250-Year-Old Probability Theorem Reimagined Through Quantum Physics

Our understanding of probability has long been guided by Bayes’ rule, a mathematical principle first proposed in 1763. At its heart, Bayes’ rule reflects how we update our expectations when new information arrives. Now, for the first time, researchers have extended this centuries-old rule into the quantum realm, opening fresh avenues for quantum computing, error correction, and machine learning.

“This is a breakthrough in mathematical physics,” said Prof Scarani. His colleague Prof Buscemi added: “Bayes’ rule has been helping us make smarter guesses for 250 years. Now we have taught it some quantum tricks.”

What Is Bayes’ Rule?

Bayes’ rule was first introduced by Thomas Bayes in his essay An Essay Towards Solving a Problem in the Doctrine of Chances. It provides a way to calculate the conditional probability of an event—that is, the probability of something happening given that we already know something else.

For example, imagine someone tests positive for flu. Their prior belief about being sick may have been uncertain, but the test result changes that belief. Bayes’ rule helps calculate the updated probability of actually having flu, taking into account both the accuracy of the test and the person’s initial expectations.

In essence, Bayes’ rule treats probability as a measure of belief rather than absolute truth. This has sparked debate for centuries, since many statisticians view probability as something objective and detached from personal beliefs. Still, Bayes’ rule has proved indispensable, finding use in medical diagnosis, weather prediction, economics, data science, and artificial intelligence.

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The Principle of Minimum Change

One of Bayes’ rule’s strengths lies in its principle of minimum change. This principle says that when new evidence appears, our beliefs should be adjusted in the smallest way possible to remain consistent with the new facts.

Take the flu test again: a negative result doesn’t prove someone is healthy, but it reduces the likelihood of them having the flu. Mathematically, this means updating probabilities while staying as close as possible to the original distribution of beliefs.

The research team extended this principle into the quantum world. Instead of using classical probability distributions, they turned to quantum states, which describe the probabilities of finding particles in various locations or conditions. The measure of change was defined using quantum fidelity, a mathematical tool that compares how similar two quantum states are.

A Quantum Bayes’ Rule

Physicists have long suspected that a quantum version of Bayes’ rule must exist, because quantum states themselves are defined in probabilistic terms. But until now, no one had succeeded in deriving such a rule from first principles.

The team achieved this by maximising the fidelity between two mathematical objects representing the “forward” and “reverse” processes—an approach analogous to comparing classical probability distributions. Maximising fidelity was shown to be equivalent to minimising change, thereby giving rise to a natural quantum Bayes’ rule.

Interestingly, in some cases their equations matched a mathematical tool known as the Petz recovery map, introduced by Hungarian mathematician Dénes Petz in the 1980s. The Petz map has long been suspected as a candidate for the quantum Bayes’ rule because of its elegant properties, but until now it lacked a rigorous derivation from a higher principle.

“This is the first time we have derived it from such a principle, which may serve as a validation for using the Petz map,” explained Prof Scarani.

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Why It Matters

The implications of this work go beyond pure mathematics. The Petz map, and now its confirmed grounding in the principle of minimum change, holds promise for quantum computing. In particular, it could help design more efficient methods for quantum error correction, ensuring that fragile quantum information is preserved during computations. It may also enhance quantum machine learning algorithms, where updating probabilities is central to training models.

Looking ahead, the researchers plan to explore whether applying the minimum change principle to other quantum measures could reveal additional rules or insights.

A Bridge Across Centuries

Bayes’ rule, once a tool for solving problems in gambling and statistics, has now found a surprising new home in quantum theory. By connecting a 250-year-old idea with the most advanced concepts in physics, this research shows how classical reasoning can continue to guide us—even in the strange, probabilistic world of quantum mechanics.

As Prof Buscemi put it: “Bayes’ rule has been helping us make smarter guesses for centuries. Now, in the quantum world, it may help us unlock the future of technology.”