[HumJanenge] Fwd: Deriving quantum theory from information processing axioms

This forward is for those few (presumably) people on this mailing list who are interested in information theory / processing in addition to simply information per se, and who can see why this is relevant to RTI..

Sarbajit
Moderator

---------- Forwarded message ----------
From: Russ Abbott <r...t@g...com>
Date: Tue, Jul 26, 2011 at 3:20 AM

From APS Physics.

We know how to use the "rules" of quantum physics to build lasers, microchips, and nuclear power plants, but when students question the rules themselves, the best answer we can give is often, "The world just happens to be that way." Yet why are individual outcomes in quantum measurements random? What is the origin of the Schrödinger equation? In a paper [1] appearing in Physical Review A, Giulio Chiribella at the Perimeter Institute inWaterloo, Canada, and Giacomo Mauro D'Ariano and Paolo Perinotti at the University of Pavia, Italy, offer a framework in which to answer these penetrating questions. They show that by making six fundamental assumptions about how information is processed, they can derive quantum theory. (Strictly speaking, their derivation only applies to systems that can be constructed from a finite number of quantum states, such as spin.) In this sense, Chiribella et al.'s work is in the spirit of John Wheeler's belief that one obtains "it from bit," in other words, that our account of the universe is constructed from bits of information, and the rules on how that information can be obtained determine the "meaning" of what we call particles and fields.

They assume five new elementary axioms—causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning—which define a broad class of theories of information processing. For example, the causality axiom—stating that one cannot signal from future measurements to past preparations—is so basic that it is usually assumed a priori. Both classical and quantum theory fulfil the five axioms. What is significant about Chiribella et al.'s work is that they show that a sixth axiom—the assumption that every state has what they call a "purification"—is what singles out quantum theory within the class. In fact, this last axiom is so important that they call it a postulate. The purification postulate can be defined formally (see below), but to understand its meaning in simple words, we can look to Schrödinger, who in describing entanglement gives the essence of the postulate: "Maximal knowledge of a total system does not necessarily include maximal knowledge of all its parts." (Formally, the purification postulate states that every mixed state ρ_A of system A can always be seen as a state belonging to a part of a composite system AB that itself is in a pure state Ψ_AB. This pure state is called "purification" and is assumed to be unique up to a reversible transformation on B).

Chiribella et al. conclude there is only one way in which a theory can satisfy the purification postulate: it must contain entangled states. (The other option, that the theory must not contain mixed states, that is, that the probabilities of outcomes in any measurement are either 0 or 1 like in classical deterministic theory, cannot hold, as one can always prepare mixed states by mixing deterministic ones.) The purification postulate alone allows some of the key features of quantum information processing to be derived, such as the no-cloning theorem or teleportation [7]. By combining this postulate with the other five axioms, Chiribella et al. were able to derive the entire mathematical formalism behind quantum theory.

 

-- Russ Abbott
_____________________________________________

  Professor, Computer Science
  California State University, Los Angeles