Replacing Nothing with Something Special: Contextuality-by-Default and Dummy Measurements

Abstract: The object of contextuality analysis is a set of random variables each of which is uniquely labeled by a content and a context. In the measurement terminology, the content is that which the random variable measures, whereas the context describes the conditions under which this content is measured (in particular, the set of other contents being measured "together" with this one). Such a set of random variables is deemed noncontextual or contextual depending on whether the distributions of the context-sharing random variables are or are not compatible with certain distributions imposed on the content-sharing random variables. In the traditional approaches, contextuality is either restricted to only consistently-connected systems (those in which any two content-sharing random variables have the same distribution) or else all inconsistently-connected systems (those not having this property) are considered contextual. In the Contextuality-by-Default theory, an inconsistently connected system may or may not be contextual. There are several arguments for this understanding of contextuality, and this note adds one more. It is related to the fact that generally not each content is measured in each context, so there are "empty" content-context pairs. It is convenient to treat each of these empty pairs as containing a dummy random variable, one that does not change the degree of contextuality in a system. These dummy random variables are deterministic ones, attaining a single value with probability 1. The replacement of absent random variables with deterministic ones, however, can only be made if one allows for inconsistently-connected systems. KEYWORDS: contextuality, dummy measurements, inconsistent connectedness, random variables.