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FQXI ARTICLE
October 16, 2017

Why Quantum?
Entropy could explain why nature chose to play by quantum rules.
by Colin Stuart
FQXi Awardees: Jon Barrett, Matt Leifer
June 11, 2014
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Jon Barrett
University of Oxford
Jon Barrett, a quantum physicist at the University of Oxford, UK, sounds like he’s setting up a joke: "What has the compression of data got to do with smashed mugs?"

Rather than generating laughs, however, the punchline could answer a far more profound question. By investigating the physical laws that govern these seemingly unrelated processes, Barrett and his colleague Matt Leifer, from the Perimeter Institute for Theoretical Physics in Ontario, Canada, hope to explain, "Why quantum?"—that is, why the subatomic realm is governed by the strange laws of quantum mechanics rather than by an alternative theory.

Quantum theory is one of the most successful frameworks in science. But it is also decidedly odd. Physicists cannot use the theory to calculate the precise outcomes of quantum experiments before they have been performed, for instance; they can only work out the probabilities of getting a certain result. But quantum mechanics is not the only probabilistic theory that could describe reality at the fundamental level. In recent years, physicists have discovered that there are a host of alternative probabilistic theories that reproduce many of the quantum world’s most exotic features—such as entanglement, interference, teleportation, and nonlocality. (See "Why Did Nature Choose Quantum Theory?") Yet these rivals seem to have been rejected in our universe. But why? "This whole research thread is motivated by the question: Why has nature chosen this particular set of rules?" says Barrett.

To get to the bottom of this issue, Barrett and Leifer have taken advice from a giant in the history of science, British physicist Sir Arthur Eddington. In his classic book The Nature of the Physical World written in 1928, Eddington described how to assess the credibility of a proposed new model of physics:

"If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations"—the equations that describe electromagnetism—"then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation —well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

Smashed Mugs

The Second Law of Thermodynamics relates to entropy, which is often described as a measure of the disorder of a system. The law states that entropy within a closed system always increases with time. Or, in terms of Barrett’s "joke," the remnants of smashed mugs do not spontaneously reconvene into their more ordered original state. That is because there are many configurations of the shattered pieces of a mug that correspond to its being broken, but only one configuration in which they are perfectly arranged to recreate the complete pristine mug, so it is far less likely that we will find that these pieces have randomly rearranged themselves back into their initial state. More technically, "thermodynamic entropy is a measure of the number of different microstates that correspond to the observed macroscopic properties of a system," Barrett explains.


Entropic Order
Entanglement (conceptualised here) is a feature of quantum theory and also
some other probabilistic theories. Could entropy explain why one realization
is favoured over another?

Credit: Art by Stephanie Simmons, © 2013 OxfordQuantum.org
Since thermodynamics deals with the science of heat flow and energy, it is not immediately obvious how it relates to quantum theory. But the crucial point is that the concept of entropy also arises in other disciplines. For instance, in information theory, "Shannon entropy" describes the compressibility of a message, that is, how much of the message you can afford to leave out and still expect the recipient to be able to reconstruct what you have said. "There’s this curious coincidence that these two things are the same—they share the same mathematical formula," says Leifer.

This brings us back to Barrett’s quip: How can the same measure tell us something about the compressibility of data and the fate of broken mugs? And could it also tell physicists something about the fundamental laws that apply to subatomic particles? With their FQXi grant of almost $120,000, the duo hope to find out by surveying the landscape of generalised probability theories, which share many properties with both quantum mechanics and classical probability. Looking for a difference between thermodynamic and information entropy in these non-quantum frameworks might open a window on whether these measures of entropy are truly the same in our physical world.

"We want to explore if there is a conceivable world in which they are different. Whether these two notions always coincide in every physical theory you can imagine," says Leifer. If they can prove that the two entropies are different in general probability theories then that implies that they are conceptually different, if not mathematically different, in our world as well.

We want to explore if there
is a conceivable world in
which they are different.
- Matt Leifer
One way to examine entropy in these alternative frameworks is to return to the way Austro-Hungarian mathematician John von Neumann put classical Gibbs entropy and statistical mechanics on a quantum footing in the 1930s, not long after Eddington’s famous words on thermodynamics were published. Von Neumann’s argument involved a quantum gas of particles in a box which can be pushed with pistons, passed through filters and immersed in a heat bath. His line of reasoning led him to derive the expression for the von Neumann entropy still in use today. "One way of getting at what thermodynamic entropy might mean in a generalised probabilistic theory would be to consider a system from that theory in a box and do similar things to it as von Neumann did and see if an expression for the entropy comes out," says Barrett.

By exploring entropy in this way Barrett and Leifer’s work strikes right at the heart of our understanding of the quantum world. If entropy works in a similar manner across the majority of generalised probability theories then the quantum rules which govern the world around us may not be particularly special. However, the work that stems from their FQXi grant may show that quantum mechanics sits in a privileged position amongst the gamut of possible scenarios. This would then explain why quantum laws rule our world.

Physical Grounding

The Second Law may also help to pick out quantum theory over its rivals in a second fashion. Physicists always strive to ground their theories in aspects of physical reality. For instance, one of the founding principles of Einstein’s special theory of relativity is the physical principle that the speed of light is a constant. At the moment, however, the phenomena associated with quantum physics can be derived from abstract mathematical postulates, but not from physical ones. "We want to see if we can make a small list of physical postulates in such a way that quantum theory is the unique thing that satisfies them," explains Barrett. So treasured is the Second Law that it is a stand out candidate for inclusion in such a list.


Matt Leifer
Perimeter Institute
It might also be possible to show that in some generalised probabilistic theories the Second Law of Thermodynamics is violated. That is anathema to physicists, so it is powerful too: If you hold the Second Law as preciously as Eddington did then you can use it to discard those theories in which it does not hold. That is useful because one day we might find that our quantum view of the world breaks down and that it is only an approximation to a more deep-seated reality. Any theory worthy of replacing quantum mechanics would still need to assign probabilities to the outcome of experiments and so would be found in the landscape of generalised probability theories that Barrett and Leifer are investigating. Physicists should be able to instantly rule out a sub-section of the choices that violate the Second Law, due to their prediction that smashed mugs of coffee can surreptitiously reform.

Tony Short, a quantum physicist at the University of Bristol, UK, agrees that our current way of presenting quantum theory is unappealing. "It’s very mathematical, it doesn’t really seem very natural," he says. "If you can find something more natural, like the Second Law, that leads you to quantum theory then perhaps you’ve understood quantum theory better." However, Short cautions that Barrett and Leifer are tackling tricky terrain. "Similar flavoured things have already be explored a little bit. It’s a difficult challenge, things involving entropy are quite complicated," he says.

It may be a steep mountain to climb but a successful ascent would provide a marvelous and potentially revolutionary view of reality from the summit. As Stephanie Wehner, a quantum physicist at the Centre for Quantum Technologies in Singapore, explains, the approach may eventually be useful for physicists who are struggling to combine quantum theory with Einstein’s theory of gravity, general relativity, in order to understand the very first moments after the Big Bang or what happens within black holes. So far these two theories have proved to be incompatible and no unified theory of quantum gravity has been found. "Using these generalised probability theories we might learn something about nature," says Wehner, particularly "where quantum mechanics cannot make predictions or where it doesn’t fit together with some other aspects of physics."

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Recent Comments


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