Phase-space elementary information content of confined Dirac spinors

TL;DR

This paper develops a covariant phase-space Wigner formalism for Dirac spinors, enabling the calculation of quantum information measures like purity and entanglement, and applies it to a charged fermion in a magnetic field.

Contribution

It introduces a novel phase-space approach for Dirac spinors that unifies discrete and continuous quantum information quantifiers, including purity and entanglement, in a covariant framework.

Findings

Derived explicit expressions for quantum purity in phase-space.

Quantified mutual information and entanglement in a magnetic field scenario.

Applied the formalism to Landau levels of a charged fermion.

Abstract

Reporting about the Wigner formalism for describing Dirac spinor structures through a covariant phase-space formulation, the quantum information quantifiers for purity and mutual information involving spin-parity (discrete) and position-momentum (continuous) degrees of freedom are consistently obtained. For Dirac spinor Wigner operators decomposed into Poincar\'e classes of $SU(2) \otimes SU(2)$ spinor couplings, a definitive expression for quantum purity is identified in a twofold way: firstly, in terms of phase-space positively defined quantities, and secondly, in terms of the {\em spin-parity traced-out} associated density matrix in the position coordinate representation, both derived from the original Lorentz covariant phase-space Wigner representation. Naturally, such a structure supports the computation of relative (linear) entropies respectively associated to discrete (spin-parity) and continuous (position-momentum) degrees of freedom. The obtained theoretical tools are used for quantifying (relative) purities, mutual information as well as, by means of the quantum concurrence quantifier, the spin-parity quantum entanglement, for a charged fermion trapped by a uniform magnetic field which, by the way, has the phase-space structure completely described in terms of Laguerre polynomials associated to the quantized Landau levels. Our results can be read as the first step in the systematic computation of the elementary information content of Dirac-like systems exhibiting some kind of confining behavior.