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In this chapter we present briefly the basic notions of classical and quantum theories of probability and information. This chapter is especially important for biologists, psychologists, experts in cognition, and sociologists who were not trained in quantum theory (but even classical theory is presented in a simple manner). We start with the presentation of the standard measure-theoretic formulation of the modern classical probability theory (Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin [1]). Then we turn to fundamentals of quantum formalism, including theory of open quantum systems and its generalizations.

We start to consider the dynamics of populations of complex multilevel systems with two well-known examples: a two-level system and a system consisting of a level and a band which is either continuous or composed of discrete levels

We now consider level grouping in multilevel systems other than the one-dimensional relay of isolated levels discussed in the previous chapter. The population dynamics of such systems can display various manifestations that are typical of the systems already considered, such as exponential decay at short times, quantum recurrences and revivals, cooperative behavior, and quantum localization. Such systems can also demonstrate diffusive behavior. Typically, a problem of this type does not correspond to any hidden symmetry resulting in specific commutation relations among different parts of the Hamiltonian that comprise a closed subalgebra, and the evolution occurs in the entire Hilbert space, being constrained only by energy conservation.

Thus far we have been considering the level–band system, for which an explicit analytic expression (3.26) for the Fourier transform of the probability amplitude exists. However, as we have seen, even for such a system, the complexity of the problem does not allow one to directly perform the inverse Fourier transformation in the generic case, and therefore in order to get physically consistent analytic results, one needs to apply a statistical description based on the idea of ensemble averages. The situation becomes more difficult still when we consider the more complex quantum system of two coupled bands.

Computation of decay time for entangled quantum systems is an important aspect of decoherence theories. Here we explore this topic from the standpoint of computing the decay time to the existence of a joint probability distribution of the entangled particles – atoms, in our case. We also analyze the problem from the viewpoint of the decay of an improper upper probability distribution, for the entangled particles and its continuous decay into a proper probability distribution. A standard quantum decoherence model and the upper-probability model have, it turns out, the same expected decay time for a familiar example of a system with a Bell state.

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) Another axiom due to Soler forces K to be the field of real, or complex numbers, or the quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's theorem fully characterizes the probability measures on the algebra of events, so that Born's rule is derived. (d) Gleason's theorem is equivalent to the existence of a certain finite set of rays, with a particular orthogonality graph (Wondergraph). Consequently, all aspects of quantum probability can be derived from rational probability assignments to finite "quantum gambles". (e) All experimental aspects of entanglement- the violation of Bell's inequality in particular- are explained as natural outcomes of the probabilistic structure. (f) We hypothesize that even in the absence of decoherence macroscopic entanglement can very rarely be observed, and provide a precise conjecture to that effect .We also discuss the relation of the present approach to quantum logic, realism and truth, and the measurement problem.

When a system is isolated and the Hamiltonian is independent of the time, the energy eigenstates are true stationary states. If the system initially is in a particular state Ψ n which is an eigenstate of the Hamiltonian H 0, it will stay there for all time. The time dependence of Ψ n is of the form $$ {\psi _n}\left( t \right) = {\psi _n}\left( 0 \right){e^{ - i{E_n}t/h}} $$ (1) where E n is the energy for the state n. If, however, the Hamiltonian is time dependent the energy is no longer a constant of motion. Knowledge of the behaviour of the system is obtained by solving the time dependent Schrodinger equation $$ i\hbar \frac{\partial }{{\partial t}}\psi = H\psi $$ (2) Unfortunately the structure of the Hamiltonian H in most cases does not permit a solution of this equation in closed form.

The story of quantum probability theory and quantum logic begins with von Neumann’s recognition1, that quantum mechanics can be regarded as a kind of “probability theory”, if the subspace lattice L(H) of the system’s Hilbert space H plays the role of event algebra and the ‘tr(WE)’-s play the role of probability distributions over these events. This idea had been completed in the Gleason theorem 2

Entropic arguments are shown to play a central role in the foundations of quantum theory. We prove that probabilities are given by the modulus squared of wave functions, and that the time evolution of states is linear and also unitary.

The simplest dynamical system is the point particle characterized by its mass. Its state is specified by its position and momentum. The dynamical law is the description of how these quantities change in time. For a free particle, the momentum remains constant while the position increases in the direction of the momentum. The increase is directly proportional to the elapsed time and inversely proportional to the mass. When such a particle is subjected to a force, the momentum also changes proportional to the force. The instantaneous state may be represented as a point in phase space which moves along a trajectory. For complicated interactions, the trajectories may form an intricate pattern but they do not intersect, and the generic dynamical evolution may be viewed as a mapping of the phase space onto itself.

A fundamental principle of quantum mechanics is that if a process can occur by more than one independent path, then the probability of that process occurring can be calculated by adding the probability amplitudes for each path and then squaring the sum. A well known example is the photoionization of an atom or a molecule. One route connecting the ground state g > with the continuum 39-1 > is direct ionization, with a probability amplitude that is proportional to 39-1

According to the complementarity principle, a photon behaves either as a particle or as a wave. That is, if it does not behave as a particle it necessarily behaves as a wave, and vice versa. It has also been argued that particle-like and wave-like behaviours are not mutually exclusive, and that a photon can also behave as a particle and as a wave. Here I propose a variant of a previous experiment in which a photon behaves neither as a particle nor as a wave. An alternative nonlocal realistic interpretation is presented, in which the notion of path followed by the photon can be meaningful.

Different versions of an experiment which allows us to prove a Bell’s theorem for two particles without using an inequality are discussed. The experiment could be used to test local realism with the introduction of additional assumptions based on direct experimental observation.

In his book More Than One Mystery,1 Mark Silverman takes issue with Feynman’s claim that “in reality it [the phenomenon of electron self-interference] contains the only mystery [of quantum mechanics]2. Silverman objects that: “As one directs attention away from systems of single particles to systems of correlated particles or particles and space together, other mysteries equally profound arise”, going on to list three such phenomena in support of his claim that

Recent results suggest that quantum mechanical phenomena may be interpreted as a failure of standard probability theory and may be described by a Bayesian complex probability theory.

Non-commutative measure theory serves as a mathematical basis to compare the classic 1 and the quantum mechanical probability scheme.

The results of the most recent efforts of Henry P. Stapp to formulate an account of physical reality compatible with relativity and the quantum theory are reviewed and remaining problems are identified. The most salient features of this quantum ontology are two kinds of realities, actual events and potentialities, and two kinds of time, Einstein time and process time. The non-local character of the “potentia” of this ontology provides in a natural manner for problems posed by Bell’s theorem. But left outstanding are difficulties with the actuality/potentiality dualism and with the two kinds of time.

The superposition law for probability amplitudes, usually considered as an empirical rule and introduced into the quantum theory as an ad hoc assumption, is shown to be the mathematical solution of almost selfevident requirements for the structural metric of the probabilities in general. When combining the general ψ metric of superposition with the postulate of a constant probability density in p-space for given p, and of a constant probability density in p-space for given q, one arrives at the mathematical result that ψ (q,/p) must necessarily have complex periodic form, exp(iqp/const). Various quantum principles can thus be reduced to a nonquantal basis.

The classical dynamics of Einstein’s closed universe (idealized for simplicity to be empty except as excited by gravitational waves) is analyzed in no way more economically than by the standard Hamilton-Jacobi method. In it the essential ideas are (1) a spacelike 3-geometry, (3) G; (2) an infinite-dimensional superspace S, each point (or collection of equivalent points) of which stands for one and only one (3) G; (3) a Hamilton-Jacobi function S (dimensions of action) or wave-phase S/ℏ (measured in radians) that propagates through superspace, S = S((3) G), according to the standard Einstein-Hamilton-Jacobi equation of A. Peres; (4) a ‘constructive interference’ between this family of wave crests and other families of wave crests (obeying the same EHJ equation) that picks out (U. Gerlach) ‘Yes’ points (constructive interference) in superspace from ‘No’ points (destructive interference); (5) definition through these ‘Yes’ points of a ‘leaf of history’ slicing through superspace; and (6) identification of this leaf of history with a particular spacetime (4) G that satisfies Einstein’s field equation. Thus, every spacelike slice through this (4) G is a ‘Yes’ (3) G, and conversely. When one goes from classical theory to quantum theory, one goes from S((3) G) to ψ((3) G), with the approximate correspondence: ψ ∼ (slowly varying amplitude) exp (iS/ℏ). When one enriches the content of the theory from pure Einstein geometrodynamics to geometry plus particle-fields and other fields, one still has every degree of freedom of the universe included in the independent variables upon which ψ depends. Therefore the observer himself would seem to be included in ψ. However, so to ‘include the observer in’ makes insuperable difficulties of principle, which are described. One has to accept that physics never permits a complete prediction, but only a forecast of a correlation between events (E. P. Wigner). One is led to recognize that a wave function ‘encompassing the whole universe’ is an idealization, formalistically perhaps a convenient idealization, but an idealization so strained that it can be used only in part in any forecast of correlations that makes physical sense. For making sense it seems essential most of all to ‘leave the observer out of the wave function’.

In his paper of 1948 [1] and in his book of 1956 [2] Louis de Broglie stressed that the ordinary relationship between joint probability and conditional probability no longer holds in wave mechanics, and that the difference lies above all in the definition of the joint probability of two physical quantities which are not simultaneously measurable. What is still more important, he showed that the double solution theory is not subject to this difficulty. He is certainly right in emphasizing this problem of joint probability and conditional probability, since it is the centre of the curious features of wave mechanics, which should be dealt with first if we wish to attempt an improvement of its ordinary interpretation.

Quantum mechanics has opened a vast sector of physics to probability calculus. In fact most of the physical interpretation of the formalism of quantum mechanics is expressed in terms of probability statements.1

Several attempts have been made in recent years to understand the theoretical foundations of quantum mechanics via axiomatization and model construction. While the philosophical and physical significance of these efforts remains largely unclear, they have produced some highly interesting methodological and formal problems. Many of these, especially the ones concerning the understanding of observables, states, and ‘logics’, are insidious and probably will bother philosophers and (mathematical) physicists for a long time.

Ideas of wave-particle duality of light are traced through various stages, from Einstein to quantum optics and recent developments in the conceptual foundations of quantum mechanics.

Interference effects produced by two sources of light are discussed within the framework of the quantum electrodynamic interpretation. Interference should disappear when all the source atoms are in the fully excited state, and this prediction may distinguish between quantum mechanics and various guided wave theories. The introduction of a light amplifier into an interferometer arm cannot unambiguously determine the path of the photon, because of spontaneous emission effects. Quantum mechanics, like classical wave optics, therefore predicts only a reduction of visibility in such interference experiments. In the special case of an interference experiment in which the two light sources consist of two single atoms, two photons cannot be found simultaneously at two positions that are separated by an odd number of half interference fringes. This quantum mechanical prediction amounts to another kind of EPR paradox, and should lend itself to another experimental test of the theory.

The fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint "distribution" derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists of some observations concerning the axiomatic foundations of quantum mechanics.

The notion and methods of calculating probabilities in quantum mechanics is strikingly different from classical mechanics and standard probability theory. Quantum mechanics manipulates mysteriously, from a classical probabilistic point of view, complex functions and operators to nevertheless end up with observable quantities which are interpreted as if they were derived from a standard type of probability theory. Whether the strange methods of obtaining the predictions of the theory are essential, in the sense that given the type of results quantum mechanics predicts implies a quantum mechanical type of mathematical formalism, is not totally clear. Certainly many of the paradoxes seem to be based on the mode of calculating probabilities. For example, the paradox of the double slit experiment arises from the fact that the probability of the particle being at a certain point on the screen is not the same probability that it went through one slit or the other. Since standard probability theory arose out of simplistic considerations it may be that it is restricted and that its future development will generalize it to a degree where quantum mechanics will fit in naturally. Weiner (1963) made some attempts along these lines by showing some examples from ordinary probability theory that yield results similar in mathematical form as in quantum mechanics. Another approach to the understanding of the probabilistic methods is to attempt to formulate quantum mechanics as a traditional probability theory. We shall describe this attempt, and although we shall show that this is strictly speaking impossible, the impossibility will illuminate certain aspects of the theory. In particular we will emphasize the question of joint distributions of non-commuting variables and show the one-to-one relationship between so-called correspondence rules and the distribution functions. As we shall see, the question as to whether there can exist a consistent rule for obtaining quantum mechanical operators (a correspondence rule) from their classical counterpart and the question whether there exists a legitimate joint distribution function are one and the same.