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How Information Theory Unifies Quantum Mechanics

A new framework ties together wave-particle duality and quantum uncertainty with bits.
​Image: Thomas Vandenberghe/Flickr

Two parallel mysteries of quantum mechanics, so-called wave-particle duality and the Heisenberg uncertainty principle, are in reality one and the same. This is according to a new framework developed by physicists at the National University of Singapore, which says that wave-particle duality relations are actually examples of quantum uncertainty.

Briefly: Wave-particle duality refers to the requirement that particles act like points, discrete packages located in one place at one time, and waves simultaneously. Simple dots vs. periodic blurs. A wave isn't a point, it's a space.


Uncertainty says, meanwhile, that we can't know both the position and velocity of a given particle at the same time. There is a fundamental limit to this knowledge.

The connection suggests a deeper principle at work behind the strangest and most fundamental physics in the universe. A unification of the quantum freakshow. This unification would bring us one step closer to understanding a realm of physics at the very boundary of understandable, yet one that's behind everything in existence.

The idea that all the myriad oddities of quantum mechanics are rooted in the same principle isn't new. It's intuitive even. In one of his famous lectures, Richard Feynman offered that, "[wave-particle duality] is a phenomenon which is impossible … to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]."

Superficially, wave-particle duality relations and quantum uncertainty actually sound quite a lot alike, but the formal connection hasn't been established, leaving them as similar but distinct-until-proven-otherwise phenomena.

Wave-particle duality has mystified physicists since the early-1800s, at least. A rippling blur of a particle-wave persists as long as we don't attempt to measure the given particle, in which case the wave is disturbed, collapsing into just a point. This "collapse" is the popular interpretation, at least, though much remains obscured.


Being both a particle and a wave is more of a paradox than it might at first sound; it can't be explained away as a particle becoming a wave or a wave becoming a particle, as if there is some before and after, like a caterpillar becoming a butterfly. It's both at once.

"This idea is manifested dramatically in the famous double slit experiment, where shooting single particles, let's say electrons, at a double-slit results in a wiggly pattern of bright spots and dark spots for the numbers of particles detected at different locations on a wall behind the slits," Patrick Coles, one of the three physicists behind the new work, explained in an email. "This pattern looks strikingly similar to what we would expect for wave interference, so it seems that electrons behave like waves."

"However, if we do the same experiment but observe which of the two slits the electron goes through, then the wiggly pattern is lost and the pattern looks more like a blob," the physicist continued. "This is what we would expect if the electron was a particle. It seems that the electron is uncertain whether it is a wave or a particle, or something in between, and this is essentially what we call wave-particle duality."

One popular interpretation of this is that the unmeasured, wave-like particle is going through both slits at the same time. Somehow.

Coles and his co-authors, ​whose work was published Friday in Nature Communications, show that the knowledge or lack thereof that we as human observers are able to glean from a physical entity or system of entities (the characteristics of a particle, such as spin or velocity) is subject to the same limits in wave-particle duality as in general quantum uncertainty. We can only know so much about either, and this limit is the same. This suggests they share the same underlying principle. Somehow.



What uncertainty says, in its most classic example, is that we can't know both the position and velocity of a given particle at the same time. Knowledge of one property proportionately limits knowledge of the other, so absolute knowledge of a particle's position means zero knowledge of its velocity. Imagine them as two quantities that, when added together, cannot equal more than 1. Complete certainty (1) can only be matched by complete uncertainty or no knowledge whatsoever (0).

This can be pretty weird too and, as with the WPDR double-slit, it implies that prior to measurement, a particle is able to be in many states at the same time. This is Schroedinger's cat, who, when shielded from observation, is permitted to be both alive and dead at the same time (given equal odds of being secretly poisoned or not).

Uncertainty is often given by a formulation known as "entropic uncertainty relations." This is a way of putting quantum physics in terms of information. It's an idea that's been around since the 1950s, but has more recently become a popular idea given the looming possibilities of quantum information within computing and cryptography. In essence: What is most that we're allowed to know about a given observable quantity?

So, in terms of information, we look at our double-slit setup as a choice between two paths, where one path is 1 and the other path is 0. This is referred to a which-path determination; particles follow paths. In wave terms, instead of particle paths, we look at phases. Phase in terms of waves is how much or how little of a periodic cycle has completed at a certain point in time. Scooting a wave forward or back is a phase shift.


As described in the paper, one phase corresponds to a "1" and the shifted version corresponds to "0," where the wave is shifted up or back by some amount (pi, specifically). We wind up with two statements: which-path and which-wave. Particle or wave.

It's just two different ways of looking at the same physics.

The Singapore physicists state the relationship between the two as a simple sum of bit-based information quantities that looks a whole lot like the relationship seen in the uncertainty principle. Remember, the uncertainty relationship constrains how much information we can acquire about a single particle/quantity where the sum of a particle's phase information and path information is limited at, again, 1. If we have 0 information about phase, we will have 1 (complete, total) information about path.

This upper limit happens to imply a crucial lower limit as well, which corresponds to information and is shared between WPDR (wave vs. particle) and quantum uncertainty (position vs. momentum, up/down spin vs. left/right spin). This lower limit is 1: a bit of information. It sounds so simple, but this is the connection. For a given quantum entity, it's not possible to describe both its lack of wave properties and its lack of particle properties with less than a 1 bit (a single bit with value 1).

"The reason why we have a lower limit in one case is that the quantities of interest are uncertainties," Coles explained. "That is, the quantities describe the lack of particle or wave behavior. In contrast, in the cases where we have an upper limit, the quantities of interest are certainties. That is, the quantities describe the presence of particle or wave behavior. But our result is that the two formulations are equivalent. It's just two different ways of looking at the same physics."



Uncertainty is a fact of information. A system can only store a finite amount of information and uncertainty, as a general concept but also applied to particles (as is usually considered), is like overflowing some toilet of binary bits.

"You can understand the uncertainty principle as a consequence of the fact that a physical system of a certain size—say dimension or energy constraint—can contain only a limited amount of information," Stephanie Wehner, another of the new study's co-authors, explained. "Very intuitively, if you had less uncertainty for some quantum measurements then you can use such a physical system to encode much more information: each measurement can be used to retrieve a portion of this information, and how well you can do that is determined by their mutual uncertainty."

Crucially, uncertainty is a statement about information, and it could correspond to a universe of other correlated phenomenon. This lower-limit has already been used, for example, to prove the security of certain quantum cryptography methods. In a separate statement, the researchers suggest that their new bridge between wave/particle relations and uncertainty could inspire new and improved encryption schemes.

Still, the practical implications can't hold a candle to what this new framework is saying. These are two features of quantum mechanics that we talk about all the time; they are the phenomena we use to explain the quantum world in the very first place.

"Physicists do not have an explanation for why the uncertainty principle is the way it is," Coles noted. "And so we say that the uncertainty principle is fundamental. And we try to understand other phenomena in terms of the basic idea stated in the uncertainty principle. One could argue that an implication of our work is that, in some sense, the wave-particle duality principle is less fundamental than the uncertainty principle. After all, we can explain it using the uncertainty principle."