Cheating the Causal Game

Floppy the dog is a hero—according to my young daughter’s early reading book, that is. He smelt smoke, barked to alert the family and saved the house from burning down. Prior to this, the story goes, Dad had accidentally left the stove on, then put a wooden tray on it. Stories like this make sense, to all age groups, because we can piece together the correct order in which the events must have occurred, even when they are presented out of sequence.

From an early age we take the cause and effect of events happening in time for granted; it’s how we think. Without cause and effect, where would science be? We could not attempt to predict the outcome of experiments to test ideas about the world, or try to formulate such theories of what will happen. Even the math that describes the atomic world—quantum theory—assumes that events take place in time in an ordered and connected fashion. Which makes it all the more strange that some physicists are trying to ditch this neat time-ordering.

This is by no means an obvious strategy to employ, notes Caslav Brukner, at the University of Vienna, Austria, one of the physicists behind the idea. "It’s simply new physics," he says. "We are asking whether space, time and causal order are truly fundamental ingredients of nature." The team hopes that by taking an approach that doesn’t rely on causal structure, it might provide a clue about where causal order comes from. Is it a necessary property of nature or can it be derived from more primitive concepts?

Uncertainty is inherent in quantum theory. It’s well established that the physical aspects of quantum experiments, such as a particle’s position or momentum, are not well defined before they are measured. But postulating that the ordering of events is also somewhat fuzzy takes this conception of uncertainty to a bold new level. Now Brukner and his colleagues, Ognyan Oreshkov and Fabio Costa, also at the University of Vienna, have calculated that time-ordering can become muddled in some situations. Even weirder, this is helpful rather than harmful, and if harnessed could potentially improve quantum information processing protocols, and help researchers trying to devise a theory of quantum gravity.

Playing the Game

Brukner illustrates his approach with what he calls a causal game. Imagine that you and I are each given a different number, either 0 or 1. The aim of the game is for each of us to guess the other’s number correctly. Now if we define the causal relations between us—say, I am before you, and I can cheat and send my number to you before you guess—then it means that you can guess perfectly, whereas I can guess correctly only half of the time. This means that together we have a 75 per cent chance of winning the game if we have well-defined causal relations between us.

But Brukner’s new findings make things a bit more complicated. "We have shown that there are certain quantum resources that would allow us to go beyond this 75 per cent if the causal relations between us are not well-defined," says Brukner. In other words, if you don’t define your time-ordering, you can win the game more often. Their work is published in Nature Communications.

That sounds great in theory, but in practice, would a situation in which the causal relationship between us is muddied ever actually arise? A key idea to achieve such a situation involves the fact that quantum mechanics allows objects to exist in superposition, so that they can be in two or more contradictory states simultaneously; for instance, an electron could be in two different places at the same time. Brukner imagines that non-causal processes are most likely to be found where you have some sort of superposition of spacetime. "If you have a well-defined spacetime then you know that everything is going to be well-defined, with well-defined causal relations, so it can’t be that," he says. He sees realising these non-causal processes in the lab as one of the research field’s key challenges—it won’t be easy mixing up spacetime itself in an experiment. "It’s one of the weakest points of the whole thing, we are not sure how you can perform these quantum resources," says Brukner. The problem is that, when you start explaining how to do an experiment using well-defined language (appropriate for the well-defined spacetime in which we live), then you have well-defined causal relations between events. "We are working on this now, it’s a new land," he says.

Brukner may not need to worry so much, however. Other independent researchers are more optimistic about creating the required conditions to replicate the team’s theoretical quantum game in the lab, in a fairly down-to-earth scenario. All that is needed is to create a situation in which two players—you and I—can send information to each other through a wire and, crucially, the causal relation of who signals to whom must be ill-defined. Physicist Matt Leifer of the Perimeter Institute (PI), in Waterloo, Ontario, who was not involved in developing the paper, says that we can imagine such a wire that is controlled by a quantum system that is in a superposition of two states. That means that in this set-up, the signal can either go from me to you or from you to me, but we don’t know which will occur.

"With suitable interactions between the quantum system and the wiring then we should be able to generate some of the correlations described by Caslav," says Leifer. Although challenging, Leifer says it may be possible to set up some sort of an experiment along these lines in quantum computing. "I expect it to be difficult but not beyond the realms of possibility," he says.

Brukner’s team is not the first to investigate these issues. Perhaps fittingly for a theory of indefinite causality, it’s difficult to pin down exactly where and when these notions began. One of the first researchers to venture forth into what Brukner described as the "new land"—devising theories using indefinite causal relations—was Lucien Hardy, who is also based at PI (see arXiv:gr-qc/0509120v1).

Then, one afternoon over coffee at the University of Pavia, Italy, a group of quantum theorists were discussing Hardy’s ideas and wondered whether casual relations could be superposed. Just as Brukner now notes that such fuzziness can increase the chances of a win in a quantum game, they realised that in a similar manner, this uncertainty could have useful applications in quantum information. Giulio Chiribella, who was in that first discussion with Benoit Valiron, FQXI member Giacomo Mauro D’Ariano and Paolo Perinotti says: "From that afternoon discussion, we suggested a way to superpose the ordering of operations in a computation, and we argued that this effect could lead to more efficient protocols for information processing."

Quantum computers, in theory, exploit superposition to perform powerful operations on data—but the idea of indefinite causal structure brings the phenomenon of superposition into a new realm, the realm of the ordering of computational operations. "Caslav’s causal game has been a key result supporting this intuition," says Chiribella, from the Center for Quantum Information, Tsinghua University, Beijing. "This is because it has provided the first concrete example of an advantage coming from indefinite causal ordering."

He describes Brukner as "one of the leading researchers in the new school of quantum foundations," and always looks forward to meeting up with him over conference dinners to chat about quantum physics and the new directions the community is exploring. "No matter which subject we pick, these chats are always fun and inspirational," says Chiribella. "I very much like his pragmatic attitude, which combines foundational ideas with applications in quantum information, always keeping an eye on the possible experimental implementations."

Chribella has made some important contributions to the new research field: in testing the properties of two black boxes. The game is to work out the contents of the box based on how it changes numbers that are input into the box. For instance, if you input 1, 2 and 3 into the box and get out 3, 5 and 7, you could calculate that the box multiplies by 2 and adds one. Chiribella discovered that the process of working out the properties of the boxes is more efficient if, instead of examining one box first then the second, you take advantage of a superposition of the two possible orderings.

"These works clearly demonstrate that harnessing the new quantum effects that arise in the absence of a definite causal structure can offer advantages and help us to save computational resources and energy costs," says Chiribella. "I expect many new examples of this kind to appear in the next few years."

Quantum Gravitational Prize

Like many theoretial physicists, Brukner has his eyes on the bigger prize. The team hopes that their new description of cause-and-effect in the quantum world will help researchers developing a theory of quantum gravity, a grand project for theorists world-over. A successful theory of quantum gravity would merge quantum theory with Einstein’s theory of general relativity to describe every interaction in the universe that we know about, from the subatomic scale to the cosmological. One of the biggest obstacles has been that general relativity and quantum mechanics treat time very differently. In the former theory, time is another dimension alongside space and can bend and stretch, speed up and slow down, in different circumstances. Quantum theories, however, usually assume that time is set apart from space and ticks at a set rate. Theories of indefinite causality tackle this mismatch head-on, by questioning what time is at a fundamental level.

Hardy is very excited by Brukner and Chiribella’s work, and hopes that over the next five years, "we will see a proto-version of quantum gravity based on this way of thinking." Hardy has already developed a mathematical framework for physical theories—including quantum theory—that allows for indefinite causality by avoiding the notion of systems evolving in time. "The causaloid framework I developed might accommodate a theory of quantum gravity," he says.

All the researchers hope their efforts will help them to formulate a more general quantum theory, in which our familiar causal structure—of dogs barking and families waking—is not assumed, but emerges in the right conditions. "I find this new direction promising because it challenges one of the key paradigms of quantum and classical mechanics: the paradigm of a state evolving in time," says Chiribella. "We are now pushing quantum theory to the extreme limits of what can be conceived by our imagination."