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A simple explanation of Born's rule

Adán Cabello

Published Jan 19, 2018• 5 citations
OpenAlex

Abstract

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.

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Oxford University Press eBooks•Jul 20, 2017•0 citations
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.
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.
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.
In a quantum-Bayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the
In a quantum-Bayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics. It seems to get quite far.
The Born rule assigns a probability to any possible outcome of a quantum measurement, but leaves open the question how these probabilities are to be interpreted and, in particular, how they relate to the outcome observed in an actual experiment. We propose to avoid this question by replacing the Born rule with two non-probabilistic postulates: (i) the projector associated to the observed outcome must have a positive overlap with the state of the measured system; (ii) statements about observed outcomes are robust, that is, remain valid under small perturbations of the state. We show that the two postulates suffice to retrieve the interpretations of the Born rule that are commonly used for analysing experimental data.
We can use quantum theory to explain an enormous variety of phenomena by showing why they were to be expected and what they depend on. These explanations of probabilistic phenomena involve applications of the Born rule: to accept quantum theory is to let relevant Born probabilities guide one’s credences about presently inaccessible events. We use quantum theory to explain a probabilistic phenomenon by showing how its probabilities follow from a correct application of the Born rule, thereby exhibiting the phenomenon’s dependence on the quantum state to be assigned in circumstances of that type. This is not a causal explanation since a probabilistic phenomenon is not constituted by events that may manifest it: but each of those events does depend causally on events that actually occur in those circumstances. Born probabilities are objective and sui generis, but not all Born probabilities are chances.