Physical theories assign values of quantities to physical systems, and they assign probabilities to the possible outcomes of measurements of those quantities. In quantum theory, probabilities are assigned using the Born rule. Here, I show that the Born rule is uniquely determined by the conditions under which quantum systems instantiate values of quantities as they are specified by the eigenstate-eigenvalue link, together with minimal theory-independent assumptions about how probabilities are sensitive to the structure of physical quantities.
Physical theories assign values of quantities to physical systems, and they assign probabilities to the possible outcomes of measurements of those quantities. In quantum theory, probabilities are assigned using the Born rule. Here, I show that the Born rule is uniquely determined by the conditions under which quantum systems instantiate values of quantities as they are specified by the eigenstate-eigenvalue link, together with minimal theory-independent assumptions about how probabilities are sensitive to the structure of physical quantities.
Identifying the property of the world that enforces the Born rule is a longstanding problem in physics. We prove that in any physical theory that assigns probabilities to the outcomes of ideal measurements, the maximal set of probability assignments for each graph of exclusivity is the one that satisfies the Born rule. Therefore, the agreement between quantum theory and experiments follows from a one-to-one correspondence between the logical possibilities and the physical possibilities and, in particular, implies that the outcomes of quantum measurements are not constrained by any physical reason.
Born's rule is the recipe for calculating probabilities from quantum mechanical amplitudes. There is no generally accepted derivation of Born's rule from first principles. In this paper, it is motivated from assumptions that link the ontological content of a proper physical model to the epistemic conditions of the experimental context. More precisely, it is assumed that all knowable distinctions should correspond to distinctions in a proper model. This principle of "ontological completeness" means, for example, that the probabilistic treatment of the double slit experiment with and without path information should differ. Further, it is assumed that the model should rely only on knowable ontological elements, and that failure to fulfill this principle of "ontological minimalism" gives rise to wrong predictions. Consequently, probabilities should be assigned only to observable experimental outcomes. Also, the method to calculate such probabilities should not rely on the existence of a precise path of the observed object if this path is not knowable. A similar principle was promoted by Born, even though he did not apply it to probability. Another crucial assumption is that the proper rule to calculate probabilities should be generally valid. It should be applicable in all experimental contexts, regardless the setup that determines which attributes of the studied object are observed, together with the probability to observe each of the associated attribute values. There is no need to refer to the Hilbert space structure of quantum mechanics in the present treatment. Rather, some elements of this structure emerge from the analysis.
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.
The Born-rule, which assigns probabilities to measurement outcomes, is one of the fundamental axioms of quantum physics. It dates back to the time of the establishment of the formalism of quantum physics in the first half of the 20th century. From the beginning, and particularly in connection with the development of different interpretations of the theory, there has been a desire/need to better understand the true nature of the Born-probabilities. Are they classical/epistemic of origin or are they irreducible and of on tic stature as a kind of intrinsic propensities of physical systems? We show that, by only using the mathematical formalism of the original theory, we find a possible answer.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the "measurement problem" of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.