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Physical Review A

Intrinsic quantum correlations for Gaussian localized Dirac cat states in phase space

Caio Fernando e Silva, Alex E. Bernardini

Published Nov 17, 2021• 5 citations
Publisher

Abstract

Following the information-based approach to Dirac spinors under a constant magnetic field, the phase-space representation of symmetric and antisymmetric localized Dirac cat states is obtained. The intrinsic entanglement profile implied by the Dirac Hamiltonian is then investigated so as to shed a light on quantum states as carriers of qubits correlated by phase-space variables. Corresponding to the superposition of Gaussian states, cat states exhibit nontrivial elementary information dynamics which include the interplay between intrinsic entanglement and quantum superposition as reported by the corresponding Dirac archetypes. Despite the involved time evolution as nonstationary states, the Wigner function constrains the elementary information quantifiers according to a robust framework which can be consistently used for quantifying the time-dependent $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ (spin projection and intrinsic parity) correlation profile of phase-space localized Dirac spinor states. Our results show that the Dirac Wigner functions for cat states---described in terms of generalized Laguerre polynomials---exhibit an almost maximized timely persistent mutual information profile which is engendered by either classical- or quantumlike spin-parity correlations, depending on the magnetic field intensity.

DOI
10.1103/physreva.104.052213

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Following the information-based approach to Dirac spinors under a constant magnetic field, the phase-space representation of symmetric and antisymmetric localized Dirac cat states is obtained. The intrinsic entanglement profile implied by the Dirac Hamiltonian is then investigated so as to shed a light on quantum states as carriers of qubits correlated by phase-space variables. Corresponding to the superposition of Gaussian states, cat states exhibit nontrivial elementary information dynamics which include the interplay between intrinsic entanglement and quantum superposition as reported by the corresponding Dirac archetypes. Despite the involved time evolution as nonstationary states, the Wigner function constrains the elementary information quantifiers according to a robust framework which can be consistently used for quantifying the time-dependent SU(2)\ensuremath⨂SU(2)\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)SU(2)\ensuremath⨂SU(2) (spin projection and intrinsic parity) correlation profile of phase-space localized Dirac spinor states. Our results show that the Dirac Wigner functions for cat states---described in terms of generalized Laguerre polynomials---exhibit an almost maximized timely persistent mutual information profile which is engendered by either classical- or quantumlike spin-parity correlations, depending on the magnetic field intensity.
Following the information-based approach to Dirac spinors under a constant magnetic field, the phase-space representation of symmetric and anti-symmetric localized Dirac cat states is obtained. The intrinsic entanglement profile implied by the Dirac Hamiltonian is then investigated so as to shed a light on quantum states as carriers of qubits correlated by phase-space variables. Corresponding to the superposition of Gaussian states, cat states exhibit non-trivial elementary information dynamics which include the interplay between intrinsic entanglement and quantum superposition as reported by the corresponding Dirac archetypes. Despite the involved time-evolution as non-stationary states, the Wigner function constrains the elementary information quantifiers according to a robust framework which can be consistently used for quantifying the time-dependent SU(2)⊗SU(2)SU(2) \otimes SU(2)SU(2)⊗SU(2) (spin projection and intrinsic parity) correlation profile of phase-space localized Dirac spinor states. Our results show that the Dirac Wigner functions for cat states -- described in terms of generalized Laguerre polynomials -- exhibit an almost maximized timely persistent mutual information profile which is engendered by either classical- or quantum-like spin-parity correlations, depending on the magnetic field intensity.
The Weyl-Wigner formalism for evaluating the intrinsic information of Dirac bispinors as correlated qubits (localized) in a magnetic field is investigated in the extension to statistical ensembles. The confining external field quantizes the quantum correlation measures implied by the spin-parity qubit structure of the Dirac equation in 3+13+13+1 dimensions, which simplifies the computation of the entanglement quantifier for mixed states in relativistic Landau levels. This allows for the evaluation of quantum and classical correlations in terms of entropy measures for Dirac structures that are eventually mixed. Our results are twofold. First, a family of mixed Gaussian states is obtained in phase space, and its intrinsic correlation structure is computed in closed form. Second, the partition function for the low-dimensional Dirac equation in a magnetic field is derived through complex integration techniques. It describes the low-temperature regime in terms of analytically continued zeta functions and the high-temperature limit as a polynomial on the temperature variable. The connection with lower dimensional systems is further elicited by mapping the spin-parity qubits to valley-sublattice bispinors of the low-energy effective Hamiltonian of graphene.
The Weyl-Wigner formalism for evaluating the intrinsic information of Dirac bispinors as correlated qubits (localized) in a magnetic field is investigated in the extension to statistical ensembles. The confining external field quantizes the quantum correlation measures implied by the spin-parity qubit structure of the Dirac equation in 3+1 dimensions, which simplifies the computation of the entanglement quantifier for mixed states in relativistic Landau levels. This allows for the evaluation of quantum and classical correlations in terms of entropy measures for Dirac structures that are eventually mixed. Our results are twofold. First, a family of mixed Gaussian states is obtained in phase space, and its intrinsic correlation structure is computed in closed form. Second, the partition function for the low-dimensional Dirac equation in a magnetic field is derived through complex integration techniques. It describes the low-temperature regime in terms of analytically continued Zeta functions and the high temperature limit as a polynomial on the temperature variable. The connection with lower dimensional systems is further elicited by mapping the spin-parity qubits to valley-sublattice bispinors of the low-energy effective Hamiltonian of graphene.
Considering the parity symmetry related to the Dirac equation, the interplay between energy localization and the temporal evolution of parity-defined quantum superpositions is investigated for fermions in a magnetic field. The unitary evolution of Dirac cat states is obtained by initializing either even or odd principal quantum numbers in the equivalent harmonic oscillator basis in relativistic Landau levels. Quantum operators feature well-defined selection rules for states thus identified, exhibiting a permanent revival structure. Our analysis is specialized for the survival probability function and for the expectation values of spinor matrix operators, which are identified as quantifiers of spin-parity correlations encoded in Dirac bispinors. In such a context, the time evolving quantum state also imprints a signature on the energy expansion. Namely, frequencies associated with revivals are doubled for each revival order, being observed up to a so-called super revival time scale. Results show that Dirac cat states exhibit a fractional revival structure, which works as a probe of suppressions and regenerations of intrinsic correlations driven by the discrete spin-parity degrees of freedom of Dirac bispinors.
Considering the parity symmetry related to the Dirac equation, the interplay between energy localization and the temporal evolution of parity-defined quantum superpositions is investigated for fermions in a magnetic field. The unitary evolution of Dirac cat states is obtained by initializing either even or odd principal quantum numbers in the equivalent harmonic oscillator basis in relativistic Landau levels. Quantum operators feature well-defined selection rules for states thus identified, exhibiting a permanent revival structure. Our analysis is specialized for the survival probability function and for the expectation values of spinor matrix operators, which are identified as quantifiers of spin-parity correlations encoded in Dirac bispinors. In such a context, the time-evolving quantum state also imprints a signature on the energy expansion. Namely, frequencies associated with revivals are doubled for each revival order, being observed up to a so-called super revival timescale. Results show that Dirac cat states exhibit a fractional revival structure, which works as a probe of suppressions and regenerations of intrinsic correlations driven by the discrete spin-parity degrees of freedom of Dirac bispinors.
Dirac equation describes the dynamics of a relativistic spin-1/2 particle regarding its spatial motion and intrinsic degrees of freedom. Here we adopt the point of view that the spinors describe the state of a massive particle carrying two qubits of information: helicity and intrinsic parity. We show that the density matrix for a gas of free fermions, in thermal equilibrium, correlates helicity and intrinsic parity. Our results introduce the basic elements for discussing the spin-parity correlation for a Fermi gas: (1) at the ultra-relativistic domains, when the temperature is quite high, T>1010 KT > 10^{10}\ KT>1010 K, the fermions have no definite intrinsic parity (50% : 50%), which is maximally correlated with the helicity; (2) at very low temperature, T≈3 KT \approx 3 \ KT≈3 K, a unique parity dominates (conventionally chosen positive), by 102010^{20}1020 to 111, while the helicity goes into a mixed state for spin up and down, and the quantum correlation decoheres. For the anti-fermions we get the opposite behavior. In the framework of quantum information, our result could be considered as a plausible explanation of why we do accept, as a fact (consistent with the experimental observation), that fermions (and anti-fermions), in our present epoch of a cool universe, have a unique intrinsic parity. The framework for constructing spin-parity entangled states is established.