The Entanglement Structure of Quantum Field Systems

Abstract

This article discusses the peculiar features of quantum entanglement and quantum non-locality within the algebraic approach to relativistic quantum field theory (RQFT). The debate on the ontology of RQFT is considered in the light of these well-known but little discussed features. In particular, this article examines the ontic structural realist understanding of quantum entanglement and quantum non-locality and its contribution to this debate.

Acknowledgements

I am grateful to the Australian Research Council (ARC, Discovery Early Career Researcher Award (DECRA), project DE120102308) and to the Swiss National Science Foundation (SNSF, Ambizione grant PZ00P1_142536/1) for financial support.

Notes

How exactly these fundamental properties might account for what we experience (e.g. definite measurement outcomes) also depends on the strategy one adopts with respect to the measurement problem: see section 8.

From an operational point of view, one can make sense of cyclicity by highlighting the need of selective operations in the considered region O (Clifton and Halvorson 2001, 17–19).

Strictly speaking, such generic entanglement is not specific to the quantum field-theoretic domain (Clifton and Halvorson 2001, 21–23).

It is rather surprising that such fact, which can be fruitful for fundamental interpretative issues in quantum theory, does not seem to have been widely exploited yet. Such discussion would obviously require separate treatment.

The terminology is Shimony's, where outcome independence is understood as the stochastic independence of the measurement outcomes on one entangled system from the outcomes on the other systems (we consider the case of two systems for simplicity); outcome independence can be convincingly interpreted as a separability condition, although it has not to be so, see Fogel (2007); for a critical view, see Maudlin (2011, 85–90).

Stochastic independence of the measurement outcomes on one system from the measurement settings on the other.

Note that this characterization of OSR does not entail that there are no relata or no objects, only that they cannot exist and have any identity independently of the structures they are part of.

This is a more restricted aim than within the original French and Ladyman's OSR, which also explicitly aims to account for theory change (Ladyman 1998; French and Ladyman 2003; French 2006).

The cases of similar elementary quantum particles and of quantum entanglement are related but clearly distinct for the point of view of OSR; the latter case has been far less discussed within the framework of OSR, see Esfeld (2004), Ladyman et al. (2007, section 3.4), and Lam and Esfeld (2012).

We do not really need the distinction between relational properties and relations insofar as the considered physical relations give rise to the relata having physical relational properties and the other way round. This latter relationship between relations and relational properties should however be reminded in order to avoid some confusion about the ontological commitment of OSR: contrary to what Ainsworth (2010) claims, OSR is not committed to saying that entangled quantum systems have no properties.

There is in any case a precise sense in which entangled quantum systems can be meaningfully considered to stand in (possibly weakly discernible) relation, see the careful discussion in Muller and Saunders (2008); therefore, the ‘third problem’ for OSR in Ainsworth (2010, 54) is clearly avoided.

As mentioned above (section 4) and further discussed below (section 8), the exact nature of the quantum entanglement relations also depends on the interpretative strategy with respect to the measurement problem. More broadly, one can here consider relations of quantum non-locality, which feature in all (realist) interpretations to the extent that these latter all have to account for the experimentally verified violation of Bell-type inequalities.

French (2012) interestingly suggests various ways to account for the existence of unitary inequivalent representations within the interpretative framework of OSR.

Conceptual and methodological worries about quantum entanglement and non-separability were most famously expressed by Einstein (1948, 321–322), quoted and translated in Howard (1985, 187–188).

For a pair of (von Neumann) algebras , the split property is basically equivalent to the existence of an isomorphism between the algebra they generate and the (von Neumann algebra) tensor product of and . In particular, this fact implies the existence of a product state on .

The split property ensures that quantum field systems can be independently and locally prepared in arbitrary states. The split property is typically satisfied in operationally interesting cases, e.g. for strictly space-like separated quantum field systems (see the discussion in Clifton and Halvorson 2001, 28–29; Summers 2009).

The trope bundle interpretation of AQFT proposed in Kuhlmann (2010) also goes along these lines to some extent.

An important and difficult project would be to further substantiate this claim by explicit investigations of the different (realist) approaches to the measurement problem in the RQFT domain, e.g. Bohmian RQFT, dynamical collapse RQFT and Everett RQFT. Contrary to what Esfeld (2013) suggests, the fact that OSR provides a general ontological framework which remains genuine and can be further specified within these different quantum interpretations does constitute a positive and convincing feature of this conception. It actually provides strong evidence that OSR captures genuine ontological features of the world as described by current fundamental physics.