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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.

Colloquia 2010

Title Date
Torsten Ekedahl: The Sato-Tate conjecture Nov 03, 2010
Jesper Grodal: Finite loop spaces Nov 10, 2010
Amol Sasane: An analogue of Serre’s Conjecture and Control Theory Oct 13, 2010
Reiner Werner: Quantum correlations - how to prove a negative from finitely many observations Sep 29, 2010
Warwick Tucker: Validated Numerics - a short introduction to rigorous computations Sep 22, 2010
Idun Reiten: Cluster categories and cluster algebras Sep 01, 2010
Stefano Demichelis: Use and misuse of mathematics in economic theory May 26, 2010
Gregory G. Smith: Old and new perspectives on Hilbert functions Apr 14, 2010
Tony Geramita: Sums of Squares: Evolution of an Idea. Mar 31, 2010
Jens Hoppe: Non-commutative curvature and classical geometry Mar 24, 2010
Margaret Beck: Understanding metastability using invariant manifolds Mar 03, 2010
Jan-Erik Björk: Glimpses from work by Carleman Feb 10, 2010