As the authors of “Separating a particle’s mass from its momentum” acknowledge in their abstract [1], their title is possibly the most extreme interpretation of their results that one could make. Despite this, they present a surprisingly nuanced discussion of their proposed extension of the quantum Cheshire-cat effect [2]. You might be wondering, though, what the quantum Cheshire-cat effect is, and how can even an extreme interpretation of it suggest such a ludicrous-sounding idea as mass separating from momentum? Time for a history lesson…

I remember first hearing about quantum Cheshire cats when I was in high school, from the popular-science media. “New weird quantum effect discovered – particles can split from their properties!” Seventeen-year-old me was intrigued; how could this be!? In my head, I had the image of quantum particles being formed of divisible properties, like how nuclei are formed of protons and neutrons, and these in turn are formed of quarks. It was only as I started studying quantum mechanics that I began to realise just how wrong this picture was. For a start, no-one was claiming the effect was caused by the sort of physical reductionism my mind had conjured up, inspired by popular-science books on high-energy particle physics: the quantum particles being considered were effectively indivisible, or “atomic”. It was just claimed that their properties could be made to separate from them. Possibly more importantly, even this claim was based on certain questionable assumptions. The paradox involved inferring about the properties of a quantum system between measurements – the very thing that the forebears of quantum mechanics had decreed was verboten, meaningless, nonsensical. This is, as I’m sure you know, because performing a measurement can change the properties of the measured quantum system – or, rather, sequentially measuring incompatible observables can lead to results which contradict one another. We don’t even need to look at anything as complex as contextuality, or Bell-inequality violations, to see this; even the sequential Stern-Gerlach experiment shows that, if we take two measurements of a certain property of a system, inserting a measurement of an incompatible property (specifically, a spin measurement in a different basis) between those two measurements can lead to them giving inconsistent results. How then do the authors of the original paper claim they can tell us about the properties of a system between two measurements, without changing the result? The answer: weak values!

Traditionally, measurements in quantum mechanics are done through strong, or projective measurement; typically, after the measurement, the measured system is left in an eigenstate of the measured operator. This means a) the system is left such that it has a definite value for the property represented by that operator (with value equal to the eigenvalue corresponding to that eigenstate of the operator), and b) doing such a measurement in general changes the quantum state of the system (e.g., if the system’s original state was a superposition of eigenstates of the observable measured). Associating definite values for properties of quantum systems with eigenvalues is part of the reason why, when a system isn’t in an eigenstate of an operator, conventional wisdom is that the system’s property corresponding to that operator doesn’t have a value (or that it is meaningless to ask about the value of that property). Note, by “value”, here we could even mean eigenvalues 1 or 0, corresponding to the system having or not having that property, respectively. Therefore, “not having a value” doesn’t mean the system doesn’t have that property – it’s even less definite, and more mysterious than that. There’s a tendency to try and treat cases like this as “we don’t know the value of the property”, but even that doesn’t work. Our classical concept of such an epistemic uncertainty (not-knowing-ness) still presupposes that there is a value to that property. In the quantum case, unless you think that there is more to the world than just that described by the quantum formalism (and, in fairness, many do), this uncertainty is more than that: it is ontic (an aspect of the world) rather than epistemic (an aspect of our knowledge).

Regardless, this idea that we couldn’t ask what value that observable had (unless a system was in an eigenstate of an observable) irritated people. They wanted to ask, “what is the value of a property of a quantum system between measurements”. Obviously, to ask this, they’d still need to do some kind of measurement – but it needed to be different from projective measurements. It needed to leave the system in its original state, or as close to that as it could. To do a projective measurement you need to couple, or link, the system with a measuring device or “meter”. Ideally, this coupling needs to be as strong as possible, and the meter needs to have as little uncertainty as possible, so you can read out the eigenvalue corresponding to the value of the system’s property as clearly as possible. Therefore, surely, to make a measurement which disturbed the system as little as possible, you needed to do the opposite: couple the system to the meter as weakly as possible, and make the meter result as uncertain as possible [3,4]. This is what Aharonov, alongside his collaborators, did [5]. They showed, as you make the coupling smaller and smaller, and the uncertainty in the meter larger and larger, the change to the state of the system by doing such a measurement got smaller and smaller. They christened the measurement with the smallest possible coupling, and largest possible uncertainty, as a “weak measurement”.

The next thing Aharonov and his collaborators noticed is that, if you start with your quantum system in a given state, do such a weak measurement of a certain observable, and then do a strong measurement of a different observable (keeping only systems which end up in a specific eigenstate of that final observable) then in the mathematics which describes the system you end up with a peculiar quantity in the same place where, if you did a strong measurement, you would see the eigenvalue. This place in the maths corresponds to the shift you measure physically in the meter system to read out the value of the property. Therefore, in this weakly-measured-then-strongly-sorted case (or weakly-measured-and-postselected case, to use the common terminology) we get something which at least appears in the same place we’d expect to see the value of the system’s property. Aharonov and his collaborators decided to call this thing the weak value. Given how large the uncertainty of the meter state needed to be to make the measurement weak, we’d need to repeat this process many times, with many systems, and then average the meter shifts we observed, in order to experimentally obtain this weak value (although recent theoretical work on feedback compensation may make it possible to get a weak value just from a single postselected system). However, Aharonov and collaborators claimed this weak value was effectively the value of the property measured between measurements. Given how this weak value appears, both in the formalism and experimentally, this claim wouldn’t seem too objectionable – except that these weak values could be very different to what we would expect for values of properties. As an example, just look at the title of the paper in which these weak values were proposed: “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100” [5]. Weak values can be huge when we would expect the values of properties to be small; they can be negative when we would expect the values of properties to be positive; and they can even be imaginary, even though all quantum observables are defined in such a way that their spectrum of eigenvalues (so the values of all possible properties) are real. What can these “anomalous” weak values mean?

Before I explain, let me first give some context. As a PhD student in the 1950s, Yakir Aharonov and his supervisor discovered something amazing. Even when confined to a region where the electric and magnetic fields are both zero, an electrically-charged particle can be affected by an electromagnetic potential, as a shift in the phase of its wavefunction: a geometric phase [6]. This proved the physical reality of the vector potential, showing potentials aren’t just a matter of gauge (free choice for calculation), and that just considering force fields (e.g., electric and magnetic field, as Maxwell’s equations do) isn’t enough to describe all physical phenomena. Had it not been for the political and personal disapproval Bohm faced during his life [7], it is likely that he and Aharonov would have won the Nobel Prize for this discovery; it regularly ranks on lists of the “Greatest Discoveries in Quantum Physics” [8]. So, what do you do when you’ve made one of the key discoveries in your field at the start of your career?

For Aharonov, the answer has been, at least as we would describe it now, to “troll”. Aharonov has dedicated his academic life to identifying paradoxical-seeming scenarios using the quantum formalism, using emphatic language to describe them, and then setting them loose on the unsuspecting quantum community. I can almost imagine him chuckling away to himself as he notices another case which is surprisingly mathematically simple, but which, considered physically, deeply challenges our intuitions: the quantum pigeonhole paradox, where supposedly three quantum pigeons can be put in two coops without any two pigeons sharing a coop [9]; Aharonov’s box, where by carefully opening a box containing a low-energy photon and inserting a mirror, a much higher-energy photon can be emitted [10]; and, of course, the quantum Cheshire-cat paradox [2].

Most of these paradoxes tend to make use of interpreting weak values, as if they described the values of properties between (strong) measurements. In the quantum Cheshire-cat paradox, the claim of particle separating from property comes from comparing two sets of weak values: the weak values for projectors on paths (for which we interpret the eigenvalues, 1 or 0, as meaning “the particle was on this path” or “the particle was not on this path” respectively); and the weak values for the tensor product of these projectors with a polarisation-discrimination operator (for which we interpret the eigenvalues, 1 or -1, as “the particle was horizontally polarised” or “the particle was vertically polarised” respectively). The paradox would seem much less paradoxical if it was just given in terms of weak values, rather than also including the interpretational leap between weak values and properties of a quantum system. However, I’m thankful that it was given in this stronger formulation as it prompted me to think about the paradox, learn about weak values, and to notice that there were deeply interesting things going on under the surface in this case. In our recent paper [11], we showed that the quantum Cheshire cat takes the standard form of many contextuality paradoxes where, depending on how we measure a system, it seems to have three properties which are mutually contradictory. This implies that the properties that the system has must, to some degree, depend on how we measure it. Further, we showed that a negative weak value of a projector in this scenario (which would normally have eigenvalues 1 or 0, meaning the system was or wasn’t in that state respectively) link to a measurement-dependent cancellation of parts of the state, showing a mechanism in the quantum formalism itself for such contextual behaviour. Without Aharonov et al’s provocative title we never would have seen this, and so would never have understood how contextuality manifests in the quantum formalism in this way.

Whilst often deeply irritating, such paradoxes use physical situations to poke at our intuitions, making them great for inspiring others to then search for the principles either underpinning them or the reasonable-seeming assumptions behind them which nature violates. Aharonov’s approach is, therefore, very different to those now common in the quantum-foundations community – for one thing, there are no polytopes in sight – and this is deeply refreshing. As quantum mechanics has become more formalised into quantum information, and mathematicians and computer scientists begin to take over, there are those who seem to forget that the framework we are looking at applies to the physical world, and that it does so in deeply baffling and counterintuitive ways. Not Aharonov.

This is why I was so excited to see Aharonov’s latest work, with regular collaborators Mordecai Waegell and Jeff Tollaksen [1]. Despite its incredibly bold title, they touch on many issues within quantum foundations as well as the link between quantum mechanics and gravity, the rather bold association between first-order perturbative effects and supposed “virtual particles” (common in quantum field theory and high-energy particle physics), and even questions of temporal and causal order. While I don’t think even its authors think that the paper does what it says “on the tin”, a large part of me honestly doesn’t think this matters. For all its quirks the paper challenges our physical intuitions and is physically motivated, giving gedanken experiments which one could imagine coming out of the realm of thought and into a real lab in the next few years. This is more than a lot of papers in quantum information do.

Acknowledgements

I thank Ellery Littlewood for useful comments on an early draft of this article.

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