No, the “long sought-after link between the theories of quantum mechanics and general relativity” has not been found

[Image source: iPerceptions.]

The Physics arXiv blog has praised a paper called “On the weight of entanglement” and claimed that the author, David Edward Bruschi, found a new link between quantum mechanics and general relativity. Unfortunately, the paper is mostly wrong, and that what isn’t wrong isn’t new.

It is well known that quantum particles too must have gravitational fields and that measuring these gravitational fields would in principle tell us something about the quantization of gravity. Whenever you have a state in a superposition of two position states, its gravitational field too should be in a superposition. However, the gravitational field of all particles, elementary or composite, that display quantum properties is way too small to be measured. Even if you take the heaviest things that have yet been brought in superpositions of location you are still about 30 orders of magnitude off. I have done these estimates dozens of times.

The only way you can find larger effects is if you exploit secondary consequences of models that are not just perturbatively quantized gravity. For example the Schrödinger-Newton equation that assumes that the gravitational field remains classical even though particles are quantized can have odd side effects like preventing particle dispersion, or reducing the Heisenberg uncertainty. These effects can be somewhat larger, but they are still much too small to be measurable. The problem is always the same: gravity is weak, really weak. Nobody has ever measured the gravitational field of an atom. We measure gravitational fields of large things: balls, mountains, planets.

In the new paper, the author argues that entanglement “has weight.” By this he seems to mean that the full entangled state couples to gravity. It would be more surprising if that wasn’t so, but the treatment in the paper is problematic for several reasons.

The biggest problem is that the author in the paper uses semi-classical gravity. That means he couples the expectation value of the stress-energy to the space-time background, not the operator, which you would do were you using perturbatively quantized gravity. It is remarkable that he doesn’t discuss this at all. He doesn’t mention any problems with this approach (discussed here), neither does he mention tests of the Schrödinger-Newton equation (discussed here). This makes me think that he must be very new to the field.

Using the semi-classical limit in the case discussed in the paper is problematic because this semi-classical approach does not only break down when you have strong gravity. It also breaks down when you have a large quantum uncertainty in the distribution of the stress-energy. Here “large” means that the uncertainty is larger than the typical width of the distribution. This can be formally shown, but it is intuitively clear: In such cases the gravitational field also must have quantum properties. While these deviations from the semi-classical limit do exist at small energies, they are too weak to be measurable. That the semi-classical limit doesn’t work in these cases has been discussed by Ford and others 30 years ago, see for example these lecture notes from 1997, page 34, and the therein mentioned reference of Ford’s1982 paper.

By using semi-classical gravity and then looking at the non-relativistic case, the new paper basically reinvents the Schrödinger-Newton limit. To make this really clear: the Schrödinger-Newton limit in this case is widely believed to be wrong for good reasons. Using it is a non-standard assumption about perturbatively quantized gravity. The author doesn’t seem to be aware of this.

He then points out that the interference terms of the state makes a contribution to the distribution of stress-energy, which is correct. This has previous been done for superposition states. I am not aware that it has previously also been done for entangled states, but since it isn’t measureable for superpositions, it seems rather pointless to look at states that are even more difficult to create.

He then argues that measuring this term would tell you something about how quantum states couple to gravity. This is also correct. He goes on to find that this is more than 30 orders of magnitude too small to be measurable. I didn’t check the numbers but this sounds plausible. He then states that “one could hope to increase the above result” by certain techniques and that “this could in principle make the effect measurable”. This is either wrong or nonsense, depending on how you look at it. The effect is “in principle” measurable, yes. Quantum gravity is “in principle measurable”, we know this. The problem is that all presently known effect aren’t measurable in practice, including the effect mentioned in the paper, as I am sure the author will figure out at some point. I am very willing to be surprised of course.

As a side remark, for all I can tell the state that he uses isn’t actually an entangled state. It is formally written as an entangled state (in Eq (4)), but the states labeled |01>; and |10> are single particle states, see Eq(5). This doesn’t look like an entangled state but like a superposition of two plane waves with a phase-difference. Maybe this is a typo or I’m misreading this definition. Be that as it may, it doesn’t make much of a difference for the main problem of the paper, which is using the semi-classical limit. (Update: It’s probably a case of details gotten lost in notation, see note added below.)

The author, David Edward Bruschi, seems to be a fairly young postdoc. He probably doesn’t have much experience in the field so the lack of knowledge is forgivable. He lists in the acknowledgements Jacob Bekenstein, who also has formerly tried his hands on quantum gravity phenomenology and failed, though he got published with it. I am surprised to find Bei-Lok Hu in the acknowledgements because he’s a bright guy and should have known better. On the other hand, I have certainly found myself in acknowledgements of papers that I hadn’t even seen, and on some instances had to insist being removed from the acknowledgement list, so that might not mean much.

Don’t get me wrong, the paper isn’t actually bad. This would have been a very interesting paper 30 years ago. But we’re not living in the 1980s. Unfortunately the author doesn’t seem to be familiar with the literature. And the person who has written the post hyping this paper doesn’t seem to know what they were talking about either.

In summary: Nothing new to see here, please move on.

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[Note added: It was suggested to me that the state |0> defined in the paper above Eq(5) was probably meant to be a product state already, so actually a |0>|0>. The creation operators in Eq(5) then act on the first or second zero respectively. Then the rest would make sense. I’m not very familiar with the quantum information literature, so I find this a very confusing notation. As I said above though, this isn’t the relevant point I was picking at.]