Friday, May 29, 2015

The unreasonable effectiveness of composition axioms

This blog is called Elliptic Composability to celebrate the method of quantum mechanics reconstruction: the invariance of the laws of Nature under composition. The majority of other attempts to derive quantum mechanics from physical principles contain composition axioms:
  • Barnum and Wilce : “composites are locally tomographic”, 
  • Dakic and Brukner : “the state of a composite system is completely determined by local measurements on its subsystems and their correlations”, 
  • Lluis Masanes and Markus Muller : “the state of a composite system is characterized by the statistics of measurements on the individual components”, 
  • Chiribella, D’Ariano, and Perinotti: “if two states of a composite system are different, then we can distinguish between them from the statistics of local measurements on the component systems”, 
  • Lucien Hardy: “Composite systems rules” (\(N_{A\otimes B} = N_A N_B\) and \(K_{A\otimes B} = K_A K_B \) where N is the dimension of the state space and K are the number of degrees of freedom). .
So the big question is WHY? Why is composition occurring so often in quantum reconstruction programs? One possible answer is this: quantum mechanics is strange because it predicts correlations larger than classical correlations. Therefore to distinguish "quantumness" from classical physics one needs to talk about correlations which by its very definition requires composites. 

However a much deeper mathematical reason is lurking about and I was very much surprised to uncover why composition requirements are so "unreasonable effective" in reconstructing quantum mechanics. Basically there is a unique connection between the tensor products and products and because of it composition axioms (which involve tensor products) have algebraic consequences.

Here is how it works:

To any bilinear product \(f\) between A and B we can associate a linear product \(\hat{f}\) such that the diagram above commutes (see 1.6 in ). This is a Universal Property

Now quantum mechanics has an underlying algebraic structure: a C* algebra. and there are several products which are of importance: the commutator which gives the time evolution in the Heisenberg picture, the Jordan product, the usual complex multiplication for the operators. So secretly when we impose composition requirements we implicitly constrain the behind the scene algebraic structure. And algebra is like playing the piano: distinct notes and distinct well defined algebraic structure. For example in Lucien Hardy's approach, Jochen Rau was able to clarify and expand the arguments by introducing dimensional arguments for the Lie groups simply because there are only a handful of classical Lie groups (unitary, orthogonal, simplectic).

Similar arguments can be made for all of the other reconstruction approaches above. 

Now once we realize the power of the composition arguments we can take full advantage of it and squeeze every ounce of mathematical consequences from it. This is what I've done in: Happy reading!

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