The Born rule for probabilities of measurement results is deduced from the set of five assumptions. The assumptions state that: (a) the state vector fully determines the probabilities of all measurement results; (b) between measurements, any quantum system is governed by that part of standard quantum mechanics, which does not refer to measurements; (c) probabilities of measurement results obey the rules of the classical theory of probability; (d) no information transfer is possible without interaction; (e) if two spin-1/2 particles are in the entangled state (sqrt(1-λ)|up>|up> + sqrt(λ)|down>|down>), and one of spins is measured by the Stern-Gerlach apparatus, then the state of the other spin after this measurement will be either |up> or |down>, corresponding to the measurement result. No one of these assumptions can be omitted. The method of the derivation is based on Zurek's idea of environment-assisted invariance [PRL 90, 120404 (2003)], though taken in rather modified form. Entanglement plays a crucial role in our approach (like in Zurek's one), so the probabilities are first considered in the case when the measured quantum system is entangled with some environment. Probabilities in pure states appear then as a particular case. The method of the present paper can be applied to both ideal and non-ideal measuring devices, irrespective to their constructions and principles of operation.
The Born rule for probabilities of measurement results is deduced from the set of five assumptions. The assumptions state that: (a) the state vector fully determines the probabilities of all measurement results; (b) between measurements, any quantum system is governed by that part of standard quantum mechanics, which does not refer to measurements; (c) probabilities of measurement results obey the rules of the classical theory of probability; (d) no information transfer is possible without interaction; (e) if two spin-1/2 particles are in the entangled state (sqrt(1-λ)|up>|up> + sqrt(λ)|down>|down>), and one of spins is measured by the Stern-Gerlach apparatus, then the state of the other spin after this measurement will be either |up> or |down>, corresponding to the measurement result. No one of these assumptions can be omitted. The method of the derivation is based on Zurek's idea of environment-assisted invariance [PRL 90, 120404 (2003)], though taken in rather modified form. Entanglement plays a crucial role in our approach (like in Zurek's one), so the probabilities are first considered in the case when the measured quantum system is entangled with some environment. Probabilities in pure states appear then as a particular case. The method of the present paper can be applied to both ideal and non-ideal measuring devices, irrespective to their constructions and principles of operation.
Zurek's derivation of Born's rule using envariance (invariance due to entanglement) is considered to capture the probability in full generality, but only as applied to measurement of a quantum observable. Contrariwise, textbook formulations of Born's rule begin with a pure state of a closed, undivided system. The task of this study is to show that a rearrangement of the Zurek approach is possible in which the latter is viewed as giving the probabilities for Schmidt states of an arbitrary composite state vector, and afterwards it is extended to probabilities in a closed, undivided system. This is achieved by determining simultaneously probability and measurement based on the fact that the physical meaning of probability and that of measurement are inextricably dependent on each other.
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 emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics.
This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption.
Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.
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.
I show how probabilities arise in quantum physics by exploring the implications of environment-assisted invariance or envariance, a recently discovered symmetry exhibited by entangled quantum systems. Envariance of perfectly entangled ``Bell-like'' states can be used to rigorously justify complete ignorance of the observer about the outcome of any measurement on either of the members of the entangled pair. For more general states, envariance leads to Born's rule ${p}_{k}\ensuremath{\propto}{\ensuremath{\mid}{\ensuremath{\psi}}_{k}\ensuremath{\mid}}^{2}$ for the outcomes associated with Schmidt states. The probabilities derived in this manner are an objective reflection of the underlying state of the system---they represent experimentally verifiable symmetries, and not just a subjective ``state of knowledge'' of the observer. This envariance-based approach is compared with and found to be superior to prequantum definitions of probability including the standard definition based on the ``principle of indifference'' due to Laplace and the relative frequency approach advocated by von Mises. Implications of envariance for the interpretation of quantum theory go beyond the derivation of Born's rule: Envariance is enough to establish the dynamical independence of preferred branches of the evolving state vector of the composite system and, thus, to arrive at the environment-induced superselection (einselection) of pointer states, which was usually derived by an appeal to decoherence. The envariant origin of Born's rule for probabilities sheds light on the relation between ignorance (and, hence, information) and the nature of quantum states.
Zurek has derived the quantum probabilities for Schmidt basis states of bipartite quantum systems in pure joint states, from the assumption that they should be not be affected by one party's action if the action can be undone by the other party (``envariance of probability'') and an auxiliary assumption. We argue that a natural generalization of the auxiliary assumption is actually strong enough to yield the Born rule itself, but that Zurek's argument and protocol can be adapted to do without this assumption, at the cost of using envariance of probability in both directions. We consider alternative motivations for envariance, one based on the no-signalling constraint that actions on one subsystem of a quantum system not allow signalling to another subsystem entirely distinct from the first, and another which is perhaps strongest in the context of a relative-state interpretation of quantum mechanics. In part because of this, we argue that the relative appeal of our version and the original version of Zurek's argument depends in part upon whether one interprets the quantum formalism in terms of relative states or definite measurement outcomes.