In this paper some consequences of the strengthened interpretative rules of quantum mechanics, which were proposed in an earlier paper, are obtained. It is also seen that, in general, the usual interpretative rules are too weak to obtain these results. For example, it is proved from the strengthened rules that if f:R → R is a Borel function which is also τ-definable, then for each observable A the procedure ``measure A and compute f(A outcome)'' is an f(A) measurement procedure. It is also shown that there exist Borel f and observables A such that the above procedure is not an f(A) measurement procedure. Two methods of measuring the sum A + B of two observables are then considered: The measurement of A and B on different systems followed by addition of the results and (if A and B commute) the simultaneous measurement of A and B on the same system followed by addition. It is proved from the strengthened rules that the first method is not a valid measurement procedure and the second is valid. Besides these, other processes such as procedures for preparing mixtures of different states, and the empirical generation of probability measures from outcome sequences are considered.
In this paper some consequences of the strengthened interpretative rules of quantum mechanics, which were proposed in an earlier paper, are obtained. It is also seen that, in general, the usual interpretative rules are too weak to obtain these results. For example, it is proved from the strengthened rules that if f:R → R is a Borel function which is also τ-definable, then for each observable A the procedure ``measure A and compute f(A outcome)'' is an f(A) measurement procedure. It is also shown that there exist Borel f and observables A such that the above procedure is not an f(A) measurement procedure. Two methods of measuring the sum A + B of two observables are then considered: The measurement of A and B on different systems followed by addition of the results and (if A and B commute) the simultaneous measurement of A and B on the same system followed by addition. It is proved from the strengthened rules that the first method is not a valid measurement procedure and the second is valid. Besides these, other processes such as procedures for preparing mixtures of different states, and the empirical generation of probability measures from outcome sequences are considered.
The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. The reason is that they do not include the intuitive requirement of randomness of the outcome sequence obtained from an infinite repetition of measuring an observable on a state. A strengthening of the rules is proposed which includes, in essence, a precise definition of randomness. The resultant rule is seen to be intuitively more satisfying than the usual rules and to include the expectation value rule and to include essentially all of the spectrum rule of the usual rules. It also suggests that the relationship between the foundations of mathematics and quantum mechanics may be quite deep and complex.
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.
We present a formulation of quantum mechanics based on a logic representing some aspects of the behaviour of the measurement process. With such an approach, we make no direct mention of quantum states, and thus avoid the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some characteristics that any model (and hence any formulation of quantum mechanics compatible with its prediction and relying on a notion of measurement) should verify. The main results we obtain are that in the case of a Hilbert space of dimension at least 3, no model can lead to the prediction with certainty of more than one atomic outcome. Moreover, if the Hilbert space is finite dimensional, then we are able to precisely describe the structure of the predictions of any model of our logic. As a consequence, we finally show that all the models of our logic make exactly the same predictions regarding whether a given sequence of outcomes is possible or not.
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.