Frequency Shifts Do Not Imply Quantum Entanglement

I’ve somehow become the go-to twitter critic (jointly with Douglas Natelson, a Rice University physics professor with an excellent blog) of a recent unpublished manuscript that has generated a lot of premature press coverage. The authors of this paper claim to have demonstrated quantum entanglement between a superconducting qubit — a special kind of electrical circuit that exhibits quantum behavior — and a remarkably hardy little creature called a tardigrade.

You can look at my twitter thread or Natelson’s blog post for a succinct summary of the paper and the flaws that many physicists were quick to point out. I’m writing this post because I recently heard Vlatko Vedral — a professor of theoretical physics at Oxford whom I greatly admire, and one of the authors — defend the claim of entanglement on the podcast of the Foundational Questions Institute FQXi. And I must say I don’t find his defense at all convincing.

Vedral mainly talked about the fact that the team observed a shift in the characteristic frequency of their superconducting qubit after placing a tardigrade nearby, and cited this frequency shift as evidence for entanglement.¹He also noted that the team demonstrated entanglement between their tardigrade-loaded qubit and another qubit. I don’t think anybody disputes this. But entangling two qubits is nothing new. The paper’s claim to novelty hinges entirely on what’s going on between the tardigrade and the single qubit close to it. So that’s all I’ll talk about here. He conceded that unambiguously verifying entanglement would require doing a good deal more than measuring the qubit’s frequency shift. But even with this disclaimer, I think he’s attributing far too much significance to the frequency shift.

Indeed, in the discourse surrounding this paper, the phrase “frequency shift” has carried a lot of weight. It sounds appropriately science-y, with perhaps a dash of quantum mysticism. With the aid of a thesaurus we can do even better: how about “the researchers observed a spectral displacement?” Amazing!

In this post I’ll try to demystify the tardigrade-induced frequency shift by showing how an analogous effect can arise in a much simpler system, and then make the analogy to the tardigrade experiment more explicit. The fact that frequency shifts arise in very mundane settings doesn’t make them uninteresting — indeed in my view frequency shifts are interesting (and useful!) precisely because they are so ubiquitous. Then I’ll address the broader question of how one should distinguish two theories that predict similar effects, and finally come to why any of this matters.

One mass, one spring, one tardigrade

A mass hanging from a spring is one of the first systems students encounter in intro physics labs, and the equations that describe its behavior recur again and again in all different branches of physics.²It’s one of the simplest examples of a system that exhibits resonance, a universal phenomenon that I’ve written about in an explainer for Quanta magazine and a blog post. I’ve linked to the video above just to give you a mental picture of the system; the first twenty seconds should suffice.

If we don’t disturb the system, the mass will hang at some constant height above the ground. When the mass is pulled away from this equilibrium position and then released, it bounces up and down with a regular frequency.

What determines the frequency of these oscillations? One relevant factor is the “springiness” of the spring.³Physicists call this the “spring constant,” but it’s my blog and nobody can stop me from saying “springiness.” If the spring is tightly coiled, even a small displacement from equilibrium exerts a large force on the mass, resulting in large acceleration and rapid oscillations. If it’s loose and floppy like a slinky, a small displacement doesn’t accelerate the mass very much.

The other important factor is the weight of the mass.⁴Technically the mass of the mass, but “springiness of the spring” was already pushing it. A heavier body has more inertia: it’s harder to get it to start moving and harder to get it to stop moving. In more precise terms, any given force imparts less acceleration to a heavier mass than it would to a lighter mass, resulting in slower oscillations.

It turns out that these two properties completely determine the characteristic frequency of the mass-on-spring system. Summarizing what we’ve learned,

• Stiffer spring and/or lighter mass → higher frequency,
• Looser spring and/or heavier mass → lower frequency.

Now let’s imagine we have a particular mass-on-spring system, with some specific characteristic frequency (the actual value doesn’t matter). Then we place a single tardigrade on top of the mass. We’ve added some mass to the system, so the frequency shifts down — the effect is very small, to be sure, but it’s measurable in principle. Mission accomplished!

The part where I address objections

“But wait!” you exclaim. “Just because you have one classical example of a frequency shift, it doesn’t follow that any frequency shift can be explained without quantum mechanics!”

Fair enough. The point of the above example was to show that there’s nothing intrinsically mysterious about a change in the characteristic frequency of an oscillator. I’ll now move on to a system that obeys the laws of quantum mechanics.

Actually, we don’t need to choose a new example, because our mass-on-spring oscillator is, in principle, a quantum system. There’s nothing in the laws of quantum mechanics that says they stop applying to big, messy, clunky objects; such objects just interact so strongly with their environment that all the quantumness gets smeared out. In principle, if we used high-quality materials and worked at very low temperature and pressure, we could resolve the oscillator’s discrete energy levels and observe other distinctly quantum effects.⁵One good thing to come out of Tardigate was a twitter thread by Rochester physics professor Machiel Blok about macroscopic quantum mechanics and quantum biology. I’ve also written a gentle introduction to experiments studying the quantum behavior of mechanical resonators specifically.

And here’s the important point: at no point in the transition from classical to quantum behavior would the frequency depend on anything other than its mass and its springiness. That is, even in the quantum regime, the value of the oscillator’s characteristic frequency can be predicted using classical physics only. Placing a tardigrade on the oscillator would shift its characteristic frequency in precisely the same way as in the classical regime, and no property of the tardigrade other than its mass would play any role. It certainly wouldn’t matter whether the tardigrade was dead or alive.

“Not so fast!” you say. “You’ve only addressed changes in the frequency of a mechanical oscillator. But a superconducting qubit is surely more quantum-y. Qubit stands for quantum bit! And superconductors are super quantum!”

Well, the situation is not as different as you might think. A superconducting qubit is a specific type of resonant circuit — one in which the electric current oscillates at a particular frequency.⁶I’ve written a bit more about how superconducting circuits work in this post. And the equations that describe resonant circuits are mathematically identical to the equations that describe mass-and-spring systems.

As in the mechanical case, the characteristic frequency of a resonant circuit is determined by two quantities: instead of mass and springiness, we have capacitance and inductance. Bleh. The capacitance in turn depends on two properties of the circuit: its geometry, and the dielectric constant of any nearby materials.

I don’t blame you if your eyes are starting to glaze over. This is a good deal more complicated than the case of the mass on a spring. But the essential point is the same: given easily measured parameters, the characteristic frequency can be calculated using only the laws of classical electromagnetism. None of this requires any knowledge of quantum mechanics, and the equation that determines the frequency doesn’t change when the circuit is sufficiently isolated from its environment that it behaves quantum mechanically.⁷There are some differences between superconducting qubits and normal resonant circuits, but they don’t affect any of these conclusions. Readers who are curious can check out the appendix. Yes, this is the type of blog that will sometimes have appendices.

The tardigrade’s body, like every other material that isn’t an electrical conductor, has a dielectric constant.⁸Different parts of the tardigrade will likely have different dielectric constants, and the overall effect on the capacitance will depend on a weighted average of the dielectric constants of these different parts. If you place a tardigrade near any resonant circuit (quantum or classical), that circuit’s characteristic frequency will shift down. No property of the tardigrade other than its dielectric constant would have any effect on the circuit. Once again, life has nothing to do with it.

Why care about frequency shifts?

Just because frequency shifts aren’t mysterious doesn’t mean we shouldn’t be amazed and delighted by their ubiquity — indeed, this is basically the whole premise of this blog. They are also extremely useful in practice, because frequencies are easier to measure precisely than many other physical quantities: we can invert the logic of the preceding sections and use frequency measurements to obtain precise values for other quantities of interest.⁹Arthur Schawlow, the co-inventor of the laser, went so far as to declare that one should “never measure anything but frequency!” In other words: set up your experiment so that the quantity you care about affects a measurable frequency. There are countless examples of this sort of thing in physics experiments, but in this section I want to offer one example of a practical application in which an oscillator’s frequency shift makes a surprising appearance. If you’re just here for the tardigrades, you can skip ahead.

Modern electronics would not be possible without processes for coating regions of silicon wafers with very thin layers of other materials. One such process is called evaporation, and the basic idea is remarkably simple.

Say you want a thin strip of aluminum to follow some specific path across your silicon wafer. First you put what amounts to a stencil on top of the wafer. Then you put the wafer in a vacuum chamber that also contains a piece of aluminum resting in a tungsten crucible, and pump the chamber down to extremely low pressure. Finally, you run an electrical current through the crucible to heat it up,¹⁰Tungsten is used for its high melting point. melting and then evaporating the aluminum, which splatters all over every surface in the vacuum chamber except for the part of the wafer shielded by the stencil. So now we have aluminum on the right part of the wafer; all the rest is collateral damage. It sounds delightfully crude, but this really is the gist of how it works.

The thickness of the deposited film is usually an important parameter than must be controlled very precisely. The longer you run the current, the more aluminum will be deposited on the wafer. But how do you know when you’ve reached the desired thickness?

This is where frequency shifts enter the story. A standard way to measure film thickness in real time during the deposition process is to use what’s called a quartz crystal deposition monitor. The crystal is an oscillator that vibrates at a very specific frequency, which depends on its mass density among other things. When the crystal is placed inside the vacuum chamber, it gets coated with aluminum like everything else. The extra mass of the aluminum film changes the crystal’s resonant frequency, which is monitored continuously using a small electrical excitation. Then the measured frequency shift together with the known density of aluminum provides a measurement of the film thickness with sub-nanometer precision! How cool is that?

Telling things apart

Let’s get back to our good friend the supposedly entangled tardigrade. There do exist quantum effects that give rise to frequency shifts that you wouldn’t expect classically.¹¹For the experts: I was thinking of vacuum Rabi oscillations here. I thank Mike Raymer for pointing me to a cavity QED paper from 1990 that showed they could also be understood classically. So how should we go about distinguishing quantum frequency shifts from classical ones? We can only do so if we have specific quantitative predictions for the expected frequency shift in the two cases. Then we need to measure the frequency shift and show both that it’s consistent with the quantum prediction and that it’s inconsistent with the classical prediction.

The authors of the tardigrade paper write down some equations for what a quantum model of the tardigrade-induced frequency shift would look like, but these equations make no quantitative predictions for the frequency shift (or for that matter any other measurable parameters). So it’s simply not correct to say that they showed the frequency shift had a quantum origin.

Jay Gambetta, one of the leaders of IBM’s research effort in superconducting quantum computing, raised an even more basic question on twitter: did the authors even prove that the tardigrade was responsible for the frequency shift they observed?

Let’s walk through how the experiment was conducted. The authors placed their superconducting qubit in a tightly sealed copper enclosure, placed this box in a cryostat, cooled it down to a temperature of 10 millikelvin, and measured the qubit’s characteristic oscillation frequency. Then they warmed the whole system back up to room temperature, opened the box, placed the tardigrade inside, and repeated the cooldown and frequency measurement. They found that the qubit’s characteristic frequency shifted down by a quarter of one percent from the value in the first cooldown.

Without context, we can’t say whether 0.25% is a small effect. It sounds like a small number by the standards of our everyday experience, but there are many cases in physics where measurements and theoretical predictions are both precise enough that a much smaller discrepancy would be immensely significant.

Gambetta’s point is that this isn’t one of those cases. It’s not uncommon for superconducting qubit frequencies to shift slightly upon thermally cycling from 10 millikelvin to room temperature and back. It’s as if you disassembled a guitar, then replaced one of the strings and put it back together. If it’s now out of tune, this might be because the new string is different, or maybe you just didn’t tighten it to the same degree. Or perhaps both effects played a role.

We need to distinguish these two possibilities — tardigrade frequency shift vs. unrelated frequency shift — for even the claim of a classical tardigrade-qubit interaction to hold up. “Unrelated frequency shift” is of course a very broad category, and we can’t hope to come up with quantitative predictions for all possible non-tardigrade causes. So in this case the best strategy is purely experimental. A simple approach would be to repeat the cooldown many times with the tardigrade and many times without it and determine whether there’s a resolvable difference between the “tardigrade” and “no tardigrade” measurements when the variation within each group is averaged out.¹²Dan Palken, my favorite statistical pedant in the world, pointed out that my original phrasing here mixed up the standard deviation and the standard error of the mean. That would be too much detail to get into here, so I’ve rephrased this to make it less misleading. Thanks, Dan!

The authors did not do this: they only cooled down the system once with the tardigrade present and once without.¹³They do have one additional data point: the frequency of the second qubit in their experimental setup changed very little between the two cooldowns. That’s a lot better than nothing, and I think it’s more likely than not that the frequency shift is mostly due to the tardigrade. But Gambetta has spent the better part of two decades in superconducting qubit research, so we should take his critique seriously. They used these two measurements to retrodict the tardigrade’s dielectric constant, i.e., they asked “what dielectric constant would the tardigrade have to have to explain the measured frequency shift?” Retrodiction can be very useful as a sanity check — in this case it allowed them to check that the dielectric constant was neither implausibly high nor implausibly low — but it should not be confused with prediction. Another use of retrodiction is to enable predictions for follow-up experiments. But the authors don’t appear to have used the retrodicted dielectric constant beyond simply reporting its value.

Singapore in 8022

I’ll leave you with one final perspective on why we shouldn’t regard this experiment as a demonstration of qubit-tardigrade entanglement.¹⁴I added this section the day after I finished the rest of the post. Fast forward to the year 8022. Our civilization collapsed a long time ago, and all historical records were totally obliterated. But things are finally on the upswing: the civilizations that succeeded us have rediscovered the theory of electromagnetism, but still know nothing of quantum mechanics.

One day, archaeologists excavating the ruins of Singapore happen upon the lab of Rainer Dumke, miraculously spared the ravages of time. They summon the world’s leading physicists, who manage to get the experiment running again. The physicists figure out how to send signals with different frequencies into the apparatus, and observe a sharp response at one frequency. That’s not so surprising; they already know about resonant electrical circuits.

When the future physicists open the experiment up and poke around, they are shocked to find a tardigrade, in cryptobiosis for six thousand years, which wakes up and waddles away. Was this small creature part of the electrical circuit? The culture of the ancient past is truly inscrutable. The physicists run the experiment again and observe that the circuit’s characteristic frequency has shifted. They plug in the numbers and conclude that tardigrade flesh has a dielectric constant of \(\epsilon_\text{r} \approx 4\). They add this number to their running list of dielectric constants and call it a day.

Did our hypothetical future researchers learn anything that challenged their classical conception of the world? Would they begin to suspect the existence of quantum entanglement? Of course not! If an observer with a deep understanding of classical physics but no knowledge of quantum mechanics would be totally unfazed by the frequency shift, how can we claim it as evidence of entanglement?

Why care about any of this?

Carl Sagan coined the memorable phrase “extraordinary claims require extraordinary evidence.” A meaningful demonstration of a living organism in an entangled quantum state would be an extraordinary claim among extraordinary claims. I hope I’ve convinced you that the authors haven’t provided the requisite extraordinary evidence, and Jay Gambetta would say they haven’t even provided ordinary evidence.

Tobey Maguire is no Carl Sagan. But his 2002 film Spiderman is widely credited¹⁵As someone who attended high school in the mid-2000s, I take great delight in citing wikipedia in support of this claim. with popularizing the phrase “with great power comes great responsibility,” which bears some resemblance to Sagan’s coinage. In a world where news of extraordinary claims becomes immediately available to readers without any relevant expertise, extraordinary claims require both extraordinary evidence and great responsibility.

Let me not overstate the case. This specific paper has stirred up a minor media kerfuffle and unfortunately convinced some number of non-scientists that physics is close to Star Trek¹⁶Formerly known, due to a spell checking slip-up, as Stark Trek. Thanks to Steven Touzard for catching this. territory; it won’t do much further damage once it drops out of the news cycle. But the collective effect of trumped-up claims like this harms the credibility of the scientific enterprise. Journalists uncritically hyping unpublished preprints doesn’t help matters,¹⁷The availability of preprints itself isn’t really the problem here: peer review is an imperfect screening process, and for better or worse the cat’s out of the bag. On balance, the existence of the physics preprint server arXiv is a very good thing. but the primary responsibility in cases like this lies with scientists themselves. We can do better than this.

Appendix: What’s special about superconducting qubits

Both the mass-on-spring and the resonant circuit I’ve discussed here are examples of harmonic oscillators: oscillators with the special property that the characteristic frequency doesn’t depend on the oscillation amplitude (i.e., how far the mass is pulled away from equilibrium in the mechanical case, or the size of the current in the electrical case). For reasons that are far beyond the scope of this post, harmonic oscillators don’t make good qubits. Rather, a qubit is a quantum version of an anharmonic oscillator.

But there’s nothing intrinsically mysterious about anharmonic oscillators. Indeed, it’s easy to modify our mass-on-spring system to make it anharmonic: all we need to do is arrange for either the mass or the springiness to depend on the oscillation amplitude. It’s hard to imagine how the mass could depend on amplitude, so let’s focus on the springiness.

A spring whose springiness doesn’t depend on amplitude is also called a linear spring: its restoring force is proportional to the displacement, and that constant proportionality factor is the springiness. If I compare the restoring force \(F(x)\) when the spring is pulled back by a small distance \(x\) to the force when it’s pulled back by a distance \(2x\) or \(4x\), I’ll obtain \(F(2x) = 2F(x)\) and \(F(4x) = 4F(x)\). A nonlinear spring doesn’t obey this property. It might be the case, say, that the force is proportional to displacement for sufficiently small displacements but then gets slightly weaker, e.g., \(F(4x) = 3.5F(x)\). That’s all it takes to make our oscillator anharmonic.

The electrical equivalent of a nonlinear spring is a nonlinear inductor, whose inductance is roughly constant for small currents but decreases as the current increases. Superconducting circuits make good qubits for two main reasons. First, superconductors have very low resistance: this is the electrical equivalent of building a mass-on-spring system out of very high-quality materials to make it easier to observe quantum behavior. Second, there’s a way to put two superconductors together to make a nonlinear inductor, and there’s no good way to do this with normal metals like gold or copper. By analogy, it’s easy to imagine that certain materials might have structural properties better suited to crafting nonlinear springs than others (after all, steel is a more suitable material for a linear spring than wax).

Neither the low resistance of superconductors nor the nonlinear inductance of certain superconducting circuits can be explained without quantum mechanics. But then again, all material properties ultimately come down to quantum mechanics, and the equations describing resonant circuits take the measured inductance as an input: they are totally agnostic as to its physical origin. We can then calculate the characteristic frequency exactly as we did for a harmonic oscillator using the low-amplitude value of the inductance.