We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.
In this paper we analyze the existence of joint probabilities for the Bell-type and GHZ entangled states. We then propose the usage of nonmonotonic upper probabilities as a tool to derive consistent joint upper probabilities for the contextual hidden variables. Finally, we show that for the extreme example of no error, the GHZ state allows for the definition of a joint upper probability that is consistent with the strong correlations.
We study transformations of conventional (`classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known `quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not…
We provide frequency probabilistic analysis of perturbations of physical systems by preparation procedures. We obtained the classification of possible probabilistic transformations connecting input and output probabilities that can appear in physical experiments. We found that so called quantum probabilistic rule is just one of possible rules. Besides the well known trigonometric transformation (for example, for the polarization of light), there exist the hyperbolic transformation of probabilities. In fact, `hyperbolic polarization' have laready been observed in experiments with elementary par…
Relativistic causality, namely, the impossibility of signaling at superluminal speeds, restricts the kinds of correlations which can occur between different parts of a composite physical system. Here we establish the basic restrictions which relativistic causality imposes on the joint probabilities involved in an experiment of the Einstein-Podolsky-Rosen-Bohm type. Quantum mechanics, on the other hand, places further restrictions beyond those required by general considerations like causality and consistency. We illustrate this fact by considering the sum of correlations involved in the CHSH in…
We study the probabilistic consequences of the choice of the basic number field in quantum formalism. We demonstrate that by choosing a number field for a linear space representation of quantum model it is possible to describe various interference phenomena. We analyse interference of probabilistic alternatives induced by real, complex, hyperbolic (Clifford) and p-adic representations.
The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigatio…
We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be define…
This paper extends our probabilistic framework for two-player quantum games to the mutliplayer case, while giving a unified perspective for both classical and quantum games. Considering joint probabilities in the standard Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting for three observers, we use this setting in order to play general three-player non-cooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities provided by the EPR-Bohm setting can change the outcome of a game, while requiring that the quantum game attains a classical interpretation for factoriza…
Aharonov and Reznik have recently (in quant-ph/0110093) argued that the form of the probabilistic predictions of quantum theory can be seen to follow from properties of macroscopic systems. An error in their argument is identified.
In this letter, two different probabilistic teleportations of a two-particle entangled state by pure entangled three-particle state are shown. Their successful probabilities are different.
By analysing probabilistic foundations of quantum theory we understood that the so called quantum calculus of probabilities (including Born's rule) is not the main distinguishing feature of "quantum". This calculus is just a special variant of a contextual probabilistic calculus. In particular, we analysed the EPR-Bohm-Bell approach by using contextual probabilistic models (e.g., the frequency von Mises model). It is demonstrated that the EPR-Bohm-Bell consideration are not so much about "quantum", but they are merely about contextual. Our conjecture is that the "fundamental quantum element" i…
The standard approach to realize a quantum repeater relies upon probabilistic but heralded entangled state manipulations and the storage of quantum states while waiting for successful events. In the literature on this class of repeaters, calculating repeater rates has typically depended on approximations assuming sufficiently small probabilities. Here we propose an exact and systematic approach including an algorithm based on Markov chain theory to compute the average waiting time (and hence the transmission rates) of quantum repeaters with arbitrary numbers of links. For up to four repeater s…
In this paper we introduce the three main notions of probability used by physicists and discuss how these are to be used when invoking spacelike separated observers in a relativistic format. We discuss a standard EPRB experiment and concentrate upon problems of the interpretation of probabilities. We promote a particularly conservative interpretation of this experiment (which need not invoke an objective notion of collapse) where probabilities are, tentatively, passively Lorentz invariant. We also argue that the Heisenberg picture is preferable in relativistic situations due to a conflict betw…
The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: ``first context and then probability.'' We start with the conventional formula…
The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.
We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely cons…
We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement and preparation procedures) we can obtain $\cosθ$-perturbation (interference term) in probabilistic rule connecting preparation procedures for purely corpuscular objects. On one hand, our investigation demonstrated that there is nothing special in so called `quantum probabilities': the right choice of statistical ensembles gives the possibility to escape all `…
In quantum operations, probabilities characterise both the degree of the success of a state transformation and, as density operator eigenvalues, the degree of mixedness of the final state. We give a unified treatment of pure-to-pure state transformations, covering both probabilistic and deterministic cases. We then discuss the role of majorization in describing the dynamics of mixing in quantum operations. The conditions for mixing enhancement for all initial states are derived. We show that mixing is monotonically decreasing for deterministic pure-to-pure transformations, and discuss the rela…
In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into perfectly distinguishable pure states. The spectral entropy, and analogues using other Schur-concave functions, can be defined as the entropy of these probabilities. We describe additional conditions under which the outcome probabilities of a fine-grained measurement are majorized by those for a sp…
Quantum mechanics may be formulated as Sensible Quantum Mechanics (SQM) so that it contains nothing probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued awareness operators in a quantum state of the universe which never jumps or collapses. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so the…
There is an incompatibility between the symmetries of causal structure in relativity theory and the signaling abilities of probabilistic devices with inputs and outputs: while time-reversal in relativity will not introduce the ability to signal between spacelike separated regions, this is not the case for probabilistic devices with space-like separated input-output pairs. We explicitly describe a non-signaling device which becomes a perfect signaling device under time-reversal, where time-reversal can be conceptualized as playing backwards a videotape of an agent manipulating the device. This …
This note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson's stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks. An extended english abstract gives most of the details.
