# Reiner Werner: Quantum correlations - how to prove a negative from finitely many observations

Reiner Werner, Leibniz Universität, Hannover

**Time: **Wed 2010-09-29 16.00 - 17.00

**Location: **Room 3721, Department of Mathematics, KTH, Lindstedtsvägen 25, 7th floor

One of the fundamental questions of quantum theory is whether the

probabilities, and specifically the correlations predicted by this theory

could alternatively be modeled by a classical probabilistic theory of yet

hidden variables. Whereas complementarity tells us that in quantum theory

there are many measurements which cannot be carried out jointly, the

possibility remains open that by being more inventive, perhaps coming up

with measurements not described by current quantum theory, a classical

description might be restored. Quantum probabilities could then be

understood as resulting from the ignorance of a finer classical microscopic

description, and our technical inability to access this level

experimentally.

Indeed, as long as we look only at the simplest scenario of systems being

prepared and measured on, such extensions are always possible. However, the

situation changes dramatically, if we consider also correlations between

distant, non-interacting parties. In this case a finite experiment,

measuring a certain set of four correlations, combined with a causality

condition, rules out all classical descriptions. The argument given in

rudimentary form by Einstein-Podolski and Rosen in 1935, and much refined by

Bell in the 1960s, will be presented in an elementary way. Moreover, some

general properties of Bell's correlation inequalities, which mark the

boundary of the classically accessible region, will be explained.

Quantum mechanics also implies linear constraints on correlations, the first

of which was established by Tsirelson. The related inequalities can be used

to verify the extremality of correlations, which is a useful property for

quantum cryptography: if such correlations are found between two parties,

quantum mechanics implies that nobody in another part of the world (i.e.,

no eavesdropper) could be correlated with the observed bits. These could

then be used for generating an absolutely private cryptographic key. Thus,

once again, a sweeping negative can be concluded from observed correlations.

In the endeavour of verifying the extremality of quantum correlations the

possibility of a curious gap arises: namely, it is possible that some

correlations allowed by algebraic quantum theory, would be impossible to

generate, even approximately, by finite dimensional systems. The negative

statement implied by the possible observation of such correlations would be

far reaching and very strange: namely, that the experiment is not described

by quantum field theory and related models, which all have good

approximations in terms of finite systems. One may be inclined to conjecture

that such a gap does not exist. Indeed, this conjecture turns out to be

equivalent to a famous an open conjecture of Alain Connes from the 1970s and

to a number of other undecided finite approximation properties, some of

which will be described in the talk.

This talk is given in connection with the program "Quantum Information

Theory" running at the Mittag-Leffler Institute from September to

mid-December.

Coffee and tea served at 15.30.