Abstract Probability is distinguished into two kinds: physical and epistemic, also, but less accurately, called objective and subjective. Simple postulates are given for physical probability, the only novel one being a locality condition. Translated into no-collapse quantum mechanics, without hidden variables, the postulates imply that the elements in any equiamplitude expansion of the quantum state are equiprobable. Such expansions therefore provide ensembles of microstates that can be used to define probabilities in the manner of frequentism, in von Mises’ sense (where the probability of P is the frequency of occurrence of P in a suitable ensemble). The result is the Born rule. Since satisfying our postulates, and in particular the locality condition (meaning no action-at-a-distance), these probabilities for no-collapse quantum mechanics are perfectly local, even though they violate Bell inequalities. The latter can be traced to a violation of outcome independence, used to derive the inequalities. But in no-collapse theory it is not a locality condition; it is a criterion for entanglement, not locality.
Abstract Probability is distinguished into two kinds: physical and epistemic, also, but less accurately, called objective and subjective. Simple postulates are given for physical probability, the only novel one being a locality condition. Translated into no-collapse quantum mechanics, without hidden variables, the postulates imply that the elements in any equiamplitude expansion of the quantum state are equiprobable. Such expansions therefore provide ensembles of microstates that can be used to define probabilities in the manner of frequentism, in von Mises’ sense (where the probability of P is the frequency of occurrence of P in a suitable ensemble). The result is the Born rule. Since satisfying our postulates, and in particular the locality condition (meaning no action-at-a-distance), these probabilities for no-collapse quantum mechanics are perfectly local, even though they violate Bell inequalities. The latter can be traced to a violation of outcome independence, used to derive the inequalities. But in no-collapse theory it is not a locality condition; it is a criterion for entanglement, not locality.
I define a notion of locality LOC, closely modelled on Bell's principle of local causality, construed as the condition that single-case probabilities cannot be modified by actions at spacelike separation. The new principle, like Bell's, forces Bell inequalities, but with two loopholes: one is violation of measurement independence, known to Bell, but the other is non-uniqueness of remote outcomes, a loophole only for LOC, not for Bell's principle. I also set out a theory of physical probability, applicable to the Everett interpretation, in which the Born rule is derived, and which therefore violates Bell inequalities. I show it is consistent with LOC. Surprisingly, both loopholes are exploited. I conclude not only that physical probability in the Everett interpretation involves no action at a distance, in the sense of LOC, but that the observed violation of Bell inequalities is powerful evidence for many worlds.
Quantum non-locality is normally defined via violations of Bell's inequalities that exclude certain classical hidden variable theories from explaining quantum correlations. Another definition of non-locality refers to the wave function collapse thereby one can prepare a quantum state from arbitrary far away. In both cases one can debate on whether non-locality is a real physical phenomenon, e.g. , one can employ formulations of quantum mechanics that do not use collapse, or one can simply refrain from explaining quantum correlations via classical hidden variables. Here we point out that there is a non-local effect within quantum mechanics, i.e. , without involving hidden variables or collapse. This effect is seen via imprecise ( i.e. , interval-valued) joint probability of two observables, which replaces the ill-defined notion of the precise joint probability for non-commuting observables. It is consistent with all requirements for the joint probability, e.g. , those for commuting observales. The non-locality amounts to a fact that (in a two-particle system) the joint imprecise probability of non-commuting two-particle observables ( i.e. , tensor product of single-particle observables) does not factorize into single-particle contributions, even for uncorrelated states of the two-particle system. The factorization is recovered for a less precise ( i.e. , the one involving a wider interval) joint probability. This approach to non-locality reconciles it with locality, since the latter emerges as a less precise description.
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.
Probabilities may be subjective or objective; we are concerned with both kinds of probability, and the relationship between them. The fundamental theory of objective probability is quantum mechanics: it is argued that neither Bohr's Copenhagen interpretation, nor the pilot-wave theory, nor stochastic state-reduction theories, give a satisfactory answer to the question of what objective probabilities are in quantum mechanics, or why they should satisfy the Born rule; nor do they give any reason why subjective probabilities should track objective ones. But it is shown that if probability only arises with decoherence, then they must be given by the Born rule. That further, on the Everett interpretation, we have a clear statement of what probabilities are, in terms of purely categorical physical properties; and finally, along lines laid out by Deutsch and Wallace, that there is a clear basis in the axioms of decision theory as to why subjective probabilities should track these objective ones. These results hinge critically on the absence of hidden-variables or any other mechanism (such as state-reduction) from the physical interpretation of the theory. The account of probability has traditionally been considered the principal weakness of the Everett interpretation; on the contrary it emerges as one of its principal strengths.
Entangled qubits are interconnected at any separation. Quantum mechanics represents the interconnection by one unfactorizable entangled state. A measurement at one station collapses the entangled state and fixes the state assigned to the other qubit, depending on the outcome. Interpreted as physics, the entangled state and wavefunction collapse conflict with Einstein's separability and locality. We resolve the conflict by Corpuscular Quantum Mechanics, which replaces the entangled state with source constraints and reproduces the quantum joint probabilities exactly. Each qubit leaves the source carrying one of two definite states and an unshared random variable. An extended Born rule assigns a generally complex coefficient to the transition from the source-assigned state to the recorded outcome, using only what the qubit carries and the locally chosen measurement direction. Averaging products of the coefficients over the source ensemble returns the joint probabilities. Nothing is fitted. An exact identity maps the amplitudes of any pure n-qubit state to the source constraints. We prove the random variables admit a positive distribution meeting the constraints. The identity yields the joint probabilities of the Bell, Greenberger-Horne-Zeilinger, Eberhard, W, Hardy, and Dicke states for arbitrary directions. The resolution requires no action at a distance, no collapse, and no outcomes for unperformed measurements. The one change is that the event-level quantity is a transition coefficient, not a probability. Bell's theorem therefore does not apply. Quantum mechanics is thus one recipe for computing the correlations, and the entangled state is a feature of the recipe, not of nature.
Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born's rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as "unitary quantum mechanics", and the assumption that ensembles on finite-dimensional Hilbert spaces are characterized by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability.