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Quantum physicists devise new tool to study classical cause-and-effect

Perimeter Institute researchers introduce a new technique called “inflation graphs” that helps unravel causal complexity and promises to prove even more powerful as computing capabilities increase.

Tobias Fritz, Robert Spekkens and Elie Wolfe

In the popular imagination, quantum physicists are probably best known for the inability to tell a living cat from a dead one. Thus, they might not be the first port of call for someone trying to sort out the tangle of cause-and-effect and confounding variables in, say, a massive drug trial.

That, it turns out, would be an unfortunate mistake because a trio of quantum foundations researchers at Perimeter Institute have made a substantial new contribution to the study of causation.

In two papers – the first published in the Journal of Causal Inference, the second currently on the arXiv – they introduce a new technique that helps unravel part of that causal complexity and has promise to prove even more powerful as computing capabilities increase.

Perimeter Faculty member Robert Spekkens and postdoctoral fellows Tobias Fritz and Elie Wolfe have used approaches inspired by fundamental quantum physics to create a new tool that helps discern plausible explanations of cause-and-effect in the classical world from impossible ones.

Wolfe and Miguel Navascués, from the Institute for Quantum Optics and Quantum Information at the Austrian Academy of Sciences, then showed that the technique can solve the problem with any desired accuracy (with the important corollary: the higher the accuracy, the greater the computational cost).

If this seems unusual territory for quantum researchers, that’s because it is. Causal inference research started in computer science and has since grown to encompass a mix of disciplines, from philosophy and statistics to epidemiology. Quantum physicists are relatively new to the field.

With this work, the Perimeter researchers show that they, too, can contribute to the discourse. The resulting tool, they hope, will prove powerful for researchers right across the spectrum.

Causal structures

Telling cause from effect is not as easy as it sounds. Often, what can look like a pretty direct relationship between two phenomena can turn out to be a peripheral thread – if they are related at all. The real connection can lie deeper: a common cause that is entirely unseen, its presence only revealed through its effect on other observables.

Perimeter Faculty member Robert Spekkens.

This is partly because, when talking about causality, one enters a realm of nearly endless probabilities. Take health research as an example. Medical researchers might test a cancer drug and find high rates of recovery for women over the age of 60. Is that because the drug works, or because women over 60 have a higher likelihood of spontaneous remission? Without careful analysis – and sometimes even with it – the data can’t tell you which connection is the right one.

In order to wade through this thicket of possible causal alternatives, researchers use causal diagrams – a visual formalism that represents each set of possible causal relationships (called a causal structure) as a graph.

The nodes in these graphs can depict two different kinds of variables: those that are observed and those that are hidden (or “latent”). An arrow connecting a pair of nodes indicates a direct cause-effect relationship between the corresponding variables.

These kinds of causal structures are already commonly used in health research, climate science, and economics. They are becoming increasingly important in other areas: today’s artificial intelligence systems are based on correlation, but for AI to replicate the way a person thinks, it will have to encompass understandings of cause and effect. For quantum theorists, causation could help distinguish quantum effects from classical.

Perhaps unsurprisingly, causal structures can be very complicated. Testing different causal hypotheses is a notoriously challenging problem in data science.

Statistical analysis can reveal significant insights that are hidden in large datasets. In the paper “The Inflation Technique for Causal Inference with Latent Variables,” Wolfe, Spekkens, and Fritz draw on probability and statistics to propose a new technique for tackling classical causality questions.

It’s called the inflation technique. It helps researchers find something called inequalities. And it all starts with a Bell.

Bell inequalities – and beyond

In the 1960s, physicist John Stewart Bell realized that there was a limit to how connected the properties of two particles could be in a classical world, i.e., a world governed by the laws of classical physics rather than those of quantum mechanics.

If, for example, the particles were separated by a great distance, then no action that is performed on one particle should be able to impact the result of any measurement performed immediately afterwards on the other particle.

Perimeter researcher Elie Wolfe.

Bell formulated this insight into a rigorous mathematical limit; any degree of correlation greater than this limit was an inequality – and a proof that researchers were measuring something non-classical (i.e., quantum mechanical).

For quantum researchers, Bell’s inequalities and others like them have become essential tools. If an inequality can be identified (by theorists) and then violated by measurements on suitable quantum systems (by experimentalists), then physicists will have pinpointed an instance where classical ideas of cause-and-effect break down and quantum ones take over.

For data scientists whose systems cannot possibly be quantum – epidemiologists, say, or economists – inequalities are also vital tools, but for different reasons. For them, an inequality is a red flag that shows that their assumptions about cause-and-effect are broken. In classical systems, any causal structure that has one of these flags is evidently an invalid causal hypothesis.

Unfortunately, there is no way to just look at a causal structure and identify all the potential inequalities it implies. Or, there wasn’t, until now.

With the inflation technique, the Perimeter researchers put forward a generic method to identify inequalities implied by any classical causal diagram. With Wolfe and Navascués’ formal hierarchy, the inflation technique can discover every constraint for statistics to admit a given causal explanation up to any desired accuracy. (To understand the extent to which any inequality can be violated within quantum theory, Wolfe and others later devised a special, quantum-specific version of the inflation technique, now on the arXiv.)

Spekkens, Fritz, and Wolfe then made a point of submitting their papers to a journal of the causal inference community, rather than a physics journal.

As Spekkens explains: “There’s a danger that, if the communities are too disconnected, the physicists could just reinvent the wheel without knowing that they’re doing so. If you can send it to the Journal of Causal Inference and they tell you that this is new, you can be confident that you’re moving the field forward.”

Perimeter Institute researcher Tobias Fritz.

That effort to reach a broader community is already having an impact, says Nicolas Gisin, a University of Geneva professor whose work spans quantum theory and experiment, with the “very original” technique generating much discussion among the causal inference experts.

“The inflation technique is a very new and promising tool,” says Gisin, who was not involved in the research. “I expect it to become a standard tool in my field.”

Still, Gisin cautioned that these were very early days for inflation graphs. “It did already lead to some discoveries, however, so far interesting only to quantum foundations,” he notes. “One needs to wait a bit and see how efficiently inflation can be programmed and how widely it can be applied.”

Apart from the potential research payoff, the work also underscores the value of applying fresh eyes to outstanding questions.

“Understanding causal structures is of paramount importance in very many fields, not limited to quantum physics,” Gisin says. “Having a new community looking at the problem in new ways will profit the society at large. New tools and new students trained somewhat differently than today’s experts in ‘big data analysis’ may lead to breakthroughs in many fields.”

Enter inflation

Generically, the inflation technique works like this. The researchers take the primary causal variables in the original causal diagram (known as the “root” variables) and imagine duplicating them. Each original, single variable is thus transformed into a plurality of identical but independently distributed copies.

It is possible to then reintroduce the dependent (non-root) variables on top of this inflated base, creating a so-called inflation graph in which every node corresponds to a copy of an original variable.

The technique allows the researcher to choose how many copies of each root variable they want to consider at a single time. There are therefore infinitely many ways to build an inflation graph from an original graph.

On left: The triangle scenario, observed nodes in yellow, latent nodes in blue.
Centre: The web inflation of the triangle scenario, where each latent node is copied once, leading to quadrupled observed nodes.
Right: Spiral inflation of the triangle scenario.

The power of the inflation technique to discover inequalities grows as the number of “root” copies increases. This technique is, in principle, capable of discovering every inequality implied by a given causal structure.

It’s a powerful result: a violated inequality is a red flag, showing that the system has diverged from what is possible in the classical world. That indicates either a quantum (non-classical) correlation, or else a mistaken assumption about cause-and-effect relationships. And that gives researchers a sieve to filter out all causal structures that are physically (classically) implausible, given some observational data.

The inflation technique can become extremely computationally demanding, however.

“We can talk about inflation level three, where we consider three copies of every root node, or inflation level four, where we consider four copies of every root node,” says Wolfe. “You get these larger and larger inflation graphs and more and more symmetry constraints which must be enforced to obtain the conclusions.”

The only limit right now is computing power. Using current computer technologies, Wolfe and Navascués can analyze causal scenarios by inflating the graph to level two or three.

Still, interest in the technique is already running high among quantum theory researchers, who are primarily interested in finding examples of quantum/classical separations: new examples of correlations that are so strong that you can’t explain them classically. “If you haven’t figured out where the classical constraints are, you might miss the fact that quantum mechanics offers an advantage,” says Spekkens.

While it will be a long time before the technique can be directly applied to massive datasets, Wolfe hopes the technique will soon become part of the standard causality toolkit, allowing researchers with big classical datasets to zoom in and probe deeply into small subsections of their causal diagrams.

Correlations between causality and quantum mechanics

The inflation technique was developed by quantum theorists, but Fritz believes its greatest use will likely be felt outside of fundamental physics.

“Causal inference is of interest to me primarily as a difficult mathematical problem with importance to problems of applied statistics,” he says. “There may be some applications to the theory of quantum information processing, and, in particular, to device-independent quantum cryptography. But in my opinion the main applications are in statistics.”

For Rob Spekkens, the work proves that the relationship between classical causality and quantum physics can be fruitful – and can work both ways. After all, these papers are entirely classical; they just happen to have been written by quantum theorists.

“The causal inference people have done all this amazing work, which we can take into the quantum context to ask questions about quantum theory,” he says.

Remember, if a new inequality is identified in a quantum system, it can help to pinpoint exactly where classical and quantum mechanics diverge. This is just one way in which the science of casual inference can move quantum mechanics forward.

These papers show that the relationship can be reciprocal – and that physicists have even more to offer. “Once you realize that Bell’s theorem is an example of this stuff, we have 50 years of expertise,” he says.

That’s why he is excited about the potential of this work, and what it could spur within the broader field of causal research.

“There’s an infinite number of these causal scenarios. We’d ideally like to have tools for deriving nontrivial inequalities for any causal scenario,” he says. “There’s a lot of work to be done.”



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