Abstract

Measurement-device-independent quantum key distribution (MDI QKD) is a promising protocol for removing all detector side channel attacks. However, the variation of reference frames, e.g., polarization and phase reference, would be an acute threat to the performance of MDI systems. Here, based on polarization scrambling units, we demonstrate a reference-frame-independent MDI QKD scheme that is inherently stable against volatile channel conditions; therefore, the final secure key rate will be insensitive to the random disturbances of polarization and the drifts of phase reference. Thus, calibrations of the primary reference frames are intrinsically removed in our scheme, which essentially reduces potential vulnerabilities as well as the resource consumption of the whole system. In addition, a proof-of-principle experiment with an improved fluctuation analysis method is demonstrated to verify the feasibility and advantages of the proposed scheme.

© 2017 Optical Society of America

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  5. A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, “Plug and play systems for quantum cryptography,” Appl. Phys. Lett. 70, 793–795 (1997).
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  6. X. F. Mo, B. Zhu, Z. F. Han, Y. Z. Gui, and G. C. Guo, “Faraday–Michelson system for quantum cryptography,” Opt. Lett. 30, 2632–2634 (2005).
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  7. Z. F. Han, X. F. Mo, Y. Z. Gui, and G. C. Guo, “Stability of phase-modulated quantum key distribution systems,” Appl. Phys. Lett. 86, 221103 (2005).
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  8. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005).
    [Crossref]
  9. H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
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  10. S. Wang, W. Chen, J. F. Guo, Z. Q. Yin, H. W. Li, Z. Zhou, G. C. Guo, and Z. F. Han, “2  GHz clock quantum key distribution over 260  km of standard telecom fiber,” Opt. Lett. 37, 1008–1010 (2012).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  33. D. S. Waddy, P. Lu, L. Chen, and X. Bao, “Fast state of polarization changes in aerial fiber under different climatic conditions,” IEEE Photon. Technol. Lett. 13, 1035–1037 (2001).
    [Crossref]
  34. Y. Namihira, Y. Horiuchi, S. Ryu, K. Mochizuki, and H. Wakabayashi, “Dynamic polarization fluctuation characteristics of optical fiber submarine cables under various environmental conditions,” J. Lightwave Technol. 6, 728–738 (1988).
    [Crossref]
  35. J. Wuttke, P. M. Krummrich, and J. Rosch, “Polarization oscillations in aerial fiber caused by wind and power-line current,” IEEE Photon. Technol. Lett. 15, 882–884 (2003).
    [Crossref]
  36. K. Yoshino, T. Ochi, M. Fujiwara, M. Sasaki, and A. Tajima, “Maintenance-free operation of WDM quantum key distribution system through a field fiber over 30 days,” Opt. Express 21, 31395–31401 (2013).
    [Crossref]
  37. J. Chen, G. Wu, L. Xu, X. Gu, E. Wu, and H. Zeng, “Stable quantum key distribution with active polarization control based on time-division multiplexing,” New J. Phys. 11, 065004 (2009).
    [Crossref]
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    [Crossref]
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    [Crossref]
  40. X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurement-device-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
    [Crossref]
  41. F. Xu, M. Curty, B. Qi, and H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
    [Crossref]
  42. X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013).
    [Crossref]
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    [Crossref]
  44. M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
    [Crossref]
  45. F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
    [Crossref]
  46. C.-M. Zhang, J.-R. Zhu, and Q. Wang, “Practical decoy-state reference-frame-independent measurement-device-independent quantum key distribution,” Phys. Rev. A 95, 032309 (2017).
    [Crossref]
  47. Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, “Statistical fluctuation analysis for measurement-device-independent quantum key distribution with three-intensity decoy-state method,” Phys. Rev. A 91, 032318 (2015).
    [Crossref]
  48. Y.-H. Zhou, Z.-W. Yu, and X.-B. Wang, “Making the decoy-state measurement-device-independent quantum key distribution practically useful,” Phys. Rev. A 93, 042324 (2016).
    [Crossref]
  49. Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, “Reexamination of decoy-state quantum key distribution with biased bases,” Phys. Rev. A 93, 032307 (2016).
    [Crossref]

2017 (1)

C.-M. Zhang, J.-R. Zhu, and Q. Wang, “Practical decoy-state reference-frame-independent measurement-device-independent quantum key distribution,” Phys. Rev. A 95, 032309 (2017).
[Crossref]

2016 (7)

Y.-H. Zhou, Z.-W. Yu, and X.-B. Wang, “Making the decoy-state measurement-device-independent quantum key distribution practically useful,” Phys. Rev. A 93, 042324 (2016).
[Crossref]

Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, “Reexamination of decoy-state quantum key distribution with biased bases,” Phys. Rev. A 93, 032307 (2016).
[Crossref]

L. C. Comandar, M. Lucamarini, B. Fröhlich, J. F. Dynes, A. W. Sharpe, S. W.-B. Tam, Z. L. Yuan, R. V. Penty, and A. J. Shields, “Quantum key distribution without detector vulnerabilities using optically seeded lasers,” Nat. Photonics 10, 312–315 (2016).
[Crossref]

Y.-L. Tang, H.-L. Yin, Q. Zhao, H. Liu, X.-X. Sun, M.-Q. Huang, W.-J. Zhang, S.-J. Chen, L. Zhang, L.-X. You, Z. Wang, Y. Liu, C.-Y. Lu, X. Jiang, X. Ma, Q. Zhang, T.-Y. Chen, and J.-W. Pan, “Measurement-device-independent quantum key distribution over untrustful metropolitan network,” Phys. Rev. X 6, 011024 (2016).

G.-Z. Tang, S.-H. Sun, F. Xu, H. Chen, C.-Y. Li, and L.-M. Liang, “Experimental asymmetric plug-and-play measurement-device-independent quantum key distribution,” Phys. Rev. A 94, 032326 (2016).
[Crossref]

H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404  km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

C. Wang, S. Wang, Z.-Q. Yin, W. Chen, H.-W. Li, C.-M. Zhang, Y.-Y. Ding, G.-C. Guo, and Z.-F. Han, “Experimental measurement-device-independent quantum key distribution with uncharacterized encoding,” Opt. Lett. 41, 5596–5599 (2016).
[Crossref]

2015 (5)

A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, S. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, “High speed prototype quantum key distribution system and long term field trial,” Opt. Express 23, 7583–7592 (2015).
[Crossref]

C. Wang, X. T. Song, Z. Q. Yin, S. Wang, W. Chen, C. M. Zhang, G. C. Guo, and Z. F. Han, “Phase-reference-free experiment of measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 115, 160502 (2015).
[Crossref]

F. Xu, “Measurement-device-independent quantum communication with an untrusted source,” Phys. Rev. A 92, 012333 (2015).
[Crossref]

S. Wang, Z. Q. Yin, W. Chen, D. Y. He, X. T. Song, H. W. Li, L. J. Zhang, Z. Zhou, G. C. Guo, and Z. F. Han, “Experimental demonstration of a quantum key distribution without signal disturbance monitoring,” Nat. Photonics 9, 832–836 (2015).
[Crossref]

Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, “Statistical fluctuation analysis for measurement-device-independent quantum key distribution with three-intensity decoy-state method,” Phys. Rev. A 91, 032318 (2015).
[Crossref]

2014 (6)

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
[Crossref]

F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014).
[Crossref]

S. Wang, W. Chen, Z.-Q. Yin, H.-W. Li, D.-Y. He, Y.-H. Li, Z. Zhou, X.-T. Song, F.-Y. Li, D. Wang, H. Chen, Y.-G. Han, J.-Z. Huang, J.-F. Guo, P.-L. Hao, M. Li, C.-M. Zhang, D. Liu, W.-Y. Liang, C.-H. Miao, P. Wu, G.-C. Guo, and Z.-F. Han, “Field and long-term demonstration of a wide area quantum key distribution network,” Opt. Express 22, 21739–21756 (2014).
[Crossref]

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509, 475–478 (2014).
[Crossref]

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, and H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 112, 190503 (2014).
[Crossref]

Z. Q. Yin, S. Wang, W. Chen, H. W. Li, G. C. Guo, and Z. F. Han, “Reference-free-independent quantum key distribution immune to detector side channel attacks,” Quantum Inf. Process. 13, 1237–1244 (2014).
[Crossref]

2013 (7)

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013).
[Crossref]

T. F. da Silva, D. Vitoreti, G. B. Xavier, G. C. do Amaral, G. P. Temporão, and J. P. von der Weid, “Proof-of-principle demonstration of measurement-device-independent quantum key distribution using polarization qubits,” Phys. Rev. A 88, 052303 (2013).
[Crossref]

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013).
[Crossref]

K. Yoshino, T. Ochi, M. Fujiwara, M. Sasaki, and A. Tajima, “Maintenance-free operation of WDM quantum key distribution system through a field fiber over 30 days,” Opt. Express 21, 31395–31401 (2013).
[Crossref]

F. Xu, M. Curty, B. Qi, and H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013).
[Crossref]

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013).
[Crossref]

Z.-W. Yu, Y.-H. Zhou, and X.-B. Wang, “Three-intensity decoy-state method for measurement-device-independent quantum key distribution,” Phys. Rev. A 88, 062339 (2013).
[Crossref]

2012 (5)

X. Ma, C.-H. F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurement-device-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012).
[Crossref]

S. Wang, W. Chen, J. F. Guo, Z. Q. Yin, H. W. Li, Z. Zhou, G. C. Guo, and Z. F. Han, “2  GHz clock quantum key distribution over 260  km of standard telecom fiber,” Opt. Lett. 37, 1008–1010 (2012).
[Crossref]

J. Z. Huang, Z. Q. Yin, S. Wang, H. W. Li, W. Chen, and Z. F. Han, “Effect of intensity modulator extinction on practical quantum key distribution system,” Eur. Phys. J. D 66, 159 (2012).
[Crossref]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012).
[Crossref]

S. L. Braunstein and S. Pirandola, “Side-channel-free quantum key distribution,” Phys. Rev. Lett. 108, 130502 (2012).
[Crossref]

2011 (2)

H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011).
[Crossref]

N. J. Muga, M. F. S. Ferreira, and A. N. Pinto, “QBER estimation in QKD systems with polarization encoding,” J. Lightwave Technol. 29, 355–361 (2011).
[Crossref]

2010 (2)

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4, 686–689 (2010).
[Crossref]

A. Laing, V. Scarani, J. G. Rarity, and J. L. O’Brien, “Reference-frame-independent quantum key distribution,” Phys. Rev. A 82, 012304 (2010).
[Crossref]

2009 (1)

J. Chen, G. Wu, L. Xu, X. Gu, E. Wu, and H. Zeng, “Stable quantum key distribution with active polarization control based on time-division multiplexing,” New J. Phys. 11, 065004 (2009).
[Crossref]

2007 (2)

C.-H. F. Fung, B. Qi, K. Tamaki, and H.-K. Lo, “Phase-remapping attack in practical quantum-key-distribution systems,” Phys. Rev. A 75, 032314 (2007).
[Crossref]

B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 073–082 (2007).

2005 (4)

X. F. Mo, B. Zhu, Z. F. Han, Y. Z. Gui, and G. C. Guo, “Faraday–Michelson system for quantum cryptography,” Opt. Lett. 30, 2632–2634 (2005).
[Crossref]

Z. F. Han, X. F. Mo, Y. Z. Gui, and G. C. Guo, “Stability of phase-modulated quantum key distribution systems,” Appl. Phys. Lett. 86, 221103 (2005).
[Crossref]

X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005).
[Crossref]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref]

2003 (1)

J. Wuttke, P. M. Krummrich, and J. Rosch, “Polarization oscillations in aerial fiber caused by wind and power-line current,” IEEE Photon. Technol. Lett. 15, 882–884 (2003).
[Crossref]

2001 (1)

D. S. Waddy, P. Lu, L. Chen, and X. Bao, “Fast state of polarization changes in aerial fiber under different climatic conditions,” IEEE Photon. Technol. Lett. 13, 1035–1037 (2001).
[Crossref]

1997 (1)

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, “Plug and play systems for quantum cryptography,” Appl. Phys. Lett. 70, 793–795 (1997).
[Crossref]

1992 (1)

C. D. Angelis, A. Galtarossa, G. Gianello, F. Matera, and M. Schiano, “Time evolution of polarization mode dispersion in long terrestrial links,” J. Lightwave Technol. 10, 552–555 (1992).
[Crossref]

1991 (1)

A. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

1988 (1)

Y. Namihira, Y. Horiuchi, S. Ryu, K. Mochizuki, and H. Wakabayashi, “Dynamic polarization fluctuation characteristics of optical fiber submarine cables under various environmental conditions,” J. Lightwave Technol. 6, 728–738 (1988).
[Crossref]

Angelis, C. D.

C. D. Angelis, A. Galtarossa, G. Gianello, F. Matera, and M. Schiano, “Time evolution of polarization mode dispersion in long terrestrial links,” J. Lightwave Technol. 10, 552–555 (1992).
[Crossref]

Bao, W.-S.

H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011).
[Crossref]

Bao, X.

D. S. Waddy, P. Lu, L. Chen, and X. Bao, “Fast state of polarization changes in aerial fiber under different climatic conditions,” IEEE Photon. Technol. Lett. 13, 1035–1037 (2001).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175–179.

Brassard, G.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of our robust MDI QKD scheme. LD, laser diode; PM, phase modulator; PG, pulse generation unit; PS, polarization scrambling unit; EPC, electronic polarization controller; FMI, Faraday–Michelson interferometer; FM, Faraday mirror; VOA, variable optical attenuator; ATT, attenuator; BS, beam splitter; PBS, polarizing beam splitter; Det, detector.

Fig. 2.
Fig. 2.

Lower secure key rate bound of our robust RFI MDI QKD scheme. “Improved” represents the final key rate of the robust MDI QKD scheme with our improved fluctuation analysis, and “Conventional” is the final key rate of the robust MDI QKD scheme with conventional fluctuation analysis. The total number of pulse pairs sent from Alice and Bob is N t = 3.5 × 10 11 , the failure probability is ε = 10 10 , and all parameters have been optimized.

Fig. 3.
Fig. 3.

Simulation comparison of the secure key rates of our polarization scrambling scheme and the polarization tracking scheme with a polarization intensity deviation of 10%, 20%, and 30%.

Tables (1)

Tables Icon

Table 1. Experimental Results of Our Robust MDI QKD Schemea

Equations (27)

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R P Z Z P Z Z μ μ { a 1 Z b 1 Z S Z Z μ μ , 11 L [ 1 I E U ] f S Z Z μ μ H ( E Z Z μ μ ) } ,
C = ( 1 2 E X X 11 ) 2 + ( 1 2 E X Y 11 ) 2 + ( 1 2 E Y X 11 ) 2 + ( 1 2 E Y Y 11 ) 2
R S Z Z 11 ( 1 I E f H ( E Z Z 11 ) ) ,
ρ μ A ω = k a k ω | k k | , ρ μ B ω = k b k ω | k k | , ρ ν A ω = k a k ω | k k | , ρ ν B ω = k b k ω | k k | , ρ o A = ρ o B = | 0 0 | .
m 11 L [ a 1 b 2 M ν ν + a 1 b 2 a 0 M o μ + a 1 b 2 a 0 M μ o ] [ a 1 b 2 M μ μ + a 1 b 2 a 0 b 0 M o o ] a 1 b 2 [ a 0 M o ν + b 0 M ν o a 0 b 0 M o o ] a 1 a 1 ( b 1 b 2 b 1 b 2 ) , m 11 U M ν ν [ a 0 M o ν + b 0 M ν o a 0 b 0 M o o ] a 1 b 1 ,
S l r = S l r ( 1 + δ l r ) , Δ N l r S l r δ l r Δ ^ N l r S l r ,
| N l r S l r δ l r + N l r S l r δ l r | Δ N l r S l r + N l r S l r .
| l r J N l r S l r δ l r | Δ l r J N l r S l r ,
S 11 L = max { S Z Z 11 L , S X X 11 L , S avg 11 L } , S 11 U = min { S Z Z 11 U , S X X 11 U , S avg 11 U } .
S avg 11 = 1 p tot l , r P Z Z P Z Z l r a 1 Z l b 1 Z r S Z Z 11 + 1 p tot l , r P X X P X X l r a 1 X l b 1 X r S X X 11 + 1 p tot l , r P Y Y P Y Y l r a 1 Y l b 1 Y r S Y Y 11 + 1 p tot l , r P X Y P X Y l r a 1 X l b 1 Y r S X Y 11 + 1 p tot l , r P Y X P Y X l r a 1 Y l b 1 X r S Y X 11 , l , r { μ , ν } ,
p tot = l , r P Z Z P Z Z l r a 1 Z l b 1 Z r + l , r P X X P X X l r a 1 X l b 1 X r + l , r P Y Y P Y Y l r a 1 Y l b 1 Y r + l , r P X Y P X Y l r a 1 X l b 1 Y r + l , r P Y X P Y X l r a 1 Y l b 1 X r
S B C 11 L = S 11 L ( 1 δ B C , L ) , S B C 11 U = S 11 U ( 1 + δ B C , U ) ,
E B C 11 L = T B C 11 L S B C 11 U , E B C 11 U = T B C 11 U S B C 11 L .
| k K β k x k | n ε k K β k , K { 1,2 , 3 , , K } ,
f max = f ( x ˜ k * ) = F ( K , n ε , V α , V β ) = n ε n = 1 K ( γ ˜ n γ ˜ n 1 ) k = n K β ˜ k ,
x ˜ k * = n ε β ˜ k ( n = k K β ˜ n n = k + 1 K β ˜ n ) ,
f min = f max = F ( K , n ε , V α , V β ) .
x k = { β K ( k = K ) , n = k K β k n = k + 1 K β k ( k < K ) ,
k K x k k K β k ,
K 2 K 1 = { 1,2 , , K } .
x 1 + x 2 + x 3 + + x K 1 + x K = β 1 + β 2 + + β K ,
x 1 + + x v 1 1 + x v 1 + 1 + + x K = β 1 + β 2 + + β K ( β v 1 + β v 1 + 1 + + β K β v 1 + 1 + β v 1 + 2 + + β K ) β 1 + β 2 + + β v 1 1 + β v 1 + 1 + + β K .
x 1 + x 2 + + x v 1 1 + x v 1 + 1 + + x v i 1 + x v i + 1 + + x K β 1 + β 2 + + β v 1 1 + β v 1 + 1 + + β v i 1 + β v i + 1 + + β K .
P v i + 1 = β v i + 1 + 1 + β v i + 1 + 2 + + β K , P v i + 1 = β 1 + β 2 + + β v 1 1 + β v 1 + 1 + + β v i 1 + β v i + 1 + + β v i + 1 1 .
P v i + 1 + β v i + 1 + P v i + 1 + P v i + 1 β v i + 1 + P v i + 1 + P v i + 1 + P v i + 1 ,
P v i + 1 + β v i + 1 + P v i + 1 ( β v i + 1 + P v i + 1 P v i + 1 ) P v i + 1 + P v i + 1
β 1 + β 2 + + β v 1 1 + β v 1 + 1 + + β v i 1 + β v i + 1 + + β v i + 1 1 + β v i + 1 + β v i + 1 + 1 + + β K ( β v i + 1 + β v i + 1 + 1 + + β K β v i + 1 + 1 + β v i + 1 + 2 + + β K ) β 1 + + β v i + 1 1 + β v i + 1 + 1 + + β K .

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