Keep on splittin'

OK, I just had to copycat this link.  Via the Quantum Pontiff, Universe Splitter, a supposed new app for the iphone that supposedly hooks up to a quantum randomness generator, allowing you to condition your decisions on quantum randomness and ensure---if you believe in the Everett interpretation of quantum mechanics---that you can "have your cake and eat it too.  I hope this thing's for real, but unless it's just my lack of App savvy, Apple may not be buying the Everett interpretation.

Keep on Splittin

Keep on Splittin'

Gross, Mueller, Colbeck, and Dahlsten: "All reversible dynamics in maximally non-local theories are trivial"

David Gross, Markus Mueller, Roger Colbeck, and Oscar Dahlsten have considered the "maximal non-signaling tensor product" of "boxlets", and shown that the reversible dynamics of this state space consists just of permutations of the systems (the boxlets) followed by reversible local transformations (i.e., ones on the individual boxlets).

What the heck does that mean, you ask?  Well, "boxlets" were introduced in several contexts.  In the "operational quantum logic" literature they're sometimes called "semiclassical test spaces".  In quantum foundations and informations, they were introduced as a generalization of a notion of Popescu and Rohrlich, who introduced the two-measurement, two-outcome-per-measurement boxlet in order to describe correlations between measurement results on distinct systems that can be stronger than quantum correlations, but still don't allow someone ("Alice") in possession of one of the systems to signal to the other ("Bob") just by making measurements on her system.

A "boxlet" is a system on which there are M distinct alternative measurements one can make, each with K outcomes.  (More complex versions allow different measurements to have different numbers of outcomes.)  The allowable states of a boxlet are given by specifying M probability distributions, each one over K alternatives:  for each measurement, the probabilities of each of its K outcomes.  All possible lists of M such distributions are allowed; this is a convex, compact subset of an MK dimensional vector space (one dimension for each probability).  The M normalization constraints mean that this set lies in an MK-M (i.e., M(K-1)) dimensional affine subspace  (higher dimensional generalization of the line, plane, etc... of high-school geometry).  The possible states of a pair of such systems are given by the "maximal tensor product" of a pair of these compact convex state spaces.  The technical definition of maximal tensor product of state spaces can be found here.  Another way of defining this is that it's the state space of the  Foulis-Randall tensor product  of the "test spaces" (definitions reviewed in Sections II and IV of this paper) describing each of the boxlets.  A test space is just a collection of subsets of some set; the elements of the set interpreted as measurement outcomes, and the subsets, called "tests", as measurements.  The semiclassical test space of a boxlet like the ones I described above just consists of a set of MK elements, partitioned into M sets of K elements.  A state on a (finite, like the ones in question) test space is a function from the set to the real numbers between zero and 1, i.e. to probabilities, such that for each test, the probabilities of the elements of the test add up to one.  The Foulis-Randall tensor product of two test spaces just takes their Cartesian product, and allows any probability assignments such that the "marginal states" obtained by fixing a measurement on one side and marginalizing all the joint distributions of this fixed measurement with measurements on the other side, is independent of which measurement is marginalized over.  That is, Alice can't signal to Bob (by affecting the probabilities of the outcomes of one or more of his measurements) just by her choice of measurement.

Now, a transformation of the state space is an affine map from the state space to itself (i.e. one that preserves convex combination, which seems only reasonable), and a reversible one is one that has an inverse that is also an affine map of the state space.  So what GMCD are saying is that, if you combine boxlets this way, there are no very interesting reversible dynamics:  just combinations of local reversible dynamics on the individual boxlets, and permuting the boxes amongst themselves.

An interesting question is, can one extend this result to maximal tensor products of *arbitrary* systems with convex state space (locally equipped, let's say, with the maximal set of possible effects)?

See the comments on the Information Causality thread at Dave Bacon's blog for a bit more discussion (and related interesting matters).