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The adventures of Alice, Bob & Eve in the Quantumland

The adventures of Alice, Bob & Eve in the Quantumland. Stefano Mancini. University of Camerino, Italy. Outline (part 3). From probabilistic to deterministic quantum cryptography From qubit to qudit up to cv Quantum Secret Sharing Beyond QKD: Quantum bit commitment Conclusion.

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The adventures of Alice, Bob & Eve in the Quantumland

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  1. The adventures of Alice, Bob & Eve in the Quantumland Stefano Mancini University of Camerino, Italy

  2. Outline (part 3) • From probabilistic to deterministic quantum cryptography • From qubit to qudit up to cv • Quantum Secret Sharing • Beyond QKD: Quantum bit commitment • Conclusion

  3. A general framework for deterministic communication Two way quantum channel Encoding by quantum operation (Alice can encode without knowing the state!) Alphabet’sstates Encoding Alice Decoding

  4. Information relies on OPERATIONS upon the state rather than on the STATE itself. This entails Eve must perturb the quantum channel twice instead of once to gain information • The protocol, for suitable alphabet’s states and encoding/decoding operations, is DETERMINISTIC. This feature also allows DIRECT COMMUNICATION. • How can Alice and Bob detect Eve? Use of two modes: MESSAGE MODE and CONTROL MODE • Possible use of non-orthogonal or entangled alphabet’s states

  5. (M. Lucamarini & S. M. PRL 2005) A qubit protocol Simple intercept-resend eavesdropping: probability of detecting Eve (d) is (0+3/4)/2 vs (0+1/2)/2 of BB84 !

  6. Quantum Direct Communication Number of message bits per protocol run = 1-c Eve wants to eavesdrop one message transfer without being detected; probability p=(1-c)+c(1-d)(1-c)+c2(1-d)2(1-c)+… Terms correspond to Eve having survived 0,1,2… controls before she gets h(d) bits of information After n successful attacks Eve gains nh(d) bits and survive with probability pn Probability to successfully eavesdrop nh(d) bits Protocol asymptotically secure!

  7. Eavesdropping success probability vs maximal eavesdropping information

  8. POVM POVM Eavesdropping in KD (individual incoherent attack) Distinguish between orthogonal subspacesDistinguish between nonorthogonal states

  9. Lemma: Optimal Eve’s incoherent attack consists in a balanced one for which x=x’

  10. Security conditions d<0.23 required vs d<0.15 of BB84

  11. POVM POVM Eavesdropping in KD (individual coherent attack)

  12. Security conditions d<0.18 required vs d<0.15 of BB84

  13. Efficiency Theoretical Efficiency E=ub/(tq+tb) E=1-c; BB84 E=1/6; Practical Efficiency E’ accounts for losses E’=EP2; BB84 E’=EP; (P being the probability to safely transmit a qubit over Alice-Bob distance) For P>1/(6(1-c)) the scheme surpasses BB84

  14. What is about entanglement ? • Bob uses a “home qubit” and a “travel qubit” in one of the Bell’s states • Alice encodes by performing {I,iY} on “travel qubit” • Bob decodes by performing Bell’s basis measurement • Analogy with DENSE CODING ! • EQUIVALENCE with previous protocol (like BB84 & E91)

  15. From qubit to qudit up to cv • Higher rates for QKD can be achieved by using larger alphabets (Hilbert spaces) • Extension of BB84 to qudit (use MUB) • Extension of E91 to qudit (generalized Bell ineq.) • QKD in infinite dimensional Hilbert space can make use of both coherent and entangled states • Development of deterministic protocol in infinite dimensional Hilbert space - multi way quantum communication (S. Pirandola, S. M. & S. Lloyd 2006)

  16. Quantum Secret Sharing

  17. GHZ States and Secret Sharing

  18. The security proof

  19. Beyond QKD ? • In a two-party computation Alice (Bob) has a private input x (y) and would like to help Bob to compute a prescribed function f(x,y) without revealing x. • Bit commitment is a primitive for implementing secure computations. • What is bit commitment ?

  20. Procedure of quantum bit commitment

  21. Information theoretic approach to QT Quantum theory can be viewed not as a mechanical theory of waves and particles but as a theory about the possibilities and impossibilities of information transfer • The impossibility of superluminal information transfer • The impossibility of perfectly broadcasting information contained in an unknown state (no cloning) • The impossibility of unconditionally secure bit commitment

  22. …the final questions • What is the exact boundary of quantum cryptography and why is there ? • How would governments regulate quantum cryptography ?

  23. A Short Bibliography • S. Singh, The Code Book (Anchor Book, 1999) • H. K. Lo, Quantum Cryptology (in Introduction to Quantum Computation & Information, World Scientific, 1998) • S. Lomonaco, How Alice outwits Eve (in Lecture Notes in Computer Science, Springer 1999) • N. Gisin et al., RMP 2002 • A. Galindo & M. Martin-Delgado, RMP 2002 Further information and research at UniCam see: http://fisica.unicam.it/stefano.mancinior contact me at: stefano.mancini@unicam.it

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