Abstract
Entangled states are a specific feature of quantum physics that neither have a counterpart in classical physics nor in the realm of our ordinary experiences. In this chapter we outline the debate about these particular states both historically and systematically. We delineate how the debate originated in an argument for the incompleteness of quantum mechanics by Einstein, Podolsky and Rosen, and we show why, on the one hand, the argument is not considered convincing today, on the other hand, however, still affects present discussions. In a second part we give a systematic overview over the contemporary debate on entanglement which focusses on Bell’s theorem and its consequences. Discerning different levels, we reconstruct the theorem and its premises in a clear way and discuss possible consequences. We analyze in detail the received view that Bell’s theorem implies non-locality and relate it to concepts such as “non-separability” and “holism”. Especially we examine the question whether the phenomena involving entangled systems can be explained causally and whether the central conflict between a non-locality and the theory of relativity can be solved.
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Notes
- 1.
- 2.
A valuable aid to this study is provided by the material and the careful analyses in Kiefer (2015).
- 3.
This somewhat misleading term is due to Bohr (1935).
- 4.
Premise A1: Either (1) is true or (2) is true. Premise A2, reformulated by contraposition, yields: (2) implies (1). The demonstration of the validity of this conclusion is a variant of the classical dilemma: Either (1) (and thus Cn) is true. Or (2) is true, but in that case, the premise A2 requires that (1) also be true.
- 5.
For details, cf. Kiefer (2015), pp. 37–39.
- 6.
This makes von Neumann’s theory of measurements important for EPR (cf. Kiefer 2015, pp. 23 and 38).
- 7.
Perhaps EPR understand this application of the quantum-theoretical state description to a single system as a completeness assumption in the sense of the negation of (1). Einstein, for example, presented his own statistical ensemble interpretation at the Solvay Congress in 1927 in opposition to an interpretation (which is evidently attributable to Bohr) according to which the wavefunction is a “complete theory of individual processes” (see Howard 1990, p. 92).
- 8.
EPR thus do not need to argue counterfactually. They explicitly deny that the location and momentum of the objects can be simultaneously predicted or measured (p. 780).
- 9.
- 10.
Although these operators are non-commuting; cf. Sect. 1.2.3.
- 11.
According to the usual convention, the angles are indicated relative to the z-axis. In principle, any other axis could be chosen as reference direction without the state \(|\phi ^+\rangle \) in (4.8) changing its basic form, since \(|\phi ^+\rangle \) is rotationally symmetric.
- 12.
Another consequence of the detector inefficiencies is the detection loophole (see the end of this section).
- 13.
While for our considerations here, we initially require only an intuitive concept of causal influences; we will render this concept more precise in a formal way in Sect. 4.3.4 (see in particular the causal Markov condition there).
- 14.
We do not accept the misleading interpretation suggested by the term “freedom-of-choice loophole”, which is unfortunately widespread, that experimenters must freely choose the settings. Rather, it is simply a question of making the settings statistically independent of the state of the photons at the source—freedom is not required.
- 15.
In this section, we print the variables \(\varvec{a}\) and \(\varvec{b}\) in bold font in order to denote them, as Bell did, to be vectors. In the other parts of this chapter, it suffices to regard a and b as scalar quantities which describe the angles of the measurement settings (wherefore they are printed non-bold there).
- 16.
See Jammer’s description of Bell’s proof (1974, p. 307).
- 17.
Measurement independence follows directly from the Markov condition. For the local factorization condition, an additional intermediate step is necessary:
The first step follows from the product rule of probability theory, and the second step follows from the Markov condition for the causal graph.
- 18.
Sometimes the condition is also called “autonomy”, “no conspiracy” or “freedom-of-choice”. Especially the latter two names already suggest a certain interpretation, which would require further assumptions; for the sake of generality, we would like to avoid this here.
- 19.
This is also suggested by the article on Bell’s theorem on the English Wikipedia pages, https://en.wikipedia.org/wiki/Bell’s_theorem (accessed on 21st Oct 2017).
- 20.
Considered precisely, \(\lambda \) must not necessarily lie between the measurement wings, even though this may seem most plausible; it must be located merely at some position outside the future light cones of \(\alpha \) and \(\beta \).
- 21.
In order to be able to send superluminal signals, there would have to be a correlation between a controllable variable in the one wing and a detectable variable in the other. This, however, is not the case: On the one hand, the measurement outcomes are correlated, but neither of them is controllable (each outcome varies statistically from one measurement cycle to the next). On the other hand, the measurement settings can indeed be controlled, but there is no correlation between the choice of a measurement setting in one wing and any variable in the other. In particular, the choice is (marginally) independent of the distant measurement outcome. (Given a local measurement outcome, there is to be sure a correlation between the choice of setting and the distant outcome, but the local outcome cannot be controlled, so that one cannot make use of this conditional dependence to send signals.)
- 22.
Two variables A and B in a causal diagram are connected if and only if A is the cause of B, or B is the cause of A, or if both have a common cause. In particular, A and B are not causally connected if they have only a common effect.
- 23.
Violations of the causal Markov condition have also been claimed (see e. g. van Fraassen 1982; Cartwright 1988), but they are controversial (see e. g. Hausman and Woodward 1999). One can also show that they would not be sufficient to explain the strong EPR/B correlations (Näger 2013; see Sect. 4.5.1 below).
- 24.
In a typical causal system, every arrow in a causal diagram is associated with a causal parameter. It describes, roughly speaking, how strongly the causal variable influences the effect variable.
- 25.
Normally, such fine-tunings are unstable with respect to external perturbations, because a perturbation as a rule affects only one of the two paths and thus destroys the balance between them. Then, cause and effect become dependent and one could send signals. The quantum-mechanical formalism, however, demonstrates how it is possible that such a fine-tuning can be stable with respect to external perturbations: In quantum-mechanical non-faithfulness, the paths are so closely interwoven that external perturbations always act on both paths, and the laws for these perturbations guarantee that both paths are always perturbed in such a way that they remain in balance (Näger 2016).
- 26.
While for experiments with the perfectly entangled Bell states (see the box in Sect. 4.3.1), the measurement outcomes are independent of each local measurement setting, this apparent independence, which results from symmetry, vanishes for partially entangled states.
- 27.
One must not, however, leave out any state which is a common cause (“causal sufficiency”).
- 28.
“Fundamental” is not supposed to mean here that the causal relation could not be analysed in terms of non-causal concepts.
- 29.
The rather technical concept of supervenience originated in the philosophy of mind and in meta-ethics. Cleland (1984) introduced a variant of the original concept into the debate over space and time, and it is this variant which is used in the debate on entangled quantum systems (cf. for example, French 1989, or Esfeld 2004). The definition is: A dyadic relation R supervenes over a determinable, non-relational property P if and only if (i) each of the relata of R instantiates the property P in a determined manner, and (ii) the instantiations of the property P determine the relation R. A simple example: The more-massive-than relation supervenes over the masses of physical objects.
- 30.
Although they are similar, these two principles do not completely overlap. Humean supervenience is, on the one hand, stronger than the principle of non-separability, since it requires that everything supervenes on the states of spacetime points, and thus not only the states of extended spacetime regions, but also entities which must not necessarily be located in space and time, such as, for example, mental states or numbers. Humean supervenience is, on the other hand, also weaker than non-separability, since, in contrast to the latter, it does not require that the supervenience be local. The question as to whether or not for example an event in region A causes another event in is determined according to Lewis not by the state of the corresponding spacetime regions, but instead by the total state of all spacetimes in all possible worlds.
- 31.
While in the general theory of relativity, all frames of reference are considered to be equivalent, this holds in the special theory of relativity only for inertial frames, i e. non-accelerated frames (the principle of special relativity).
- 32.
A simultaneity plane of a frame of reference through a point P is the set of all spacetime points which are simultaneous with P in this frame of reference. Simultaneity in the theory of relativity is, in contrast to its role in classical theories, not an absolute, objective fact, but rather it depends upon the frame of reference. Every inertial frame of reference can be uniquely determined by specifying one of its simultaneity planes.
- 33.
Note that also the preparation of the quantum state is controlled by an experimental apparatus, which, if it is correctly constructed, can be started by a corresponding macroscopic mechanism (e. g. a pushbutton, a laser beam, an electrical pulse)—and thus falls within the area of validity of the intervention assumption.
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Exercises
Exercises
- 1. :
-
In the EPR article, there is an assumption of major significance which we have called the “locality assumption”: “Since at the time of measurement, the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system”. How does this assumption relate to the other concepts of locality which we introduced in Sect. 4.4.5: Does it imply global and causal Einstein locality as well as spatiotemporal separability?
- 2. :
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Assume that an EPR/B experiment were correctly described by a local causal structure with hidden variables \(\lambda \) (see Fig. 4.5). One can then show that the existence of perfect correlations implies that measurements must proceed deterministically. Try to formulate a suitable argument.
- 3. :
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List the minimal set of assumptions which are required to derive a Bell inequality and sketch out what they state.
- 4. :
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Apply the causal Markov condition to the local causal structure in Fig. 4.5 and note the resulting statistical independencies.
- 5. :
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Outline the four fields of conflict of a non-local theory with the theory of relativity.
- 6. :
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Discuss the two known possibilities for non-locality which are not in conflict with the principle of relativity. Take into account both physical and ontological consequences.
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Näger, P.M., Stöckler, M. (2018). Entanglement and Non-locality: EPR, Bell and Their Consequences. In: The Philosophy of Quantum Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78356-7_4
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