66. []C.H.Bennett, G.Brassard, N.D.Mermin: Phys. Rev. Lett. 68 (99) 557 [3]J.F.Clauser, M.A.Horne: Phys. Rev. D 0 (974) 56 [4]A.Garg, N.D.Mermin: Phys. Rev. D 35 (987) 383 [5]J-A.Larsson: Phys. Rev. A 57(998)
52 ()--b)e9 (public channel) Bell (Bell's inequality) Bell (Eve) Bell (Eve) E9 E9 Bell detection loophole []A.K.Ekert: Phys. Rev. Lett. 67 (99) 66. []C.H.Bennett, G.Brassard, N.D.Mermin: Phys. Rev. Lett. 68 (99) 557 [3]J.F.Clauser, M.A.Horne: Phys. Rev. D 0 (974) 56 [4]A.Garg, N.D.Mermin: Phys. Rev. D 35 (987) 383 [5]J-A.Larsson: Phys. Rev. A 57(998) 345 5
53 ()- ()--a) GV ()--a)-. no-cloning 995 Goldenberg Vaidman [] GV no-cloning Eve Bob Alice Bob Eve Bob Alice Bob Eve GV i.) ii) a b a a = b b =, a b = 0 Ψ Ψ ( a + 0 = b ( a = b ), ), Ψ0 Ψ a b 5
54 a b τ Alice Bob θ τ > θ b a Bob ( τ ) Alice Bob SR D 0 S 0 C C S SR D ()--a)-.gv Mach-Zehnder Alice S0S S0 Ψ S Ψ Alice 0 S0 S C a SR b τ τ θ ( ) C Ψ 0 Ψ a SR SR Ψ 0 D0 Ψ D 53
55 Alice S0S t Bob tr s Alice Bob i) ii) t t t s r = t s +τ +θ Eve a b r Ψ0 Ψ a (τ ts t r t t τ > ) BB84 ()--a)-. GV GV Eve 997 Alice Bob S A0 0 C C 0 D 0 S A SR a SR b D ()--a)-. GV GV CC 5050 TR(T+R=) Bob 54
56 A0 A ab C S0S Φ 0 = 0 a b T i R 0 a b Φ = 0 a b T i R 0 a b 0 a a a 0 0 b b 0 b a 0 b 0 a b C 0 T 0 R a b i ( T 0 i R ) a b 0 0 S0S Φ0 Φ Ψ 0 = 0 0 Ψ = 0 0 D0D T=R Φ0 Φ GV GV Eve Eve Bob Alice Alice Bob 55
57 Eve SR C3 C4 SR ()--a)-3eve Eve Alice Φ0 Φ Bob SR' Bob Alice GV Bob Ψ f = ( i T e θ + R ) i ( i Re θ T ) 0 ( T R ) = TR 0 ()--a)-3. BB84 B9 E9 GV [3,4] no-cloning Alice Bob 56
58 B9 ()--a)- BB84,E9 GV []L. Goldenberg and L. Vaidman, Phys. Rev. Lett. 75, (995), pp.39. []M. Koashi and N. Imoto, Phys. Rev. Lett. 79, (997), pp.383 [3] 56, (00),pp.7. [4] 8, (000), pp
59 (3) (3)- (3)--a) (3)--a)-. Alice Bob Alice Bob Alice 0 Bob Alice Bob Alice Bob Alice Bob Alice Bob Alice Bob Alice Bob Alice Alice Alice Bob BC(b,r) b r (P)(P) BC:{0,} {0,} {0,} l (P) BC(b,r)=BC(b',r') (b',r'), bb' (P) X X=BC(b,r) b (P)Alice (P)Bob BC(b,r) ) A bu{0,} r X=BC(b,r) A X B ) B cu{0,} c A 58
60 3) A (b,r)b 4) B X=BC(b,r) (3)--a)-. AliceBob (P)(P) (P)(P) Alice Bob AliceBob A B B A BC(b,r)=g b h r mod p (h=g a ) BC(b,r)=g b r mod p A B A B 59
61 [] (997) [] J. Gruska: Quantum computing, Mc Graw Hill (999). 60
62 (3)--b) (3)--b)-. AliceBob (3)--b)-. Lo and Chau [] Mayers[] Alice Bob s Alice b {0,} b=0 b= r Bob Bob s Bob Alice b Alice r b Alice b Bob Alice (EPR pair)s Bob Bob 6
63 b b r Bob Bob Alice 6
64 (3)- (3)--a) (3)--a)-. AliceBob (3)--a)-. [] [] J. Muller-Quade and H. Imai: ISITA000 (000) pp.665. [] 56 (00) pp.7. 63
65 (3)--b) (3)--b)-. (3)--b)- Salvail[] n n b,,bm n (3)--b)-3. Yao [] Yao 64
66 [] L. Salvail: Crypto'98 (999) pp.338. [] A. Yao: Proc. of the 7th Symposium on the Theory of Computing (995) pp.67. [3] J. Muller-Quade and H. Imai: ISITA000 (000) pp
67 OT 66
68 b 0 b b w 67
69 n n q = q, q, L, q ) r = r, r, L, r ) ( n ( n } Θ = ( θ, θ, L, θ n ), θi {0 45 ( ri 45 + qi 90 ) θ i r n i =,, L, n r = 0 i 0 r = i 45 ),( i =,, L, n q i x 0 x x d ( d = 0 or ) x 0 x w = d w = d x b, x ) w = d x b, x ) ( 0 0 b x b d w ( 0 b0 0 cos (.5 ) q x0 x i 68
70 b0 b c q d µ a ε ε ( µ q+ d) a a = ( e µ q) ε d / a d / µ q Η( p) = p log + ( p)log p p µ µ H (ε) < ( e µ e ) / a µ a ε N / a a 69
71 x 0 x ĉ c = cˆ c c = ˆ x b, x ) x b, x ) ( 0 0 b b c ( 0 b0 70
72 []M.O.Rabin, How to exchange secrets by oblivious transfer, Technical Memo TR-8, Aiken Computation Laboratory, Harvard University, 98. []S. Even, O. Goldreich and A. Lempel, A randomized protocol for signing contracts, Advances in Cryptology: Proceedings of Crypto 8, August 98, Plenum Press, pp [3]C. Crépeau, Equivalence between two flavors of oblivious transfer (abstract), Advances in Cryptology: Proceedings of Crypt 87, August 987, Springer-Verlag, pp [4]C. H. Bennett, F. Bessette, G. Brassard, L. Salvail and J. Smolin, Experimental Quantum Cryptography, Journal of Cryptology, Vol.5, pp.3-8, 99. [5]C. Crépeau and J. Kilian, Achieving oblivious transfer using weakened security assumptions, Proceedings of 9th IEEE Symposium on the Foundations of Computer Science, October, 988, pp [6]C. H. Bennett, G. Brassard, C. Crépeau and M-H. Skubiszewska, "Practical quantum oblivious transfer", Advances in Cryptology, Crypto '9 Proceedings, August 99, Springer - Verlag, pp [7]A. Yao, Security of Quantum Protocols Against Coherent Measurements, in Proc. of 6th Annual ACM Symposium on Theory of Computing,
73 [8]J. Kilian, Proc. of 988 ACM Annual Symposium on Theory of Computing, 988. [9]D. Mayers, On the Security of the Quantum Oblivious Transfer and Key Distribution Protocols, Proc. of Crypto 95, LNCS963, Springer-Verlag, pp. 4-35, 996. [0]D. Mayers, Quantum Key Distribution and String Oblivious Transfer in Noisy Channels, arxiv: quant-ph/
74 Polarizer Quantum Channel Calcite wollaston prism APDs Laser Aperture Color Filter Pockels cells Pockels cells cos (.5 ) 73
75 74
76 75
77 (4)- (4)--a) (4)--a)-. 98 Blum [] Alice Bob [] i) 3 ii) Alice Bob iii) Alice Bob iv) Bob [] Alice x y = H(x) H ( ) Alice y Bob Bob x Alice Bob Alice x Bob Bob y = H(x) (4)--a) Bennett Brassard [3] BB-cointoss Alice b0 BB-commit( b 0 ) Bob Bob b Alice Alice Bob BB-open( b 0 Bob b = b 0 BB-commit( b ) 76
78 Alice s B = ( b, b, K, bs) Bob s ) Θ = ( θ, θ, K, θs θi, 45 Alice s b b = 0 0 at b = 0 90 at b = b = 45 at b = 0 Bob 35 at b = Bob Θ 0 45 b = b i = Bob Θ B = b, b, K, b ) ( s Alice b B BB-open( b ) i Alice Bob b B Bob b b = Bob Alice BB-cointoss [4,5] EPR ( ) Alice Bob i i i i i b i (4)--a)-3. MayersSalvail [6,7] 4 i) ii) iii) 77
79 iv) [8] MayersSalvail Slow Coin Tossing Alice Bob Alice Bob m a, j b j m z = a b ψ( 0), ψ() ψ ( 0) = c 0 + s, ψ() = c 0 s j j j ψ(0) θ θ ψ() 0 (4)--a)- 0, c = sin θ, s = sin θ 78
80 n n Φ 0) = ψ(0), Φ() = () ( k = k = ψ Θ cos Θ = cos n θ n Φ(0) Φ() Φ(0) Φ() POVM ( E0, E0 ) E E 0 0 = [ + cosθ] = E 0 Φ(0) Φ(0) ( E, E ) E = [ + cosθ] E = E Φ() Φ() Φ(0) Φ() E Φ(0) 0 E Φ() Alice m ( j =, Km) a j Bob m ( j =, K, m) b j Alice mn ( i =, K n, j =, Km) mn c ij ψ c ) ψ( c ) Bob c c ( ij ij Bob mn ( i =, K n, j =, Km) d ij 79
81 mn ψ d ) ψ( d ) Alice ( ij ij Alice Bob i =, Kn Alice m e ij = a j c ij (i )Bob Bob e m Alice ij m e = ψ c )( = ψ( a )) Alice ij ( ij j ψ c )( = ψ( a )) Bob ( ij j e = ψ c )( = ψ( a )) Alice ij ( ij j ψ c )( = ψ( a )) Bob ( ij j i a e Bob m f ij = b j d ij (i )Alice j ij Alice f ij m Bob m f = ψ d )( = ψ( b )) Bob ij ( ij j ψ d )( = ψ( b )) Alice ( ij j f = ψ d )( = ψ( b )) Bob ij ( ij j ψ d )( = ψ( b )) Alice ( ij j i b f j ij 80
82 Alice n Φ( b j ) Φ( a j) Bob n Φ a ) Φ b ) ( j ( j Alice a Bob Bob j j n Φ a ) POMV ( E, E ) Φ a ) ( j a j a j Alice E, E ) E, E ) a ~ ( 0 0 ( Bob b Alice Alice j j ( j n Φ b ) POMV ( E, E ) Φ b ) ( j bj b j Bob E, E ) E, E ) ( 0 0 ( ( j b ~ j Alice Bob n Φ a ) POMV ( j ( E, E a j a j ) Φ a ) ( j Bob Bob Alice n Φ b ) POMV ( E, E ) ( j bj b j Φ b ) Bob ( j Alice ~ A B = ( j a ) ( j j ~ b ) Bob ~ A B = ( j ~ a ) ( j j j b ) j 8
83 m 3 / O( m ) m []M. Blum, Proc. of COMPCON, IEEE, (98), pp.37 []B. Schneier, Applied Cryptography nd Ed., John Wiley & Sons, Inc., (996), pp.89. [3]C. H. Bennett and G. Brassard, Proc. of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India., (984), pp.67. [4]D. Mayers, Phys. Rev. Lett.78, (997), pp.344. [5]H. -K. Lo and H. F. Chau, Phys. Rev. Lett.78, (997),pp.340. [6]D. Mayers, L. Salvail, and, Y. Chiba-Kohno, quanta-ph/ , (999). [7] D. Mayers, L. Salvail, and, Y. Chiba-Kohno, QIT99-40, (999), pp.5 [8] H. -K. Lo and H. F. Chau, quanta-ph/97065, (997). 8
84 (4)--b) (4)--b)-. a) MayersSalvail [,] b)-3 BB-cointoss[3] BB-cointoss BB84 air BB84 [4] PBS I II Alice Bob (4)--b)-. BB-cointoss air AliceBob 83
85 Alice Bob Alice ( m) cm Alice Bob Bob 0 45 PBS I 0 45 II Bob (4)--b)-. (3)--b)-. Alice Bob Alice Bob Alicea a,, a, K, b, K Bobb, b s s Alice a0 Alice a0 a i (4)--b)-. 84
86 i = a 0 0 (4)--b)-Alice a i Bob Alice b (4)--b)-. Bob i b i (4)--b)-.Bob Bob a, a,, K a s I II φ() Bob b0 Bob Alice Alice a a Bob 0 i Bob φ φ b i = a 0 85
87 Bob Alice ai a = OK NG i a i NG b = 0 a 0 Bob Alice (4)--b)-3.BB-cointoss i ai bi bit:a' bit rate R6/5=40% bit OK OK NG OK bit error rate BER=/4=5% (4)--b)-3.BB-cointoss s =5 Alice Bob b Bob 0 a 0 (4)--b)-3. BB-cointoss BB-cointoss BB-cointoss MayersSalvail 86
88 Alice Bob i) Alice Bob 50Km 50sec ii) Alice Bob []D. Mayers, L. Salvail, and, Y. Chiba-Kohno, quanta-ph/ , (999). [] D. Mayers, L. Salvail, and, Y. Chiba-Kohno, QIT99-40, (999), pp.5 [3]C. H. Bennett and G. Brassard, Proc. of IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India., (984), pp.67. [4]B. C. Jacobs and J. D. Franson, Optics Lett., (996), pp
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90 89 (4)--a)-. [ ] + z z (spin eigenstate) z }, { + z z Bell state 4 Bell state ) ( B A B A z z z z = + φ ) ( B A B A z z z z + + = φ ) ( B A B A z z z z = + ψ ) ( B A B A z z z z + + = ψ x-spin (x-spin eigenstate) ) ( + + = + z z x ) ( + = z z x x-spin ) ( B A B A x x x x = + φ ) ( B A B A x x x x = φ ) ( B A B A x x x x + + = + ψ ) ( B A B A x x x x + + = ψ
91 Bell + + Ψ = ( φ + ψ ) = ( z + x + + z x ) A B A B = ( x + A z + B + x A z B ) + Φ ( φ ψ ) = ( z + x z x + ) A B A B = ( x + z x z + A B A B ) + + { ψ, φ, Ψ, Φ } ψ + φ = 0 + Ψ Φ = 0 (nonorthogonal) ψ + Ψ + 0 ψ + Φ 0 φ Ψ + 0 φ + Φ 0 90
92 + + { ψ, φ, Ψ, Φ } step. Trent + + { 0, } { ψ, φ } { 0', ' } { Ψ, Φ } Alice Bob z x step. Alice Bob (measurement outcome) (measurement outcome) Alice Bob step3. Trent Alice Bob Trent Alice Bob Trent Alice Bob step4. Step3 Alice Bob Alice Bob Trent (4)--a)- (4)--a)-. Alice Bob Alice/Bob z+ z- x+ x- z+ + z- ψ x+ x- φ + + ψ Ψ + Ψ φ + Φ Ψ + Φ ψ Φ + Φ Ψ φ φ + ψ 9
93 [3 ] 3 GHZ []M.Hillery, V.Buzek, A.Berthiaume: quant-ph/ (998) []A.Karlsson, M.Koashi, N.Imoto: Phys. Rev. A 59 (999) 6 9
94 (4)--b) (4)--b)-. 3 Alice Trent ( ) Bob PBS PBS source (4)--b)-. Alice Bob AliceBobTrent 3 [Trent] Trent Alice Bob + + { 0, } { ψ, φ } { 0 ', ' } { Ψ, Φ } [Alice]Alice PBS z x [Bob] Bob Alice PBS 93
95 z x (4)--b)-. Alice Trent ( ) Bob PBS PBS source (4)--b)-. stepstep3 step 94
96 Alice Bob step3 Trent Alice Bob []M.Hillery, V.Buzek, A.Berthiaume: quant-ph/ (998) []A.Karlsson, M.Koashi, N.Imoto: Phys. Rev. A 59 (999) 6 95
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