Marcos Curty1,2

1
On One-way and Two-way Classical Post-Processing
Quantum Key Distribution

• Marcos Curty1,2
• Coauthors Tobias Moroder2,3, and Norbert
  Lütkenhaus2,3

• Center for Quantum Information and Quantum
  Control (CQIQC), University of Toronto
• Institute for Quantum Computing, University of
  Waterloo
• Max-Plank-Forschungsgruppe, Institut für Optik,
  Information und Photonik, Universität
  Erlangen-Nürnberg

2
Overview

• Quantum Key Distribution (QKD)
• Precondition for secure QKD (Two-way One-way)
• Witness Operators (Two-way One-way QKD)
• Semidefinite Programming
• Evaluation

3
Quantum Key Distribution (QKD)
Phase I Physical Set-Up
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Quantum Key Distribution (QKD)
Phase II Classical Communication Protocol

• Advantage distillation (e.g. announcement of
  bases in BB84 protocol)
• Error Correction (? Alice and Bob share the same
  key)
• Privacy Amplification (? generates secret key
  shared by Alice and Bob)

5
Quantum Key Distribution (QKD)
Phase II Classical Communication Protocol

• Advantage distillation (e.g. announcement of
  bases in BB84 protocol)
• Error Correction (? Alice and Bob share the same
  key)
• Privacy Amplification (? generates secret key
  shared by Alice and Bob)

6
Quantum Key Distribution (QKD)
Phase II Classical Communication Protocol

• Advantage distillation (e.g. announcement of
  bases in BB84 protocol)
• Error Correction (? Alice and Bob share the same
  key)
• Privacy Amplification (? generates secret key
  shared by Alice and Bob)

7
Quantum Key Distribution (QKD)
8
Precondition for Secure QKD
Theorem (Two-way QKD)
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Precondition for Secure QKD
Theorem (One-way QKD)
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Precondition for Secure QKD
?AB with symmetric extension to two copies of
system B
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Witness Operators (Two-way QKD)
?AB separable?
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Witness Operators (One-way QKD)
?AB symmetric extension?
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Witness Operators (Two-way QKD)
Evaluation 4-state QKD protocol
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Witness Operators (Two-way QKD)
Evaluation 4-state QKD protocol
(only parameter combinations leading to negative
expectation values are marked)
Tr?W??AB? ?ij cij P(Ai,Bj )
MC, O. Gühne, N. Lewenstein, N. Lütkemhaus, Phys.
Rev. A 71, 022306 (2005) MC, O. Gühne, N.
Lewenstein, N. Lütkemhaus, Proc. SPIE Int. Soc.
Opt. Eng. 5631, 9-19 (2005). J. Eisert, P.
Hyllus, O. Gühne, MC, Phys. Rev. A 70, 062317
(2004).
Other QKD protocols (including higher
dimensional QKD schemes)
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Witness Operators (Two-way and One-way QKD)
Advantages Witnesses operators
One witness Sufficient condition as a first
step towards the demonstration of the feasibility
of a particular experimental implementation of
QKD. This criterion is independent of any chosen
communication protocol in Phase II.
All witnesses Systematic search for quantum
correlations (or symmetric extensions) for a
given QKD setup. Ideally the main goal is to
obtain a compact description of a minimal
verification set of witnesses (Necessary-and
Sufficient).
Disadvantages Witnesses operators
How to find them? To find a minimal
verification set of EWs, even for qubit-based QKD
schemes, is not always an easy task, and it seems
to require a whole independent analysis for each
protocol.
Too many tests To guarantee that no secret key
can be obtained from the observed data it is
necessary to test all the members of the minimal
verification set.
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Semidefinite Programming (SDP)
SDPs can be efficiently solved
Qubit-based QKD (with losses) ?AB ? H2?H3
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Semidefinite Programming (SDP)
Two-way QKD
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Semidefinite Programming (SDP)
Dual problem (one way two-way) ? Witness
operator optimal for Pr(Ai,Bj)
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Evaluation
We need experimental data ? Pr(Ai,Bj)
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Evaluation
QBER 33
QBER 16.66
H. Bechmann-Pasquinucci, and N. Gisin, Phys.
Rev. A 59, 4238 (1999).
QBER 25
QBER 14.6
C. A. Fuchs, N. Gisin, R. B. Griffiths, C.-S.
Niu, and A. Peres, Phys. Rev. A 56, 1163 (1997)
J. I. Cirac, and N. Gisin, Phys. Lett. A 229, 1
(1997).
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Evaluation
Limit USD p?1-2?2 e0
Inflexion point e constant p1-2?2 (USD)
Other QKD protocols ?
MC, T. Moroder, and N. Lütkenhaus, in preparation
(2006)
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Summary

• Interface Physics Computer Science
  Classical Correlated Data with a Promise
• Necessary condition for secure QKD
  (Two-way One-way).
• Relevance for experiments show the
  presence of entanglement (states without
  symmetric
• 
  extension)
• No need to enter details of classical
  communication protocols
• Prevent oversights in preliminary analysis
• One properly constructed proof suffices
• Evaluation Semidefinite programming
  (qubit-based QKD protocols in the presence of
  loss).
• Task for Theory Develop practical tools
  for realistic experiments ( for given
  measurements).