## Abstract

Quantum entanglement is widely believed to be a feature of physical reality with undoubted (though debated) metaphysical implications. But Schrödinger introduced entanglement as a theoretical relation between representatives of the quantum states of two systems. Entanglement represents a physical relation only if quantum states are elements of physical reality. So arguments for metaphysical holism or nonseparability from entanglement rest on a questionable view of quantum theory. Assignment of entangled quantum states predicts experimentally confirmed violation of Bell inequalities. Can one use these experimental results to argue directly for metaphysical conclusions? No. Quantum theory itself gives us our best explanation of violations of Bell inequalities, with no superluminal causal influences and no metaphysical holism or nonseparability—but only if quantum states are understood as objective and relational, though prescriptive rather than ontic. Correct quantum state assignments are backed by true physical magnitude claims: but backing is not grounding. Quantum theory supports no general metaphysical holism or nonseparability; though a claim about a compound physical system may be significant and true while similar claims about its components are neither. Entanglement may well have have few, if any, first-order metaphysical implications. But the quantum theory of entanglement has much to teach the metaphysician about the roles of chance, causation, modality and explanation in the epistemic and practical concerns of a physically situated agent.

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## Notes

By this word choice he unconsciously anticipated the monogamy of (maximal) entanglement.

Interestingly he also used the word ‘verheddert’ on the same page to describe the way a system’s \(\psi \)-function gets tangled up with that of an apparatus during a measurement. He uses only ‘Verschränkung’ and ‘verschränkt’ throughout the rest of the paper.

Or even to its adequacy:

Indubitably the situation described here is, in present quantum mechanics, a necessary and indispensable feature. The question arises, whether it is so in Nature too. I am not satisfied about there being sufficient experimental evidence for that. (Schrödinger 1936, p. 451)

Entanglement is still a necessary and indubitable feature of quantum theory—a feature that has by now received exhaustive experimental confirmation.

The concept is now regularly applied to states of two or more systems that have never interacted, and even retrospectively to systems that no longer exist (e.g. in entanglement-swapping scenarios, some of which include delayed choice). It has been generalized to apply to other kinds of representations of states, also in relativistic quantum mechanics and quantum field theory.

As the example shows, a Hilbert space generally has multiple inequivalent decompositions as a tensor product of Hilbert spaces.

See (Fine 1986, chap. 3).

Einstein was right to consider Dirac, von Neumann and others following them “orthodox” in his sense because they accepted the descriptive completeness of the theory.

Rigorously, they can never be localized at a point and the eigenstate-eigenvalue link cannot be applied to either of these states. Physicists commonly assume a vague but stronger principle inferring spatial location from a “suffficiently” localized spatial \(\psi \)-function. That is why I use the phrase “or something like it”. It is interesting to note that the entangled spatial state to which the authors of the EPR argument appealed did not require rigorous localization of component spatial states. But they did apply the eigenstate-eigenvalue link to states without regard for mathematical niceties!

Indeed this interactive interpretation was what motivated the analysis of holism and nonseparability in my Healey (1991), not some more “orthodox” interpretation.

Here are two instances among many.

...the state vector is only a shorthand expression of that part of our information concerning the past of the system which is relevant for predicting (as far as possible) the future behavior thereof. ...the laws of quantum mechanics only furnish probability connections between results of subsequent observations carried out on a system. (Wigner 1963, p.166).

...the quantum state of a Qbit or a collection of Qbits is not a property carried by those Qbits, but a way of concisely summarizing everything we know that has happened to them, to enable us to make statistical predictions about the information we might then be able to extract from them. (Mermin 2007, p. 109)

In the “paradox” of Wigner’s friend it is the absence of such channels that places Wigner (outside the laboratory) in a different agent situation from his friend before he joins him in the lab.

Here is another way to see the problem. Fine (1982) showed that the CHSH (1969) inequalities express necessary and sufficient conditions for probability distributions of variable pairs generated by applying the Born rule to a state vector in a Hilbert space of 4 or more dimensions to equal marginals of a joint probability distribution over all canonical magnitude claims about it. Gisin’s theorem shows those inequalities are violated by any entangled state of such a system.

Seevinck (2010) provides examples in a nice review of work relating the monogamy of entanglement to the shareability of non-classical correlations.

Giustina et al. (2015) report a recent advance on seminal previous experiments to which they refer.

See http://www.colorado.edu/physics/2000/bec/how_its_made.html for a more complete elementary description.

This would happen most of the time in any realistic experiment, since detector efficiencies are usually well below 100 %.

Correspondence with Ismael confirms this is indeed the kind of modal connection intended in Ismael and Schaffer (2016).

I use the “tenseless present” rather than the more idiomatic future tense here for reasons that will soon become clear.

See Ismael (2008). I have slightly altered her notation to avoid conflict with my own. Here ‘

*e*’ ambiguously denotes both an event and the proposition that it occurs.*Cr*stands for credence: an agent’s degree of belief in a proposition, represented on a scale from 0 to 1 and required to conform to the standard axioms of probability.Friederich makes a similar point in Friederich (2015).

Since he cannot predict the actual value of \(Ch_{q}(e_{A})\) following his hypothetical polarization measurement, he would have to decide on his best estimate of \(Ch_{q}(e_{A})\) in accordance with Ismael’s (2008) Ignorance Principle:

Where you’re not sure about the chances, form a mixture of the chances assigned by different theories of chance with weights determined by your relative confidence in those theories.

A unitary evolution \(\Phi ^{+}\Rightarrow \Xi ^{+}\) corresponding to a

*local*interaction there would still yield \(Pr_{\Xi ^{+}}(H_{B})={\frac{1}{2}}\).Einstein’s formulation of a principle of local action also appeals to intervention:

The following idea characterizes the relative independence of objects far apart in space (

*A*and*B*): external influence on*A*has no*immediate*(“unmittelbar”) influence on*B*; this is known as the ‘principle of local action’ (Einstein 1948, pp. 321–322)As does Price (2011) who advocates subject naturalism as an alternative rather than a contribution to metaphysics as currently practised.

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## Acknowledgments

This paper benefited from discussions with Jenann Ismael and David Glick and helpful suggestions from two reviewers. I thank the Yetadel Foundation for support during the time it was written.

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Healey, R. A pragmatist view of the metaphysics of entanglement.
*Synthese* **197**, 4265–4302 (2020). https://doi.org/10.1007/s11229-016-1204-z

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DOI: https://doi.org/10.1007/s11229-016-1204-z