Inverse spin-s portrait and representation of qudit states by single probability vectors

Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j+1)(4j+1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits (j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and (2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s portraits. We also address an auxiliary problem of the optimum reconstruction of qudit states, where the optimality implies a minimum relative error of the density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian Laser Researc

Qubit portrait of the photon-number tomogram and separability of two-mode light states

In view of the photon-number tomograms of two-mode light states, using the qubit-portrait method for studying the probability distributions with infinite outputs, the separability and entanglement detection of the states are studied. Examples of entangled Gaussian state and Schr\"{o}dinger cat state are discussed.Comment: 20 pages, 6 figures, TeX file, to appear in Journal of Russian Laser Researc