Formalizing the separability condition in Bell's theorem

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Abstract

The nonseparability of physical systems is often invoked in philosophical analyses of what has come to be known as Bell's theorem. Until recently, the formalization of the notion of separability was assumed to be unproblematic, equivalent to that of outcome independence (Jarrett incompleteness). Although this equivalence has been called into question, an alternative has not yet been specified with sufficient precision, leading to confusion as to what kinds of models should be considered separable. I identify four plausible candidates for the proper formalization of the separability condition, understood in terms of part–whole determination, and then discuss the relative merits of each. I show that three of these are, in fact, equivalent to outcome independence under most conditions, and that one is not, raising the possibility of a new decomposition of Bell locality. I also question whether part–whole determination should be considered sufficient for separability.

Section snippets

Historical background

Bell (1964) considered systems apparently composed of smaller subsystems capable of being subjected to independent measurement (e.g., a pair of electrons). He assumed what he termed a “locality” condition, meant to capture the notion that only a subsystem's local state should determine the outcomes of measurements performed on it, and then derived inequalities that quantum mechanics could be shown to violate (similar inequalities were subsequently confirmed by experiment). If two subsystems

Definitions

In this section, I offer definitions of the terms needed to state the separability condition as well as some related conditions.

Conceptual definition

I will follow Winsberg and Fine in adopting Howard's conceptual definition of separability, which consists of two subconditions:

Definition

Howard's conceptual definition of separability:

  • 1.

    Each (system) possesses its own, distinct physical state.

  • 2.

    The joint state of the two systems is wholly determined by these separate states. (Howard, 1989, p. 226).

In a footnote on the same page, Howard clarifies the second condition: “To say that the joint state is wholly determined by the separate states is to say that the

Separability does and does not imply factorizability

In this section, I show that the three pointwise conditions imply factorizability, whereas the only viable functionwise one does not.

Review

Careful analysis of the formalization of the separability condition in Bell's theorem brings several ambiguities to the fore, only one of which resolves itself to any degree of satisfaction. The first and most important ambiguity concerns the conceptual definition of separability itself; Winsberg and Fine are content to consider only the way in which the properties of an ostensibly composite system are related to those of its apparent subsystems (i.e., the whole to its parts). If the subsystems

Acknowledgment

I am deeply grateful and indebted to Don Howard for his help and helpful criticisms in the preparation of this text. I would also like to thank Claus Beisbart, Jeremy Butterfield, Steven French, and Holger Lyre, as well as the other participants in the 2004 Bonn–Cologne Philosophy of Physics Spring School, and Nick Huggett, Jon Jarrett, Steven Leeds, as well as the other members of the Chicagoland philphys discussion group, for their encouragement and helpful feedback after presentations of

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