QCCC07, Aschau, October 2007

Miguel Navascués

Stefano Pironio

Antonio AcínICFO-Institut de Ciències Fotòniques (Barcelona)

Cryptographic properties of nonlocal correlations

Characterization of quantum correlations

Motivation

Information is physical.

If it is guaranteed that there is not causal influence between the parties:

xapyxbapb

),,( No-signalling principle

Alice Bob

y

a b

),,( yxbap

x

Motivation

If the correlations have been established using classical means:

,,),,( ybqxappyxbap

This constraint defines the set of “EPR” correlations. Independently of fundamental issues, these are the correlations achievable by classical resources. Bell’s inequalities define the limits on these classical correlations.

Clearly, classical correlations satisfy the no-signalling principle.

Motivation

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

ybxaAB MMtryxbap ,,

1

1

b

yb

a

xa

M

M

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?

Motivation

Physical principles impose limits on correlations

Example: 2 inputs of 2 outputs

1,0,,, yxba

CHSH inequality

NL machine

QM

Local points

22Q

2L

4NBLMPPR

SNSQSL Popescu-RohrlichBell

Motivation

• What are the allowed correlations within our current description of Nature?

• How can we detect the non-quantumness of some observed correlations? Quantum Bell’s 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

Necessary conditions for quantum correlations

It is assumed there exists a quantum state and measurements reproducing the observed probability distribution

''''''

'' 11111 y

bABxa

ybAB

xaB

xa MMMMMX

A set consisting of product of the measurement operators is considered

Then, is such that ABjiij XXtr 0

Given two sets X, X’ if

X is at least as good as X’

j

jjii XcX '

Example ybAB

xa MMX 11Given p(a,b|x,y), take

1

BxaM 1 y

bA M1 yxbapMMtr y

bxaABij ,,

ybpMtr ybAABii 2

1

011 ' ybA

ybAABij MMtr

ijybA

ybAABij xMMtr '

'11

Do there exist values for the unknown parameters such that ?

Recall: if p(a,b|x,y) is quantum, the answer to this question is yes.ijx 0

SDP techniques

Hierarchy of necessary conditions

2

BxaM 1 y

bA M1

1yb

xa MM B

xa

xaMM 1'

'

Constraints: AByb

xaAB

yb

xa

yb

xa MMtrMMMMtr

Hierarchy of necessary conditions

We can define the set X(n) of product of n operators and the corresponding matrix γ(n). If a probability distribution p(a,b|x,y) satisfies the positivity condition for γ(N), it does it for all n ≤ N.

1

NO

2

NO

YES YES

NO

YES

Is the hierarchy complete?

YES

Hierarchy of necessary conditions

If some correlations satisfy all the hierarchy, then:

ybxaMMtryxbap ,, with

a

xa

yb

xa

M

MM

1

0,

?

ybxaAB MMtryxbap ,,

Rank loops: If at some point rank(γ(N))=rank(γ(N+1)), the distribution is quantum.

ApplicationsApplication 1: Quantum correlators in the simplest 2x2 case

We restrict our considerations to the correlation values c(x,y)=p(a=b|x,y)- p(a≠b|x,y)

In the quantum case,

AByxtryxc , c

c(0,0) c(0,1)

c(1,0) c(1,1)

c(0,0) c(0,1)

c(1,0) c(1,1)

1 x

x 1

1 y

y 1When do there exist values for x and y

such that this matrix is positive?

1,1arcsin0,1arcsin1,0arcsin0,0arcsin cccc

Cirelson, Landau and Masanes

1,11,1,,0,01,1,,0,00,0,, pppyxbap

0a 1a 0b 1b

Applications

Application 2: Maximal Quantum violations of Bell’s inequalities

yxbapc yxba ,,,,, nBtr 0n

max

such thatnT

12 TTT

Applications

Examples:

CHSH

cBtrcccc 1,10,11,00,0 B

1 1

1 -1

0 0

0 0

0 0

0 0

1 1

1 -1221 T Cirelson’s bound

CGLMP (d=3) 1547.39149.2 12 TT

Quantum value! ADGL

The same results hold up to d=8

Our results provide a definite proof that the maximal violation of the CGLMP inequalities can be attained by measuring a nonmaximally entangled state

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 β the amount of violation of a Bell’s inequality, and local randomness, measured by pL and defined as . xapp xaL ,max

Clearly, if β=0 → pL=1. 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).

Intrinsic Quantum Randomness

Intrinsic Quantum RandomnessAlice

$

QRG

Eve

Colbeck & KentTrusted Random

Number Generator:

Ask for a device able to get the maximal quantum violation of the CHSH inequality.

The same result is valid for other inequalities of larger alphabets. Using one random bit, one gets a random dit.

Intrinsic Quantum Randomness

Q M

LP

Is maximal non-locality needed for perfect randomness? What about the other extreme correlations?

1,1arcsin0,1arcsin1,0arcsin0,0arcsin cccc

For any point in the boundary → Bell-like inequality.

Its maximal quantum violation gives perfect local randomness.

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?• What’s 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.

Thanks for

your attention!

Miguel Navascués, Stefano Pironio and Antonio Acín, PRL07