Feynman Festival, Olomouc, June 2009

Antonio Acín

N. Brunner, N. Gisin, Ll. Masanes, S. Massar, M. Navascués, S. Pironio, V. Scarani

Quantum correlations and device-independent quantum information protocols

Scenario

Alice Bob

y=1,…,m

a=1,…,r b=1,…,r),,( yxbap

x=1,…,m

mmrrprrpppyxbap ,,,,1,1,,,1,12,1,1,11,1,,

Vector of m2 r2 positive components satisfying m2 normalization conditions

Distant parties performing m different measurements of r outcomes.

r

ba

yxyxbap1,

,1,,

Physical Correlations

1) Classical correlations: correlations established by classical means.

,,),,( ybqxappyxbap

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.

Physical principles translate into limits on correlations

For a finite number of measurements and results, these correlations define a polytope, a convex set with a finite number of extreme points.

Physical Correlations

yb

xaAB MMtryxbap ,,

xaaa

xa

xa

a

xa

MMM

M

''

1

xapyxbapb

),,(

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.

Physical Correlations

Example: 2 inputs of 2 outputs

1,0,,, yxba

CHSH inequality

NL machine

QM

Local points

22Q

2L

4NBLMPPR

NSQC Popescu-RohrlichBell

Trivial facets

Motivation

Is p(a,b|x,y) a quantum probability?

yb

xaAB MMtryxbap ,,

xaaa

xa

xa

a

xa

MMM

M

''

1

Example:

32,32,32,328

10,1,1,0,0,0, bapbapbap

245.0,255.0,255.0,245.01,1, bapAre these correlations

quantum?

No constraint on the dimension → pure states and projective measurements

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

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.

Alice Bob

y=1,…,m

a=1,…,r b=1,…,r),,( yxbap

x=1,…,m

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

Hierarchy of necessary conditions

Given a probability distribution p(a,b|x,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.

01

NO NO

YES YES

NO

YES

Is the hierarchy complete?

YES

002

Convergence of the hierarchy

1) If some correlations satisfy all the hierarchy, then:

yb

xa MMtryxbap ,, with

a

xa

yb

xa

M

MM

1

0,

?

yb

xaAB MMtryxbap ,,

2) Rank loops:

If n s.t. the distribution is quantum. 1 nn rankrank

Device-Independent QKD

Standard QKD protocols based their security on:1. Quantum Mechanics: any eavesdropper, however

powerful, must obey the laws of quantum physics.2. No information leakage: no unwanted classical

information must leak out of Alice's and Bob's laboratories.

3. Trusted Randomness: Alice and Bob have access to local random number generators.

4. 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?

),,( yxbap

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 Eve’s information as a function of the observed error and Bell inequality violation.

The obtained critical QBER is of approx 7.1%

Can the presence of randomness be guaranteed by

any physical mechanism?

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?

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.

50%

50%

T

R

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

Private Randomess

• Many applications require private randomness.• Untrusted scenario: can one be sure that nobody

has a deterministic model for the observed randomness?

50%

50%

T

R

1r2r

nr

.

.

.

Classical Memory 1r2rnr …

Random Numbers from Bell’s 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.

Random Numbers from Bell’s Theorem

We want to explore the relation between non-locality, measured by the violation β of a Bell inequality, and local randomness, quantified by . Clearly, if β =0 → r=1. xapr xa,max

y=1,2

a=+1,-1 b=+1,-1),,( yxbap

x=1,2

NOTE: In all what follows, loopholes are not analyzed. They are important when considering the practical implementation of these ideas.

Results

All the region above the curve

is impossible within Quantum

Mechanics.

Statement of the problem

yxbapc

Qyxbap

xapr

xyab ,,

,,

max

We have developed an asymptotically convergent series of sets approximating the quantum set.

yxbapc

yxbap

xapr

xyab

n

n

,,

,,

max

Other Bell inequalities

The same computation can be done for other inequalities, such as the CGLMP Bell inequality.

y=1,2

a=1,…,r b=1,…,r),,( yxbap

x=1,2

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

Randomness Witnesses

Q MLP

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.

The maximal quantum violation of any of these inequalities always guarantees perfect randomness.

Arbitrarily small amounts of quantum non-locality give perfect randomness.

22CHSH

I

212211 BBABBAI 12

2

2

I

I classical

quantum

Device-Independent Quantum Randomness Expanders

A device violating a Bell’s 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 α 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.

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?

Random Numbers from Bell’s 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.

(C or Q)RNG

Specifications: it passes all statistical randomness tests.

DIQRNE

Specifications:

It won’t pass all the existing randomness tests!

Which device is more random?

Take-home question

Thanks for

your attention!

Post-doc and PhD positions available in the group, see www.icfo.es