Superposition Principle and Born’s Rule in the Probability Representation of Quantum States

The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.