We express the matrix elements of the density matrix of the qutrit state in terms of probabilities associated with artificial qubit states. We show that the quantum statistics of qubit states and observables is formally equivalent to the statistics of classical systems with three random vector variables and three classical probability distributions obeying special constrains found in this study. The Bloch spheres geometry of qubit states is mapped onto triangle geometry of qubits. We investigate the triada of Malevich's squares describing the qubit states in quantum suprematism picture and the inequalities for the areas of the squares for qutrit (spin-1 system). We expressed quantum channels for qutrit states in terms of a linear transform of the probabilities determining the qutrit-state density matrix. arXiv:1804.04886v1 [quant-ph] 13 Apr 2018 matrix [2]. The qutrit states can be realized as the states of a spin-1 particle or as the states of the three-level atom. The density matrix of the spin state in the spin tomographic probability representation [3,4] is determined by a fair probability distribution of spin projections on arbitrary directions in the space called the spin tomogram. The von Neumann entropy [5] of the qutrit state was shown [6] to satisfy the entropic inequality, which is the subadditivity condition analogous to the subadditivity condition for bipartite systems of two qubits. Recently [7,8], the triangle geometry of qubit states, in which the density matrix of the spin-1/2 particle was associated with the triada of Malevich's squares, was investigated. The areas of the Malevich's squares are determined by three tomographic probabilities of spin projections m = 1/2 onto three perpendicular directions in the space.The aim of this work is to construct the triada of Malevich's squares associated with the density matrix of qutrit states using the approach connecting the qutrit states with the states of two artificial qubits found in [6] and extra artificial qubit associated with the permutation of the axes x ←→ z in the three-dimensional space. We review the probability description of qubit states [7,8,9] and derive compact formulas for spin tomograms of these states. We use the relation of quitrit states to the states of artificial qubits to express the density matrix elements of the qutrit state in terms of probabilities of the spin-1/2 projection. This paper is organized as follows.In Sec. 2, we review the quantum suprematism picture of spin-1/2 particle states suggested in [7,8]. In Sec. 3, we discuss the statistical properties of the quantum spin-1/2 observable. In Sec. 4, we consider the qutrit-state density matrix and express its matrix elements in terms of probabilities of spin-1/2 projections related to three artificial qubit states connected with the given indivisible qutrit system. In Sec. 5, we discuss the triangle geometry of the qutrit state and study the inequalities for the tomographic probabilities determining the state density matrix. We present our conclusions and prospectives ...

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.developed to obtain the formulation of quantum states more similar to the formulation of the states in classical statistical mechanics.Recently, the tomographic probability representation of quantum states was suggested [12]; in this representation, the quantum states are identified with fair probability distributions connected with density matrices in its phase-space representations by integral transforms; e.g., the Radon transform [13] of the Wigner function provides the optical tomogram [14,15], which is a standard probability distribution of continuous homodyne quadrature of photon depending on an extra parameter called the local oscillator phase, which can be measured [16].The probability distributions determining the spin states were considered in [17,18,19,20,21,22,23,24], and the tomographic probability representation of quantum states was studied in [25,26,27,28,29,30,31,32,33,34].The tomographic probabilities identified with quantum states can be associated with density operators, in view of the formalism of star-product quantization [35,36,37,38,39] analogous to the procedure where the phase-space quasidistributions of quantum states, like the Wigner function, are presented within the star-product framework in [40] (see also recent reviews [41,42]). On the other hand, quantum observables associated with Hermitian operators are presented within the star-product framework by symbols of the operators, which are some functions on the phase space, say, in the Wigner-Weyl representation or the functions of discrete variables in the spin-tomographic description of qudit states.The aim of this work is to extend the probability representation of quantum states to describe the quantum observables in conventional quantum mechanics by fair probability distributions depending on extra parameters. Formally, we address the problem of constructing the invertible map of Hermitian matrices (not only nonnegative trace-class ones) onto sets of probability distributions depending on random variables and extra parameters. We construct such probability representation of quantum observables for systems with finite-dimensional Hilbert spaces of states which are spin-1/2 systems or systems of qubits.

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A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. In view of the qubit-portrait method, the Bell inequalities for two qubits and two qutrits are discussed within the framework of the probability-representation of quantum mechanics. A semigroup of stochastic matrices is associated with tomographic-probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as an ansatz to provide a necessary condition for the separability of quantum states.

We map the density matrix of the qubit (spin-1/2) state associated with the Bloch sphere and given in the tomographic probability representation onto vertices of a triangle determining Triada of Malevich's squares. The three triangle vertices are located on three sides of another equilateral triangle with the sides equal to √ 2. We demonstrate that the triangle vertices are in one-to-one correspondence with the points inside the Bloch sphere and show that the uncertainty relation for the three probabilities of spin projections +1/2 onto three orthogonal directions has the bound determined by the triangle area introduced. This bound is related to the sum of three Malevich's square areas where the squares have sides coinciding with the sides of the triangle. We express any evolution of the qubit state as the motion of the three vertices of the triangle introduced and interpret the gates of qubit states as the semigroup symmetry of the Triada of Malevich's squares. In view of the dynamical semigroup of the qubit-state evolution, we constructed nonlinear representation of the group U (2).