Quantum Cryptography

1
Quantum correlations and device-independent
quantum information protocols
Antonio Acín N. Brunner, N. Gisin, Ll. Masanes,
S. Massar, M. Navascués, S. Pironio, V. Scarani
Feynman Festival, Olomouc, June 2009
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Scenario
Distant parties performing m different
measurements of r outcomes.
y1,,m
x1,,m
a1,,r
b1,,r
Bob
Alice
Vector of m2 r2 positive components satisfying m2
normalization conditions
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Physical Correlations
Physical principles translate into limits on
correlations
1) Classical correlations correlations
established by classical means.
These are the standard EPR correlations.
Independently of fundamental issues, these are
the correlations achievable by classical
resources. Bell inequalities define the limits
on these correlations.
For a finite number of measurements and results,
these correlations define a polytope, a convex
set with a finite number of extreme points.
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Physical Correlations
2) Quantum correlations correlations established
by quantum means.
The set of quantum correlations is again convex,
but not a polytope, even if the number of
measurements and results is finite.
3) No-signalling correlations correlations
compatible with the no-signalling principle, i.e.
the impossibility of instantaneous communication.
The set of no-signalling correlations defines
again a polytope.
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Physical Correlations
Popescu-Rohrlich
Bell
BLMPPR
Example 2 inputs of 2 outputs
CHSH inequality
Trivial facets
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Motivation
Is p(a,bx,y) a quantum probability?
Example
Are these correlations quantum?
No constraint on the dimension ? pure states and
projective measurements
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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 analogues of Bell
  inequalities.
• What are the limits on correlations associated to
  the quantum formalism?
• To which extent Quantum Mechanics is useful for
  information tasks?

Previous work by Tsirelson and Wehner
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Device-Independent Quantum Information protocols
Goal to construct information protocols where
the parties can see their devices as quantum
black-boxes ? no assumption on the devices.
y1,,m
x1,,m
a1,,r
b1,,r
Alice
Bob
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Outline of the talk

• Introduction
• Hierarchy of necessary conditions for quantum
  correlations
• Definition of the hierarchy
• Discussion of convergence
• Device-Independent Quantum Information protocols
• Quantum Key Distribution
• Randomness Generation
• Conclusions / Open questions

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Hierarchy of necessary conditions
Given a probability distribution p(a,bx,y), we
have defined a hierarchy consisting of a series
of tests based on semi-definite programming
techniques allowing the detection of
supra-quantum correlations.
YES
YES
YES
YES
NO
NO
NO
Is the hierarchy complete?
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Convergence of the hierarchy
1) If some correlations satisfy all the
hierarchy, then
with
?
2) Rank loops If ? n s.t.
the distribution is
quantum.
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Device-Independent QKD

• Standard QKD protocols based their security on
• Quantum Mechanics any eavesdropper, however
  powerful, must obey the laws of quantum physics.
• No information leakage no unwanted classical
  information must leak out of Alice's and Bob's
  laboratories.
• Trusted Randomness Alice and Bob have access to
  local random number generators.
• Knowledge of the devices Alice and Bob require
  some control (model) of the devices.

Is there a protocol for secure QKD based on
without requiring any assumption on
the devices?
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Security against collective attacks

• Device-Independent protocol based on the CHSH
  Bell inequality.
• Collective attacks Eve prepares always the same
  state and measurements (identical and independent
  realizations).
• Bound on Eves information as a function of the
  observed error and Bell inequality violation.

The obtained critical QBER is of approx 7.1
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Can the presence of randomness be guaranteed by
any physical mechanism?
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Randomness tests

• Good randomness is usually verified by a series
  of statistical tests.
• There exist chaotic systems, of deterministic
  nature, that pass all existing randomness tests.
  Uchida et al., Nat. Phot. 2, 728 (2008)
• Do these tests really certify the presence of
  randomness?

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Known solutions

• Classical Random Number Generators (CRNG) all of
  them are of deterministic Nature.
• Quantum Random Number Generators (QRNG) all the
  existing solution require some knowledge of the
  devices. The provider has to be trusted.
• In any case, all the solutions guarantee the
  randomness using standard statistical randomness
  tests.

The standard solution crucially depends on the
details of the device used for the random number
generation.
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Private Randomess

• Many applications require private randomness.
• Untrusted scenario can one be sure that nobody
  has a deterministic model for the observed
  randomness?

...
Classical Memory

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Random Numbers from Bells Theorem

• Randomness can be certified in the quantum world
  by means of non-local correlations, i.e. the
  violation of a Bell inequality.
• The obtained randomness is private.
• It represents a novel application of Quantum
  Information Theory, solving a task whose
  classical realization is, at least, unclear.
• Our findings can be used to design
  Device-Independent Quantum Randomness Expanders.

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Random Numbers from Bells Theorem
We want to explore the relation between
non-locality, measured by the violation ß of a
Bell inequality, and local randomness, quantified
by . Clearly, if ß 0
? r1.
y1,2
x1,2
a1,-1
b1,-1
NOTE In all what follows, loopholes are not
analyzed. They are important when considering the
practical implementation of these ideas.
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Results
All the region above the curve is impossible
within Quantum Mechanics.
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Statement of the problem
We have developed an asymptotically convergent
series of sets approximating the quantum set.
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Other Bell inequalities
y1,2
x1,2
The same computation can be done for other
inequalities, such as the CGLMP Bell inequality.
a1,,r
b1,,r
One gets a perfectly random trit. The only known
way of obtaining this point is by measuring a
non-maximally entangled state.
The more non-local ? the more random
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Randomness Witnesses
The CHSH inequality at the maximal quantum
violation defines a tangent to the quantum
boundary. We consider other hyper-planes tangent
to the quantum set.
classical
quantum
The maximal quantum violation of any of these
inequalities always guarantees perfect
randomness. Arbitrarily small amounts of quantum
non-locality give perfect randomness.
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Device-Independent Quantum Randomness Expanders
A device violating a Bells inequality can be
used to generate random numbers. However,
randomness is needed for the Bell test ?
Randomness Expander! Kent Kollbeck In these
devices, the two outcomes contain randomness and
are useful.
In the limit of very large a one gets two random
bits. Quantum Theory is as random as
possible. This is not the case for general
no-signaling theories, where the number of random
bits is at most one.
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Quantum correlations

• Hierarchy of necessary condition for detecting
  the quantum origin of correlations.
• Each condition can be mapped into an SDP problem.
• Is this hierarchy complete for tensor product
  measurements?
• How does this picture change if we fix the
  dimension of the quantum system?
• Are all finite correlations achievable measuring
  finite-dimensional quantum systems?

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Random Numbers from Bells Theorem

• Randomness can be certified in the quantum world
  by means of non-local correlations, i.e. the
  violation of a Bell inequality.
• The obtained randomness is private.
• It represents a novel application of Quantum
  Information Theory, solving a task whose
  classical realization is, at least, unclear.
• Our findings can be used to design
  Device-Independent Quantum Randomness Expanders.

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Take-home question
(C or Q)RNG
DIQRNE
Specifications it passes all statistical
randomness tests.
Specifications
It wont pass all the existing randomness tests!
Which device is more random?
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Thanks for your attention!
Post-doc and PhD positions available in the
group, see www.icfo.es