Born's Rule from the Principle of Unitary Equivalence

Sabine Hossenfelder

Zhen Su

The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.

Lev Vaidman A

Gaurang Agrawal, Matt Wilson

A basic postulate of modern compositional approaches to generalised physical theories is the generalised Born rule, in which probabilities are postulated to be computable from the composition of states and effects. In this paper we consider whether this postulate, and the strength of the identification between scalars and probabilities, can be argued from basic principles. To this end, we first consider the most naive possible process- theoretic interpretation of textbook quantum theory, in which physical processes (unitaries) along with states and effects (kets and bras) and a probability function from states and effects satisfying just some basic compatibility axioms are identified. We then show that any process theory equipped with such structure is equivalent to an alternative process theory in which the generalised Born rule holds. We proceed to consider introduction of noise into any such theory, and observe that the result of doing so is a strengthening of the identification between scalars and probabilities; from bare monoid homomorphisms to semiring isomorphisms.

Peter Nyman

In this paper we continue to study so called “inverse Born’s rule problem”: to construct representation of probabilistic data of any origin by a complex probability amplitude which matches Born’s rule. The corresponding algorithm – quantum-like representation algorithm (QLRA) was recently proposed by A. Khrennikov [1]–[5]. Formally QLRA depends on the order of conditioning. For two observables a and b, b|a- and a|b conditional probabilities produce two representations, say in Hilbert spaces H b|a and H a|b. In this paper we prove that under natural assumptions these two representations are unitary equivalent. This result proves consistency QLRA. Keywords quantum-like representation algorithm, inverse Born’s rule problem, order of conditioning, unitary equivalence of representations. 1

Dimiter Prodanov