Quantum probabilities as Bayesian probabilities

Eprint Arxiv 1011 6331, 2010

In the following we revisit the frequency interpretation of probability of Richard von Mises, in order to bring the essential implicit notions in focus. Following von Mises, we argue that probability can only be defined for events that can be repeated in similar conditions, and that exhibit 'frequency stabilization'. The central idea of the present article is that the mentioned 'conditions' should be well-defined and 'partitioned'. More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve a wide variety of classic paradoxes of probability theory. As a corollary, we arrive at the surprising conclusion that at least one central idea of the orthodox interpretation of quantum mechanics is a direct consequence of the meaning of probability. More precisely, the idea that the "observer influences the quantum system" is obvious if one realizes that quantum systems are probabilistic systems; it holds for all probabilistic systems, whether quantum or classical.

It is argued that quantum mechanics does not have merely a predictive function like other physical theories; it consists in a formalisation of the conditions of possibility of any prediction bearing upon phenomena whose circumstances of detection are also conditions of production. This is enough to explain its probabilistic status and theoretical structure. Published in: Collapse, 8, 87-121, 2014

Lecture Notes in Physics

From its very birth in the 1920s, quantum theory has been characterized by a certain strangeness: It seems to run counter to the intuitions that we humans have about the world we live in. According to these “realistic” intuitions all things have their definite place and sharply determined qualities, such as speed, color, and weight. Quantum theory, however, refuses to precisely pinpoint them. With respect to this apparent shortcoming of the theory different points of view can be taken. It could be suspected that quantum theory is incomplete, in that it gives a coarse description of a reality that is actually more refined. This is the viewpoint once taken by Einstein, and it still has adherents today. It calls for a search for finer mathematical models of physical reality, based on classical probability, often referred to as “hidden variable models” (see chapter “Photonic Realization of Quantum Information Protocols”). One such attempt is Bohm’s theory of non-relativistic quantum mec...

2001

The acquisition and representation of basic experimental information under the probabilistic paradigm is analysed. The multinomial probability distribution is identified as governing all scientific data collection, at least in principle. For this distribution there exist unique random variables, whose standard deviation becomes asymptotically invariant of physical conditions. Representing all information by means of such random variables gives the quantum mechanical probability amplitude and a real alternative. For predictions, the linear evolution law (Schrödinger or Dirac equation) turns out to be the only way to extend the invariance property of the standard deviation to the predicted quantities. This indicates that quantum theory originates in the structure of gaining pure, probabilistic information, without any mechanical underpinning.

arXiv (Cornell University), 2007

Quantum experiments yield random data. We show that the most efficient way to store this empirical information by a finite number of bits is by means of the vector of square roots of observed relative frequencies. This vector has the unique property that its dispersion becomes invariant of the underlying probabilities, and therefore invariant of the physical parameters. This also extends to the complex square roots, and it remains true under a unitary transformation. This reveals quantum theory as a theory for making predictions which are as accurate as the input information, without any statistical loss. Our analysis also suggests that from the point of view of information a slightly more accurate theory than quantum theory should be possible.

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2007

I outline an argument for a subjective Bayesian interpretation of quantum probabilities as degrees of belief distributed subject to consistency constraints on a quantum rather than a classical event space. I show that the projection postulate of quantum mechanics can be understood as a noncommutative generalization of the classical Bayesian rule for updating an initial probability distribution on new information, and I contrast the Bayesian interpretation of quantum probabilities sketched here with an alternative approach defended by Chris Fuchs.

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2007

In the Bayesian approach to quantum mechanics, probabilities-and thus quantum states-represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preëxisting system property. The idea that measurement outcomes occurring with certainty correspond to preëxisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [A. Stairs, Phil. Sci. 50, 578 (1983)], which applies the Kochen-Specker theorem to an entangled bipartite system.

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.

2014

In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the 'general tension-reduction' (GTR) model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a 'universal measurement'. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by 'uniform' fluctuations on the measurement situation. Since quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge ab...

2004

The probability 'measure' for measurements at two consecutive moments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We demonstrate that there are only two ways to resolve this problem. The first solution places emphasis on the precise use of the concept of conditional probability for successive measurements. The physically correct conditional probabilities define additive probabilities for two-time measurements. These probabilities depend explicitly on the resolution of the physical device and do not, therefore, correspond to a function of the associated projection operators. It follows that quantum theory distinguishes between physical events and propositions about events, the latter are not represented by projection operators and that the outcomes of two-time experiments cannot be described by quantum logic.