Saturday, January 19, 2008

Contextuality

The Quantum Pontiff has a nice post on The Contextuality of Quantum Theory in Ten Minutes. Contextuality is important in understanding the relationship between quantum models and classical models.

He has a useful footnote that I will reproduce:

A postscript for those who want to read more: the example described above is taken from David Mermin's article, "Hidden variables and the two theorems of John Bell" Rev. Mod. Phys. 65, 803 - 815 (1993). Unfortunately this may be unaccessible to those outside of academia. A nice summary with links to more information is The Kochen-Specker Theorem at the Standford Encyclopedia of Philosophy.
There are some other interesting perspectives on contextuality from the same period. For example, Emilio Santos proves that a contextual model for a quantum mechanical system is always possible, in "Problem of Hidden Variables", Int. J. Theor. Phys. 31, 1909-1913 (1992).

Abstract: The problem of hidden variables in quantum mechanics is formalized as follows. A general or contextual (noncontextual) hidden-variables theory is defined as a mapping f: Q×M rarr C (f: QrarrC) where Q is the set of projection operators in the appropriate (quantum) Hilbert space, M is the set of maximal Boolean subalgebras of Q and C is a (classical) Boolean algebra. It is shown that contextual (noncontextual) hidden-variables always exist (do not exist).
This paper gives a mathematically nice definition of a contextual hidden-variable theory, where almost all definitions are given in words. As well as saying what quantum mechanical measurement operator O is to be used, we specify a maximal Boolean subalgebra B of Q as well, which defines an equivalence class of experimental apparatuses that might be used to make the measurement O, and tells us that any measurement in B could potentially be made, using one of the class of measurement apparatuses defined by B, without disrupting the O measurement.
Santos observes that "Contextual hidden-variable theories are unsatisfactory in many respects. In particular, they are nonlocal, in Bell's (1965) sense, as can be shown without difficulty". The question here is what definition of classical locality Santos intends?

At about the same time, Rob Clifton, "
Getting contextual and nonlocal elements-of-reality the easy way", Am. J. Phys. 61, 443-447 (1993), says that
"if we endorse the spirit of the EPR program --- that all observables possess values that are independent of what measurements locally or at space-like separation are performed on the system --- then a contradiction with quantum predictions ensues. So these predictions make contextualism (dependence on the local measurement context) and nonlocality (dependence on the space-like separated measurement context) inescapable features of any interpretation of quantum theory that postulates elements-of-reality."
This is obviously a different definition of contextuality and of nonlocality. Both this definition and that of Santos make no distinction between dynamical nonlocality (which is classically anathema except in thermodynamics, such as solutions of the heat equation) and the nonlocality of initial conditions (which is not a problem in classical Physics -- parts of the experimental apparatus are at space-like separation from one another).

There are also different kinds of Contextuality. Claudio Garola, Jarosław Pykacz, and Sandro Sozzo, in "Quantum Machine and Semantic Realism Approach: a Unified Model", Found. Phys. 36, 862-882 (2006), (or, arXiv:quant-ph/0507133, 14 Jul 2005) give two distinct definitions,
"According to a standard viewpoint, a physical theory is contextual whenever the value of an observable A in a given state of a physical system depends on the set of (compatible) measurements that are simultaneously performed on the system [Here, a citation to the Mermin article cited above]. We call this kind of contextuality here contextuality1, and note that no reference is made in its definition to individual differences between apparatuses measuring A, which are thus implicitly considered ideal and identical. On the contrary, according to the GB [Geneva-Brussels] approach the contextuality of the quantum machine follows from the fact that each individual experiment introduces a different set of hidden variables of the measuring apparatus, so that different measurements of the same observable may yield different results[Here, a citation to an article by Aerts, who is a principal of the Geneva-Brussels approach]. This provides implicitly a different definition of contextuality, that we call here contextuality2, which makes reference to the differences that unavoidably exist between individual apparatuses measuring A."
Contextuality2 is clearly different from the logical kind of contextuality given by Santos, which is driven by the concerns of hidden-variable models for quantum theory. Santos's definition is essentially a mathematically clear definition of Contexuality1, which Garola et al. could usefully have cited as well as Mermin's paper.

There will be more, but enough for now.