Intrinsic correlations for statistical ensembles of Dirac-like structures

Abstract

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.

Figures (6)

• Figure 1: (Color online) Total intrinsic information for stationary Landau levels. The amount of information is computed through the mutual information quantifier (cf. Eq. \ref{['mutualinfo']}; black lines) for which it is assumed that $m=0$ (cf. Eq. \ref{['parameters']}). (Left plot) Results are for pure superpositions of spin up and down states from Eq. \ref{['superposition']}, for coefficients $\sin^2(\theta) = 0$ (solid), $1/4$ (dashed), $1/2$ (dot-dashed), and $3/4$ (dotted). (Right plot) Results are for mixtures of spin up and down states from \ref{['mixture1']}, for coefficients $\sin^2 (\phi) = 0$ (solid), $1/4$ (dashed), $1/2$ (dot-dashed), and $3/4$ (dotted), i.e. with the same line patterns. Results for the intrinsic concurrence (orange lines) are also depicted in both plots, with the same line patterns for the related coefficients. Whereas some amount of information is lost in mixed states for strong magnetic fields ($B_n =1$), randomness creates correlations in weak magnetic fields ($B_n = 0$). In addition, all mixtures share the same value of quantum concurrence.
• Figure 2: (Color online) Delocalization of mixed Gaussian states in phase space. Localization is measured in terms of the probability density $\frac{\mathop{\mathrm{\hbox{Tr}}}\nolimits[\mathcal{W} \gamma_0]}{\sqrt{e\mathcal{B}}}$ of states obtained in Eqs. (\ref{['mixedGaussian1']})-(\ref{['mixedGaussian2']}).The so-called coherent state (left plot), with $z=0$, covers the minimum area, whereas the Gaussian states with $z=0.5$ (center plot) and $z=0.9$ (right plot) are smoothed over phase space.
• Figure 3: (Color online) Information balance for mixed Gaussian states. Linear entropies, $\mathcal{I}_{\{x,k_x\}}$ (cf. Eq. \ref{['entropy1']}; dotted yellow line) and $\mathcal{I}^{SP}$(cf. Eq. \ref{['entropy2']}; dashed magenta line), with respect to phase-space and spin-parity Hilbert spaces are depicted alongside the estimated concurrence (cf. Eq. \ref{['maximalconcurrence']}; solid gray line), and quantum purity (cf. Eq. \ref{['puritymixed']}; dot-dashed blue line). The total mutual information considering all these quantifiers (cf. Eq. \ref{['mutualmixed']}; solid black line) is also included. It is equal to the spin-parity entropy for a maximally mixed state, with $z=1$, which has zero purity and maximum phase-space entropy.
• Figure 4: Dimensionless internal energy (left plot) and specific heat (right plot) in high temperatures. In natural units, $\mu = \frac{\sqrt{2 \,e\mathcal{B}}}{ T}$ is the expansion parameter for the partition function. Results are for $\kappa = 0$ (solid line), $1$ (dot-dashed line), and $10$ (dashed line) when some few terms from the infinite expansion (cf. Eqs. \ref{['eq1a']} and \ref{['exp']}) are considered. At $\mu=0$, the result is analytic and independent of $\kappa = \frac{m^2+ k_z ^2}{2 \,e\mathcal{B}}$ (cf. Eq. \ref{['thermalparameters']}).
• Figure 5: Quantum purity of uneven Landau levels in finite temperatures. As in Fig. \ref{['thermofunctions']}, plots are for $\kappa = 0$ (solid line), $1$ (dot-dashed line), and $10$ (dashed line). At absolute zero ($T \to 0$), $\mu \rightarrow \infty$, and all ensembles approach asymptomatically $\mathcal{P} = 1$, a pure state. In the infinite-temperature limit ($T \to \infty$), $\mu = 0$, all states are maximally mixed.