Tabletop Experiments for Quantum Gravity Are Also Tests of the Interpretation of Quantum Mechanics

Abstract

Recently there has been a great deal of interest in tabletop experiments intended to exhibit the quantum nature of gravity by demonstrating that it can induce entanglement. In order to evaluate these experiments, we must determine if there is any interesting class of possibilities that will be convincingly ruled out if it turns out that gravity can indeed induce entanglement. In particular, since one argument for the significance of these experiments rests on the claim that they demonstrate the existence of superpositions of spacetimes, it is important to keep in mind that different interpretations of quantum mechanics may make different predictions about superpositions of spacetimes. $\psi$-complete interpretations of quantum mechanics, like the Everett interpretation, almost universally predict the existence of superpositions of spacetimes, whereas in $\psi$-incomplete, $\psi$-nonphysical interpretations it seems more natural to predict that spacetime superpositions are not possible. Meanwhile $\psi$-incomplete, $\psi$-supplemented interpretations present us with a more complex picture where we may or may not end up predicting that spacetime superpositions are possible, depending on the particular way in which the coupling between spacetime and matter is constructed. This line of reasoning suggests that what would be ruled out by a successful result to these tabletop experiments is a class of quantum gravity models that we refer to as $\psi$-incomplete quantum gravity (PIQG)—i.e. models of the interaction between quantum mechanics and gravity in which gravity is coupled to non-quantum beables rather than quantum beables. It follows that the results of tabletop experiments can also be regarded as giving us new evidence about the interpretation of quantum mechanics: roughly speaking, a positive result to these experiments should increase our confidence in $\psi$-complete interpretations, whilst a negative result should instead increase our confidence in $\psi$-incomplete interpretations. We introduce these ideas in Sect. 1, and then in Sect. 2 we make the reasoning more precise by presenting a set of inferences that may be made about the ontology of quantum mechanics based on the results of tabletop experiments. In Sect. 3 we discuss some existing PIQG models and consider what more needs to be done to make these sorts of approaches more appealing. There are two competing paradigms for the interpretation of these experiments, which have been dubbed the ‘Newtonian’ paradigm and the ‘tripartite’ paradigm: here we largely work within the tripartite paradigm, because the tripartite view is specifically concerned with ontological aspects of the mediating gravitational interaction and that makes it a suitable setting for enquiries about the ontology of quantum mechanics, but in Sect. 4 we consider what conclusions can be drawn if one does not presuppose the tripartite view. Finally in Sect. 5 we discuss a cosmological phenomenon which could be regarded as providing evidence for PIQG models.

Appendix 1: Arguments that Gravity must be Quantum

1.1 Eppley and Hannah

An argument due to Eppley and Hannah dating to 1977 [27] purports to show that the gravitational field must be quantized. Eppley and Hannah consider the interaction of a classical gravitational wave of small momentum with a quantum particle described by a wave function. If we suppose that this interaction does not collapse the wavefunction whereas ordinary measurements do collapse the wavefunction, it follows that this setup can be used to perform superluminal signalling, as the wavefunction of both particles will collapse when Bob performs a measurement, and Alice can then use the gravitational wave to probe the state of her particle to see whether Bob has performed his measurement or not. But if we suppose this interaction does collapse the wavefunction, then Eppley and Hannah argue that there will be problems with momentum conservation or violations of the Heisenberg uncertainty relation. So assuming that we are not willing to accept violations of no-signalling, momentum conservation of the Heisenberg uncertainty relation, it follows that the gravitational field must be quantized.

As noted by Huggett and Callender [22], the first horn of the dilemma makes interpretational assumptions by assuming that nongravitational measurements lead to a wavefunction collapse; evidently interpretations which don’t tell us that measurements cause wavefunction collapses won’t have this problem. In particular, $\psi$-incomplete models don’t typically postulate a collapse of the wavefunction upon measurement, because definite classical results for measurements are already assured by the fact that measurement results supervene directly on the nonquantum sector, so there’s no need for a wavefunction collapse. For example, the Bell flash ontology does postulate ‘flashes’ which are roughly similar to collapses, but they are not caused by measurement: they’re spontaneous and uncontrollable and thus can’t be used for signalling. Moreover, in a PIQG model spacetime is coupled to the nonquantum sector rather than the quantum sector, so gravitational waves can’t directly probe the quantum sector. Thus as long as the nonquantum sector is defined in such a way that the knowledge Alice can gain about the nonquantum sector via a gravitational probe doesn’t give her too much information about the quantum sector, no violation of no-signalling will ever occur.

The second horn of the dilemma is also suspect—as Mattingly points out ‘it may be that the uncertainty relations can be violated. They haven’t really been tested in this way.’ [53] In any case this horn is not really relevant to PIQG models, as if gravity is coupled directly to the nonquantum sector there is never any direct interaction between gravity and the quantum sector, so presumably the wavefunction will not be collapsed by gravity.

Therefore the argument of Eppley and Hannah does give us valuable information—it tells us that in formulating a theory of quantum gravity in which gravity is not quantized, we should probably avoid approaches where measurements and/or gravitational interactions cause a collapse of the wavefunction—but it certainly doesn’t rule out PIQG models.

1.2 DeWitt

An argument due to DeWitt dating to 1962 [25] argues that the quantization of any system, as expressed by the uncertainty principle, implies the quantization of all other systems to which it can be coupled. This is a generalization of an argument used by Bohr and Rosenfeld to show that the electromagnetic field must be quantized [59]. However, the argument relies on understanding the uncertainty principle in terms of ‘disturbance,’ and it has since been recognised that this approach is not correct [14]. And in any case, in a PIQG model gravity is not coupled to the quantum sector, so DeWitt’s argument does not apply. Of course, in a PIQG model typically we will have the quantum sector coupled to the nonquantum sector, so one might try to invoke DeWitt’s argument twice over to show first that the nonquantum sector must be quantized and then that gravity must be quantized. However, it is clearly not the case that a nonquantum sector coupled to a quantum sector must be quantized—numerous counterexamples exist, including the de Broglie Bohm interpretation [34, 38]. DeWitt’s arguments do not apply to this kind of case, because he proceeds by supposing that the measurement of a system A by another quantized system B will create a disturbance which leads to A becoming quantized, whereas the coupling between the de Broglie–Bohm particles and the quantum state does not proceed by measurement and hence is not subject to uncertainty relations.

1.3 Marletto and Vedral

Recently Marletto and Vedral presented a novel information-theoretic argument for the quantization of gravity [51]. The aim here is to show that if a system C is coupled to a quantum system Q with two non-commuting variables, then C must also have at least two non-commuting variables. Rather than making assumptions about the dynamics of the coupling as in previous proofs, Marletto and Vedral work entirely with information-theoretic concepts within the constructor theory approach. In particular, they assume ‘interoperability’ (i.e. that classical information on Q can be copied to C) and they make an assumption which we will refer to as $A*$, which requires that an operation which measures the variable $x_1$ on Q and then copies the result to C will also be a ‘distinguisher’ for the non-commuting variable $x_2$ on Q, i.e. it will map the possible values of Q to states of C, Q which are perfectly distinguishable.

The assumption $A*$ may seem quite specific, but Marletto and Vedral justify it by the claim that it is true in the case of quantum mechanics. And indeed, this is correct if C and Q are both quantum systems. For example, suppose Q is a qubit and C is a second qubit which we denote by $Q'$. Now if Q is prepared in the computational basis (variable $x_1$) and $Q'$ is prepared in the state $|0 \rangle _{Q'}$, we can use a CNOT with Q as the control to ‘measure’ Q and copy the result to $Q'$. Meanwhile if Q is prepared in the Hadamard basis (variable $x_2$) and $Q'$ is again prepared in the state $|0 \rangle _{Q'}$, and we apply a CNOT with Q as the control, if Q was in state $| + \rangle _{Q}$ the result will be the Bell state $\frac{1}{\sqrt{2}} ( | 00 \rangle _{QQ'} + |11 \rangle _{QQ'})$, whereas if Q was in the state $| - \rangle _{Q}$ the result will be the Bell state $\frac{1}{\sqrt{2}} ( | 00 \rangle _{QQ'} - |11 \rangle _{QQ'})$, so the two possible values of $x_2$ are indeed perfectly distinguished.

However, what we actually want is for C to be the gravitational field, so in this example it is necessary that the state of the second qubit $Q'$ should be mapped to a state of the gravitational field. One option is to simply measure $Q'$ in the computational basis after applying the CNOT gate; then if the result is 0 we move a large mass to the left (producing state $|L \rangle _M$) and if the result is 1 we move a large mass to the right (producing state $| R \rangle _M$). Thus in the case where we are measuring the variable $x_1$, the final state of the gravitational field can be regarded as a record of the result of the measurement of $x_1$. Now let us consider what happens if instead we attempt to perform the $x_2$ distinguishing operation. The argument of Marletto and Vedral requires that we perform exactly the same operations as in the case where we are measuring $x_1$, so we must apply the CNOT gate and then measure $Q'$ in the computational basis and then move a large mass around conditioned on the result of that measurement. However, according to the usual interpretation of measurement, measuring the second qubit destroys the coherence between the qubits; therefore we will just end up with either state the state $|0 \rangle _{Q'} | L \rangle _{M}$ or the state $|1 \rangle _{Q'} | R \rangle _{M}$ with fifty percent probability for each regardless of whether Q was originally in state $|+ \rangle _{Q}$ or $| - \rangle _{Q}$, meaning that the distinguishing property is lost. Of course, we could retain the distinguishing property if we instead measured the pair of qubits $Q, Q'$ in the Bell basis and then moved around the mass conditional on the results of that measurement; but the argument of Marletto and Vedral does not allow that, as we are required to use the same operation in both cases. Thus the assumption $A*$ does not hold if the intention is for us to map the qubit state to the gravitational field by means of measurement.

The other option is to make both of the qubits $Q,Q'$ massive objects, with state $|0\rangle$ corresponding to one spatial position and the state $| 1 \rangle$ corresponding to a different spatial position, such that C can be dentified with the gravitational field sourced by these massive objects. Then, if it is the case that spatial superpositions of massive objects give rise to superpositions of spacetimes, it follows that the two Bell states $\frac{1}{\sqrt{2}} ( | 00 \rangle _{QQ'} + |11 \rangle _{QQ'})$, $\frac{1}{\sqrt{2}} ( | 00 \rangle _{QQ'} - |11 \rangle _{QQ'})$ will give rise to different spacetime superpositions, thus retaining the coherence and preserving the distinguishing property on which the argument relies. But this begs the question: whether or not massive objects in superpositions produce spacetime superpositions is precisely the point at issue, so we can’t simply assume that the distinguishing property is retained when information is mapped to a state of the gravitational field.

Here we have considered just one possible implementation of the scenario envisioned by Marletto and Vedral, but it seems likely that the same kinds of problems will hold for all possible implementation of the scenario. For according to the textbook account of quantum measurement, a coupling implemented by measuring qubits is capable only of copying information in a single basis, so we cannot have a coupling of this kind which preserves both the computational basis states for the $x_1$ measurement and also the Bell states for the $x_2$ distinguishing operation. So assumption $A*$ will hold only if the coupling is achieved by some kind of quantum operation which transfers the whole quantum state of the qubits into a state of the gravitational field, which is possible only if the gravitational field is itself quantum and spacetime superpositions are possible. Thus $A*$ does not hold in any PIQG model, since such models do not allow the existence of spacetime superpositions.

Note that Marletto and Vedral might respond to this objection by rejecting the idea that measurement destroys coherence and instead adopting an Everettian picture. In that case, the result of measuring the qubit and transferring the result to the gravitational field would not in principle destroy the distinguishability of the two Bell states, although they would now only be distinguishable in an abstract external sense, since no person within any individual branch of the wavefunction would be able to distinguish them. So the argument of Marletto and Vedral arguably does succeed in an Everettian context—but then this is not very surprising, as we have already noted that the existence of spacetime superpositions seems more or less inevitable in an Everettian picture.

1.4 Belenchia et al

Ref [8] provides an interesting new argument for the quantisation of gravity. They consider a case where one particle A is able to obtain information about the spatial position of another particle B by probing B’s gravitational field; then if B is in a superposition of spacetime positions, it follows that if A is present the state of B must become less pure as A becomes entangled with it, whereas if A is not present the state of B remains pure, and thus we seem to have a case of superluminal signalling. The resolution to this apparent paradox is to perform a more careful analysis which takes account of vacuum fluctuations and the quantum properties of radiation. Thus it is argued that gravity must at the last possess the properties of a quantum field with regard to vacuum fluctuations and the quantum properties of radiation, so it must be quantum.

Now, this is not an argument for quantisation per se—rather it is an argument to the effect that if gravity can be used to produce entanglement as in the BMV experiment, then it must indeed have quantum features. In the scenario described in the article, a PIQG model would simply deny that A could become entangled at all: for example, a PIQG model might hold that there are actually no beables present in the region where B is in a spacetime superposition, so B does not source any gravitational fields during this time, so A can’t obtain any information about its position from the gravitational field and no paradox can arise. So the approach of ref [8] is not at all inconsistent with our approach in this article—indeed, this argument may be understood as supporting our choice to assume the correctness of the tripartite paradigm in our analyses in Sect. 2.

That said, the argument isn’t completely conclusive. For what has been shown is simply that in the case where gravity is a quantum field, the resolution to the paradox is to consider vacuum fluctuations and the quantum properties of radiation. It doesn’t necessarily follow that there is no possible resolution to the paradox if gravity is not quantum; it would simply have to be a different resolution. However, insofar as no classical resolution is currently available, and a well-motivated quantum resolution is available, this argument does seem to offer solid reasons to adopt the quantum approach. Moreover, ref [8] notes that a very similar paradox can be set up using electromagnetic rather than gravitational interactions, and it would seem somewhat unsatisfying to have one resolution of the paradox in the case where the interaction is electromagnetic and a completely different resolution in the case where the interaction is gravitational: if the electromagnetic and gravitational interactions lead to exactly the same kinds of behaviour, the natural conclusion is that the two interactions are also of the same kind, i.e. they are both quantum. PIQG models would of course deny that electromagnetic and gravitational interactions do lead to exactly the same kinds of behaviour, since PIQG models do not allow superposed particles to source gravitational fields, but this argument is primarily directed at the case where superposed particles can gravitate, and from that standpoint it does indeed paint a convincing picture to the effect that that if superposed particles gravitate, gravity must possess the properties of a quantum field.

1.5 Unification

As noted in Sect. 1, arguments for the quantisation of gravity frequently appeal to the need for unification. Of course one possible rebuttal would be to argue for a pluralist or instrumentalist approach which denies that unification is always desirable or necessary, but there is no need for us to make that argument here, because we consider that PIQG models do in fact count as a form of unification.

In particular, we observe that ‘unification’ can mean several different things. Unifying theories from two different regimes may sometimes take the form of reducing one theory to another, or reducing them both to a single deeper underlying theory. These sorts of unifications typically lead us to conclude that the objects postulated by the theories are in fact one and the same kind of object—for example, showing that the gravitational field is in fact a quantum field. But sometimes unification may just involve giving a consistent account of the relationship between two theories which explains how the different types of objects that they postulate interact with one another and how transitions between the two regimes work. If we are committed to the former kind of unification then it may seem reasonable to conclude that the gravitational field must be quantum (although of course we could also choose to insist that all of physical reality is non-quantum), but if we are committed only to the latter kind of unification then there is nothing compelling us to say that the gravitational field must be quantum.

Moreover, although there are certain sorts of metaphysical commitments that might lead someone to prefer the former sort of unification, the latter is perfectly adequate if our interest in unification simply stems from the simple realist position that ultimately there exists a single universe and so all the parts of that universe must fit together into one consistent schema. Nothing about this entails that there can only be one kind of of ‘physical stuff’—of course it may be easier to figure out how to fit everything together into a consistent schema if we have only one type of stuff, but the universe is not obliged to make things easy for us, and therefore the conjecture that there is only one type of stuff is subject to confirmation or disconfirmation by empirical data just like any other hypothesis. The preference for only one type of stuff seems in some cases to be almost an aesthetic preference, but of course if the empirical data strongly indicated that there is more than one kind of physical stuff then it would be bad practice to disregard that evidence on purely aesthetic grounds. At present the empirical data does not seem wholly conclusive, but convincingly demonstrating either the existence or the nonexistence of spacetime superpositions would be a major step towards reaching a conclusion on this point.

Tilloy also mentions a related kind of argument [68]: the enthusiasm for quantizing gravity stems in part from the hope that successfully quantizing gravity will also solve some of our other theoretical problems—e.g. it may regularize the UV divergences of Quantum Field Theory and tame the singularities of General Relativity. But as Tilloy points out, these hopes have not yet been realised, so at present these arguments look a lot like wishful thinking. Moreover, one might well feel that there is just as good reason to think that finding the correct solution to the measurement problem will solve these problems, particularly if it turns out that solution is not $\psi$-complete; so it does not seem methodologically sound to insist that quantizing gravity is necessary in order to solve these problems.

1.6 Success of QG so far

An argument that does not appear much in the literature but which is often brought up informally begins from the observation that all currently existing well-developed approaches to quantum gravity postulate spacetime superpositions, and proceeds to the conclusion that we have good reason to believe in spacetime superpositions. Indeed, given the vast size of the literature on string theory [11] and the rapidly increasing literature on Loop Quantum Gravity [60], it may seem tempting to conclude that we could not have made so much theoretical progress on quantizing gravity if quantizing gravity were not in some sense the right thing to do.

We do agree that the progress that has been made on these approaches into account is significant and should be assigned due weight when making assessments about the quantization of gravity. On the other hand, theoretical progress is not the same as empirical confirmation: after all mathematicians routinely discover all sorts of interesting structure associated with abstract mathematical objects which as far as we know don’t represent anything in reality, so the mere fact that we have discovered interesting structure within theories of quantized gravity doesn’t in and of itself entail that they represent anything in reality. Thus until empirical confirmation becomes available it would be unwise to shut off all other possible routes.

Moreover, most well-developed approaches to QG—including canonical quantum gravity, LQC and string theory—took as their starting point the need to quantize the gravitational field (which is then standardly identified with the structure of spacetime), and therefore it is no coincidence that they predict spacetime superpositions. So the theoretical success of these research programmes does not necessarily give us reason to believe in spacetime superpositions unless we are convinced that similarly serious attempts have been made to formulate a theory of interactions between quantum systems and gravity without quantizing the gravitational field, and that those attempts have failed. It doesn’t seem that this is the case: throughout the history of the field, ‘quantizing gravity’ has largely been the favoured approach, whereas semiclassical gravity and PIQG have received comparatively little attention. Of course, as we discuss in Sect. 3.1, there are a number of very good reasons to be sceptical about semiclassical gravity, but nonetheless the fact that non-quantized gravity approaches have received significantly less energy and attention makes it hard to draw any strong conclusions from the fact that quantized gravity has made more theoretical progress than non-quantized gravity.

A different version of this argument would appeal specifically to the success of low energy quantum gravity. For unlike string theory, LQC and so on, low energy quantum gravity is quite well confirmed [74], and thus it might seem reasonable to extrapolate from the success of low energy quantum gravity in certain regimes to the overall correctness of the theory. Since low energy quantum gravity assumes that gravity is quantized and that spacetime superpositions can exist, this would entail that gravity must indeed be quantized. That said, the theory of ‘low energy quantum gravity plus dynamical or gravitational collapse’ makes roughly the same predictions as standard low energy quantum gravity in the regimes that we have tested but diverges in the so-far untested regime where superpositions of spacetimes become insignificant, so the empirical evidence does not obviously distinguish between these two alternatives. Similarly, the Bohmian PIQG approach discussed in Sect. 3.2 can reproduce much of the success of low energy quantum gravity, and indeed it seems to give rise to better semiclassical approximations in certain regimes [67], so the evidence doesn’t seem to rule Bohmian views out either. Thus the success of low energy quantum gravity doesn’t rule out PIQG approaches, because most PIQG approaches would be expected to coincide with low energy quantum gravity in the regimes in which it has been tested.