Approches opérationnelles de la mécanique quantique

entrée au 25 rue de la Montagne Sainte-Geneviève, Paris 5ème

Organisation: Alexei Grinbaum

10:00 *Alexander Wilce* (Susquehanna University). Aspects of entanglement in general probabilistic theories

11:15 *Robert Spekkens* (University of Cambridge). Liouville's mechanics with an epistemic restriction and Bohr's response to EPR

14:30 *Howard Barnum *(Los Alamos). Information processing in convex operational theories: Toward a characterization of quantum mechanics

15:45 *Bob Coecke* (Oxford University). Monoidal categories as an operational foundation for physical theories

17:00 Discussion

In a well-known generalization of classical probability theory, arbitrary compact convex sets serve as “state spaces” for probabilistic models (with classical models corresponding to simplices). Using a reasonably general notion of a tensor product for such abstract state spaces, many familiar properties of quantum-mechanical entanglement turn out to be generically non-classical, rather than specifically quantum. These include the distinction between proper and improper mixtures, entanglement monogamy, no-cloning and no-broadcasting theorems (Barnum, Barrett, Leifer and Wilce, quant-ph/061195), and even a version of the measurement problem. After sketching the results just mentioned, I’ll discuss what I take to be their implications for the project of providing a realist, “no-collapse” interpretation of quantum mechanics — or, indeed, of any non-classical probabilistic theory.

**Robert Spekkens (****University**** of ****Cambridge****)
**

I will discuss a toy theory that reproduces a wide variety of qualitative features of quantum theory for degrees of freedom that are continuous. The ontology of the theory is that of classical particle mechanics, but it is assumed that there is a constraint on the amount of knowledge that an observer may have about the motional state of any collection of particles — Liouville mechanics with an epistemic restriction. The formalism of the theory is determined by examining the consequences of this “classical uncertainty principle” on state preparations, measurements, and dynamics. The result is a theory of hidden variables, although it is not a hidden variable model of quantum theory because it is both local and noncontextual. Despite admitting a simple classical interpretation, the theory also exhibits the operational features of Bohr’s notion of complementarity. In fact, it includes all of the features of quantum mechanics to which Bohr appeals in his response to EPR. This theory demonstrates, therefore, that Bohr’s arguments fail as a defense of the completeness of quantum mechanics.

**Howard Barnum (****Los Alamos**** National Laboratory)
**

I give an exposition of a convex operational framework for possible physical theories, and review work and work-in-progress, primarily by myself and collaborators, on the information-processing properties of theories in this framework. The main results reviewed are the fact that the only information that can be obtained in the framework without disturbance is inherently classical, the probable existence of exponentially secure bit commitment in non-classical theories without entanglement, and the consequences for theories of the existence of a conclusive teleportation scheme. I’ll also discuss results on the “remote steering” of ensembles using entanglement, in particular a sufficient condition for all states to be remotely steerable, thereby rendering insecure bit commitment protocols of the form shown to be secure in the unentangled case.

I discuss these results in light of the broader program of obtaining information-theoretic characterizations of quantum mechanics, commenting on their potential philosophical and physical significance as part of an as-yet-unavailable full characterization, and also in light of the beautiful *mathematical* characterization of the unnormalized state spaces of finite-dimensional quantum theory over the reals, complex numbers, and quaternions, as three of the four infinite families of homogeneous self-dual cones (equivalently, state spaces of Jordan algebras).

Much of this is joint work with various groups of collaborators including Jon Barrett, Matt Leifer, Alexander Wilce, Oscar Dahlsten, and Benjamin Toner.

**Bob Coecke (****Oxford ****University**** Computing Laboratory)
**

We provide the motivation and discuss the conceptual significance of a particular research program which aims to equip quantum physics with an operational foundation, a genuine logic, an intuitive purely diagrammatic calculus, and where all structural elements should represent established empirical facts. The background structure for this program are symmetric monoidal categories: objects are types of systems, morphisms are operations thereon, composition is sequential application of operations, and the tensor structure represents joint systems. Conceptually this means that the tensor product is a primitive ingredient of the axiomatics. Hence our approach substantially differs from most other operational approaches where the primal concepts are typically measurement related. Computer Scientists welcome this particular axiomatics since it induces an extremely powerful logical system which enables automated design and verification. Some informal introductionary texts are:

· Introducing Categories to the Practicing Physicist

· Kindergarten Quantum Mechanics

The main recent development in this research program is the ability to capture quantum measurements (pure and mixed), classical data manipulations, phase data, and quantum informatic quantities and concepts within the language which was initially designed to reason about quantum entanglement. We are for example able to distinguish between classical non-determinism, stochastic processes, reversible classical processes, etc. At the core of all this lies an analysis of the abilities to clone and delete data in the classical world `from the perspective of the quantum world’. In this view, the classical world looks surprisingly complicated as compared to the very simple quantum world. This is a situation which computer scientists already encountered a while ago: classical logic becomes much easier to manipulate if you decompose it as:

Ø classical logic = linear logic + (copying, deleting).

Our stance is:

Ø classical world = quantum world + (copying, deleting).

Some examples of things we can do within our purely graphical calculus are:

· derive dense coding and teleportation-like schemes including classical control;

· prove Naimark's extension theorem for POVMs;

· establish informatic resource inequalities involving coherent communication;

· prove universality of measurement based computational schemes;

· compute the quantum Fourier transform.

Several people contributed to this research program including Samson Abramsky (Oxford), Ross Duncan (Oxford), Eric Paquette (Universite de Montreal, Canada), Dusko Pavlovic (Kestrel Institute, Palo Alto, US), Simon Perdrix (PPS, Paris), Peter Selinger (Dalhousie, Canada), Jamie Vicary (Imperial College).

#872 - Last update : 09/04 2007