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Register a new userRenyi and von Neumann entropies of thermal state in Generalized Uncertainty Principle-corrected harmonic oscillator
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The R{e}nyi and von Neumann entropies of the thermal state in the generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of the GUP parameter $alpha$. While the von Neumann entropy with $alpha = 0$ exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes the decreasing behavior of the von Neumann entropy at the large temperature region. As a result, the von Neumann entropy is maximized at the finite temperature if $alpha eq 0$. The R{e}nyi entropy $S_{gamma}$ with nonzero $alpha$ also exhibits similar behavior at the large temperature region. In this region the R{e}nyi entropy exhibit decreasing behavior with increasing the temperature. The decreasing rate becomes larger when the order of the R{e}nyi entropy $gamma$ is smaller.
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We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Renyi 2-entropy at least as great as $K_{1,n-1}$ and for $alpha>1$, $K_n$ maximizes Renyi $alpha$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.
An alternative method is presented for extracting the von Neumann entropy $-operatorname{Tr} (rho ln rho)$ from $operatorname{Tr} (rho^n)$ for integer $n$ in a quantum system with density matrix $rho$. Instead of relying on direct analytic continuation in $n$, the method uses a generating function $-operatorname{Tr} { rho ln [(1-z rho) / (1-z)] }$ of an auxiliary complex variable $z$. The generating function has a Taylor series that is absolutely convergent within $|z|<1$, and may be analytically continued in $z$ to $z = -infty$ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in $n$. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.
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 principle (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 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.
Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discrete sets of these generalized coherent states are given. Several examples are discussed in detail.
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