From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations

The concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to statistically complete observables is revisited, and its limitation in view of a classical reformulation of the statistical scheme of quantum mechanics is discussed. In particular, on the basis of a theorem concerning a non-denseness property of a set of coexistent effects, it is shown that an injective classical embedding of the quantum states cannot be supplemented by an at least approximate classical description of the quantum mechanical effects. As an alternative approach, the concept of quasi-probability representations of quantum mechanics is considered.Comment: 35 page

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed

Moment operators of the Cartesian margins of the phase space observables

The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the $x$-margin are polynomials of the position operator and those of the $y$-margin are polynomials of the momentum operator.Comment: 14 page

Semispectral measures as convolutions and their moment operators

The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables.Comment: 7 page

Optimal measurements in quantum mechanics

Four common optimality criteria for measurements are formulated using relations in the set of observables, and their connections are clarified. As case studies, 1-0 observables, localization observables, and photon counting observables are considered.Comment: minor correction

Quantum particles from coarse grained classical probabilities in phase space

Quantum particles can be obtained from a classical probability distribution in phase space by a suitable coarse graining, whereby simultaneous classical information about position and momentum can be lost. For a suitable time evolution of the classical probabilities and choice of observables all features of a quantum particle in a potential follow from classical statistics. This includes interference, tunneling and the uncertainty relation.Comment: 19 page

How many orthonormal bases are needed to distinguish all pure quantum states?

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem

Quantization and noiseless measurements

In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable $f:\R^2\to \R$ is associated with a unique positive operator measure (POM) $E^f$, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly, we notice that the noiseless measurements are the ones which are determined by a selfadjoint operator. The POM $E^f$ in our quantization is defined through its moment operators, which are required to be of the form $\Gamma(f^k)$, $k\in \N$, with $\Gamma$ a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical \emph{questions}, that is, functions $f:\R^2\to\R$ taking only values 0 and 1. We compare two concrete realizations of the map $\Gamma$ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.Comment: 15 pages, submitted to Journal of Physics