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TOPIC: The Convexity Club [refresh]
Blogger Matthew Saul Leifer wrote on Aug. 15, 2007 @ 17:31 GMT
Plenty of people have been writing about the recent fqxi conference, which was excellent by the way, so I'll write instead about another fqxi-funded event that happened at the beginning of July in St. Catherine's college, Cambridge.
The two-week workshop was entitled Operational Probabilistic Theories as Foils for Quantum Theory and organized by Rob Spekkens, Jonathan Barrett and...
St. Catherine's College
The two-week workshop was entitled Operational Probabilistic Theories as Foils for Quantum Theory and organized by Rob Spekkens, Jonathan Barrett and Tony Short. It's aim was to try and understand quantum theory by setting it in a wider context of probabilistic theories that follow from more directly intuitive axioms.
This is what the people at the workshop looked like.
Preparing to punt. L to R: Me, Mana, Appleby, d'Ariano, Wootters, Short
Hard at work. L to R: Barrett, Dahlsten, Mana, Toner, Spekkens, arm of Barnum, Short
Much of this work takes place in the "Convex Sets Framework" that has become something of an addiction for a small band of researchers in quantum information recently, including myself. The basic idea is that you assume that states, whatever else they are, should be objects that assign probabilities to all possible measurements that can be made on a system. Since preparation procedures can be mixed, you assume that these form a convex set. Classical and quantum theories provide examples of this. In classical probability theory the state space is a simplex and in quantum theory it is the set of density operators on a Hilbert space.
However, the framework also contains much more general things, e.g. theories in which the Bell inequalities can be violated to a greater extent than in quantum theory whilst still not permitting signaling. This class of theories has attracted considerable attention recently because it is vastly more powerful than quantum theory for certain information processing tasks, e.g. it trivializes communication complexity. Questions that were addressed at the workshop include:
- How can we understand why the world obeys quantum theory rather than any of the other theories?
This has lead to a few new axiomatizations of quantum theory recently, e.g. the work of Hardy and d'Ariano.
- Which features of quantum theory are special to quantum theory and which of them hold generically in all convex theories?
Surprisingly, considering the paucity of assumptions that go into the framework, many of our most cherished ideas of what makes something "genuinely quantum" turn out to hold generically. Examples include no-cloning, no-broadcasting, existence of indistinguishable pure states, measurement-disturbance, violation of Bell inequalities, and many more.
I should also mention that a variety of other approaches and topics were discussed at the workshop, such as toy theories that are not convex (Spekkens), axiomatizations of quantum theory not in the convex sets framework (Goyal), theories in the decoherent histories framework (Dowker), operational approaches to quantum gravity (Hardy), generalizations of the quantum de Finetti theorem (Toner, Renner) and the concept of negative information (Oppenheim).
There was also plenty of free time for discussion at the workshop, and several papers should appear based on them in the coming months.
this post has been edited by the author since its original submission
paul valletta wrote on Aug. 16, 2007 @ 03:02 GMT
"How can we understand why the world obeys quantum theory rather than any of the other theories?"
One can start with the H.U.P?..can one really locate (measure) a particle..anywhere?
If you know a particle's position, you may not know it's momentum. Relative to the process of measure is what one is asking about the process of measure and measurer. The history of a particle WRT time, is "fixed". You can know a particles path in a past history, it is "fixed", without knowing it's path in a future trajectory, "random" and uncertain.
Now WRT the H.U.P, one can make assumptions based on position and momentum, thus:If one knows a particles future path, then it's location history is unknown. (this is my intepretation).
Seems straight forward for observers, time dictates an observer to be the measurer in a "now" context, if the measurer tries to observe a particles future, then the observation will fail to make sense?
Thinking about the "contact" needed for measurement, how does one locate something that has not yet reached there?..I mean I am trying to pinpoint an objects position of where it is "not" ( it's future location ), by determining where it is ( where it HAS been ), the possible way I can determine a full observation measurment, is to perform the measuments at a very fast rate,(signaling the results performed, to confirm measures taken) faster than the devise is capable of?
The is a limit of observation, random variables operate differently for every measure needed, if say one random variable becomes known (random variable of particles future location), then the corrsponding momentum (which is really nothing more than the particles history), will become unknown, or in the context of particle interactions, become a changed (as opposed to fixed, perminant) factor.
This can be translated to the appearance of "unmeasured" particles, and dissapearance of "measured" particles, as QM shows.
I think this is a fairly conventional view, but I don't think the HUP can really be used as a founding principle for quantum theory, at least not in its usual form. In particular, it is not strong enough to entail the canonical commutation relations. In fact, having something like a HUP is another thing that's going to be generic in the framework I described.
Bee wrote on Aug. 20, 2007 @ 13:33 GMT
off topic: there's something wrong with the picture placement using Internet Explorer (pics are on top of each other and cover the text).
Ian Durham wrote on Aug. 27, 2007 @ 01:18 GMT
Actually, the commutation relations *lead* to HUP (well, to Schrödinger's generalization of HUP) so, in a sense, you'd almost be better off building everything on that (the commutation relations). In essence, what I think Matt is describing is a generalization that contains sets of inequalities, one class of which are Schrödinger's generalized HUP, and so forth. Reminds me a bit of something I tried to do once with set theory (http://arxiv.org/abs/quant-ph/0508076). I don't think I was terribly successful, but I'm sure these insanely smart folks will succeed.
Ian, are you the same I T Durham, as cited in this co-incedetally topic relevant recent paper? :http://arxiv.org/abs/0708.3519