Stefano Mancini

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The adventures of Alice, Bob Eve in the
Quantumland

• Stefano Mancini

University of Camerino, Italy
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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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A general framework for deterministic
communication
Two way quantum channel Encoding by quantum
operation (Alice can encode without knowing the
state!)
Alphabetsstates
Encoding
Alice
Decoding
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• 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 alphabets 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
  alphabets states

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A qubit protocol
(M. Lucamarini S. M. PRL 2005)
Simple intercept-resend eavesdropping
probability of detecting Eve (d) is (03/4)/2 vs
(01/2)/2 of BB84 !
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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!
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Eavesdropping success probability vs maximal
eavesdropping information
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Eavesdropping in KD (individual incoherent
attack)
Distinguish between orthogonal
subspacesDistinguish between nonorthogonal
states
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Lemma Optimal Eves incoherent attack consists
in a balanced one for which xx
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Security conditions
dlt0.23 required vs dlt0.15 of BB84
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Eavesdropping in KD (individual coherent attack)
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Security conditions
dlt0.18 required vs dlt0.15 of BB84
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Efficiency
Theoretical Efficiency Eub/(tqtb) E1-c BB84
E1/6 Practical Efficiency E accounts for
losses EEP2 BB84 EEP (P being the
probability to safely transmit a qubit over
Alice-Bob distance) For Pgt1/(6(1-c)) the scheme
surpasses BB84
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What is about entanglement ?

• Bob uses a home qubit and a travel qubit in
  one of the Bells states
• Alice encodes by performing I,iY on travel
  qubit
• Bob decodes by performing Bells basis
  measurement
• Analogy with DENSE CODING !
• EQUIVALENCE with previous protocol (like BB84
  E91)

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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)

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Quantum Secret Sharing
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GHZ States and Secret Sharing
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The security proof
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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 ?

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Procedure of quantum bit commitment
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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

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the final questions

• What is the exact boundary of quantum
  cryptography and why is there ?
• How would governments regulate quantum
  cryptography ?

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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_at_unicam.it