Marco Piani course @ UFF – Lecture 1

To inaugurate the blog, we’ll post summaries of the lectures of the mini-course on “Non-classical correlations: from quantum non-locality to quantum discord” , given by Prof. Marco Piani (University of Waterloo, Canada) at UFF from 26 – 30 Aug 2013.

Lecture 1 – Mon 26 Aug

Marco kicked off by introducing a general scenario that he will explore during much of the course. Two participants, Alice and Bob, initially share some joint physical system which they can each separately probe. The question is: what are the possible correlations between the information that each of them can extract, given the resources available to them? One of the main themes of the course is that the answer may be very different depending on whether these resources are classical or quantum (or even, conceivably, something else!)

He first considered the situation where the system can indeed be described by quantum mechanics, and where the information is extracted by quantum measurements. He pointed out that in this case there are two different levels at which our question can be addressed: on the first, within the usual quantum formalism, we can study the properties of the density operator  \rho_{AB} that describes the joint system. For example, we can ask whether it is separable, ie, a convex combination of product states. If it is, then it can be assembled by Alice and Bob from scratch using only local quantum operations and classical communications. Separable states form a small subset of the wider set of inseparable, or entangled states, and we can study the geometry of this set.

On a second level, we can look at the correlations between the results of actual measurements that Alice and Bob perform on their local quantum subsystems. More formally: using the letters x,y to denote Alice and Bob’s respective choices of measurements then quantum mechanics  provides rules for calculating the conditional probabilities p(a,b | x,y) that they obtain, respectively the measurement results a,b.

Marco then pointed out that, from a purely abstract point of view,  the classical parameters x,y and a,b can be seen simply as the possible inputs and outputs of a random process. Analogous probability distributions may be defined for any abstract theory – not necessarily quantum- that might seek to describe the results of the measurements by Alice and Bob. As it turns out, though, the kinds of distribution that may appear can vary greatly, depending on the underlying theory. In particular,  in Lecture 3 we will see that, for certain choices of initial state, the distributions obtained from quantum mechanics have correlation properties that cannot be reproduced by any local-realistic classical theory. Similarly, we will also see that there are limits to the correlations produced by quantum theory, and that it is possible to construct non-trivial supra-quantum theories that nevertheless preserve some of its properties.

Finally, Marco mentioned another idea closely linked to the theme of quantum vs classical correlations, that of ‘quantum discord’. He introduced this concept by means of a game-like scenario: Alice and Bob again begin sharing a quantum state \rho_{AB}, but now Alice’s task is to convey instructions to a third party, Charlie, so that he can share the same state \rho_{CB} with Bob. The catch is that these instructions must be sent in the form of classical information only (ie, via a classical channel). It turns out that this task is generally impossible, for nearly all states \rho_{AB} ! In fact, Marco showed that this can be done only for a very small subset of the separable states, known as “Classical-Quantum states”. These states are defined as having zero ‘quantum discord’. In Lecture 4, we will see a formal definition of this quantity, valid for any state, and discuss its meaning.

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