Experimenters Freedom in Bells Theorem and Quantum Cryptography

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Experimenters Freedom in Bells Theorem and
Quantum Cryptography
archivquant-ph/0510167, accepted for Phys. Rev. A

• Johannes Kofler, Tomasz Paterek, and Caslav
  Brukner

Non-local Seminar ViennaBratislava Vienna,
February 3rd 2006
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Contents

• Bells Theorem
• Local Realism and the Freedom of Choice
• The CHSH and the Mermin Inequality
• Quantum Cryptography

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Bells Theorem
Locality The result of a measurement performed
on system A is unaffected by operations on a
space-like separated system B Realism Measurement
results are determined by hidden variables
which exist prior to and independent of
observation Bell Theorem Bell (1964) Local
realism is in conflict with quantum
mechanics Famous experiments Freedman and
Clauser (1972) Clauser and Horne
(1974) Aspect (1981, 1982) Tittel et al.
(1997) Weihs et al. (1998)
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Loopholes in Bell Tests
1. Fair Sampling the detected pairs are
statistically significant (fair) representatives
of all the emitted pairs experimental problem
detection efficiency 2. Locality no causal
mechanism whatsoever can bring information from
one side to the other experimental problem
space-like separation 3. Freedom the
experimenter is free to choose the measurement
settings
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Agenda

• assume local realism
• for consistency with experiments give up freedom
• characterize (quantify) the insane consequences
  of such a program
• show consequences in quantum cryptography

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Freedom of Choice

• 2 separated partners, A and B, space-like
  separated experiments
• A settings k 1,2 B settings l 1,2
• outcomes X 1,1 outcomes Y 1,1
• measured probability for correlation P(XYkl)
• local realism assumes a quadruple q
  X1,X2,Y1,Y2 produced by the source for each run
  (pair), existing independently of whether any or
  which measurements are performed
• X ? X1,X2, Y ? Y1,Y2
• mathematical probability for correlation P(XkYl)

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Freedom of Choice

• measured probability for correlation P(XYkl)
• mathematical probability for correlation P(XkYl)
  
• Freedom Gill et al. (2003)
• The setting choice (k,l) is statistically
  independent of
• the quadruple q X1,X2,Y1,Y2

in many thought repetitions of the experiment the
probability of every possible value of q remains
the same for any choice of the settings
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Lack of Freedom
Imagine freedom is an illusion both choice of
settings and results are consequences of a
common local realistic mechanism experimenters
choice is determined in advance or, e.g., the
parity of the number of cars passing the
laboratory within the next n seconds, where n is
given by the cube of the fourth decimal of the
actual temperature in degrees Fahrenheit, is
correlated with the local realistic source which
emits the particles Measure of the lack of
freedom
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The CHSH Inequality
The mathematical probabilities satisfy a
set-theoretical constraint CHSH inequality
Clauser et al. (1969) In terms of measured
probabilities, with where
adapted bound
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CHSH Inequality with No Freedom

• Example Local realistic mechanism with no
  freedom
• source knows (in advance) the measurement
  settings of A and B
• whenever A and B both (will) measure their second
  setting (k 2, l 2), the source sends
  perfectly anti-correlated pairs
• P(XY22) 0
• in the other three cases it sends perfectly
  correlated pairs
• P(XY11) P(XY12) P(XY21) 1
• then the logical bound of 3 can be reached
• still satisfied as

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CHSH Inequality with total Freedom

• Example Local realistic mechanism with total
  freedom
• complete freedom
• the two inequalities become identical

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Abandonment of Freedom

• consider experiment with maximally entangled
  state
• for perfect setting angles the measured CHSH
  expression can become
• therefore, to keep a local realistic view which
  is in agreement with the experimental results, we
  have to have
• hence the freedom has to be restricted
  (abandoned) at least by the amount
• if we assume
• then

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Generalization to N parties

• N space-like separated parties
• settings kj ? 1,2
• outcomes X(j) ? 1,1
• Local realism assumes existence of
  X1(1),X2(1),,X1(N),X 2(N)
• mathematical probability for correlation
• measured probability for correlation
• lack of freedom

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Mermin Inequality
The mathematical probabilities satisfy a
set-theoretical constraint, equivalent to the
Mermin inequality Mermin (1990), Zukowski et al.
(2002) In terms of the measured
probabilities with
adapted bound
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Abandonment of Freedom

• take maximally entangled N-party GHZ state
• gap between non-adapted bound and experimental
  result increases exponentially with the number of
  partners
• to explain experimental result local
  realistically, the freedom of each party has to
  be abandoned by the (exponentially fast
  saturating) amount

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Intermediate Summary

• Local realistic picture with restricted
  experimenters freedom
• local realistic bound increases
• why is the (experimental) bound in the CHSH
  inequality 2.414 and not, e.g., the logical bound
  3?
• we had to introduce purely theoretical and
  experimentally not accessible entities, i.e., the
  mathematical probabilities P(XkYl)
• Occams razor?
• Poppers falsifiability principle?

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Quantum Cryptography

• Motivation
• the violation of Bells inequality is necessary
  and sufficient for efficient extraction of a
  secret key Gisin et al. (2002)
• if an eavesdropper Eve has (partial) knowledge
  about the settings of Alice and Bob (e.g., bad
  random-number generator), she can simulate a
  violation of Bells inequality and successfully
  eavesdrop Hwang (2005)
• Connection
• lack of freedom in a local realistic picture
• is equivalent to the fact that an
• eavesdropper has partial setting knowledge in a
  quantum experiment

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The BBMCHSH Protocol
(?1,?1), (?2,?2) key establishing
measurement (?1,?3), (?2,?3), (?1?4),
(?2,?4) CHSH measurement orthogonal
combinations ignore
CHSH
?
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Eves Setting Knowledge

• source sends a (maximally entangled)
• singlet pair in each run
• Eve has partial setting knowledge
• model
• before each run Eve gets the probabilities for
  all 8 settings to be chosen
• qij is the probability for (?i,?j) with i 1,2
  j 1,2,3,4
• for simplicity
• one setting always has high probability Q 1/8
• the others have equal low probability (1Q)/7
• Q 1 E total knowledge, A and B no freedom
• Q 1/8 E no knowledge, A and B total freedom
• Lack of freedom

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Eves Attack Algorithm

• if qij Q, Eve attacks with the product state
• maximizes P(XY13) ? 1
• P(XY23) ? 1
• P(XY24) ? 1
• know the key for 11 and 22
• exception q14 Q
• minimizes P(XY14) ? 0
• Eve always sends product states local realism
  with restricted freedom

or
or
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CHSH Violation

• CHSH can be violated for Q gt 0.44
• but what about the secret key?

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Bit Error Rate and Mutual Informations

• bit error rate
• mutual informations
• best mutual information for a given
• D under the condition of no setting
• knowledge
• secret key agreement iff (never fulfilled)

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• Secret key extraction is never possible
• Critical error rate
• D0 14.6 at Q Q0 0.63
• Q gt Q0
• protocol insecure IAB IBE and
• D lt D0 and
• CHSH violated
• Q Q0
• protocol secure IAB lt IBE but
• D D0
• Alice and Bob will not use their key because
  they find D D0

D gt D0
D lt D0
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Consequences

• the one-to-one correspondence between secret key
  extraction and violation of a Bells inequality
  is lost
• depending on the amount of setting knowledge (Q)
  which is leaking out of the laboratories of A and
  B, they have to calculate (via an optimization
  procedure over all possible attacks) a new bound
  for CHSH
• a violation of this adapted bound corresponds
  again to the possibility of secret key extraction
  (as this is equivalent to a violation of the
  original CHSH expression)

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Conclusions

• Bells theorem
• a local realistic description of the world can
  only be maintained if the experimenters freedom
  is partly abandoned
• we quantified the degree of the lack of freedom
  for the CHSH and the Mermin inequality
• quantum cryptography
• the lack of freedom in a local realistic world
  is equivalent to a situation in which the
  eavesdropper has setting knowledge
• if a certain knowledge threshold is beaten, the
  eavesdropper can find out the key without being
  revealed (neither by the error rate nor by the
  CHSH inequality)
• the one-to-one correspondence between secret key
  extraction and violation of the CHSH inequality
  can only be restored by adapting the bound of the
  inequality