Spin observables and spin structure functions: Inequalities and dynamics

Abstract

Model-independent identities and inequalities which relate the various spin observables of collisions in nuclear and particle physics are reviewed in a unified formalism. Their physical interpretation and their implications for dynamical models are also discussed. These constraints between observables can be obtained in several ways: from the explicit expression of the observables in terms of a set of helicity or transversity amplitudes, a non-trivial algebraic exercise which can be preceded by numerical simulation with randomly chosen amplitudes, from anticommutation relations, or from the requirement that any polarisation vector is less than unity. The most powerful tool is the positivity of the density matrices describing the spins in the initial or final state of the reaction or its crossed channels. The inequalities resulting from positivity need to be projected to single out correlations between two or three observables. The quantum aspects of the information carried by spins, in particular entanglement, are considered when deriving and discussing the constraints Several examples are given, with a comparison with experimental data in some cases. For the exclusive reactions, the cases of the strangeness-exchange proton-antiproton scattering and the photoproduction of pseudoscalar mesons are treated in some detail: all triples of observables are constrained, and new results are presented for the allowed domains. The positivity constraints for total cross-sections and for the simplest observables of single-particle inclusive reactions are reviewed. They also apply to spin-dependent structure functions and parton distributions, both integrated or transverse-momentum dependent. The corresponding inequalities are shown to be preserved by the evolution equations of quantum chromodynamics.Inequalities on spin observables 3

Classical positivity, quantum positivity and entanglement of a multi-partite density matrix, with the example of polarized reactions

“…More details can be found in Ref. [1]. The concept of classical positivity and its dual relationship with separability can help to find efficient entanglement criteria, for instance in quantum cryptography.…”

“…The R matrix is semi-positive definite, that is to say ⟨φ|R|φ⟩ ≥ 0 for any, separable or entangled, multiparticle spin state |φ⟩ (we will use the inexact but simpler word "positive"). Physically it means that the cross section is positive for all independent or entangled polarizations [1]. An equivalent statement is Tr {R ρ} ≥ 0 for any ρ , separable or entangled.…”

“…In the first field it is important to know if a given multi-partite density matrix is separable or entangled. In spin physics the attention is focused on the positivity constraints on the spin correlations [1]. These correlations are encoded in a cross section matrix R which is a density matrix except for a trace differing from unity.…”

“…In Ref. [1], some properties of R related to quantum information, like entropy and entanglement are also discussed. In this paper we will review the concepts of classical and quantum positivity of R introduced in Refs.…”

“…In this paper we will review the concepts of classical and quantum positivity of R introduced in Refs. [2,1] with the help of graphical representations. We will also review a duality between the domains of separability and classical positivity of R. This property can be usefull in quantum information theory.…”