Statistics is a core pillar of data science, yet its assumptions are not always fully tested. This is exacerbated by the rise of quantum computing, where even statistical axioms can be violated. In this article, we explore just how quantum physics breaks statistics, and uncover ways to understand it using data science analogies.

Let’s play a coin-toss game: toss three coins, and try to have them all land differently. This is a seemingly impossible task, because no matter how rigged a coin is, it can only have two sides. There simply aren’t enough possibilities for all three tosses to land differently.

Yet, with the power of quantum physics, such an impossible feat can be achieved statistically: three coin tosses can all land differently. And the reward for winning? 2022’s Nobel Prize in Physics, which was awarded to Alain Aspect, John Clauser, and Anton Zeilinger on 2022-10-04.

According to nobelprize.org, their achievements were

“for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science.”

This sentence is filled with jargon: *entangled photons*, *Bell inequalities,* and *quantum information* science. We need a simpler, plain English description for such an important feat. Here’s a translation:

Scientists showed that our statistical view of the world is flawed, by showing that quantum physics can defy seemingly impossible odds.

The details of these impossible odds are captured by mathematical formulae called *Bell inequalities*. Instead of flipping coins, researchers demonstrated these impossible odds by playing with lasers (using beams of *entangled photons*)*.*

How is this relevant to data science? Since our quantum mechanical world is the ultimate source of data, flaws in our statistical laws could disrupt the very foundation of data science. If statistics is indeed incomplete, we wouldn’t be able to trust conclusions derived from it.

Fortunately, in our Universe, these statistical flaws tend to be very tiny and negligible. Nevertheless, it is important to understand how classical statistics needs to be modified, as data science in the distant future may need to incorporate these flaws (e.g., in quantum computers).

Before answering how quantum physics defies the laws of statistics, we first need to understand how statistics works as an effective description for our world.

Flip a coin, you get heads/tails. Yet coins aren’t exactly random: A robot with perfect control can seriously rig a coin-toss.

What does a 50/50 probability mean? A coin’s orientation is very sensitive to the minute details of its surrounding. This makes it difficult to predict a coin’s landing orientation. So instead of solving very complicated equations to come up with a deterministic outcome, we opt for a nondeterministic one. How? We observe that typical coins are pretty symmetrical with respect to heads/tails. In the absence of any particular bias, 50/50 odds would be a great approximation (although studies have shown these odds can be altered, e.g., Clark MP et al.).

To summarize,

Probabilities are approximations for modeling details of a complex system. Complicated physics is traded for uncertainties in order to simplify the mathematics.

From weather patterns to economics and healthcare, uncertainties can be traced back to complex dynamics. Mathematicians have converted these approximations into rigorous theorems based on axioms, to help us manipulate and derive insights from unpredictable outcomes.

How does quantum physics break the laws of statistics? It violates the *Additivity Axiom*.

How does this Axiom work? Let’s consider some common scenarios where we use statistics to make decisions:

- When it’s rainy 🌧 outside, we bring an umbrella ☔️.
- When we get sick, doctors prescribe medications 💊 to help us get better.

In the rainy scenario, while there could be trillions of ways raindrops could fall, the majority of these possibilities make us wet and cold, so we bring an umbrella.

In the doctor scenario, there are multiple possibilities given a diagnosis: different disease progressions, side-effects, recovery rates, quality of life, or even misdiagnosis… etc. We choose the treatment that will lead to the best overall outcome.

The Additivity Axiom is the formalized statement that we can break probability down into possibilities:

This Axiom makes sense because statistics is created to quantify our ignorance of a system. Just like how we assign 50/50 to a coin flip, we use the Additivity Axiom to derive properties of a system by averaging out all the possible trajectories of its constituents.

While all this sounds intuitive, is it really how nature works? Through experiments, we can confirm that macroscopic objects work this way, but what happens when we zoom in on the microscopic? Is it the same as the macroscopic world, with subatomic actors moving from one scene to the next? Or is it more like a movie screen, where abstract pixels are blinking on/off, creating the illusion of a story?

It turns out, the pixel analogy is more accurate. The distinct paths of possibilities become more ill-defined as we zoom in. As a consequence, the Additivity Axiom is violated.

What is the replacement for our Axiom? It’s the laws of quantum physics.

While quantum physics is quite complicated, we can understand its gists through data science analogies. Quantum physics is based on linear algebra, and thus can be thought of as a special ML model.

Below are the key quantum axioms linked to ML analogies:

- The world is described by giant list of (complex) numbers, called a
*quantum state —*analogous to the pixel values of an image, or more abstract embedding vectors in ML. - As time goes on, this quantum state changes. This update can be computed by passing our quantum state through a neural network like function, called an
*operator*(a unitarity matrix technically):

Continuing our ML analogy, we can think of the Universe as a giant neural network. Each operator represents a (linear) network layer. Through this network, every interaction that has occurred has been imprinted onto the quantum state of our Universe. Without pause, this computation has been continuously running since the beginning of time. This is a profound way of viewing our world:

Our coherent reality emerges from isolated groupings in our quantum state.

Our macroscopic feeling of an object’s existence emerges from the specific neural network linkages of our operators.

It all sounds a bit abstract, so let’s consider an explicit example: how does quantum physics describe raindrops falling on our heads?

- The data of the air molecules and us in the open are captured in a quantum state.
- As water molecules feel the Earth’s gravity, the quantum state gets updated by the corresponding operators.
- After going through many layers in this neural-network-like update, the quantum state picks up some particular numerical values.
- Laws of physics dictates that these numbers tend to form clusters. Some of these clusters translate into a consistent existence for these raindrops, which ultimately link to our neurons feeling these raindrops.

In this modern viewpoint, there is no reason why the Additivity Axiom should hold. Because

Similar to an ML blackbox, it is not always possible to track all the physical properties of a quantum state. Therefore, a physical outcome doesn’t always come with a list of intermediate possibilities.

In the raindrop scenario, this means that we can’t always find the specific numbers in the quantum state that leads to a specific water molecule falling. In fact, the quantum state generally contains data of the molecules in multiple locations (e.g., superpositions), and our perception of its physical location could be a complicated sum of all these data.

This may seem paradoxical, as we do we not sense weird discrepancies and superpositions in our daily lives at all! The reason though is that these discrepancies are tiny, and their tininess can be proved using the technical theory of decoherence, which is well beyond our scope (although here is one of my articles that may help shed some light).

Still, being tiny isn’t the same as being zero. Quantum effects can at times be significant, and they can lead to seemingly impossible statistics.

How? Let’s find out.

In order to invalidate ordinary laws of statistics, we need to consider simple but impossible scenarios. The simplest of which involves 3 coins.

Imagine 3 robots performing 3 separate coin-tosses. In classical statistics, we can use the Additivity Axiom to fully specify the statistics: by listing all 8 outcomes and their probabilities (Note: the robots/coins could be rigged):

Experimentally, we can measure these probabilities by repeating these coin-tosses.

Regardless of the choice of probabilities, there is a sanity constraint: A coin only has 1+1 = 2 sides, so when we flip 3 coins, there are bound to be at least 2 of them that land the same. So if we randomly (uniformly) choose one pair of coins to examine, we should expect at least 1/3 chance to observe that they are equal.

Let’s try out some examples, label the three coins as *A*, *B*, *C*

- If all 3 coins are fair and independent, then the chance that we pick an equal pair is 1/2.
- If
*A**= B*, but*A*≠*C*. Regardless of how*A*is tossed, there is only one equal pair. The chance to pick this pair is 1/3.

We see that the same-pair probability is always at least 1/3. This can be summarized into a *Bell inequality *(following this paper by L. Maccone)

While it might seem ridiculous to test something so obvious, it would turn out that this inequality can in fact be *violated — *a testament that they are not so obvious after-all.

In order to observe violation of Bell inequality, physicists can’t just rely on conventional coins. Instead they need to utilize quantum coins made of lasers, which has all the ingredients for coin-tosses:

- Flipping a coin: sending a laser down a beam
- Observing Head/Tail: getting a reading on one of two detectors*
- Randomness: readings are generally unpredictable unless manipulated

(* there could be faulty readings if no detector observes anything)

Now, we can setup the lasers in different orientations to mimic 3 different coin-tosses. So how exactly can quantum coins manage the impossible? If we observe the literal result of three coin-tosses, seeing three different outcomes is logically impossible.

This is where our Bell inequality comes in: it breaks down a logical statement about 3 coins into a probability statement that involves only 2 coins per term. So if we toss 3 coins, but only observe 2 at a time, then it is possible to violate statistical laws while preserving logic. In quantum physics, tossing a coin vs observing a coin follows two distinct interactions:

**Quantum**: tossing a coin and observing it are governed by two different operators. A coin-toss that hasn’t been observed yet does not need to be assigned a definitive outcome*.

This is in contrast with classical statistics

**Classical**: heads/tails are determined when the coins are tossed. This is guaranteed by the Additivity axiom. It doesn’t matter whether we decide to observe it or not.

(*This is where “spooky action-at-a-distance” comes in, since at any moment anyone can turn on a detector to observe the third coin and ruin our results.)

How to perform our experiment then? We need to prepare our coins to be in a particular quantum state. Here, we cook up a system where the 3 coins quantum state can be denoted by three vectors on a plane, like the one shown below*:

(* Technically the quantum state involves more complicated entangled photons, but we’ll skip the details for brevity)

What is the probability that two coin-tosses would yield the same result? The answer comes from physics, and is engineered to be the cosine similarity squared:

Now, if we randomly select a pair of quantum coins to examine*, there is only a 1/4 chance that they’d be the same; this is lower than the logical 1/3 guarantee!

(*The experiment needs to be set up such that this choice is chosen after the coins have been tossed, so that one can rule out spooky collusion between the particles and the apparatus)

Rephrasing this in terms of our Bell inequality, we have

Our sanity check is violated! If we pretend that classical statistics still applies, this would imply that that at least 1/4 of the time, all three coin-tosses land differently!

Note that while our three-coin experiment is simple to understand, there are experimental difficulties and potential loopholes in its results. Thus, typical experiments tend to involve more coin-tosses and more convoluted observations (e.g., GHZ experiment by Jian-Wei Pan et el.).

So, we see that quantum probabilities sometimes lead to unexpected results, what is the big deal, and why should we care?

First, let’s start with the practical. As technology pushes toward packing more computational power in a smaller size, quantum physics will become more important. Eventually, our computational paradigms will need to be overhauled in order to take full advantage of quantum devices. So while violations of Bell inequalities may be subtle, it signals that we need to think carefully when designing quantum algorithms.

Second, these violations expose a fundamental limit on conventional statistical reasoning. For example, if someone wins the lottery, it is perfectly reasonable to attribute the cause to the lottery balls coming out in a particular way. However, we cannot zoom in and causally link winning lottery to the (quantum) state of all the molecules in the room. So our statistical theory of causal inference has a physical limit!

Lastly, quantum effects challenge us to rethink our Universe. While quantum physics has been validated repeatedly, it could still just be an approximation. In the future, we may yet discover its succession by even more abstract fundamental laws.

As a historical lesson, even Einstein was dissuaded by quantum physics’s weirdness, so much so that he rejected it by proclaiming “god does not play dice”. Yet quantum physics continued to triumph and was fundamental in advancing much of our modern technology and understanding of the world (see my article).

In summary, quantum physics rules the world, and 2022’s Physics Nobel highlights its deep connection to statistics and data science. While quantum physics isn’t commonly taught, we should all strive to understand and embrace its significance.