In Quantum Mechanics the transition from a deterministic description to a probabilistic one is done using a simple rule termed the Born rule. This rule states that the probability of an outcome (a) given a state (Ψ) is the square of their inner products ((a ⊤ Ψ) 2). In this paper, we will explore the use of the Born-rule-based probabilities for clustering, feature selection, classification, and for comparison between sets. We show how these probabilities lead to existing and new algebraic algorithms for which no other complete probabilistic justification is known.

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the nonprobabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics. Key words: Many worlds, quantum probability, rational decision theory. 1

Information-theoretic derivations of the formalism of quantum theory have recently attracted much attention. We analyze the axioms underlying a few such derivations and propose a conceptual framework in which, by combining several approaches, one can retrieve more of the conventional quantum formalism.

We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason's theorem under stronger assumptions.

The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there for simple systems always is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantummechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability / quantum inference theory in the simple case. Planck’s constant, and the Schrödinger equation are also derived from this conceptual framework. The theory is illustrated by one and

We argue that the intractable part of the measurement problem—the ‘big ’ measurement problem—is a pseudo-problem that depends for its legitimacy on the acceptance of two dogmas. The first dogma is John Bell’s assertion that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is the view that the quantum state has an ontological significance analogous to the significance of the classical state as the ‘truthmaker ’ for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. We show how both dogmas can be rejected in a realist information-theoretic interpretation of quantum mechanics as an alternative to the Everett interpretation. The Everettian, too, regards the ∗E-mail address:

A simple model of quantum particle is proposed in which the particle in a macroscopic rest frame is represented by a microscopic d-dimensional oscillator, s=(d-1)/2 being the spin of the particle. The state vectors are defined simply by complex combinations of coordinates and momenta. It is argued that the observables of the system are Hermitian forms (corresponding uniquely to Hermitian matrices). Quantum measurements transforms the equilibrium state obtained after preparation into a family of equilibrium states corresponding to the critical values of the measured observable appearing as values of a random quantity associated with the observable. Our main assumptions state that: i) in the process of measurement the measured observable tends to minimum, and ii) the mean value of every random quantity associated with an observable in some state is proportional to the value of the corresponding observable at the same state. This allows to obtain in a very simple manner the Born rule. 1

What is probability? Physicists, mathematicians, and philosophers have been engaged with this question since well before the rise of modern physics. But in quantum mechanics, where probabilities are associated only with measurements, the question strikes to the heart of other foundational problems: what distinguishes measurements from other physical processes? Or in more formal terms: when are the unitary dynamical equations suspended in favour of probabilistic ones? This is the problem of measurement in quantum mechanics. The most clearcut solutions to it change the theory: they either add hidden variables (as in the pilot-wave theory), or they give up the unitary formalism altogether (as in state-reduction theories). The two strategies are tied to different conceptions of probability: probability as in classical statistical mechanics (as formulated by Boltzmann, Gibbs and Einstein), and probability as in Brownian motion (with the dynamics given by a stochastic process, as formulated by Einstein and Smulochowski). The former is sometimes called epistemic probability, as classical

It is often objected that the Everett interpretation of QM cannot make sense of quantum probabilities, in one or both of two ways: either it can’t make sense of probability at all, or it can’t explain why probability should be governed by the Born rule. David Deutsch has attempted to meet these objections. He argues not only that rational decision under uncertainty makes sense in the Everett interpretation, but also that under reasonable assumptions, the credences of a rational agent in an Everett world should be constrained by the Born rule. David Wallace has developed and defended Deutsch’s proposal, and greatly clarified its conceptual basis. In particular, he has stressed its reliance on the distinguishing symmetry of the Everett view, viz., that all possible outcomes of a quantum measurement are treated as equally real. The argument thus tries to make a virtue of what has usually been seen as the main obstacle to making sense of probability in the Everett world. In this note I outline some objections to the Deutsch-Wallace argument, and to related proposals by Hilary Greaves about the epistemology of Everettian QM. (In the latter case, my arguments include an appeal to an Everettian analogue of the Sleeping Beauty problem.) The common thread to these objections is that the symmetry in question remains a very significant obstacle to making sense of probability in the Everett interpretation. 1

During the last ten years or so, derivations of the Born rule based on decision theory have been proposed and developed, and it is claimed that these are valid in the context of the Everett interpretation. This claim is critically assessed and it is shown that one of its key assumptions is a natural consequence of the principles underlying the Copenhagen interpretation, but constitutes a major additional postulate in an Everettian context. It is further argued that the Born rule, in common with any interpretation that relates outcome likelihood to the expansion coefficients connecting the wavefunction with the eigenfunctions of the measurement operator, is incompatible with the purely unitary evolution assumed in the Everett interpretation. Key words: