Quantum Cryptography

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Cryptographic properties of nonlocal correlations
Characterization of quantum correlations
Miguel Navascués Stefano Pironio Antonio
Acín ICFO-Institut de Ciències Fotòniques
(Barcelona)
QCCC07, Aschau, October 2007
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Motivation
Information is physical.
y
x
a
b
Bob
Alice
If it is guaranteed that there is not causal
influence between the parties
No-signalling principle
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Motivation
If the correlations have been established using
classical means
This constraint defines the set of EPR
correlations. Independently of fundamental
issues, these are the correlations achievable by
classical resources. Bells inequalities define
the limits on these classical correlations.
Clearly, classical correlations satisfy the
no-signalling principle.
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Motivation
Is p(a,bx,y) a quantum probability?
Example
Are these correlations quantum?
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Motivation
Physical principles impose limits on correlations
Popescu-Rohrlich
Bell
BLMPPR
Example 2 inputs of 2 outputs
CHSH inequality
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Motivation

• What are the allowed correlations within our
  current description of Nature?
• How can we detect the non-quantumness of some
  observed correlations? Quantum Bells
  inequalities.
• What are the limits on correlations coming
  associated to the quantum formalism?
• To which extent Quantum Mechanics is useful for
  information tasks?

Previous work by Cirelson, Landau and Wehner
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Necessary conditions for quantum correlations
It is assumed there exists a quantum state and
measurements reproducing the observed probability
distribution
A set consisting of product of the measurement
operators is considered
Then,
is such that
Given two sets X, X if X is at least as good as
X
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Example
Given p(a,bx,y), take
SDP techniques
Do there exist values for the unknown parameters
such that ? Recall if
p(a,bx,y) is quantum, the answer to this
question is yes.
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Hierarchy of necessary conditions
Constraints
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Hierarchy of necessary conditions
We can define the set X(n) of product of n
operators and the corresponding matrix ?(n). If a
probability distribution p(a,bx,y) satisfies the
positivity condition for ?(N), it does it for all
n N.
YES
YES
YES
YES
NO
NO
NO
Is the hierarchy complete?
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Hierarchy of necessary conditions
If some correlations satisfy all the hierarchy,
then
with
?
Rank loops If at some point rank(?(N))rank(?(N1
)), the distribution is quantum.
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Applications
Application 1 Quantum correlators in the
simplest 2x2 case
We restrict our considerations to the correlation
values c(x,y)p(abx,y)- p(a?bx,y) In the
quantum case,
1
c(0,0)
c(0,1)
x
c(1,0)
c(1,1)
x
1
c(0,0)
c(0,1)
1
y
When do there exist values for x and y such that
this matrix is positive?
c(1,0)
c(1,1)
y
1
Cirelson, Landau and Masanes
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Applications
Application 2 Maximal Quantum violations of
Bells inequalities
max such that
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Applications
Examples CHSH
1
1
0
0
1
-1
0
0
0
0
1
1
Cirelsons bound
0
0
1
-1
CGLMP (d3)
The same results hold up to d8
Quantum value!
ADGL
Our results provide a definite proof that the
maximal violation of the CGLMP inequalities can
be attained by measuring a nonmaximally entangled
state
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Intrinsic Quantum Randomness
Unfortunately, God does play dice!
The existence of non-local correlations implies
the non-existence of hidden variables ? randomness
We would like to explore the relation between
non-locality, measured by ß the amount of
violation of a Bells inequality, and local
randomness, measured by pL and defined as
.
The correlations can be mimicked by classical
variables. The observed randomness is only
fictitious, only due to the ignorance of the
actual classical instructions (or hidden
variables).
Clearly, if ß0 ? pL1.
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Intrinsic Quantum Randomness
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Intrinsic Quantum Randomness
Eve
Alice
Trusted Random Number Generator
Colbeck Kent

Ask for a device able to get the maximal quantum
violation of the CHSH inequality.
QRG
The same result is valid for other inequalities
of larger alphabets. Using one random bit, one
gets a random dit.
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Intrinsic Quantum Randomness
Is maximal non-locality needed for perfect
randomness? What about the other extreme
correlations?
For any point in the boundary ? Bell-like
inequality. Its maximal quantum violation gives
perfect local randomness.
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Conclusions

• Hierarchy of necessary condition for detecting
  the quantum origin of correlations.
• Each condition can be mapped into an SDP problem.
• Is this hierarchy complete?
• How do resources scale within the hierarchy?
• Whats the complexity of the problem? Recall
  separability is NP-hard.
• How does this picture change if we fix the
  dimension of the quantum system?
• Are all finite correlations achievable measuring
  finite-dimensional quantum systems?
• Optimization of observed data over all quantum
  possibilities, e.g. estimation of entanglement.
• Quantum Information Theory with untrusted devices.

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Thanks for your attention!
Miguel Navascués, Stefano Pironio and Antonio
Acín, PRL07