Oxford Study Finds Five Famous Derivations of Quantum Probability All Quietly Assume Probability
Jiaxuan Zhang traced the assumptions behind five attempts to derive the Born rule and found each depends on additivity. The work does not rule out a probability-free derivation.
A new analysis from the University of Oxford suggests that physicists cannot get quantum probabilities out of quantum theory without first putting probability in. Jiaxuan Zhang examined five prominent attempts to derive the Born rule and found that every one of them depends on a probability-like assumption called additivity, sometimes stated openly and sometimes hidden inside other assumptions. The paper, "Summing to uncertainty: On the necessity of additivity in deriving the Born rule," is published in Physical Review A.
The Born rule is the recipe that gives the probability of each outcome when a quantum system is measured. It is central to quantum mechanics, yet it is usually added to the theory as an extra postulate. For decades, physicists have tried to derive it from the theory's other, non-random rules. Each attempt, however, has relied on its own assumptions about what probability is, which makes the attempts hard to compare.
Zhang took a different approach. Rather than assuming probabilities from the start, she assigned each measurement outcome a plain number and traced the assumptions behind the five derivations. All five, she found, rely on additivity: the idea that the number for "one outcome or another" equals the sum of the numbers for each outcome separately. The article describes this as an inherently probability-like property.
The role of additivity varies. In some derivations it is explicit. In others it is buried within other assumptions. In at least one argument, the derivation works only after additivity is added by hand.
Zhang also proved that additivity is distinct from another assumption that often appears in these arguments, noncontextuality, the idea that an outcome's number does not depend on how it is measured or which other outcomes are measured alongside it. The two are sometimes treated as the same thing, and the proof shows they are not.
The result does not close the question. The work does not prove that a probability-free derivation is impossible. It supports the view that some probability-like assumption is needed, and it sets out a common yardstick against which future derivations can be measured. Zhang's broader point is that "Quantum probabilities can't simply be extracted from a theory that contains none to begin with."
The finding bears on a long-running debate in the foundations of physics about whether randomness is fundamental or can be explained from deeper rules. The study is theoretical and reports no experimental data. For anyone claiming to have derived quantum probability from scratch, it supplies a concrete test: show where additivity entered.
