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

TL;DR

This paper introduces a bijective mapping of qudit states onto a single probability vector using the inverse spin-portrait method, enabling easier measurement and visualization of quantum states.

Contribution

It proposes a new probability representation for qudit states, linking density operators to a single, physically meaningful probability vector with practical measurement advantages.

Findings

Quantum states form a convex subset of a high-dimensional probability simplex.

The approach relates to existing spin-portrait probability vectors.

Optimal reconstruction minimizes errors in density matrix estimation.

Abstract

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.