The quantum illumination is examined by making use of the three-mode maximally entangled Gaussian state, which involves one signal and two idler beams. It is shown that the quantum Bhattacharyya bound between $rho$ (state for target absence) and $sig
ma$ (state for target presence) is less than the previous result derived by two-mode Gaussian state when $N_S$, average photon number per signal, is less than $0.295$. This indicates that the quantum illumination with three-mode Gaussian state gives less error probability compared to that with two-mode Gaussian state when $N_S < 0.295$.
We discuss classical electrodynamics and the Aharonov-Bohm effect in the presence of the minimal length. In the former we derive the classical equation of motion and the corresponding Lagrangian. In the latter we adopt the generalized uncertainty pri
nciple (GUP) and compute the scattering cross section up to the first-order of the GUP parameter $beta$. Even though the minimal length exists, the cross section is invariant under the simultaneous change $phi rightarrow -phi$, $alpha rightarrow -alpha$, where $phi$ and $alpha$ are azimuthal angle and magnetic flux parameter. However, unlike the usual Aharonv-Bohm scattering the cross section exhibits discontinuous behavior at every integer $alpha$. The symmetries, which the cross section has in the absence of GUP, are shown to be explicitly broken at the level of ${cal O} (beta)$.
The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in $D$-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.
Recently in [Phys. Rev. D $99$ $(2019)$ 104010] the non-relativistic Feynman propagator for harmonic oscillator system is presented when the generalized uncertainty principle is employed. In this short comment it is shown that the expression is incorrect. We also derive the correct expression of it.
The non-relativistic quantum mechanics with the generalized uncertainty principle (GUP) is examined when the potential is one-dimensional $delta-$function. It is shown that unlike usual quantum mechanics, the Schr{o}dinger and Feynmans path-integral
approaches are inequivalent at the first order of GUP parameter.
The R{e}nyi and von Neumann entropies of various bipartite Gaussian states are derived analytically. We also discuss on the tripartite purification for the bipartite states when some particular conditions hold. The generalization to non-Gaussian states is briefly discussed.
Uncertainties $(Delta x)^2$ and $(Delta p)^2$ are analytically derived in an $N$-coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state. When $N = 2$,
those uncertainties are shown as just arithmetic average of uncertainties of two single harmonic oscillators. We call this property as sum rule of quantum uncertainty. However, this arithmetic average property is not generally maintained when $N geq 3$, but it is recovered in $N$-coupled oscillator systems if and only if $(N-1)$ quantum numbers are equal. The generalization of our results to a more general quantum system is briefly discussed.
We derive explicitly the thermal state of the two coupled harmonic oscillator system when the spring and coupling constants are arbitrarily time-dependent. In particular, we focus on the case of sudden change of frequencies. In this case we compute p
urity function, R{e}nyi and von Neumann entropies, and mutual information analytically and examine their temperature-dependence. We also discuss on the thermal entanglement phase transition by making use of the negativity-like quantity. Our calculation shows that the critical temperature $T_c$ increases with increasing the difference between the initial and final frequencies. In this way we can protect the entanglement against the external temperature by introducing large difference of initial and final frequencies.
In order to explore the effect of external temperature $T$ in quantum correlation we compute thermal entanglement and thermal discord analytically in the Heisenberg $X$ $Y$ $Z$ model with Dzyaloshinskii-Moriya Interaction term ${bm D} cdot left( {bm
sigma}_1 times {bm sigma}_2 right)$. For the case of thermal entanglement it is shown that quantum phase transition occurs at $T = T_c$ due to sudden death phenomenon. For antiferromagnetic case the critical temperature $T_c$ increases with increasing $|{bm D}|$. For ferromagnetic case, however, $T_c$ exhibits different behavior in the regions $|{bm D}| geq |{bm D_*}|$ and $|{bm D}| < |{bm D_*}|$, where ${bm D_*}$ is particular value of ${bm D}$. It is shown that $T_c$ becomes zero at $|{bm D}| = |{bm D_*}|$. We explore the behavior of thermal discord in detail at $T approx T_c$. For antiferromagnetic case the external temperature makes the thermal discord exhibit exponential damping behavior, but it never reaches to exact zero. For ferromagnetic case the thermal entanglement and thermal discord are shown to be zero simultaneously at $T_c = 0$ and $|{bm D}| = |{bm D_*}|$. This is unique condition for simultaneous disappearance of thermal entanglement and thermal discord in this model.
The dynamics of mixedness and entanglement is examined by solving the time-dependent Schr{o}dinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of
oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially we introduce three quenched models. In the quenched models both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It is shown that the mixedness for the case of one inaccessible oscillator is larger than that for the case of two inaccessible oscillators in the most time interval. Contrary to the mixedness entanglement for the case of one inaccessible oscillator is smaller than that for the case of two inaccessible oscillators in the most time interval.
