A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical properties and measurement problem is suggested. A quantum definition of classical states is proposed.Keywords Intrinsic vs. extrinsic properties of quantum systems · Structural and conditional intrinsic properties · Classicality · Classical property of quantum linear chain · Quantum measurement

We consider the grading of sl(n,C) by the group ⌸ n of generalized Pauli matrices. The grading decomposes the Lie algebra into n 2 Ϫ1 one-dimensional subspaces. In the article we demonstrate that the normalizer of grading decomposition of sl(n,C) in ⌸ n is the group SL(2,Z n ), where Z n is the cyclic group of order n. As an example we consider sl(3,C) graded by ⌸ 3 and all contractions preserving that grading. We show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in ⌸ 3 .

The paper gives a systematic review of the basic ideas of (non-relativistic) quantum mechanics including all improvements that result from previous work of the authors. The aim is to show that the new theory is self-consistent and that it reproduces all measurable predictions of standard quantum mechanics. The most important improvements are: 1) A new realist interpretation of quantum mechanics assumes that all properties that are uniquely determined by preparations can be viewed as objective. These properties are then sufficient to justify the notion of quantum object. 2) Classical systems are defined as macroscopic quantum objects in states close to maximum entropy. For classical mechanics, new states of such kind are introduced, the so-called maximum-entropy packets, and shown to approximate classical dynamics better than Gaussian wave packets. 3) A new solution of quantum measurement problem is proposed for measurements that are performed on microsystems. First, it is assumed that readings of registration apparatuses are always signals from detectors. Second, an application of the cluster separability principle leads to a locality requirement on observables and to the key notion of separation status. Separation status of a microsystem is shown to change in preparation and registration processes. This gives preparation and registration new meaning and enhances the importance of these notions. Changes of separation status are alterations of kinematic description rather than some parts of dynamical trajectories and thus more radical than 'collapse of the wave function'. Standard quantum mechanics does not provide any information of how separation status changes run, hence new rules must be formulated. As an example of such a new rule, Beltrametti-Cassinelli-Lahti model of measurement is modified and shown then to satisfy both the probability-reproducibility and the objectification requirements.

Abstract. Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of N + 1 mutually unbiased bases in Hilbert spaces of prime dimension N is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of SL(2, Z N ) on finite phase space Z N × Z N implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime N , Z N is also a finite field.

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