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 Schroedinger equation are also derived from this conceptual framework. The theory is illustrated by one and by two spin 1/2 particles; in particular, a statistical discussion of Bell's inequality is given. Several paradoxes and related themes of conventional quantum mechanics are briefly discussed in the setting introduced here.
There is presented a contextual statistical model of the probabilistic description of physical reality. Here contexts (complexes of physical conditions) are considered as basic elements of reality. There is discussed the relation with QM. We propose a realistic analogue of Bohr's principle of complementarity. In the opposite to the Bohr's principle, our principle has no direct relation with mutual exclusivity for observables. To distinguish our principle from the Bohr's principle and to give better characterization, we change the terminology and speak about supplementarity, instead of compleme…
We test Bohr's complementary relation, which captures the most counterintuitive difference of a classical and a quantum world, for single and bipartite neutral kaons. They present a system that is naturally interfering, oscillating and decaying. Moreover, kaons break the CP symmetry (C...charge conjugation, P...parity). In detail we discuss the effect of the CP violation on Bohr's relation, i.e. the effect on the "particle-like" information and the "wave-like" information. Further we show that the quantity that complements the single partite information for bipartite kaons is indeed concurrenc…
We generalize Bohr's complementarity principle for wave and particle properties to arbitrary quantum systems. We begin by noting that a particle-like state is represented by a spatially-localized wave function and its narrow probability density is displaced by spatial translations. In contrast a wave-like state is represented by a spatially-delocalized wave function and the corresponding broad position probability density is invariant to spatial translations. The wave-particle dichotomy can therefore be seen as a competition between displacement and invariance of the state with respect to spat…
The sum of the Holevo quantity (that bounds the capacity of quantum channels to transmit classical information about an observable) and the quantum discord (a measure of the quantumness of correlations of that observable) yields an observable-independent total given by the quantum mutual information. This split naturally delineates information about quantum systems accessible to observers -- information that is redundantly transmitted by the environment -- while showing that it is maximized for the quasi-classical pointer observable. Other observables are accessible only via correlations with …
Modern applications in quantum computation and quantum communication require the precise characterization of quantum states and quantum channels. In practice, this means that one has to determine the quantum capacity of a physical system in terms of measurable quantities. Witnesses, if properly constructed, succeed in performing this task. We derive a method that is capable to compute witnesses for identifying deterministic evolutions and measurement-induced collapse processes. At the same time, applying the Choi-Jamiolkowski isomorphism, it uncovers the entanglement characteristics of biparti…
We analyze the results of an experimental setup that consist of two statistically independent laser beams that cross, interfere and end at detectors. At the beam intersection we place a thin wire at the center of a dark interference fringe and analyze the complementarity inequality. We find that the complementarity inequality is fulfilled provided we include a scattering interaction. We find that this interaction is implicit in the formalism of quantum theory; however, this interaction is active only when the conservation laws are satisfied.
We consider dynamics of hidden variables for measurements in a generalized bell-type model for a single spin using natural assumptions. The evolution of the system, which can be expressed as dynamic chaos is studied. The equilibrium state that the system evolves to asymptotically is consistent with the predictions of quantum theory. The thesis of incompleteness of quantum mechanics in dynamic interpretation and completeness in statistical interpretation are developed. Conceptual problems of quantum mechanics such as violations of Bell inequalities, negative probabilities, complementarity princ…
We present critical arguments against individual interpretation of Bohr's complementarity and Heisenberg's uncertainty principles. Statistical interpretation of these principles is discussed in the contextual framework. We support the possibility to use Statistical Contextual Realist Interpretation of quantum formalism. In spite of all {\bf no-go} theorems (e.g., von Neumann, Kochen and Specker,..., Bell,...), recently (quant-ph/0306003 and 0306069) we constructed a realist basis of quantum mechanics. In our model both classical and quantum spaces are rough images of the fundamental {\bf presp…
We approach the physical implications of the non-commutative nature of Complementary Observable Algebras (COA) from an information theoretic perspective. In particular, we derive a general \textit{entropic certainty principle} stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, wher…
We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman'…
Motivated by Quantum Bayesianism I give background for a general epistemic approach to quantum mechanics, where complementarity and symmetry are the only essential features. A general definition of a symmetric epistemic setting is introduced, and for this setting the basic Hilbert space formalism is arrived at under certain technical assumptions. Other aspects of ordinary quantum mechanics will be developed from the same basis elsewhere.
We focus on characterizing entanglement of high dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by Negativity is explored by analytically relating it to the widely used statistical correlators viz. mutual predictability, mutual information and Pearson Correlation coefficient for different types of bipartite arbitrary dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutu…
A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over physical states that can be prepared. This is termed an "epistemic restriction" because it implies a fundamental limit on the amount of knowledge that any observer can have about the physical state of a classical system. This article provides an overview of epistricted theories, that is, theories that start from a classical statistical theory and apply an epistemic restriction. We consider bo…
We discuss a procedure of measurement followed by the reproduction of the quantum state of a three-level optical system - a frequency- and spatially degenerate two-photon field. The method of statistical estimation of the quantum state based on solving the likelihood equation and analyzing the statistical properties of the obtained estimates is developed. Using the root approach of estimating quantum states, the initial two-photon state vector is reproduced from the measured fourth moments in the field . The developed approach applied to quantum states reconstruction is based on the amplitudes…
In multiparticle quantum interference, bosons show rather generally the tendency to bunch together, while fermions can not. This behavior, which is rooted in the different statistics of the particles, results in a higher coincidence rate $P$ for fermions than for bosons, i.e. $P^{(bos)}<P^{(ferm)}$. However, in lossy systems such a general rule can be violated because bosons can avoid lossy regions. Here it is shown that, in a rather general optical system showing passive parity-time ($\mathcal{PT}$) symmetry, at the $\mathcal{PT}$ symmetry breaking phase transition point the coincidence proba…
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory. A total parameter space $Φ$, equipped with a group $G$ of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter $ϕ$ can be estimated. Choose an experiment/ question $a$, and get from this a parameter space $Λ^{a}$, perhaps after some model reduction compat…
A version of quantum theory is derived from a set of plausible assumptions related to the following general setting: For a given system there is a set of experiments that can be performed, and for each such experiment an ordinary statistical model is defined. The parameters of the single experiments are functions of a hyperparameter, which defines the state of the system. There is a symmetry group acting on the hyperparameters, and for the induced action on the parameters of the single experiment a simple consistency property is assumed, called permissibility of the parametric function. The ot…
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Center for Quantum Information and Quantum Biology, The University of Osaka（大阪大学量子信息与量子生物学中心）
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The quantum Hilbert space is constructed from the parameters of the various experiments using group representation theory. It is shown under natural assumptions that a subset of the set of unit vectors of this space, the generalized coherent state vectors, can be put in correspondence with questions of the kind: What is the value of the (complete) parameter? - t…
We derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The basic setting is a set $\mathcal{A}$ of incompatible experiments, and a transformation group $G$ on the cartesian product $Π$ of the parameter spaces of these experiments. The set of possible parameters is constrained to lie in a subspace of $Π$, an orbit or a set of orbits of $G$. Each possible model is then connected to a parametric Hilbert space. The spaces of different experiments are linked unitarily, thus defining a common Hilbert space $\mathbf{H}$. A …
We study the quantum state transfer (QST) of a class of tight-bonding Bloch electron systems with mirror symmetry by considering the mode entanglement. Some rigorous results are obtained to reveal the intrinsic relationship between the fidelity of QST and the mirror mode concurrence (MMC), which is defined to measure the mode entanglement with a certain spatial symmetry and is just the overlap of a proper wave function with its mirror image. A complementarity is discovered as the maximum fidelity is accompanied by a minimum of MMC. And at the instant, which is just half of the characteristic t…
