QCCC07, Aschau, October 2007 Miguel Navascués Stefano Pironio Antonio Acín ICFO-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: No-signalling principle Alice Bob y a b x

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. 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? Example: Are these correlations quantum?

Motivation Physical principles impose limits on correlations Example: 2 inputs of 2 outputs CHSH inequality BLMPPR 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 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’

Example Given p(a,b|x,y), take 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. SDP techniques

Hierarchy of necessary conditions Constraints:

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. NO YES NO YES Is the hierarchy complete? YES

Hierarchy of necessary conditions If some correlations satisfy all the hierarchy, then: with ? Rank loops: If at some point rank(γ(N))=rank(γ(N+1)), the distribution is quantum.

Applications Application 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, 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 x1 1 y y1 When do there exist values for x and y such that this matrix is positive? Cirelson, Landau and Masanes

Applications Application 2: Maximal Quantum violations of Bell’s inequalities max such that

Applications Examples: CHSH Cirelson’s bound CGLMP (d=3) 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 p L and defined as. Clearly, if β=0 → p L =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

Alice $ QRG Eve Colbeck & Kent Trusted 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 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.

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